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The Ring (revista) — revista sobre boxe
Nürburgring — autódromo alemão também conhecido como The Ring
The Ring (South Park) — episódio da série de desenhos animados
Filmes
The Ring (filme de 1927) — dirigido e escrito por Alfred Hitchcock
The Ring (2002) — de terror; dirigido por Gore Verbinski
The Ring Two — remake do anterior, dirigido por Hideo Nakata
Rings — sequela do anterior de 2017, dirigido por F. Javier Gutiérrez
Desambiguações de cinema
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/mental-models.pdf}
\caption{Don Norman's mental model framework \cite{Norman1987} describes how designers use their mental models to implement systems.
End-users then interact with the systems and develop their own mental models of how they believe the systems work.
While this process is similar for AI models, a key difference is that an AI is not a direct representation of a developer's intent, but a stochastic model learned from data.
This means that \textbf{(1)} AI developers \textit{themselves} have to make sense of what an AI system has learned through testing and iteration.
Subsequently, they can encode these insights as \textbf{(2)} \textit{behavior descriptions}, details of how an AI performs on subgroups of instances, that can be shown to end-users to improve human-AI collaboration.
}
\label{fig:model}
\end{figure*}
Human-AI collaboration is finding real-world use in tasks ranging from diagnosing prostate cancer \cite{Cai2019} to screening calls to child welfare hotlines \cite{de-arteaga_case_2020, kawakami_improving_2022}.
To effectively work with AI aids, people need to know when to either accept or override an AI's output.
People decide when to rely on an AI by using their \textit{mental models} \cite{Kulesza2012, Bansal2019}, or internal representations, of how the AI tends to behave: when it is most accurate, when it is most likely to fail, etc.
A detailed and accurate mental model allows a person to effectively complement an AI system by appropriately relying \cite{Lee2004} on its output, while an overly simple or wrong mental model can lead to blind spots and systematic failures~\cite{Bansal2019, bucinca_trust_2021}.
At worst, people can perform worse than they would have unassisted, such as clinicians who made more errors than average when shown incorrect AI predictions \cite{jacobs_how_2021, Bussone2015}.
Mental models are an inherently incomplete representation of any system, but numerous factors make it especially challenging to develop adequate mental models of AI systems.
First, modern AI systems are often black-box models for which humans cannot see how or why the model made a prediction \cite{Rudin2019}.
Second, black-box models are also often stochastic and can provide different outputs for slightly different inputs without human-understandable reasons \cite{Akhtar2018}.
Lastly, people often expect AI models to behave as humans do, which often does not match their actual behavior \cite{Luger2016} and makes people unaware of how an AI may fail \cite{Kang2015, zhang_ideal_2021}.
These factors make it challenging for people to develop appropriate mental models of AI systems and effectively rely on them to improve their decision making.
To help people collaborate more effectively with AI systems, we propose directly showing end-users insights of AI behavior (\cref{fig:model}).
We term these insights \textbf{behavior descriptions}, details of an AI's performance (metrics, common patterns, potential failures, etc.) on subgroups of instances (subsets of a dataset defined by different features, e.g. ``images with low exposure'').
Behavior descriptions can take many forms, but should help end-users \textit{appropriately rely} \cite{Lee2004} on an AI, using its output when it is most likely correct and overriding it when it is most likely incorrect.
The goal of behavior descriptions is similar to that of explainable AI (xAI), but differs in what type of information is provided to end-users.
Explainable AI attempts to describe \textit{why} an AI system produced a certain output, while behavior descriptions describe \textit{what} patterns of output a model tends to produce.
Thus, xAI and behavior descriptions can be used together to support effective human-AI collaboration.
We hypothesize that behavior descriptions will help people better detect systematic AI failures and improve AI-assisted decision making.
Additionally, we hypothesize that people will both trust an AI aid more and find it more helpful when shown behavior descriptions.
To test these hypotheses, we conducted human-subject experiments in three distinct domains: fake review detection, satellite image classification, and bird classification.
These three domains cover a range of data types, classification tasks, and human and AI accuracies to isolate the effect of behavior descriptions from domain-specific effects.
We found that behavior descriptions can significantly improve the overall accuracy of human-AI teams through two distinct mechanisms.
First, for instances with a behavior description, users can directly correct AI failures.
Second, people tend to rely more on the AI system for instances without behavior descriptions when behavior descriptions are shown for other underperforming subgroups.
We additionally found that showing behavior descriptions had no significant impact on people's qualitative judgements, such as trust and helpfulness, of the AI.
Despite the potential benefits of behavior descriptions, their effects are not universal and depend both on how obvious AI failures are to a person and on a person's ability to correct the output once they know the AI is wrong.
In summary, this work introduces \textbf{behavior descriptions}, details shown to people working with AI systems of how an AI performs (metrics, common patterns, potential failures, etc.) for specific subgroups of instances.
We show how behavior descriptions can improve the performance of human-AI collaboration in three human-subject experiments.
These results indicate that knowing the mental models of end-users in human-AI collaboration is essential to understand how a human-AI team will perform and which interventions and decision aids will be most successful.
\section{Background and Related Work}
We review three main areas of work related to both creating and applying behavior descriptions.
First, we explore existing research on understanding and improving human-AI collaboration.
Second, we discuss methods for improving end-user AI understanding using explainable and interpretable AI.
Lastly, we describe tools and visualizations for analyzing AI behavior that can be used to create behavior descriptions.
\subsection{Human Factors and AI Aids}
When a person interacts with an AI system, they develop a \textit{mental model} of how it behaves - when it performs well, when it fails, or when it has quirky results \cite{Kulesza2012}.
Mental models let people effectively work with AIs by helping them decide whether to rely on, modify, or override an AI's output.
Therefore, it is important for people to have adequate mental models of AI systems so they can appropriately rely on an AI and override it when it is likely to fail \cite{Lee2004}.
There are various methods for encouraging \textit{appropriate reliance} of AI systems, often called \textit{trust calibration} \cite{Tomsett2020} techniques.
Studies have explored what factors influence people's mental models of AI systems.
\citet{Kulesza2013} found that people with more complete mental models were able to collaborate more effectively with a recommendation system.
\citet{Bansal2019} focused on attributes of AI systems and found that systems with \textit{parsimonious} (simple), and \textit{non-stochastic} (predictable) error boundaries were the easiest for humans to work with.
Other factors such as stated and perceived accuracy \cite{kocielnik_will_2019}, confidence values \cite{zhang_effect_2020}, and model personas \cite{Khadpe2020} can also influence people's mental models and their reliance on AI.
Behavior descriptions can complement these existing findings on human-AI collaboration, further helping people better collaborate with AI aids.
Recent methods have explored improving people's mental models, including tutorials explaining a task \cite{Lai2020}, tuning a model to better match human expectations \cite{martinez_personalization_2020}, or adaptive trust calibration \cite{Okamura2020}.
Some methods for improving mental models, such as adaptive trust calibration, use model details such as calibrated confidence scores that can be used in conjunction with behavior descriptions.
The technique most similar to our work is \citet{mozannar_teaching_2021}'s exemplar-based teaching of model behavior.
Their method learns nearest neighborhood regions around model failures to help validate people's mental models.
Although similar, behavior descriptions are \textit{semantic}, high-level insights of model behavior discovered and validated by developers.
Example-based training can fill gaps that were not previously identified by behavior descriptions.
Like model tutorials and trust calibration, behavior descriptions aim to improve human-AI collaboration by updating people's mental models of AI behavior.
Behavior descriptions provide additional information to end-users about \textit{when} to rely on an AI, further enriching their mental models and complementing these existing techniques.
\subsection{Explainable and Interpretable AI}
AI systems can be designed and explained to help people understand model outputs.
For example, interpretable model architectures allow users to inspect how a model makes predictions, while explanations can surface factors that affect model behavior.
One approach to improving end-user model understanding is to design inherently interpretable, or ``glass-box'', models.
Glass-box models can be easier to debug and trace, improving people's ability to understand AI predictions \cite{Rudin2019}.
Despite their benefits, glass-box models can be similarly difficult to collaborate with, for example, a study by \citet{poursabzi-sangdeh_manipulating_2021} found that more ``clear'' models can actually lead to information overload and hinder people's ability to detect AI errors.
Additionally, glass-box models may not work for more complex domains or data types such as images and videos.
Instead, another approach is model-agnostic, post-hoc explanations of model outputs.
LIME \cite{Ribeiro2016} is one such method, which fits linear models to a neighborhood of instances to highlight which features most influenced a model's prediction.
Instead of using all input features to explain a prediction, follow-up techniques such as Anchors \cite{Ribeiro2018} and MUSE \cite{lakkaraju_faithful_2019} learn sets of rules on features that are sufficient to explain predictions for a subset of instances.
Behavior descriptions also show details about a model for subsets of instances, but instead of approximating \textit{why} a model produces certain outputs, focuses on describing \textit{what} the pattern of outputs is.
Despite the insights post-hoc model explanations can provide end-users, they have been found to have a small or even detrimental effect on the performance of AI-assisted decision making \cite{Lai2019, chu_are_2020}.
Explanations have been shown to potentially increase people's reliance on an AI, whether it is right or wrong, leading to worse team performance than a human or AI alone \cite{Bansal2020, Kaur2020, zhang_effect_2020, yang_how_2020}.
By telling users how a model performs, behavior descriptions specifically target people's mental models and inform users when to rely on or override an AI's prediction.
Behavior descriptions can be combined with existing techniques to improve human-AI collaboration.
For example, behavior descriptions can still be used with interpretable models or shown in conjunction with black-box explanations.
\subsection{Behavioral Analysis of AI Systems}\label{sec:analysis}
There are numerous tools and techniques that help developers discover and fix model failures, especially those with real-world effects such as safety concerns and biases.
Behavioral analysis enables developers to go beyond aggregate metrics, such as accuracy, to discover specific and important behaviors \cite{Rahwan2019, cabrera_what_2022}.
Interactive tools, visualizations, and algorithmic techniques have shown promise in helping developers discover AI errors.
The tools most relevant to this work are focused on exploring model performance on a subset of data.
These include systems such as FairVis \cite{Cabrera2019}, a visual analytics system for discovering intersectional biases, AnchorViz \cite{chen_anchorviz_2018}, a visualization for semantically exploring datasets, and Slice Finder \cite{chung_slice_2019}, a method for automatically surfacing subsets of data with large model loss.
There are also tools tailored to specific domains, for example, Errudite \cite{Wu2019}, a visual analytics system for discovering and testing behaviors in NLP models.
More recently, crowd-based systems such as Pandora \cite{Nushi2018}, Deblinder \cite{Cabrera2021Deblinder}, and P-BTM \cite{Liu2020} have shown how crowdsourcing can augment behavioral analysis tools by surfacing previously unknown model failures.
Lastly, there are algorithmic testing methods for discovering AI behaviors.
A common technique is \textit{metamorphic testing} \cite{he_structure-invariant_2020}, which verifies the relationship between changes to an input and expected changes in an output.
Checklist \cite{Ribeiro2020} is one such testing tool for NLP models that creates sets of modified instances to test if language models understand basic logic and grammar.
These model debugging techniques can be used to create behavior descriptions, statements of model performance on subgroups of data.
Additionally, analyses can be rerun on updated models to update behavior descriptions and directly inform end-users how a model has changed \cite{bansal_updates_2019}.
\section{Behavior Descriptions}
We define \textbf{behavior descriptions} as details of how an AI performs (metrics, common patterns, potential failures, etc.) for a subgroup of instances.
Behavior descriptions should be semantically meaningful and human-understandable but can vary significantly between datasets and tasks.
For domains like binary classification, e.g. spam detection, they can be simple statements of accuracy like \textit{our system incorrectly flags marketing emails as spam 53\% of the time.}
In more complex domains like image captioning, they can describe specific behaviors like \textit{our system often describes mountain climbing as skiing} or \textit{our system can produce run-on sentences.}
The goal of behavior descriptions is to help end-users develop better mental models of an AI and thus \textit{appropriately rely} on the model --- using the AI output when it is more accurate and overriding the AI when it is more likely to fail.
In addition to what behavior descriptions contain, \textit{how} they are presented to end-users can vary and impact their effectiveness.
For example, \textit{when} behavior descriptions are shown can vary, from only showing them for extreme edge cases to every instance.
Similarly, behavior descriptions may only be shown during training or for the first few uses of an AI aid.
Due to the broad diversity of potential behavior descriptions, we do not attempt to enumerate all possible forms, and focus instead on testing some core assumptions of their efficacy.
\subsection{Principles for Effective Behavior Descriptions} \label{sec:props}
To design the behavior descriptions used in this work, we drew from existing studies of human-AI collaboration.
From these studies, we derived three properties behavior descriptions should have to maximize their effectiveness.
These are not the only principles, but an initial set used to design the behavior descriptions used in this study.
The first property comes directly from the goal of behavior descriptions, to help users appropriately rely on AI output.
Therefore, behavior descriptions should provide information that helps end-users decide both \textit{whether} they should rely on an AI and, if the AI is wrong, \textit{how} they should override it.
We summarize this principle as \textbf{actionable} behavior descriptions that provide end users with concrete information they can act on when using an AI aid.
The second principle comes from studies of AI error boundaries and explainable AI.
\citet{Bansal2019}'s study of people's mental models of AI systems found that models with errors that were \textit{parsimonious} (e.g. simple to define) led to more effective mental models.
Separately, \citet{poursabzi-sangdeh_manipulating_2021} found that showing end-users more details about an AI reduced their ability to detect AI errors due to information overload.
Thus, behavior descriptions should aim to be \textbf{simple}, short, and easy to interpret and remember.
The third and final principle comes from findings on alert fatigue and cognitive load.
When alerts or messages are shown continuously, people can suffer from \textit{alert fatigue} and begin to ignore the messages \cite{cash_alert_2009}.
Additionally, showing people more information increases their cognitive load, which can lead to decreased learning and performance \cite{van_gog_effects_2011, sweller_element_2010}.
To avoid these pitfalls, behavior descriptions should be limited and focus on the subgroups of instances with the highest impact.
Thus, the third principle is to aim for \textbf{significant} behavior descriptions, either common behaviors or those with the most serious consequences.
In summary, the three design principles that we followed to create the behavior descriptions in this work are the following:
\begin{enumerate}
\item \textbf{Actionable}, suggesting both whether and how a person should override an AI output.
\item \textbf{Simple}, aiming to be as parsimonious and easy to remember as possible.
\item \textbf{Significant}, limited to behaviors that are common and/or have serious consequences.
\end{enumerate}
\subsection{Why Not Just Fix AI Failures?}
A key question surrounding the utility of behavior descriptions is why developers would not directly fix the systematic model failures or behaviors they discover.
Modern AI systems are often stochastic, black-box models like neural networks that cannot be deterministically ``fixed'' or ``updated''.
ML practitioners interviewed by \citet{holstein_improving_2019} would often try to fix one problem that would cause the model to start failing in other unrelated ways.
In another empirical study, \citet{Hopkins2021} found that practitioners would avoid or limit model updates due to concerns about breaking their models and introducing more failures.
Challenges to fixing known model failures are also present in research.
This is most apparent with natural language processing models, which are approaching human aptitude in many domains such as question answering and programming \cite{Brown2020}.
Despite the growing capability of NLP models, many state-of-the-art systems still encode serious biases and harms \cite{Caliskan2017} and fail basic logical tests \cite{Ribeiro2020} that developers have been aware of for years, but have not been able to fix.
Fixing model failures can also require significant amounts of new training data, which tends to be expensive and time consuming.
Additionally, the specific instances needed to improve certain model failures can be difficult to get, such as globally diverse images or accents \cite{Kuznetsova2020}.
Behavior descriptions can be an important intermediate solution for model issues; end-users can effectively work with an imperfect AI that developers are working to improve and fix.
Lastly, behavior descriptions can be helpful when models are updated.
Model updates can be incompatible with end-users' existing mental models and violate their expectations of how an AI behaves, leading to new failures \cite{bansal_updates_2019}.
Updated behavior descriptions can be deployed with a new model to directly update end-user's mental models and avoid decreased performance.
\section{Experimental Design}
To directly test how behavior descriptions impact human-AI collaboration, we conducted a set of human-subject experiments across three different tasks.
The tasks range across dimensions such as human accuracy, AI accuracy, and data type to reduce the chance that our results are confounded by domain-specific differences.
\subsection{Experimental Setup}
To test the effect of behavior descriptions, we conducted experiments across three different classification tasks.
All three tasks shared the same core setup and only varied in the type of classification task (binary or multiclass) and what participants were asked to label.
For each task, we tested three between-subjects conditions to isolate the effect of behavior descriptions:
\begin{itemize}
\item \textbf{No AI:} Participants were asked to classify instances without any assistance.
\item \textbf{AI:} Participants were asked to classify instances with the help of an AI they were told was 90\% accurate.
\item \textbf{AI + Behavior Descriptions (BD):} Participants were asked to classify instances with the help of an AI they were told was 90\% accurate.
Additionally, for instances that were part of a behavior description group (10/30 instances), participants were shown a behavior description stating the AI accuracy for that type of instance (see \cref{sec:groups} for details about behavior description groups).
\end{itemize}
Participants in every condition and task were first shown a consent form, introduced to the task, and shown example instances and labels.
Those in the condition with the AI were also shown a screen before labeling that introduced the AI and stated its overall accuracy of 90\%.
The participants were then shown and asked to label 30 instances (see \cref{fig:ui} for an example UI), 20 instances from the overall dataset, and 5 instances each from two subsets of the data, \textit{behavior description groups}, where the AI performance was significantly worse than for the overall model (see \cref{sec:groups} for details).
After labeling the 30 instances, participants completed a short questionnaire with Likert scale questions, open-ended text responses, and an attention check question.
The experiments were conducted on Amazon Mechanical Turk (AMT) with participants from the United States.
We selected participants who had completed more than 1,000 tasks and had an approval rating of more than 98\% to ensure high-quality responses.
Participants that failed the attention check, a question asking which step they were currently on, were still paid, but were excluded from the results and analysis.
Although we considered providing bonuses as an incentive for accurate responses, we found that the incentive to have the task approved was sufficient to get good results without a bonus.
We confirmed this by finding similar average accuracy on the control condition for the reviews task, 56\%, to that reported by \citet{Lai2019} on the same task using a bonus, 51\%.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/exp-setup.pdf}
\caption{\textbf{Experimental setup}.
Each participant was randomly assigned to a condition and dataset (3x3 between-subjects study, 25 participants per condition, 225 participants total).
For each dataset, participants saw 30 instances, 20 instances from the whole dataset with an AI accuracy of 95\%, and 5 instances each from two subsets of the dataset with an AI accuracy of 40\% and 20\% respectively (simulating subgroups behavior descriptions would be useful for).
In the \textit{AI + Behavior Description} condition, participants were shown behavior descriptions for instances in group 1 and 2.
While the instances shown to participants were randomly chosen from a larger subset of data, each participant saw the same number of AI errors to ensure they observed the same AI accuracies.
}
\label{fig:conditions}
\end{figure*}
The study was approved by an Institutional Review Board (IRB) process.
We had a total of 225 participants, 25 per task/condition pair.
The number of participants per condition was chosen using a power analysis on the mean and standard error of the accuracy in the initial usability studies for the interface.
Of the 225 participants, we removed 13 for failing the attention check.
Additionally, we removed three other participants, across two conditions, who had an accuracy of less than 10\% (the next highest accuracy being 35\%), the same as guessing randomly.
We paid participants \$2 for the task, which lasted 15 minutes for an hourly compensation of \$8 an hour.
\subsubsection{Behavior descriptions and wizard-of-oz AI}\label{sec:groups}
For this work, we used behavior descriptions stating the model accuracy for subgroups of the data for which the model performs significantly worse than average \cite{Cabrera2019,Nushi2018,chung_slice_2019,Ribeiro2020,wu_local_2019}.
Depending on whether the task was binary or multiclass classification, the behavior descriptions resembled text sentences such as "the model is 20\% accurate for this type of instance" or "the model confuses these two classes 80\% of the time".
These types of behavior descriptions are straightforward to create, calculating accuracy on a subset of data, and are actionable for end-users, informing them of how likely it is that they need to override the AI.
In order to control the distribution of instances and AI accuracy each participant saw, we used a wizard-of-oz AI system, mock AI outputs, with a fixed observed accuracy.
Of the 30 instances each participant labeled, 20 were randomly chosen from the overall dataset, and the final 10 split into two groups of 5 instances randomly selected from two subsets of the dataset which we term \textit{behavior description groups}.
The AI accuracy for instances in the behavior descriptions groups was significantly lower than the overall model to simulate the types of subgroups behavior descriptions would be used for.
Lastly, while we fixed the distribution of correct/incorrect model outputs per behavior description group, the instances each participant saw were randomized, both sampled from a larger set of images and randomly ordered.
The accuracy breakdown for the subgroups was the following: 95\% accuracy (1/20 misclassified) for the \textit{main group} of 20 instances, 40\% accuracy (3/5 misclassified) for the first behavior description group, \textit{group 1} and 20\% accuracy (4/5 misclassified) for the second group, \textit{group 2}.
The behavior description groups for each task are detailed in \cref{fig:conditions} and \cref{sec:tasks}.
The purpose of the behavior description groups is to have two concrete subsets representing the type of instances for which a behavior description would be used - these are the instances for which we show behavior descriptions in the AI + BD condition.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/ui.pdf}
\caption{
UI screenshots for the fake reviews (left) and satellite image classification (right) tasks.
Each participant labeled 30 instances, distributed according to the instance groups described in Figure \ref{fig:conditions} and Section \ref{sec:tasks}.
Both screenshots are shown on a labeling step for the \textit{AI + Behavior Description (BD)} condition and on instances that are part of a behavior description group.
In the \textit{AI} condition participants are not shown the additional text for instances in a BD group, and in the \textit{No AI} condition participants are not shown the AI output.
The bird classification task used the same format as the satellite classification task shown.}
\label{fig:ui}
\end{figure*}
The accuracy breakdown above gives an actual AI accuracy of 73.33\%, not the 90\% accuracy stated to the participants, since the task would have been too long to have both an actual 90\% overall accuracy and significantly low accuracies for the two behavior description groups.
We wanted to simulate a situation where an AI is reasonably accurate, e.g. > 90\%, so that a human would want to work with it.
Existing work has explored the impact of stated versus observed accuracy and found that there is a small decrease in agreement with the AI the lower the observed accuracy \cite{yin_understanding_2019}.
Since each condition had the same stated vs. observed accuracy discrepancy (90\% vs. 73.33\% respectively), it should not impact our relative findings on the efficacy of behavior descriptions.
\subsubsection{Classification tasks}\label{sec:tasks}
We chose three distinct tasks for the study to ensure that our findings are not tied to a specific dataset or task.
The three tasks vary by data type (text/image), task type (binary/multiclass classification), and human accuracy (human better/worse than AI).
For each task description below, we also detail what types of instances make up \textit{group 1} and \textit{group 2}, the subsets of instances (behavior description groups) for which the AI is less accurate and for which behavior descriptions are shown in the BD condition (see \cref{fig:conditions}).
The tasks are the following:
\begin{enumerate}[leftmargin=0cm,itemindent=.5cm,labelwidth=\itemindent,labelsep=0cm,align=left]
\setlength\itemsep{0.5em}
\item[] \noindent\textbf{Fake Review Detection.} The dataset of deceptive reviews from \citet{Ott2011, Ott2013} has 800 truthful reviews and 800 deceptive reviews for hotels in Chicago.
The truthful reviews were collected from online sites such as TripAdvisor, while false reviews were collected from Amazon Mechanical Turk workers.
The objective of the task is to determine whether the review is ``truthful'', written by someone who actually stayed at the hotel, or ``deceptive'', written by someone who has not.
We chose this task since it has been used in previous studies of human-AI collaboration to test the effect of explanations~\cite{Lai2019} and tutorials~\cite{Lai2020}.
BD groups were chosen from research on common failures in language models:
\\
\textit{Group 1} - Reviews with less than 50 words \cite{Ribeiro2020}.
\\
\textit{Group 2} - Reviews with more than 3 exclamation marks \cite{Teh2015}.
\item[] \noindent\textbf{Satellite Image Classification.} The satellite images come from the NWPU-RESISC45 dataset \cite{Cheng2017}, which has 31,500 satellite images across 45 classes.
The task for the dataset is multiclass classification, labeling each square satellite image with a semantic class.
We selected a subset of 10 classes for the task in order to show participants at least a couple instances per class.
This task was inspired by real-world human-AI systems for labeling and segmenting satellite images \cite{logar_pulsesatellite_2020}.
The BD groups were chosen from areas of high error in the confusion matrices from the original paper \cite{Cheng2017}:
\\
\textit{Group 1} - Cloud and glacier images
\\
\textit{Group 2} - Meadow and golf course images
\item[] \noindent\textbf{Bird Classification.} The bird images came from the Caltech-UCSD Birds 200 dataset \cite{WelinderEtal2010}, made up of 6,033 images of 200 bird species.
As in the satellite image task, we chose a subset of 10 classes from the dataset for multiclass classification.
The task was inspired by numerous apps and products for classifying bird species \cite{van_horn_building_2015}.
The BD groups were chosen from birds in the same family, classes that are the most similar and difficult to distinguish:
\\
\textit{Group 1} - Cactus Wrens and House Wrens
\\
\textit{Group 2} - Red Bellied Woodpeckers and Red Headed Woodpeckers
\end{enumerate}
\subsection{Hypotheses}
From the primary goal of behavior descriptions, helping end-users appropriately rely on an AI, we formulated the following hypotheses of how we expect behavior descriptions to affect human-AI collaboration.
The hypotheses focus both on quantitative measures of performance and qualitative opinions from participants.
Our first hypothesis is that behavior descriptions will improve the overall accuracy of human-AI teams.
We hypothesize that behavior descriptions will help end-users more appropriately rely on the AI \cite{Lee2004}, leading to improved performance.
\begin{itemize}
\item[\textbf{H1.}] Showing participants behavior descriptions (BD) results in higher overall accuracy than just showing the AI prediction or no AI.
\end{itemize}
\noindent We hypothesize that this improved performance will primarily come from participants overriding the systematic failures identified by behavior descriptions.
By providing actionable descriptions of when the model is most likely to be wrong, for instances in BD groups, we hypothesize that the participants will be able to better identify and correct errors when shown behavior descriptions.
\begin{itemize}
\item[\textbf{H2.}] The higher accuracy from showing BDs is due to a higher accuracy on instances that are part of BD groups.
\end{itemize}
\noindent Lastly, we have a set of hypotheses on how we expect people's perception of the AI to change when they are shown behavior descriptions.
We hypothesize that participants will find the AI to be \textit{more helpful}, will be \textit{more likely to want to use the AI}, and will \textit{trust the AI more} when they are shown behavior descriptions.
These hypotheses come from \citet{yin_understanding_2019}'s study on accuracy and trust in AI systems, which found that observed accuracy significantly affected trust and reliance on AI systems.
\begin{itemize}
\item[\textbf{H3a.}] Participants shown BDs trust the AI more than when just shown the AI output.
\item[\textbf{H3b.}] Participants shown BDs find the AI more helpful than when just shown the AI output.
\item[\textbf{H3c.}] Participants shown BDs are more likely to want to use the AI in the future than when just shown the AI output.
\end{itemize}
\section{Results}
To assess the significance of different conditions on participant accuracy, we used ANOVA tests with Tukey post-hoc tests to correct for multiple comparisons.
For the Likert scale questions, we used Mann-Whitney U tests with Bonferroni corrections.
Lastly, we used linear models to test for learning effects, also using a Bonferroni correction for multiple comparisons.
We used a $p$ value of 0.05 as the cutoff for significance.
\subsection{Overall Accuracy}
To test \textbf{H1} we can compare the average human-AI team accuracy for each task across the three conditions.
Overall, we found that the \textit{AI} and \textit{AI + behavior descriptions (BD)} interventions have a different effect on team performance in each task (\cref{fig:acc-results}).
For the reviews task, there was a significant difference in accuracy across the three conditions $(F_{2, 64}=9.18, \ p<0.001)$.
The only significant pairwise difference was between the \textit{No AI} and \textit{AI + BD} conditions $(p < 0.001, \ 95\% \ C.I. = [0.06, 0.22]$).
While the AI by itself did not significantly improve the accuracy of the participants, AI supplemented with behavior descriptions led to significantly higher team accuracy, supporting \textbf{H1}.
Despite the increased performance, there was no human-AI complementarity -- higher accuracy than either the human or AI alone -- as participant accuracy at every condition was lower than the baseline AI accuracy.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/acc-results.pdf}
\caption{\textbf{Average participant accuracy by task and condition.}
The vertical orange bar indicates the AI accuracy, what would be the participant's accuracy if they picked the AI response every time.
The blue shaded area indicates \textit{complementarity}, the region where the human+AI accuracy is higher than either the human or AI alone.
We find that behavior descriptions led to higher accuracy in the reviews and birds tasks, with complementarity in the birds task (red point in rightmost chart).
The error bars represent standard error.
}
\label{fig:acc-results}
\end{figure*}
In the satellite classification task there was no significant difference between conditions $(F_{2, 67}=0.63, \ p=0.534)$.
The baseline human accuracy without AI support was the highest across tasks, around 90\%, so there was a smaller margin to improve the accuracy of the participants using an AI with a significantly worse accuracy.
Lastly, in the birds classification task, there was a significant difference in participant accuracy $(F_{2, 65}=3.98, \ p=0.023)$.
As in the reviews task, the only pairwise difference was between the \textit{No AI} and \textit{AI + BD} conditions $(p = 0.048, \ 95\% \ C.I. = [0.00, 0.16]$), showing how behavior descriptions can lead to significant increases in participant accuracy using an AI and supporting \textbf{H1}.
This increased performance also led to complementary human-AI accuracy, higher than that of both the AI or human alone.
In sum, these results \textbf{partially support H1}, with behavior descriptions leading to significantly higher accuracy in two of the three tasks.
This suggests that while behavior descriptions will not universally improve the accuracy of human-AI teams, they can lead to significant improvements in certain tasks and domains.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/groups-results.pdf}
\caption{\textbf{Average team accuracy by task, condition and instance group.}
We further break down accuracy by instance type (see \cref{sec:groups}): the main group (20 instances), and two behavior description groups (5 instances each).
The average human-AI accuracy across the three groups of instances gives us an idea of \textit{how} behavior descriptions improve the performance of human-AI teams.
We find that participants relied more on the AI when shown BDs in every task.
Participant performance on the different behavior description groups was mixed, from no effect to significant improvement in group 2 birds (bottom right).
These results highlight the two effects of behavior descriptions, increasing human reliance on a more accurate AI and overriding systematic AI errors.
The error bars represent standard error.
}
\label{fig:groups}
\end{figure*}
\subsection{Accuracy by Behavior Description Group}
To better understand the ways in which behavior descriptions impact performance, we can look at participant accuracy across both conditions and instances in the three different behavior description groups described in \cref{sec:groups} (\cref{fig:groups}).
This allows us to directly test \textbf{H2} by seeing if participants in the \textit{AI + BD} condition perform significantly better on the two subsets of instances that have behavior descriptions.
The results can be seen in \cref{fig:groups}.
We found that this is partially true, as the increased accuracy of the \textit{AI + BD} condition was due both to higher accuracy on the behavior description groups \textit{and} instances in the main group.
In the reviews task, there was only a significant difference between conditions for instances in the normal group $(F_{2, 64}=11.82, \ p<0.001)$, with no differences between conditions for either of the behavior description groups.
The \textit{AI} $(p = 0.007, \ 95\% \ C.I. = [0.03, 0.24]$) and \textit{AI + BD} $(p < 0.001, \ 95\% \ C.I. = [0.10, 0.31] $) conditions were significantly more accurate than the \textit{No AI} condition for instances in the main group.
These results do not support \textbf{H2}, as the higher overall accuracy of the \textit{AI + BD} condition was primarily due to greater reliance on the AI for instances in the main group, not higher accuracy on instances with behavior descriptions.
Despite there being no difference in overall accuracy between conditions for the satellite task, there were differences when looking at the behavior description groups.
As with the reviews task, there was a significant difference between conditions for instances in the main group $(F_{2, 67}=5.34, \ p=0.007)$.
Participants in both the \textit{AI} $(p = 0.026, \ 95\% \ C.I. = [0.00, 0.09]$ and \textit{AI + BD} $(p = 0.007, \ 95\% \ C.I. = [0.01, 0.10]$ conditions had a higher accuracy than participants in the \textit{No AI} condition for instances in the main group.
This is the same result we found in the reviews task, where there were significant accuracy differences for instances in the main group.
Despite this difference, the increased accuracy did not translate to a higher overall accuracy for the \textit{AI} and \textit{AI + BD} conditions.
Since we did not find any difference between conditions for the two BD groups, these findings do not support \textbf{H2}.
Lastly, we found significant differences between conditions for multiple behavior description groups in the bird classification task.
As with the other two tasks, there is a significant difference in accuracy between conditions for instances in the main group $(F_{2, 65}=12.22, \ p<0.001)$.
Both \textit{AI} and \textit{AI + BD} have a significantly higher accuracy than \textit{No AI}, but there is no significance between \textit{AI} and \textit{AI + BD}.
Although there is no significance for instances in group 1, we do see a difference between conditions for instances in group 2 $(F_{2, 65}=6.81, \ p=0.002)$.
Both the \textit{No AI} and \textit{AI + BD} conditions both have a higher accuracy than just \textit{AI}.
This shows that while participants were able to distinguish between the two types of woodpeckers in group 2 without the AI or when informed about the AI failures, participants trusted the AI and performed significantly worse when just shown the AI prediction.
These results support \textbf{H2}, as the increased accuracy on group 2 led to higher accuracy in the AI + BD condition.
In summary, these results \textbf{partially support H2}.
Depending on the task, the higher accuracy in the \textit{AI + BD} condition was due to a higher accuracy on instances that were part of BD groups \textit{and} and a greater reliance on the AI for instances in the main group.
\subsection{Qualitative Results}
After labeling the 30 instances, participants were shown a questionnaire page asking about their opinions and feelings of the AI aid using Likert scale questions (\cref{fig:likert}).
Since these questions were directly related to the AI output, they were only shown to participants in the \textit{AI} and \textit{AI + BD} conditions.
The questions were the following: (1) How \textbf{helpful} was the AI for this task? (2) How \textbf{likely} are you to use this AI again in a future task? (3) How much do you \textbf{trust} the AI?
We did not find significant differences between the \textit{AI} and \textit{AI + BD} conditions for any of the Likert scale questions across tasks.
These results reject \textbf{H3a, H3b, H3c}, and indicate that behavior descriptions do not significantly impact participant perceptions of AI despite differences in how participants use AI predictions.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/likert.pdf}
\caption{\textbf{Likert-scale responses on perception of AI.}
The diverging stacked bar chart centered around the neutral response shows that participants across all conditions and subjective measures overwhelmingly viewed the AI favorably.
There were no significant differences in user's perception of the AI when they were give behavior descriptions.
}
\label{fig:likert}
\end{figure*}
\subsection{Additional Findings}
In addition to the accuracy metrics and Likert scale responses, we collected and analyzed additional measurements and qualitative responses to further unpack the dynamics of behavior descriptions.
Although these are post hoc, exploratory results for which we did not have hypotheses, they can serve as inspiration for further, more formal studies.
One such factor was the time it took participants to label each instance, which can potentially surface interesting insights when compared between conditions and behavior description groups.
Unfortunately, the time per instance had high variance and was inconsistent, with numerous outliers.
This is likely due to the way AMT workers complete tasks, as they often take breaks or search for new HITs while working on a task \cite{lascau_monotasking_2019}.
Future studies could incentivize quick responses to gather more accurate time data and measure the speed of participant responses across conditions.
We also tracked in which round each instance was labeled, $(n/30)$, to detect any learning effects (\cref{fig:learning}).
To measure learning effects, we fit a linear model of average participant accuracy for each condition and each step.
We found that for the two domains in which behavior descriptions were effective, reviews and birds, there was also a significant learning effect for the \textit{AI + BD} condition (reviews/AI + BD: $\beta = 0.0062, p = 0.020$; birds/AI + BD: $\beta = 0.0031, p = 0.035$).
The endpoints of the learning curves also show some interesting patterns.
For the reviews task, participants in the \textit{AI} and \textit{AI + BD} conditions started at similar accuracies and only over time did participants in the \textit{AI + BD} condition learn to effectively work with the AI and make use of the behavior descriptions.
In contrast, in the birds classification task, participants in the \textit{AI + BD} condition were consistently better than the \textit{AI} condition and improved at a similar rate over time.
Interestingly, for the satellite domain, the \textit{AI + BD} condition learning curve ends at a point similar to the \textit{No AI} condition but starts significantly higher.
This suggests that, while over time participants learned the correct satellite labels or when the AI tended to fail, the behavior descriptions helped bootstrap the learning process.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{images/learning.pdf}
\caption{\textbf{Learning curves by task and condition.}
We fit linear models of accuracy on round number to measure learning effects.
We found that the two conditions in which BDs significantly improved performance also had significant learning effects, the \textit{AI + BD} conditions in the reviews and birds tasks (denoted by *).
}
\label{fig:learning}
\end{figure*}
Lastly, we collected qualitative free response answers from participants about patterns of AI behavior they noticed and general comments they had about the task.
As expected, participants in both the \textit{AI} and \textit{AI + BD} noticed that the AI failed in the behavior description groups.
While more participants described failures in the \textit{AI + BD} condition than the \textit{AI} condition, the comments were inconsistent and did not show many significant differences between conditions.
We found an interesting pattern from comments in the birds task, where participants in the \textit{AI} condition described more general but correct patterns of AI failure.
Specifically, a participant found that \textit{``the AI is good at predicting the main class of birds, but might get the sub class incorrect,''} which was the common failure reason between the two BD groups.
\section{Discussion}
These results show that directly informing people about AI behaviors can significantly improve human-AI collaboration.
We hope these initial insights spark future work on understanding people's mental models and developing new types of behavior-based decision aids.
\subsection{Effectiveness of Behavior Descriptions}
Overall, the results generally supported our primary hypothesis that behavior descriptions can significantly improve the accuracy of participants working with AI aids.
Surprisingly, however, we found that the improved accuracy came from two complementary effects: people overriding systematic AI failures \textit{and} relying more often on the AI for instances without behavior descriptions.
The first effect, helping participants fix systematic AI failures, was the initial goal of behavior descriptions, but we found that it was inconsistent and varied significantly between conditions and behavior description groups.
For example, in the reviews task, there was no significant accuracy difference for instances in either behavior description group.
This is likely due to the behavior descriptions in the reviews domain not being sufficiently \textit{actionable} and the participants not knowing whether or not to override the AI when they knew it was likely wrong.
On the contrary, there \textit{was} a significant difference in accuracy for instances in group 2 of the bird classification task, where participants in the BD condition were able to fix AI failures that most participants with just the AI did not notice.
Thus, while we did find some support for this effect, participants fixing AI failures in BD groups was often not the main driver of increased overall accuracy.
In contrast, the more consistent effect was participants in the \textit{AI + BD} condition relying on the AI more often for instances not in the BD groups (main group) compared to the \textit{AI} condition.
This increased reliance on the AI when it was more accurate, on instances in the main group, contributed to the higher overall accuracy in the \textit{AI + BD} conditions for the reviews and birds tasks.
This was an unexpected effect which was a factor in the higher overall accuracy for participants in the \textit{AI + BD} condition.
Although unexpected, the effect is corroborated by studies on trust in AI systems that found that low reliability, AI output that violates people's expectations over time, decreases trust and reliance in an AI \cite{Glikson2020}.
By isolating common AI errors into well-defined, predictable subgroups, the AI appears more reliable to people and can increase their trust and reliance.
Although two of the three domains showed an overall increase in accuracy with behavior descriptions, there was no significant increase in the satellite classification task.
We believe that this is due to the high baseline accuracy of humans performing the task, with approximately 90\% accuracy, which left little room for improvement with a significantly less accurate AI.
The high accuracy of the participants also allowed them to detect and fix the AI errors in the BD groups without any prompting in the \textit{AI} condition.
In summary, while behavior descriptions can improve performance, they are not the panacea to human-AI collaboration --- AI aids must still provide value and complementarity to human decision makers.
\subsection{Learning and Behavior Descriptions}
We found that participants improved more quickly when using behavior descriptions, learning to effectively complement the AI.
The primary pattern we noticed was the significant learning rate in the \textit{AI + BD} condition for the reviews and birds task, as participants quickly learned when they should override the AI.
This could be an important property for AI systems that are updated often, as new behavior descriptions could be used to directly update end-users mental models and avoid failures from incompatible updates \cite{bansal_updates_2019}.
A secondary effect we found was a higher initial accuracy in the \textit{AI + BD} condition for the satellite and birds task.
While it took time for participants in the \textit{AI} condition to notice how a model tended to fail, the participants with behavior descriptions were aware of the failures right from the start.
Even if people's accuracy converges over time, as in the reviews task, behavior descriptions can speed up end-user onboarding and improve early-stage performance.
\subsection{Authoring Behavior Descriptions}
The AI debugging techniques described in Section \ref{sec:analysis} can be used to create behavior descriptions, but tools designed specifically for creating BDs could provide important benefits.
For example, behavior descriptions do not \textit{have} to be AI failures, but could highlight consistent output patterns or areas where the model performs much better than humans.
Bespoke tools could also optimize for creating behavior descriptions with effective properties such as those described in Section \ref{sec:props}.
Tools customized to create behavior descriptions could optimize for these different properties.
The types of people who create behavior descriptions can also be much broader than AI/ML developers.
With the right tools, stakeholders ranging from quality assurance engineers to domain-specific teams could discover and deploy their own behavior descriptions.
In the future, techniques from crowdsourcing could be used to harness end-user's own insights for generating behavior descriptions.
This could be, for example, prompting end-users to report consistent failures and patterns that could then be aggregated and voted on \cite{Cabrera2021Deblinder}.
The insights could be processed to automatically generate up-to-date behavior descriptions.
\subsection{Understanding and Improving Mental Models of AI}
These experiments also implicitly tested the more general effect of human mental models on how they collaborate with AIs.
We found that when humans have more detailed mental models of how an AI performs, they are more likely to rely on the AI in general.
This result is interesting regardless of the use of behavior descriptions, as more experienced end-users will likely develop better mental models of AI aids and gradually change how they rely on the AI.
Developing better techniques for quantifying end-user's mental models of AI systems can help researchers design effective decision aids such as behavior descriptions.
This work could take inspiration from studies in HCI and psychology on \textit{cognitive modeling}, mathematical models of human cognition \cite{anderson_act-r_1997}.
Cognitive models have been used in education by simulating how people learn math to dynamically teach students \cite{ritter_cognitive_2007}.
For human-AI collaboration, cognitive models of how people learn AI behavior could inform the design of decision aids such as behavior descriptions.
Our results can also guide the design of machine learning models that can more directly provide behavior descriptions.
For example, sparse decision trees could be used to directly generate behavior descriptions to show to end-users.
New algorithmic methods that are more amenable to finding clusters of high error could lead to better human-AI collaborative systems.
\section{Limitations and Future Work}
The participants in our study were Amazon Mechanical Turk workers with limited domain expertise in the tasks they completed.
Domain experts and professionals, e.g., doctors or lawyers, have deeper expertise in their fields and may develop different mental models of AI systems they work with.
For example, they may notice AI failures more often or be less influenced by the information provided by behavior descriptions.
Future studies can explore the dynamics of using BDs with domain experts.
Although we selected domains that reflect potential real-world examples of human-AI collaboration, our experiment was a controlled study in a simulated setting.
The impact of behavior descriptions may vary when applied to real-world situations with consequential outcomes \cite{zhang_effect_2020}, such as a radiologist classifying tumors.
When classification errors have a much higher cost, people may update their mental models differently or act more conservatively when shown BDs.
Our study used one specific type of behavior description, subgroup accuracy.
There are countless variations of BDs that can be explored further, such as highlighting subgroups with high accuracy, describing what features of an instance are correlated with failure, or suggesting alternative labels.
Future experiments could test these variations to disentangle which features of behavior descriptions are the most effective in improving people's performance.
Lastly, our study focused on the relatively simple domain of classification.
Modern AI systems are used in much more complex tasks such as image captioning, human pose estimation, and even image generation.
The behavior descriptions for these domains will likely look significantly different from those tested in this work.
For example, BDs for a captioning model might focus on grammatical issues and object references rather than statistical metrics.
The impact of behavior descriptions will likely vary significantly in these domains, and specific studies could explore both their effect and optimal designs.
\section{Conclusion}
We introduce \textbf{behavior descriptions}, details shown to people in human-AI teams of how a model performs for subgroups of instances.
In a series of user studies with 225 participants in 3 distinct domains, we find that behavior descriptions can significantly improve human-AI team performance by helping people both correct AI failures and rely on the AI when it is more accurate.
These results highlight the importance of people's mental models of AI systems and show that methods directly improving mental models can improve people's performance when using AI aids.
This work opens the door to designing behavior-based AI aids and better understanding how humans represent, develop, and update mental models of AI systems.
\begin{acks}
This material is based upon work supported by an Amazon grant, a National Science Foundation grant under No. IIS-2040942, and the National Science Foundation Graduate Research Fellowship Program under grant No. DGE-1745016. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of Amazon or the National Science Foundation.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
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{"url":"https:\/\/math.stackexchange.com\/questions\/739059\/orbits-of-action-of-sl-m-mathbbz-on-mathbbzm","text":"# Orbits of action of $SL_m(\\mathbb{Z})$ on $\\mathbb{Z}^m$\n\nI'm considering the action of $SL_m(\\mathbb{Z})$ on $\\mathbb{Z}^m$: if $A\\in SL_m(\\mathbb{Z})$ and $v\\in\\mathbb{Z}^m$, then $Av\\in\\mathbb{Z}^m$.\n\nMy question is: what are the orbits of this action? I'm especially interested in the case $m=3$.\n\nFor $m=2$, we have the following:\n\nIf $a$ and $b$ are (positive) relatively prime integers, then you can always find integers $c$ and $d$ so that $ad-bc=1$, so that $\\begin{pmatrix} a&c \\\\ b&d \\end{pmatrix}$ is in $SL_2(\\mathbb{Z})$. That means that $\\begin{pmatrix} a \\\\ b \\end{pmatrix}$ is in the orbit of $\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}$ under this action. Conversely it's easy to see that nothing else can be in that orbit. More generally, the orbits of this action are in bijection with the nonnegative integers: the orbit corresponding to $n>0$ consists mainly of the lattice points $\\begin{pmatrix} a \\\\ b \\end{pmatrix}$ with $\\gcd(|a|,|b|)=n$. And if $n=0$, then the orbit consists of $\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}$ only.\n\n\u2022 Isn't it the same for general $n$. Aren't they just in the same orbit if and only if their entries have the same gcd? \u2013\u00a0Derek Holt Apr 4 '14 at 7:20\n\u2022 I suspect it is the same, but how can I even begin to explicitly show that? \u2013\u00a0Isabelle Apr 4 '14 at 13:59\n\nApply Euclid's algorithm to the set of entries of a nonzero integer vector $v$. The operations of the algorithm amount to multiplying by elementary integer matrices (with all 1's on the diagonal and $\\pm 1$ in one off-diagonal entry), which are, therefore, in $GL(m,Z)$. After finitely many steps (which will amount to acting by a product of elementary matrices), you convert $v$ to a vector of the form $(d, 0,...,0)$, where $d$ is the gcd of the entries of $v$. In order to get a matrix in $SL(m,Z)$, just multiply by a matrix of the form $Diag(1,-1,1...)$ if needed. The conclusion is that there is only one orbit of the action for each absolute value of $gcd$ of the vector entries.\n\u2022 This is correct (except for $m=1$, where the sign is also an invariant...) \u2013\u00a0YCor Apr 7 '14 at 21:49\n\u2022 True: I was implicitly assuming that $m>1$. \u2013\u00a0Moishe Kohan Apr 7 '14 at 22:19","date":"2020-04-08 16:29:01","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\": 0.9386254549026489, \"perplexity\": 94.3781905517258}, \"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-2020-16\/segments\/1585371818008.97\/warc\/CC-MAIN-20200408135412-20200408165912-00330.warc.gz\"}"}
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\section{Introduction}
\label{sec:intro}
Molecular communication is a promising approach to realise communications among nano-scale devices \cite{Akyildiz:2008vt,Hiyama:2010jf,Nakano:2012dv,Nakano:2014fq}. There are many possible applications with these networks of nano-devices, for example, in-body sensor networks for health monitoring and therapy \cite{Atakan:2012ej,Nakano:2012dv}. This paper considers diffusion-based molecular communication networks.
In a diffusion-based molecular communication network, transmitters and receivers communicate by using signalling molecules or ligands. The transmitter uses different time-varying functions of concentration of signalling molecules (or emission patterns) to represent different transmission symbols. The signalling molecules diffuse freely in the medium. When signalling molecules reach the receiver, they react with chemical species in the receiver to produce output molecules. The counts of output molecules over time is the receiver output signal which the receiver uses to decode the transmitted symbols.
Two components in diffusion-based molecular communication system are modulation and demodulation. A number of different modulation schemes have been considered in the literature. For example, \cite{Atakan:2010bj,Mahfuz:2011te} consider Concentration Shift Keying (CSK) where different concentrations of signalling molecules are used by the transmitter to represent different transmission symbols. Other modulation techniques that have been proposed include Molecule Shift Keying (MSK) \cite{ShahMohammadian:2012iu,Kuran:2011tg}, Pulse Position Modulation (PPM) \cite{Kadloor:ua},
Amplitude Shift Keying (ASK)\cite{Mahdavifar15}, Frequency Shift Keying (FSK) \cite{Chou:2012ug}, and token communication \cite{Rose:hn}. This paper assumes that the transmitter uses different chemical reactions to generate the emission patterns of different transmission symbols. The motivation to use this type of modulation mechanism is that chemical reactions are a natural way to produce signalling molecules, e.g. the papers \cite{Ni:2010de,Jovic:2010kb} study a number of molecular circuits (which are sets of chemical reactions) that can produce oscillating signals, and the paper \cite{Tostevin:2010bo} discusses a number of signalling mechanisms in living cells.
We assume the receiver consists of receptors. When the signalling molecules (ligands) reach the receiver, they can react with the receptors to form ligand-receptor complexes (which are the output molecules in this paper). We consider the problem of using the continuous-time history of the number of complexes for demodulation assuming that the transmitter and receiver are synchronised. The ligand-receptor complex signal is a stochastic process with three sources of noise because the chemical reactions at the transmitter, the diffusion of signalling molecules and the ligand-receptor binding process are all stochastic. We derive a continuous-time Markov process (CTMP) which models the chemical reactions at the transmitter, the diffusion in the medium and the ligand-receptor binding process. By using this model and the theory of Bayesian filtering, we derive the maximum a posteriori (MAP) demodulator using the continuous-time history of the number of complexes as the input.
This paper makes two key contributions: (1) We propose to use a CTMP to model a molecular communication network with chemical reactions at the transmitter, a diffusive propagation propagation medium and receptors at the receiver. The CTMP captures all three sources of noise in the communication network. (2) We derive a closed-form expression for the MAP demodulation filter using the proposed CTMP. The closed-form expression gives insight into the important elements needed for optimal demodulation, these are the timings at which the receptor bindings occur, the number of unbound receptors and the mean concentration of signalling molecules around the receptors.
The rest of the paper is organised as follows. Section \ref{sec:related} discusses related work. Section \ref{sec:model} presents the system assumptions, as well as a mathematical model from the transmitter to the ligand-receptor complex signal based on CTMP. We derive the MAP demodulator in Section \ref{sec:MAP} and illustrate its numerical properties in Section \ref{sec:eval}. Finally, Section \ref{sec:con} concludes the paper.
\section{Related work}
\label{sec:related}
There is a growing interest to understand molecular communication from the communication engineering point of view. For recent surveys of the field, see \cite{Akyildiz:2008vt,Hiyama:2010jf,Nakano:2012dv,Nakano:2014fq,Akyildiz:hy}. We divide the discussion under these headings: transmitters, receivers, models and others.
{\bf Transmitters.} A number of different types of transmission signals have been considered in the molecular communication literature. The papers \cite{ShahMohammadian:2013jm,Mahfuz:2011te} assume that the transmitter releases the signalling molecules in a burst which can be modelled as either an impulse or a pulse with a finite duration. A recent work in \cite{Mosayebi:2014dm} assumes that the transmitter releases the molecules according to a Poisson process. In this paper, we instead assume that the transmitter uses different sets of chemical reactions to generate different transmission symbols and we use CTMP to model these transmission symbols. Since a Poisson process can also be modelled by a CTMP, the transmission process in this paper is more general than that of \cite{Mosayebi:2014dm}. Our CTMP model can also deal with an impulsive input by using an appropriate initial condition for the CTMP. The use of CTMP as an end-to-end model --- which includes the transmitter, the medium and the receiver --- does not appear to have been used before.
{\bf Receivers. }
Demodulation methods for diffusion-based molecular communication have been studied in \cite{Noel:2014hu,Mahfuz:2014vs}. Both papers also use the MAP framework with discrete-time samples of the number of output molecules as the input to the demodulator. Instead, in this paper, we consider demodulation using continuous-time history of the number of complexes.
The demodulation from ligand-receptor signal has also been considered in \cite{ShahMohammadian:2013jm}. The key difference is that \cite{ShahMohammadian:2013jm} uses a linear approximation of the ligand-receptor process while we use a non-linear reaction rate.
The capacity of molecular communications based on ligand-receptor binding has been studied in \cite{Einolghozati:2011ge,Einolghozati:2011cj} assuming discrete samples of the number of complexes are available. A recent work \cite{Thomas:2014tf} considers the capacity of such systems in the continuous-time limit. Instead of focusing on the capacity, our work focuses on demodulation.
Receiver design is an important topic in molecular communication and has been studied in many papers, some examples are \cite{Noel:2014fv,Noel:2014hu,Kilinc:2013by,Mahfuz:2014vs,Meng:2014hh}. These papers either use one sample or a number of discrete samples on the count of a specific molecule to compute the likelihood of observing a certain input symbols. This paper takes a different approach and uses continuous-time signals.
Another approach of receiver design for molecular communication is to derive molecular circuits that can be used for decoding. An attempt is made in \cite{Chou:2012ug} to design a molecular circuit that can decode frequency-modulated signals. However, the work does not take diffusion and reaction noise into consideration. A recent work in \cite{Pierobon:2014iu} analyses end-to-end molecular communication biological circuits from linear time-invariant system point of view. The work in \cite{Chou:2014jca} compares the information theoretic capacity of a number of different types of linear molecular circuits. This paper differs from the previous work in that it uses a non-linear ligand-receptor binding model.
The noise property of ligand-receptor for molecular communication has been characterised in \cite{Pierobon:2011ve}. The case for non-linear ligand-receptor binding does not appear to have an analytical solution and \cite{Pierobon:2011ve} derives an approximate characterisation using a linear reaction rate assuming that the number of signalling molecules around the receptor is large. This paper uses a non-linear ligand-receptor binding model and no approximation is used in solving the filtering problem.
{\bf Models.} This paper uses the Reaction Diffusion Master Equation (RDME) \cite{Gillespie:2013kk} framework to model the reactions and diffusion in the molecular communication networks. RDME assumes that time is continuous while the diffusion medium is discretised into voxels. This results in a CTMP with finite number of (discrete) states. RDME has been used to model stochastic dynamics of cells in the biology literature \cite{Lawson:2013fe}. An attraction of RDME is that it has the Markov property which means that one can leverage the rich theory behind Markov process.
The author of this paper has previously used an extension of the RDME model, called the RDME with exogenous input (RDMEX) model, to study molecular communication networks in \cite{Chou:rdmex_tnb,Chou:rdmex_nc,Chou:tnano_impact,Chou:tnano_analog}. The RDMEX assumes that the times at which the transmitter emits signalling molecules are deterministic. This results in a stochastic process which is piecewise Markov or the Markov property only holds in between two consecutive emissions by the transmitter. In this paper, we assume the transmitter uses chemical reactions to generate the signalling molecules. Therefore, the emission timings are not deterministic but are governed by a stochastic process.
In this paper, we assume that the propagation medium is discretised in the voxels. An alternative modelling paradigm that has been used in a number of molecular communication network papers \cite{Mahfuz:2011te,ShahMohammadian:2013jm,Mosayebi:2014dm} is that the transmitter or receiver has a non-zero spatial dimension (commonly modelled by a sphere) while the propagation medium is assumed to be continuous. (Note that though \cite{Mosayebi:2014dm} does not explicitly state the dimension of the receiver, one can infer from the fact that the receiver must have a non-zero dimension because it has a non-zero probability of receiving the signalling molecules.) We believe the technique in this paper can be adapted to this alternative modelling paradigm and we do not expect this alternative modelling paradigm will change the results in this paper; we will explain this in Section \ref{sec:map:css}.
There is a rich literature in the modelling of biological systems discussing the difference between: (1) The particle approach which has a continuous state space because the state of a particle is its position; and (2) The mesoscopic approach (the approach in this paper) which discretises the medium into discrete voxels and consider the number of molecules in the voxels as the state. The first approach is more accurate but the computation burden can be high \cite{Burrage:2011te}, while the second approach is accurate for appropriate discretisation \cite{Hellander:2011ub,Gillespie:2013kk}. There are also hybrid approaches too. An overview of various modelling and simulation approaches can be found in \cite{Burrage:2011te}.
{\bf Others:} The results of this paper may also be of interest to biologists who are interested to understand how living cells can distinguish between different concentration levels \cite{Siggia:2013dd,Kobayashi:2011dh}. The result of this paper can be viewed as a generalisation of \cite{Siggia:2013dd} which studies how cells can distinguish between two constant levels of ligand concentration.
\section{End-to-end communication models}
\label{sec:model}
This paper considers diffusion-based molecular communication with one transmitter and one receiver in a fluid medium. Figure \ref{fig:overall} gives an overview of the setup considered in this paper. The transmitter uses different chemical reactions to generate the emission patterns of different transmission symbols. The transmitter acts as the source and emitter of signalling molecules. The signalling molecules diffuses in the fluid medium. The front-end of the receiver consists of a ligand-receptor binding process and the back-end consists of the demodulator with the number of complexes as its input.
In this section, we first describe the system assumptions in Section \ref{sec:model_basic}. We then present, in Section \ref{sec:e2e:g}, an end-to-end model which includes the transmitter, the transmission medium and the ligand-receptor binding process in the receiver, see the dashed box in Figure \ref{fig:overall}. The end-to-end model is a CTMP which includes chemical reactions in the transmitter, diffusion in the medium and the ligand-receptor binding process in the receiver.
\subsection{Model assumptions}
\label{sec:model_basic}
We assume that the medium (or space) is discretised into voxels while time is continuous. This modelling framework results in a RDME \cite{Gillespie:2013kk,Gardiner,vanKampen}, which is a CTMP commonly used to model systems with both diffusion and reactions. In addition, we assume the communication uses only one type of signalling molecule (or ligand) denoted by $S$. We divide the description of our model into three parts: transmission medium, transmitter and receiver. We begin with the transmission medium. Table \ref{tab:notation} summaries the frequently used notation and chemical symbols.
\begin{table}
\begin{tabularx}{\textwidth}{|c|X|} \hline
Symbol & Meaning \\ \hline
$W$ & Dimension of one side of a voxel \\
$D$ & Diffusion constant \\
$d$ & Diffusion rate between neighbouring voxels \\
$N_v$ & Total number of voxels \\
$M$ & Total number of receptors \\
$\tilde{\lambda}$ & Reaction rate constant for the binding reaction \\
$\lambda$ & $\lambda = \frac{\tilde{\lambda}}{W^3}$ \\
$\mu$ & Reaction rate constant for the unbinding reaction \\
$s$ & A transmission symbol \\
$b(t)$ & Number of complexes at time $t$ \\
$n_i(t)$ & Number of signalling molecules in voxel $i$ at time $t$\\
$N(t)$ & Equation \eqref{eqn:state}. A vector containing the number of signalling molecules in each voxel, the counts of intermediate
chemical species in the transmitter and the cumulative count of the number of molecules that have left the system \\
$\sigma_s(t)$ & The mean number of signalling molecules in the receiver voxel at time $t$ if the transmitter sends symbol $s$\\
$U(t)$ & The cumulative number of times the receptors have switched from the unbound to bound state at time $t$\\
$E$ & An unbound receptor \\
$S$ & A signalling molecule \\
$C$ & A complex \\
\hline
\end{tabularx}
\caption{Notation and chemical symbols}
\label{tab:notation}
\end{table}
\subsubsection{Transmission medium}
We model the transmission medium as a three dimensional (3-D) space and partition the space into cubic {\sl voxels} of volume $W^3$. Figure \ref{fig:model} shows an example of a medium which has a dimension of 4 voxels along both the $x$ and $y$-directions, and 1 voxel in the $z$-direction. (Note that Figure \ref{fig:model} should be viewed as a projection onto the $x-y$ plane.) In general, we assume the medium to have $N_x$, $N_y$ and $N_z$ voxels in the $x$, $y$ and $z$ directions where $N_x, N_y$ and $N_z$ are positive integers. In Figure \ref{fig:model}, $N_x = N_y = 4$ and $N_z = 1$. We also use $N_v = N_x N_y N_z$ to denote the total number of voxels.
We refer to a voxel by a triple $(x,y,z)$ where $x$, $y$ and $z$ are integers or by a single index $\xi \in [1,N_v]$. Figure \ref{fig:model} shows the triples for some of the voxels. The single index $\xi$ is calculated from the triple $(x,y,z)$ by using $\xi(x,y,z) = x + N_x (y-1) + N_x N_y (z-1)$. The single indices for voxels are shown in the top right-hand corner of the voxels in Figure \ref{fig:model}.
Diffusion is modelled by molecules moving from one voxel to a neighbouring voxel. For examples, in Figure \ref{fig:model}, molecules can diffuse from Voxel 1 to Voxels 2 or 5, from Voxel 2 to Voxels 1, 3 and 6, and so on. The diffusion of molecules between neighbouring voxels is indicated by the two-way arrows in Figure \ref{fig:model}.
We assume that the signalling molecule $S$ is the only diffusible chemical species in our model and the diffusion coefficient for $S$ is $D$. This means the signalling molecules diffuse from one voxel to a neighbouring voxel at a mean rate of $d$ where $d = \frac{D}{W^2}$. In other words, within an infinitesimal time $\Delta t$, the probability that a signalling molecule diffuses to a neighbouring voxel is $d \; \Delta t$. Note that the expression for $d$ can be derived from spatially discretising the diffusion equation into regular cubic voxels of volume $W^3$, see \cite[p.341]{Gardiner} or \cite[Section 3]{Erban:2007we}.
The dashed lines in Figure \ref{fig:model} indicate the boundary of our transmission medium. Many different boundary conditions are used by engineers and physicists to model what happens when a molecule reaches the boundary of a medium. Two typical boundary conditions are absorbing and reflecting boundaries \cite{Gardiner}. An absorbing boundary means that a molecule can leave the transmission medium and once the molecule has left, it will not return to the medium. For example, in Figure \ref{fig:model}, we allow molecules to leave the medium via one surface of Voxel 16 as indicated by the one-way arrow. Mathematically, this is modelled by a rate of leaving the medium, similar to that of modelling the diffusion between the voxels. A reflecting boundary means that a molecule cannot leave the medium, i.e. a molecule hitting a reflecting boundary will stay in the voxel. Our model can capture these boundary conditions.
It has been shown in \cite{Hellander:2011ub,Gillespie:2013kk} that in order for RDME to produce physically meaningful results, the voxel dimension $W$ cannot be too small and in order for RDME to reduce the discretisation error, $W$ cannot be too large. In this paper, we assume that $W$ comes from a valid range. The choice of $W$ is beyond the scope of the paper and the reader can refer to \cite{Hellander:2011ub,Gillespie:2013kk} for further discussion.
For simplicity, we assume that the medium is homogeneous with a constant diffusion coefficient $D$. It is straightforward to extend the framework to cover inhomogeneous medium \cite{Chou:tnano_impact}. It is also possible to use non-cubic voxels, see \cite{Isaacson:2007ui,Engblom:2009tr}.
\subsubsection{Transmitter}
\label{sec:model:transmitter}
We assume the transmitter occupies one voxel. However, it is straightforward to generalise to the case where a transmitter occupies multiple voxels. We limit our consideration to one symbol interval without inter-symbol interference (ISI) at this moment. We will discuss the multiple symbol interval case with ISI in Section \ref{sec:isi}.
We assume that the transmitter can send $K$ different symbols $s = 0,1,..,K-1$ where each symbol $s$ is characterised by an emission pattern $u_s(t)$. The role of emission pattern in molecular communication is the same as that of transmitted signal in electromagnetic communication. If a transmitter uses a deterministic emission pattern $u_s(t)$ to represent symbol $s$, it means the transmitter emits $u_s(t)$ signalling molecules into the transmitter voxel at time $t$. We use an example to illustrate the meaning of emission pattern. Consider an emission pattern $u_1(t)$ for Symbol 1 where $u_1(t) = \delta_{t,1.2} + \delta_{t,5.6} + 2 \delta_{t,8.1}$ where $\delta_{i,j}$ denotes the Kronecker delta, which means $\delta_{i,j} = 1$ if and only if $i = j$. The emission pattern $u_1(t)$ means that, for Symbol 1, the transmitter emits one signalling molecule at times 1.2 and 5.6, two signalling molecules at time 8.1 and does not emit any molecules at any other times.
In this paper, we assume that the emission pattern for each symbol is produced by a set of chemical reactions located in the transmitter voxel. Given $K$ symbols, the transmitter uses $K$ different reactions to generate these symbols, see Figure \ref{fig:overall}. As an example, a class of chemical reactions inside living cells \cite{Tkacik:2011jr} is
\begin{align}
\cee{
RNA & ->[\kappa] RNA + A
}
\label{cr:rna}
\end{align}
where ribonucleic acid (RNA, which is a molecule commonly found in living cells) produces the chemical species $A$. This class of chemical reactions can be modelled by a Poisson process where molecules of $A$ are produced at a mean rate of $\kappa$ \cite{Shahrezaei:2008bp}. Note that the emission patterns produced by chemical reactions are {\sl not} deterministic, but {\sl stochastic}. The mean emission pattern of this chemical reaction is ${\bf E}[u(t)] = \kappa$.
Following on from the above example, one can realise Amplitude Shift Keying (ASK) in molecular communication by using different chemical reactions that can produce signalling molecules at different mean rates. For example, if there are four different reactions that can produce signalling molecules at four different mean rates of $\kappa_0$, $\kappa_1$, $\kappa_2$ and $\kappa_3$, then one can use these four different reactions to produce 4 different symbols. Note that it is possible for the four chemical reactions to produce the same emission pattern (or realisation), though with different probabilities.
A standard result in physical chemistry shows that the dynamics of a set of chemical reactions can be modelled by a CTMP \cite{Gillespie:1992tq}. Therefore, we will model the transmitter by a CTMP. Note that, in this paper, we will not specify the sets of chemical reactions used by the transmitter except for simulation because the MAP demodulator does not explicitly depend on the sets of chemical reactions that the transmitter uses.
\subsubsection{Receiver}
\label{sec:model:rec}
We assume the receiver occupies one voxel and we use $R$ to denote the index of the voxel at which the receiver is located. In Figure \ref{fig:model}, we assume the receiver is at Voxel 7 (light grey) and hence $R = 7$ for this example. In addition, we assume that the transmitter and receiver voxels are distinct.
We assume that the receiver has $M$ non-interacting receptors and we use $E$ as the chemical name for an unbound receptor. These receptors are fixed in space and do not diffuse, and they are only found in the receiver voxel. Furthermore, these receptors are assumed to be uniformly distributed in the receiver voxel.
The receptor $E$ can bind to a signalling molecule $S$ to form a ligand-receptor complex (or complex for short) $C$, which is a molecule formed by combining $E$ and $S$. This is known as ligand-receptor binding in molecular biology literature \cite{Alberts}. The binding reaction can be written as the chemical equation:
\begin{align}
\cee{
S + E & ->[\tilde{\lambda}] C
}
\label{cr:on}
\end{align}
where $\tilde{\lambda}$ is the reaction rate constant. Since the receptors are only found in the receiver voxel, the binding reaction occurs in a volume of $W^3$, which is the volume of a voxel. The rate at which the complexes $C$ is formed is given by the product of $\frac{\lambda}{W^3}$, the number of signalling molecules in the receiver voxel and the number of unbound receptors\footnote{\label{fn:bind} This footnote explains how $\frac{\lambda}{W^3}$ comes about. Consider a chemical reaction where reactants S and E react to form product C. We assume the reactions are taking place within a volume of $W^3$. Let $c_{\rm S}$, $c_{\rm E}$ and $c_{\rm C}$ be, respectively, the concentration of S, E and C in the volume. The law of mass action says that $\frac{d c_{\rm C}}{dt} = \tilde{\lambda} \; c_{\rm E} \; c_{\rm S}$. In the case of the CTMP or RDME in this paper, we want to keep track of the number of molecules in a volume (the voxel) instead. Let $n_{\rm S}$, $n_{\rm E}$ and $n_{\rm C}$ be, respectively, the number of S, E and C molecules in the volume. Since concentration and molecule counts are related by $c_{{\rm C}} W^3 = n_{{\rm C}}$ etc, we will, in a mathematically loose way, write $\frac{d n_{\rm C}}{dt} = \frac{\tilde{\lambda}}{W^3} \; n_{{\rm S}} \; n_{{\rm E}}$. Since $n_{\rm C}$ is a discrete quantity, the derivative $\frac{d n_{\rm C}}{dt}$ is not defined but we can interpret it as the production rate of molecules $C$. This explains how to convert the law of mass action, which is in terms of concentration, to the rate law used in RDME which is in terms molecular counts. This conversion is also discussed in \cite{Erban:2009us,Erban:2007we}.}. We define $\lambda = \frac{\tilde{\lambda}}{W^3}$ and will use $\lambda$ in the CTMP. Note that this is equivalent to ligand-receptor binding model used in \cite[Section V-B]{Pierobon:2011ve}.
A ligand-receptor complex $C$ can dissociate into an unbound receptor $E$ and a signalling molecule $S$. This can be represented by the chemical equation
\begin{align}
\cee{
C & ->[\mu] E + S
}
\label{cr:off}
\end{align}
where $\mu$ is the reaction rate constant. The rate at which the complexes are dissociating is given by the product of $\mu$ and the number of complexes\footnote{The law of mass action for the dissociation reaction is $\frac{d c_{\rm C}}{dt} = - \mu \; c_{\rm C}$ where $c_{\rm C}$ is the concentration of the complexes. We can use the same argument in Footnote \ref{fn:bind} to show that the dissociation rate of $C$ is $\mu \; n_{\rm C}$ where $n_{\rm C}$ is the number of complexes. In particular, note that no scaling by volume $W^3$ of the voxel is required.
}.
Since a receptor can either be in an unbound state $E$ or in a complex $C$, we have the following conservation relation: the number of unbound receptors plus the number of complexes is equal to the total number of receptors $M$.
\subsection{General end-to-end model}
\label{sec:e2e:g}
In order to derive the MAP demodulator, we need an end-to-end model which includes the transmitter, the medium and the ligand-binding process, see Figure \ref{fig:overall}. Since chemical reactions (which includes the chemical reactions in the transmitter as well as the ligand-receptor binding process in the receiver) and diffusion can be modelled by CTMP, it is possible to use a CTMP as an end-to-end model. In this section we present a general end-to-end model that includes the transmitter, diffusion and the ligand-receptor process in the receiver. An excellent tutorial introduction to the modelling of chemical reactions and diffusion by using CTMP can be found in \cite{Erban:2007we}. We have also included an example in Appendix \ref{app:e2e_ex}.
The aim of the end-to-end model is to determine the properties of the receiver signal from the transmitter signal. The receiver signal in our case is the number of complexes over time. Since the transmitter uses $K$ symbols, the transmitter signal is generated by one of the $K$ sets of chemical reactions. This means that we need $K$ end-to-end models with a model for each of the $K$ symbols or sets of chemical reactions. The principle behind building these $K$ models is identical so without loss of generality, we will assume that the model here is for Symbol 0. We begin with a few definitions.
Let $n_i(t)$ (where $1 \leq i \leq N_v$) be the number of signalling molecules $S$ in Voxel $i$ at time $t$. In particular, since we have defined $R$ to be the index of the receiver voxel, $n_R(t)$ is the number of signalling molecules in the receiver voxel. We assume the transmitter is a set of chemical reactions which uses $H$ intermediate chemical species $Q_1$, $Q_2$, ... and $Q_H$ and these intermediate species remain in the transmitter voxel. Let $n_{Q_i}(t)$ be the number of chemical species $Q_i$ in the transmitter voxel at time $t$. Molecules may also be degraded or leave the system forever if absorbing boundary condition is used. We use $n_A(t)$ to denote the cumulative number of molecules that have left the system. Note that since $n_i(t), n_{Q_i}(t)$ and $n_A(t)$ are molecular counts, they must belong to the set of non-negative integers ${\mathbb Z}_{\geq 0}$. We define the vector $N(t) \in {\mathbb Z}^{N_v+H+1}_{\geq 0}$ to be:
\begin{align}
N(t) =
\left[ \begin{array}{ccccccc}
n_{1}(t) & ... & n_{N_v}(t) & n_{Q_1}(t) & ... & n_{Q_H}(t) & n_{A}(t)
\end{array}
\right]^T
\label{eqn:state}
\end{align}
where ${}^T$ denotes matrix transpose.
Let $b(t)$ denote the number of complexes or bound receptors at time $t$ and ${\mathbb Z}_{[0,M]}$ denote the set of integers between $0$ and $M$ inclusively. We require that a valid $b(t)$ must be an element of ${\mathbb Z}_{[0,M]}$. Since there are $M$ receptors, the number of unbound receptor is $M-b(t)$.
The state of the end-to-end model is the tuple $(N(t),b(t))$. We will now specify the transition probabilities from state $(N(t),b(t))$ to state $(N(t+\Delta t),b(t+\Delta t))$. State transitions can be caused by any one of these events: a chemical reaction in the transmitter, the diffusion of a signalling molecule from a voxel to neighbouring voxel, and the binding or unbinding of a receptor in the receiver. We know from the theory of CTMP that the probability of two events taking place in an infinitesimal duration of $\Delta t$ is of the order of $o(\Delta t^2)$. Intuitively, this means only one event can occur within $\Delta t$. We can divide the transition probabilities from $(N(t),b(t))$ to $(N(t+\Delta t),b(t+\Delta t))$ into 2 groups depending on whether the number of complexes has changed or not in the time interval $(t,t+\Delta t)$. If the number of complexes has changed from time $t$ to $t + \Delta t$, i.e. $b(t+\Delta t) \neq b(t)$, this means either a binding reaction \eqref{cr:on} or a unbinding reaction \eqref{cr:off} has occurred.
If a binding reaction \eqref{cr:on} has occurred, then the number of signalling molecules in the receiver voxel is decreased by 1 and the number of complexes $b(t)$ is increased by 1. This reaction occurs at a mean rate of $\lambda \; n_R(t) \; (M-b(t))$. We use $\mathbb{1}_i$ to denote the standard basis vector with a `1' at the $i$-th position. We can write the state transition probability of the receptor binding reaction \eqref{cr:on} as:
\ifsingle
\begin{align}
{\bf P}[N(t+\Delta t) = N(t)-\mathbb{1}_R, b(t+\Delta t) = b(t) + 1 | N(t), b(t)]
= \lambda \; n_R(t) \; (M-b(t)) \; \Delta t
\label{eqn:tp:r1}
\end{align}
\else
\begin{align}
& {\bf P}[N(t+\Delta t) = N(t)-\mathbb{1}_R, b(t+\Delta t) = b(t) + 1 | N(t), b(t)] \nonumber \\
& = \lambda \; n_R(t) \; (M-b(t)) \; \Delta t
\label{eqn:tp:r1}
\end{align}
\fi
Recalling that $R$ is the index of the receiver voxel and $n_R(t)$ is the $R$-th element of $N(t)$ in \eqref{eqn:state}, the expression $N(t+\Delta t) = N(t)-\mathbb{1}_R$ is equivalent to $n_R(t+\Delta t) = n_R(t) - 1$, which means the number of signalling molecules in the receiver voxel has decreased by 1. Similarly, the expression $b(t+\Delta t) = b(t) + 1$ says the number of complexes has increased by 1. The right-hand side (RHS) of \eqref{eqn:tp:r1} is the transition probability and is given by the product of mean reaction rate and $\Delta t$.
Similarly, the transition probability of the unbinding reaction is given by:
\ifsingle
\begin{align}
{\bf P}[N(t+\Delta t) = N(t)+\mathbb{1}_R, b(t+\Delta t) = b(t) - 1 | N(t), b(t)] = \mu \; b(t) \; \Delta t
\label{eqn:tp:r2}
\end{align}
\else
\begin{align}
& {\bf P}[N(t+\Delta t) = N(t)+\mathbb{1}_R, b(t+\Delta t) = b(t) - 1 | N(t), b(t)] \nonumber \\
& = \mu \; b(t) \; \Delta t
\label{eqn:tp:r2}
\end{align}
\fi
where RHS of \eqref{eqn:tp:r2} is the transition probability.
We now specify the second group of transition probabilities with $b(t+\Delta t) = b(t)$. These transitions are caused by either a reaction in the transmitter or diffusion of signalling molecules between neighbouring voxels. Let $\eta_i, \eta_j \in {\mathbb Z}^{N_v+H+1}_{\geq 0}$ be two valid $N(t)$ vectors; let also $\beta \in {\mathbb Z}_{[0,M]}$. For $\eta_i \neq \eta_j$, we write
\ifsingle
\begin{align}
{\bf P}[N(t+\Delta t) = \eta_i, b(t+\Delta t) = \beta | N(t) = \eta_j, b(t) = \beta]
= d_{ij} \; \Delta t
\label{eqn:tp:eta}
\end{align}
\else
\begin{align}
& {\bf P}[N(t+\Delta t) = \eta_i, b(t+\Delta t) = \beta | N(t) = \eta_j, b(t) = \beta] \nonumber \\
& = d_{ij} \; \Delta t
\label{eqn:tp:eta}
\end{align}
\fi
where $d_{ij}$ is the transition rate from state $(\eta_j,\beta)$ to state $(\eta_i,\beta)$. Since this transition is due to either a reaction in the transmitter or diffusion, $d_{ij}$ is independent of the number of complexes $\beta$. Depending on the type of transition, the value of $d_{ij}$ can depend on the reaction constants in the transmitter, diffusion rate and some states of $\eta_j$. For example, if the transition from $\eta_j$ to $\eta_i$ is caused by the diffusion of a signalling molecule from Voxel 1 to Voxel 2, we have $\eta_i = \eta_j - {\mathbb 1}_1 + {\mathbb 1}_2$ at a rate of $d \eta_{j,1}$ where $\eta_{j,1}$ is the first element in $\eta_j$ or equivalently the number of signalling molecules in Voxel 1 in state $\eta_j$; so, for this example, $d_{ij} = d \eta_{j,1}$.
The main advantage of using Equation \eqref{eqn:tp:eta} is that it allows us a cleaner abstraction to solve the Bayesian filtering problem when deriving the MAP demodulator. We also remark that we will not specify the exact expression of $d_{ij}$ because $d_{ij}$'s do not appear explicitly in the demodulator.
Equations \eqref{eqn:tp:r1}, \eqref{eqn:tp:r2} and \eqref{eqn:tp:eta} specify all the possible state transitions. The probability of no state transition is therefore:
\begin{align}
& {\bf P}[N(t+\Delta t) = \eta_j, b(t+\Delta t) = b(t) | N(t) = \eta_j, b(t)] \nonumber \\
& =
1 - d_{jj} \; \Delta t - \lambda \; n_R(t) \; (M-b(t)) \; \Delta t - \mu \; b(t) \; \Delta t
\label{eqn:tp:no}
\end{align}
where
\begin{align}
d_{jj} = & \sum_{i \neq j} d_{ij}
\label{eqn:djj}
\end{align}
We have now specified all the state transition probabilities for Symbol 0. If a different symbol is used, the value of $H$, the dimension of $N(t)$ and the $d_{ij}$ parameters can change. However, the state transition probabilities still can be summarised by Equations of the form \eqref{eqn:tp:r1}, \eqref{eqn:tp:r2} and \eqref{eqn:tp:eta}. In any case, the derivation of the MAP demodulator only requires us to work with one symbol at a time. Hence, we will use Equations \eqref{eqn:tp:r1}-\eqref{eqn:djj} for any transmission symbol.
\subsection{Discussion}
Note that the CTMP includes all the three sources of noise in our system, due to chemical reactions in the transmitter, random diffusive movements in the medium and the ligand-receptor binding process at the receiver. Some of these noise components have also been characterised in earlier literature. For example, \cite{Pierobon:2011vr} discusses sampling noise at the transmitter and counting noise at the receiver. One can study these noises using the derived CTMP and let us take sampling noise as an example. For the moment, let us isolate the transmitter voxel from the propagation medium, i.e. we do not allow the signalling molecules to leave or enter the transmitter voxel. In this case, the number of signalling molecules in the transmitter voxel is due entirely to chemical reactions and we use $n_{\rm isolated}(t)$ to denote the number of signalling molecules in the isolated transmitter voxel at time $t$. Now, let us consider the transmitter voxel again but we allow signalling molecules to diffuse in and out of the voxel; we use $n_{\rm connected}(t)$ to denote the number of signalling molecules in the transmitter voxel in this case. It is natural to consider $n_{\rm connected}(t)$ as the transmitter signal because this is the number of signalling molecules in the transmitter voxel. However, $n_{\rm connected}(t)$ can be different from $n_{\rm isolated}(t)$ because signalling molecules can diffuse in and out of the transmitter voxel, which is the cause of sampling noise.
Equations \eqref{eqn:tp:r1} to \eqref{eqn:tp:no} hold for any valid state $(N(t),b(t))$. If we collect all the transition probability equations for all valid states, then we can form the infinitesimal generator of the CTMP. For a given initial probability distribution of the initial state $(N(0),b(0))$, one can in principle solve the first order ordinary differential equation (ODE) associated with the infinitesimal generator to compute the probability of the number of complexes $b(t)$, or the property of the receiver signal. However, in practice, this ODE is of a very high dimension and it is an active area of research to derive algorithms to solve this ODE efficiently and accurately \cite{Munsky:2006es}.
\section{The MAP demodulator}
\label{sec:MAP}
This section aims to derive the optimal demodulator using the CTMP derived in the previous section. We assume the input to the demodulator is the continuous-time signal $b(t)$ which is the number of complexes at time $t$. There are a number of reasons why we choose to work with the continuous-time signal $b(t)$, rather than its sampled version. First, the signal $b(t)$ may {\bf not} be strictly band limited in the frequency domain. Second, our results show that the optimal demodulator needs to know the time instances at which a receptor is switching from the unbound to bound state. This timing information, which is essentially an impulse, is unfortunately lost by sampling $b(t)$. Third, the solution of the proposed decoding problem can be used to benchmark molecular circuit \cite{Chou:2014jca} based decoders. Since molecular circuits use chemical reactions for computation, they are fundamentally analogue circuits. Fourth, there is an increasing interest in the circuit design community to design low-power analogue signal processing circuits \cite{Barsakcioglu:2014kd}.
An intermediate step to derive the MAP demodulator is to solve a continuous-time filtering problem. In a filtering problem, one uses all the observations available up till time $t$ to predict the system state at time $t$. For our case, the observations are the number of complexes $b(t)$ in the continuous interval $[0,t]$. (This means the number of observations is infinite because we are considering the continuous-time signal $b(t)$ in a non-zero time interval $[0,t]$.) We use ${\cal B}(t) = \{ b(\tau); 0 \leq \tau \leq t \}$ to denote the section of the signal $b(t)$ in the time interval $[0,t]$. (Note that the $\{ \}$ is not a set notation. Here ${\cal B}(t)$ is a realisation of the number of complexes in $[0,t]$ of the CTMP. This notation is used in some non-linear filtering literature, e.g. \cite{Haykin:1997il}.) We can think of ${\cal B}(t)$ as the history of the number of complexes up till time $t$. The demodulation problem is to use the history ${\cal B}(t)$ to determine which symbol the transmitter has sent.
In Sections \ref{sec:map1} to \ref{sec:map3}, we consider only one symbol interval and do not consider ISI. We consider the ISI case in Section \ref{sec:isi}.
\subsection{The MAP framework}
\label{sec:map1}
We adopt a MAP framework for detection. Let ${\mathbf P}[s | {\cal B}(t)]$ denote the posteriori probability that symbol $s$ has been sent given the history ${\cal B}(t)$. If the demodulation decision is to be done at time $t$, then the demodulator decides that symbol $\hat{s}$ has been sent if
\begin{align}
\hat{s} = {\arg\max}_{s = 0, ..., K-1} {\mathbf P}[s | {\cal B}(t)]
\end{align}
Instead of working with ${\mathbf P}[s | {\cal B}(t)]$, we will work with its logarithm. Let
\begin{align}
L_s(t) = \log ({\mathbf P}[s | {\cal B}(t)])
\end{align}
The first step is to determine $L_s(t+\Delta t)$ from $L_s(t)$. Given ${\cal B}(t)$ is the section of $b(t)$ in the time interval $[0,t]$, one can consider ${\cal B}(t+\Delta t)$ as the concatenation of ${\cal B}(t)$ and the section of $b(t)$ in the time interval $(t,t + \Delta t]$. Over an infinitesimal $\Delta t$, we can consider the signal $b(t)$ is a constant in the time interval $(t,t + \Delta t]$; we therefore abuse the notation and use $b(t + \Delta t)$ to denote the section of $b(t)$ in the time interval $(t,t + \Delta t]$. By using Bayes' rule, it can be shown that
\ifsingle
\begin{align}
L_s(t+\Delta t) =& L_s(t) + \log (\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)])
- \log (\mathbf{P}[b(t+\Delta t) | {\cal B}(t)])
\label{eqn:logpp_prelim}
\end{align}
\else
\begin{align}
L_s(t+\Delta t) =& L_s(t) + \log (\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]) \nonumber \\
& - \log (\mathbf{P}[b(t+\Delta t) | {\cal B}(t)])
\label{eqn:logpp_prelim}
\end{align}
\fi
where $\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]$ is the probability that there are $b(t+\Delta t)$ complexes given that the transmitter has sent the symbol $s$ and the previous history ${\cal B}(t)$. The last term on the RHS of \eqref{eqn:logpp_prelim}, i.e. $\mathbf{P}[b(t+\Delta t) | {\cal B}(t)]$, is independent of the transmission symbol so we do not need it for the purpose of detection. We will focus on determining $\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]$.
\subsection{Computing $\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]$}
The problem of determining the probability $\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]$ is essentially a Bayesian filtering problem. Recall that the complete state of the system is $(N(t),b(t))$ and the receiver can only observe $b(t)$, therefore the task of the receiver is to use the history ${\cal B}(t)$ and the system model to do prediction. Standard method can be used to derive $\mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)]$ but the derivation is long, especially because of the diffusion terms; the derivation can be found in Appendix \ref{app:proofA}. The result is
\ifsingle
\begin{align}
& \mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)] \nonumber \\
= & \delta_{b(t+\Delta t), b(t) + 1} \; \lambda (M - b(t)) \; \Delta t \; \mathbf{E} [n_R(t) | s, {\cal B}(t)] +
\delta_{b(t+\Delta t), b(t) - 1} \; \mu b(t) \; \Delta t \; + \nonumber \\
& \delta_{b(t+\Delta t), b(t)} \;
(1 - \lambda (M - b(t)) {\mathbf E}[n_R(t) | s, {\cal B}(t)] \; \Delta t - \mu b(t) \; \Delta t)
\label{eqn:predictb}
\end{align}
\else
\begin{align}
& \mathbf{P}[b(t+\Delta t) | s, {\cal B}(t)] \nonumber \\
= & \delta_{b(t+\Delta t), b(t) + 1} \lambda (M - b(t)) \; \Delta t \; \mathbf{E} [n_R(t) | s, {\cal B}(t)] + \nonumber \\
& \delta_{b(t+\Delta t), b(t) - 1} \mu b(t) \; \Delta t \; + \nonumber \\
& \delta_{b(t+\Delta t), b(t)} \times \nonumber \\
& (1 - \lambda (M - b(t)) {\mathbf E}[n_R(t) | s, {\cal B}(t)] \; \Delta t - \mu b(t) \; \Delta t)
\label{eqn:predictb}
\end{align}
\fi
Note that only one of the three terms on the RHS of Equation \eqref{eqn:predictb} is non-zero depending on whether the observed $b(t+\Delta t)$ is one more, one less or equal to that of $b(t)$.
The term $\mathbf{E} [n_R(t) | s, {\cal B}(t)]$ is the expected number of signalling molecules in the receiver voxel given the history and the symbol $s$. The meaning of this term is that the receiver uses the history to predict what the expected number of signalling molecules in the receiver voxel is. Note that only the chemical kinetic parameters $\lambda$ and $\mu$ of the receptor appear explicitly in Equation \eqref{eqn:predictb}. Other parameters, such as the set of chemical reaction that generate Symbol $s$ and the diffusion coefficient, do not appear explicitly in Equation \eqref{eqn:predictb} but influence the system behaviour via the term $\mathbf{E} [n_R(t) | s, {\cal B}(t)]$.
\subsection{The demodulation filter}
\label{sec:map3}
By substituting Equation \eqref{eqn:predictb} into Equation \eqref{eqn:logpp_prelim} and let $\Delta t$ go to zero, we show in Appendix \ref{app:proofB} that
\ifsingle
\begin{align}
\frac{dL_s(t)}{dt} =& \frac{dU(t)}{dt} \log( {\mathbf E}[n_R(t) | s, {\cal B}(t)] ) -
\lambda (M - b(t)) {\mathbf E}[n_R(t) | s, {\cal B}(t)] + \tilde{L}(t)
\label{eqn:logpp_dd}
\end{align}
\else
\begin{align}
\frac{dL_s(t)}{dt} =& \frac{dU(t)}{dt} \log( {\mathbf E}[n_R(t) | s, {\cal B}(t)] ) - \nonumber \\
& \lambda (M - b(t)) {\mathbf E}[n_R(t) | s, {\cal B}(t)] + \tilde{L}(t)
\label{eqn:logpp_dd}
\end{align}
\fi
with $L_s(0)$ initialised to the logarithm of the prior probability that Symbol $s$ is sent.
Equation \eqref{eqn:logpp_dd} is the {\sl optimal} demodulation filter.
The term $U(t)$ is the cumulative number of times that the receptors have turned from the unbound to bound state. The meaning of $U(t)$ is illustrated in Figure \ref{fig:ut} assuming there are two receptors. The top two pictures in Figure \ref{fig:ut} show the state transitions for the two receptors. The third picture shows the function $U(t)$ which is increased by one every time a receptor switches from the unbound to bound state. The bottom picture shows $\frac{dU(t)}{dt}$ which is the derivative of $U(t)$. Note that $\frac{dU(t)}{dt}$ consists of a train of impulses (or Dirac deltas) where the timings of the impulses are the times at which a receptor binding event occurs. Loosely speaking, one may also view $\frac{dU(t)}{dt}$ as $\max(\frac{db(t)}{dt},0)$.
The function $\tilde{L}(t)$, which is the last term on the RHS of \eqref{eqn:logpp_dd}, contains all the terms that are independent of Symbol $s$. Since $L_s(t)$ does not appear on the RHS of \eqref{eqn:logpp_dd}, this means that $\tilde{L}(t)$ adds the same contribution to all $L_s(t)$ for all $s = 0, ..., K-1$. We can therefore ignore $\tilde{L}(t)$ for the purpose of demodulation.
The term ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ in Equation \eqref{eqn:logpp_dd} is the prediction of the mean number of signalling molecules in the receiver voxel using the history of receptor state. This is a filtering problem which requires extensive computation. Instead, we assume that the receiver has prior knowledge that if Symbol $s$ is transmitted, then the mean number of signalling molecules in the receiver voxel is $\sigma_s(t) = {\mathbf E}[n_R(t) | s]$ and the receiver uses $\sigma_s(t)$ for demodulation. We can view $\sigma_s(t)$ as internal models that the demodulator uses. The use of internal models is fairly common in signal processing and communication, e.g. a matched filter correlates the measured data with an expected response.
After making the modifications described in the last two paragraphs, we are now ready to describe the demodulator. Using $b(t)$ as the input, the demodulator runs the following $K$ continuous-time filters in parallel:
\begin{align}
\frac{dZ_s(t)}{dt} = \frac{dU(t)}{dt} \log(\sigma_s(t)) - \lambda (M - b(t)) \sigma_s(t)
\label{eqn:logmap_s}
\end{align}
where $Z_s(0)$ is initialised to the logarithm of the prior probability that the transmitter sends Symbol $s$. If the demodulator makes the decision at time $t$, then the demodulator decides that Symbol $\hat{s}$ has been transmitted if
\begin{align}
\hat{s} = {\arg\max}_{s = 0, ..., K-1} Z_s(t)
\end{align}
The demodulator structure is illustrated in Figure \ref{fig:demod}. By comparing Equations \eqref{eqn:logpp_dd} and \eqref{eqn:logmap_s}, it can be shown that $L_{s_1}(t) - L_{s_2}(t) = Z_{s_1}(t) - Z_{s_2}(t)$ for any two symbols $s_1$ and $s_2$. An interpretation of the demodulation filter output $Z_s(t)$ is that $\exp(Z_s(t))$ is proportional to the posteriori probability ${\mathbf P}[s | {\cal B}(t)]$.
Note that the replacement of ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ in Equation \eqref{eqn:logpp_dd} by $\sigma_s(t)$ means that the demodulation filter \eqref{eqn:logmap_s} is {\sl sub-optimal}. If ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ and $\sigma_s(t)$ are close to each other, we expect the performance degradation to be small. It is an open problem what the difference between ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ and $\sigma_s(t)$ is. The difficulty in answering this problem is due to the fact that ligand-receptor binding is a nonlinear process. In Appendix \ref{app:C}, we motivate the closeness between
${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ and $\sigma_s(t)$ by studying an analogous problem in linear time-invariant (LTI) systems. We will compare the performance of the optimal and sub-optimal demodulation filters in Section \ref{sec:prop_cf_opt_bf}.
In order to understand Equation \eqref{eqn:logmap_s}, we consider the situation where Symbol 1 generates a lot more signalling molecules than Symbol 0 such that it results in more signalling molecules in the receiver voxel, or $\sigma_1(t) > \sigma_0(t)$ for all $t$. If the transmitter sends Symbol 1, then more signalling molecules are expected to reach the receiver voxel. The consequence is that there are more receptor binding events and the number of unbound receptors $(M-b(t))$ is smaller. Therefore, in Equation \eqref{eqn:logmap_s}, we expect a big positive contribution from the first term on the RHS and a small negative contribution from the second term. The net effect is a big $Z_1(t)$. On the other hand, if the transmitter sends Symbol 0, the number of receptor binding events is smaller and $(M-b(t))$ is big. This results in a smaller $Z_0(t)$. Therefore, $Z_1(t)$ is likely to be bigger than $Z_0(t)$, which means correct detection.
\subsection{Discussions}
\subsubsection{Implementation issues}
The implementation of the demodulator is an open research problem. The demodulation filter \eqref{eqn:logmap_s} is an analogue filter and it requires the internal model $\sigma_i(t)$. The design of analogue circuits using chemical reactions for calculations is an active research area, see \cite{Sarpeshkar:2014dg} for a recent overview. The demodulation filter requires logarithm, multiplication, subtraction, integration and counting the number of times the receptors have switched from the unbound to bound state. An analogue molecular circuit for calculating logarithm is presented in \cite{Daniel:2013ke}. There are also circuits that can perform multiplication and subtraction \cite{Sarpeshkar:2014dg}. Living cells are known to use integration \cite{Yi:2000tj}. It may be possible to implement the counting using a chemical reaction \cite{Siggia:2013dd}. It may be possible to approximate the internal models by using some lower order chemical reactions. This discussion shows that some components to implement the demodulator exist but the exact implementation remains an open problem.
In order to bypass the difficulty of computing ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$, we have proposed to use internal models $\sigma_s(t)$. An open research problem is to study sub-optimal estimation of ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$.
Note that \cite{Sarpeshkar:2014dg} shows that analogue computation is more efficient than digital computation if high precision is not required. An interesting problem is to study the impact of low precision analogue calculations on the demodulation performance.
\subsubsection{Continuous transmission medium versus voxels}
\label{sec:map:css}
We mention in Section \ref{sec:related} that some papers in molecular communications assume a continuous medium (rather than discretising the medium into voxels) and a non-zero receiver size. If a continuous medium is assumed, the state of a signalling molecule in the transmission medium is its position. Let $p(\vec{x},t)$ be the probability that a molecule is at position $\vec{x}$ at time $t$. The time evolution of $p(\vec{x},t)$ can again be modelled by a CTMP, which is continuous in both time and space. The time evolution of $p(\vec{x},t)$ is described by the differential Chapman-Kolmogorov Equation (CKE) \cite{Gardiner}. It is possible to derive an alternative CTMP by replacing the diffusion of signalling molecules in Equation \eqref{eqn:tp:eta} by differential CKE as well as by adding equations to describe how the signalling molecules enter or leave the receiver voxel. We can then apply Bayesian filtering to this alternative CTMP. We expect this alternative CTMP will give the same demodulator filter because in our derivation based on discrete voxels, the MAP demodulator depends only on the number of the signalling molecules in the receiver voxel and the diffusion parameters do not appear explicitly in the demodulation filter. Further support of this argument is given in the derivation in Appendix \ref{app:proofB} where we show that the $d_{ij}$ parameters in \eqref{eqn:tp:eta} are cancelled out in deriving the Bayesian filter.
\subsection{Inter-symbol interference}
\label{sec:isi}
It is in principle possible to use the demodulation filters \eqref{eqn:logmap_s} to deal with the case with ISI. We use $T_x$ to denote one symbol duration and we assume that ISI only lasts for a finite amount of time. Specifically, consider a symbol sent in $[kT_x,(k+1)T_x]$, we assume that the effect of this transmission can be neglected after the time $ (k+n) T_x$. This can be realised by appropriately choosing the transmitter and receiver parameters, $T_x$ and $n$.
In order to make the explanation here a bit more concrete, we assume that the transmitter uses $K = 2$ symbols and $n = 3$. Over a duration of $n = 3$ symbols, the possible sequences sent by the transmitter are 000, 001, 010, 011, ... , 111. Let $\sigma_{0,0,0}(t)$ denote the mean number of signalling molecules at the receiver voxel if the sequence 000 is sent. We can similarly define $\sigma_{0,0,1}(t), ..., \sigma_{1,1,1}(t)$. Consider the transmission of three consecutive symbols $s_{k-2}, s_{k-1}$ and $s_k$. Assuming that we have an estimation of the first two symbols $\hat{s}_{k-2}$ and $\hat{s}_{k-1}$, then the decoding of $s_k$ can be done by using the demodulation filter \eqref{eqn:logmap_s} by replacing $\sigma_s(t)$ by $\sigma_{\hat{s}_{k-2},\hat{s}_{k-1},s}$. For example, if $\hat{s}_{k-2} = 1$ and $\hat{s}_{k-1} = 0$, then one can decode what $s_k$ is by using the demodulator filters $\sigma_{1,0,1}(t)$ and $\sigma_{1,0,1}(t)$. Although the decision feedback based method can solve the ISI problem, the number of internal models increases exponentially with the memory length parameter $n$.
The reason why we need to consider all $2^n$ possible transmission sequences is that the ligand-receptor binding process has a non-linear reaction rate. A method to reduce the number of internal models is to design the system so that $\sigma_{0,0,0}(t)$ etc. can be decomposed into a sum. Let $\sigma_s(t)$ ($s = 0,1$) be the mean number of signalling molecules at the receiver voxel if the symbol $s$ is sent for one symbol duration and in the absence of ISI. If
\begin{align}
\sigma_{s_1,s_2,s_3}(t) \approx \sigma_{s_1}(t - 2 T_x) + \sigma_{s_2}(t - T_x) + \sigma_{s_3}(t)
\label{eqn:sigma_sum}
\end{align}
holds for all $s_1, s_2$ and $s_3$, then one can again make use of decision feedback to decode the ISI signal. However, this time, only $K$ internal models are needed. Equation \eqref{eqn:sigma_sum} can be made to hold approximately if the number of receptors is large. This can be explained as follows. First of all, if ligand-receptor binding is absent, this means there is only free diffusion then Equations \eqref{eqn:sigma_sum} holds because the mean number of signalling molecules obeys the diffusion equation which is linear. This means that we need to create an environment that ``looks like" free diffusion even when ligand-receptor binding is present. This can be realised if the number of signalling molecules that are bound to the receptors is small compared to those that are free. A method to achieve this is to increase the number of receptors. We will demonstrate this with a numerical example in Section \ref{sec:eval}. However, it is still an open problem to solve the ISI in the general case.
\section{Properties of the demodulator}
\label{sec:eval}
The aim of this section is to study the properties of the MAP demodulator numerically. We begin with the methodology.
\subsection{Methodology}
We assume the diffusion coefficient $D$ of the medium is 1 $\mu$m$^2$s$^{-1}$. The receptor parameters are $\tilde{\lambda}$ = 0.005 $\mu$m$^3$ s$^{-1}$, $\lambda = \frac{\tilde{\lambda}}{W^3}$, and $\mu = 1$ s$^{-1}$. These values are similar to those used in \cite{Erban:2009us} and
\cite{Pierobon:2011ve}~\footnote{
The paper \cite{Pierobon:2011ve} considers ligand-receptor binding in the chemical master equation setting. In our notation, the parameter values in \cite{Pierobon:2011ve} are $D$ = 100 $\mu$m$^2$s$^{-1}$, $\tilde{\lambda}$ = 0.2 $\mu$m$^3$ s$^{-1}$ and $\mu = 10$ s$^{-1}$. These parameters are 10--100 times faster than ours and can be considered as a time-scaling. Note that \cite{Pierobon:2011ve} uses $k_+$ and $k_-$ instead of, respectively, $\tilde{\lambda}$ and $\mu$.
}.
The above parameter values will be used for all the numerical experiments.
For each experiment, the transmitter uses either $K = 2$ or $K = 3$ symbols. Each symbol is generated by a different sets of chemical reactions. Different experiments may use different sets of chemical reactions and will be described later. The number of receptors also varies between the experiments.
We use the Stochastic Simulation Algorithm (SSA) \cite{Gillespie:1977ww} to obtain realisations of $b(t)$ which is the number of complexes over time. SSA is a standard algorithm in chemistry to simulate diffusion and reactions; it is essentially an algorithm to simulate a CTMP.
In order to use Equation \eqref{eqn:logmap_s}, we require the mean number of signalling molecules $\sigma_s(t)$ in the receiver voxel when Symbol $s$ is sent. Unfortunately, it is not possible to analytically compute $\sigma_s(t)$ from the CTMP because of moment closure problem which arises when the transition rate is a non-linear function of the state \cite{Smadbeck:2013fk}. We therefore resort to simulation to estimate $\sigma_s(t)$. Each time when we need an $\sigma_s(t)$, we run SSA simulation 500 times and average the results to obtain $\sigma_s(t)$. Note that these simulations are different from those that we use to generate $b(t)$ for the performance study. In other words, the simulations for estimating $\sigma_s(t)$ and for performance study are completely independent.
Once $b(t)$ and $\sigma_s(t)$ are obtained, we use numerical integration to calculate $Z_s(t)$ using Equation \eqref{eqn:logmap_s}. We assume that all symbols appear with equal probability, so we initialise $Z_s(0) = 0$ for all $s$.
\subsection{Optimal filter \eqref{eqn:logpp_dd} versus sub-optimal filter \eqref{eqn:logmap_s}}
\label{sec:prop_cf_opt_bf}
The optimal demodulation filter \eqref{eqn:logpp_dd} requires the term ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ which can be obtained by solving a Bayesian filtering problem. Since filtering problems are computationally expensive to solve, we propose the sub-optimal demodulation filter \eqref{eqn:logmap_s} which uses $\sigma_s(t) = {\mathbf E}[n_R(t) | s]$ as an internal model. The aim of this section is to compare the performance of these two demodulation filters.
In this comparison, we consider a medium of 1$\mu$m $\times$ $\frac{1}{3}$$\mu$m $\times$ $\frac{1}{3}$$\mu$m. We assume a voxel size of ($\frac{1}{3}$$\mu$m)$^{3}$ (i.e. $W = \frac{1}{3}$ $\mu$m), creating an array of $3 \times 1 \times 1$ voxels. The voxel co-ordinates of transmitter and receiver are, respectively, (1,1,1) and (3,1,1). A reflecting boundary condition is assumed.
The reason why we have chosen to use such a small number of voxels is because of the dimensionality of the filtering problem. For example, if each voxel can have a maximum of 100 signalling molecules at a time, then there are 10$^6$ possible $N(t)$ vectors and the filtering problem has to estimate the probability ${\mathbf P}[N(t) | s, {\cal B}(t)]$ for each possible $N(t)$ vector. Although there are approximation techniques to solve the Bayesian filtering problem, that would introduce inaccuracies. The use of small number of voxels will allow us to compute ${\mathbf P}[n_R(t) | s, {\cal B}(t)]$ precisely.
For this experiment, we use $K = 2$ symbols and two values of $M$ (the number of receptors): 5 and 10. Both Symbols 0 and 1 use Reaction \eqref{cr:rna} such that Symbols 0 and 1 causes, respectively, 10 and 50 signalling molecules to be generated per second on average by the transmitter. The simulation time is about 1.8 seconds.
We first show that ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ and $\sigma_s(t)$ are rather similar. Figure \ref{fig:prop:opt_mean} shows $\sigma_s(t)$ and a trajectory of ${\mathbf E}[n_R(t) | s, {\cal B}(t)]$ (obtained from one realisation of $b(t)$) are pretty similar. This result is obtained from using $M = 10$ and Symbol 1. The results for other choices of $M$, transmission symbols or other realisations of $b(t)$ are similar.
Figure \ref{fig:prop:opt_ser} shows the mean symbol error rates (SERs), for both optimal and sub-optimal demodulation filters, if the detection is done at time $t$ = 1, 1.05, 1.1, ..., 1.8. The SERs is obtained from 400 realisations of $b(t)$. The difference in SERs between the optimal and sub-optimal filter is less than 1\%. We have also checked that the two demodulators make the same decoding decision on average 99.3\% of the time.
In the rest of this section, we will use the sub-optimal demodulation filter \eqref{eqn:logmap_s} because of its lower computational complexity.
\subsection{Properties of the demodulator output}
\label{sec:prop_demod}
We consider a medium of 2$\mu$m $\times$ 2$\mu$m $\times$ 1 $\mu$m. We assume a voxel size of ($\frac{1}{3}$$\mu$m)$^{3}$ (i.e. $W = \frac{1}{3}$ $\mu$m), creating an array of $6 \times 6 \times 3$ voxels. The transmitter and receiver are located at (0.5,0.8,0.5) and (1.5,0.8,0.5) (in $\mu$m) in the medium. The voxel co-ordinates are (2,3,2) and (5,3,2) respectively. We assume an absorbing boundary for the medium and the signalling molecules escape from a boundary voxel surface at a rate of $\frac{d}{50}$. This configuration will be used for the rest of this section.
For this experiment, we use $K = 2$ symbols and $M = 50$ receptors. Both Symbols 0 and 1 use Reaction \eqref{cr:rna} such that
Symbols 0 and 1 causes, respectively, 40 and 80 signalling molecules to be generated per second on average by the transmitter. The simulation time is about 3 seconds.
Figure \ref{fig:prop_demod_raw_u0} shows the demodulation filter outputs $Z_0(t)$ and $Z_1(t)$ if the transmitter sends a Symbol 0. It can be seen that $Z_0(t) > Z_1(t)$ most of the time after $t = 1.2$, which means the detection is likely to be correct after this time. The sawtooth like appearance of $Z_0(t)$ and $Z_1(t)$ is due to the fact that every time when a receptor is bound, there is a jump in the filter output according to Equation \eqref{eqn:logmap_s}. Figure \ref{fig:prop_demod_raw_u1} shows the filter outputs $Z_0(t)$ and $Z_1(t)$ if the transmitter sends a Symbol 1; the behaviour is similar.
Figure \ref{fig:prop_demod_mean_u0} shows the {\sl mean} filter outputs $Z_0(t)$ and $Z_1(t)$ if the transmitter sends a Symbol 0. The mean is computed over 200 realisations of $b(t)$. It can be seen that the mean filter output of $Z_0(t)$ is greater than that of $Z_1(t)$. Similarly, if Symbol 1 is sent, then we expect of the mean of $Z_1(t)$ to be bigger. The figure is not shown for brevity.
Figure \ref{fig:prop_demod_mean_ser} shows the mean SERs for Symbols 0 and 1 if the detection is done at time $t$. The SER for Symbol 1 is high initially but as more information is processed over time, the SER drops to a low value.
This experiment shows that it is possible to use the analogue demodulation filter \eqref{eqn:logmap_s} to compute a quantity that allows us to distinguish between two emission patterns at the receiver.
\subsection{Impact of number of receptors}
We continue with the setting of \ref{sec:prop_demod} but we vary the number of receptors between 1 and 20. We assume the demodulator makes the decision at $t = 2.5$ and calculate the mean SER for both symbols at $t = 2.5$. Figure \ref{fig:prop_nrec} plots the SERs versus the number of receptors. It can be seen that the SER drops with increasing number of receptors.
We have used $K = 2$ symbols so far. We retain the current Symbols 0 and 1, and add a Symbol 2 which is also of the form of Reaction \eqref{cr:rna} but its mean rate of production of signalling molecules is 3 times that of Symbol 0. The number of receptors $M$ used are: 1, 10, 20, ..., 150.
We compute the average SER at $t = 2.5$ assuming each symbol is transmitted with equal probability. We plot the logarithm of the average SER against $\log(M)$ in Figure \ref{fig:prop_nrec_3s}.
It can be seen that the SER drops with increasing number of receptors $M$. The plot in Figure \ref{fig:prop_nrec_3s} suggests that, when the number of receptors $M$ is large, the relationship between logarithm of SER and $\log(M)$ is linear. We perform a least-squares fit for $M$ between 50 and 150. The fitted straight line is shown in Figure \ref{fig:prop_nrec_3s} and it has a slope of $-1.13$. A possible explanation is that, because the receptors are non-interacting, each receptor provides an independent observation. The empirical evidence suggests that the average SER scales according to $\frac{1}{M}$ asymptotically provided that the voxel volume can contain that many receptors.
\subsection{Distinguishability of different chemical reactions}
Equation \eqref{eqn:logmap_s} suggests that if the transmitter uses two sets of reactions which have almost the same mean number of signalling molecules in the receiver voxel, then it may be difficult to distinguish between these two symbols. In this study, Symbol 0 is generated by Reaction \eqref{cr:rna} with a rate of $\kappa$ while Symbol 1 is generated by:
\begin{align}
\cee{
[RNA]_{\rm ON} & <-> [RNA]_{\rm OFF}
} \\
\cee{
[RNA]_{\rm ON} & ->[2\kappa] [RNA]_{\rm ON} + S
}
\label{eqn:r1}
\end{align}
where we assume that RNA can be in an ON or OFF state, and signalling molecules $S$ are only produced when the RNA is in the ON-state. We assume that the there is an equal probability for the RNA to be in the two states and the reaction rate constant for the production of signalling molecule $S$ from ${\rm [RNA]}_{\rm ON}$ is $2\kappa$. This means that the mean rate of production of signalling molecules $S$ by Symbols 0 and 1 are the same. This gives rise to very similar $\sigma_0(t)$ and $\sigma_1(t)$. Figure \ref{fig:equal} shows the demodulation filter outputs $Z_0(t)$ and $Z_1(t)$ for one simulation. It can be seen that the two outputs are almost indistinguishable. Consequently, the SER is pretty high. This shows that symbols generated by reactions which have similar mean number of signalling molecules at the receiver voxel can be hard to distinguish.
\subsection{Inter-symbol interference}
The aim of this experiment is to study the performance of the demodulator in the presence of ISI. We use the decision feedback method described in Section \ref{sec:isi} together with the approximation decomposition in \eqref{eqn:sigma_sum}. We vary the number of receptors from 25 to 150. We use two different memory lengths $\ell$ of 4 and 5. If the memory length is $\ell$, we express the mean number of output molecules at the current symbol interval as a sum of $\ell$ terms, i.e. a generalisation of \eqref{eqn:sigma_sum} to $\ell$ terms. We carry out the simulation for a duration of 20 symbol lengths and compute the average SER over 20 symbols. Figure \ref{fig:isi} shows the average SER versus the number of receptors. It can be seen that an increasing number of receptors can also be used to deal with ISI.
\section{Conclusions and future work}
\label{sec:con}
This paper studies a diffusion-based molecular communication network that uses different sets of chemical reactions to represent different transmission symbols. We focus on the demodulation problem. We assume the receiver uses a ligand-receptor binding process and uses the continuous history of the number of ligand-receptor complexes over time as the input signal to the demodulator. We derive the maximum a posteriori demodulator by solving a Bayesian filtering problem.
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class NfcAdapter {
public:
NfcAdapter(PN532Interface &interface);
~NfcAdapter(void);
void begin(void);
boolean tagPresent(); // tagAvailable
NfcTag read();
boolean write(NdefMessage& ndefMessage);
// FUTURE boolean share(NdefMessage& ndefMessage);
// FUTURE boolean unshare();
// FUTURE boolean erase();
// FUTURE boolean format();
private:
PN532* shield;
byte uid[7]; // Buffer to store the returned UID
unsigned int uidLength; // Length of the UID (4 or 7 bytes depending on ISO14443A card type)
unsigned int guessTagType();
};
#endif
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Ett interkantonalt konkordat (ty. interkantonales Konkordat, fr. concordat intercantonal) är i Schweiz ett traktat mellan två eller flera kantoner. Sådana överenskommelser träffas inom områden som inte, eller delvis, omfattas av förbundsrätt, exempelvis skol-, polis- och byggnadsväsen.
Historia
Mediationsakten från 1803 upphävde alla tidigare fördrag mellan kantoner. Snart krävde dock kantonen Bern att överenskommelser med Solothurn om de kyrkliga förhållandena i det reformerta distriktet Bucheggberg i Solothurn skulle fortsätta gälla. Samma år beslöt förbundsförsamlingen (ty. Tagsatzung) att tillåta interkantonala fördrag inom vissa områden. Förbundsförsamlingen måste dock informeras om dessa. Beteckningen konkordat användes eftersom det första fallet gällde ett konfessionellt spörsmål.
Möjligheten att träffa konkordat bibehölls vid de konstitutionella förändringarna 1815 och 1848.
Vissa konkordat, exempelvis det gällande vapenhandel, har ersatts av förbundsbestämmelser.
Källor
Andreas Kley
Noter
Politik i Schweiz
Schweiz rättsväsen
|
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{"url":"https:\/\/www.freemathhelp.com\/forum\/threads\/vector-centroid-of-triangle.116264\/","text":"# Vector, centroid of triangle\n\n#### burgerandcheese\n\n##### Junior Member\n\nWhat do they mean by \"point of trisection on the median AD closer to D\"? Where is that? I know that vector d is vector AD, but what does vector a represent?\nBy the way, please tell me how to write vectors on a computer without coding. Should I have to bold the letters like b or is there a way to write vector b with the squiggly line? maybe b_~? Haha\n\n#### tkhunny\n\n##### Moderator\nStaff member\nWhat do they mean by \"point of trisection on the median AD closer to D\"? Where is that? I know that vector d is vector AD, but what does vector a represent?\nIt means only that length(DG) is length(DA)\/3 and that the same relationship holds for the other two medians. I'm not a fan of the terminology, since there are two points on each median that could be called the \"point of trisection\". Perhaps \"a point of trisection\" would be better.\n\n#### Dr.Peterson\n\n##### Elite Member\nView attachment 12244\n\nWhat do they mean by \"point of trisection on the median AD closer to D\"? Where is that? I know that vector d is vector AD, but what does vector a represent?\nBy the way, please tell me how to write vectors on a computer without coding. Should I have to bold the letters like b or is there a way to write vector b with the squiggly line? maybe b_~? Haha\nI believe they are using a, b, c, and d to mean the position vectors of points A, B, C, and D. So d is not AD, unless A is the origin. Didn't they state what the vectors are?\n\nThere are two \"points of trisection\" on AD, namely the points that divide it into three equal parts. G is the one that is closer to D, so that DG = 1\/3 DA, and AG = 2 GD.\n\nBold works fine.\n\n#### burgerandcheese\n\n##### Junior Member\nDidn't they state what the vectors are?\nThey did not. Perhaps I'm not reading properly\n\n#### Dr.Peterson\n\n##### Elite Member\nIf it was not explicitly stated in connection with this problem, they may have a convention in general that the position vector of a point is named with the corresponding lower-case letter.","date":"2019-06-17 11:46:12","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\": 0.8343579173088074, \"perplexity\": 1066.610107382208}, \"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-2019-26\/segments\/1560627998473.44\/warc\/CC-MAIN-20190617103006-20190617125006-00046.warc.gz\"}"}
| null | null |
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Hr business partner / generalist people support specialist
MindSource Austin, TX, USA
Title: HR Business Partner/Generalist II (People Support Sr. Specialist) Location: Austin, TX or Raleigh, NC Duration: 12 Months Responsibilities: Responds to inquiries from employees, managers, and People team members through phone or email all through case management system Uses a variety of tools and resources such as knowledge bases to handle inquires within service level agreements (SLA) Collaborates with stakeholders, leadership and peers on a range of activities to resolve unique and complex employee support needs and recommended process improvements Records all inquiries and resolutions in a customer relationship management system (Service Now) using the defined way of working according to established guidelines Proven ability to drive resolution on a wide range of complex benefits and time away related topics Ability to build strong partnerships and work collaboratively with partners and People team colleagues...
Title: HR Business Partner/Generalist II (People Support Sr. Specialist) Location: Austin, TX or Raleigh, NC Duration: 12 Months Responsibilities: Responds to inquiries from employees, managers, and People team members through phone or email all through case management system Uses a variety of tools and resources such as knowledge bases to handle inquires within service level agreements (SLA) Collaborates with stakeholders, leadership and peers on a range of activities to resolve unique and complex employee support needs and recommended process improvements Records all inquiries and resolutions in a customer relationship management system (Service Now) using the defined way of working according to established guidelines Proven ability to drive resolution on a wide range of complex benefits and time away related topics Ability to build strong partnerships and work collaboratively with partners and People team colleagues Observes case trends and share insights and...
Corebridge Financial Houston, TX, USA
Who We Are Corebridge Financial helps people make some of the most meaningful decisions they're ever going to make. We help them plan and take action to protect the future they envision and respond to some of life's most difficult moments through the solutions and services we provide. We do this through our broad portfolio of life insurance, retirement, and institutional products, offered through an extensive, multi-channel distribution network. We provide solutions for a brighter future through our client-centered service, breadth of product expertise, deep distribution relationships, and outstanding team of hardworking and passionate employees. About The Position The Human Resources Business Partner is responsible for taking an outside-in approach on current and future human capital challenges. This requires business credibility, the ability to partner with leaders, and a deep HR and OD functional knowledge. This role will partner with leaders on strategic business...
Specialist - hr business partner
ApTask Tampa, FL, USA
Role: Specialist- HR Business Partner Location: Tampa, FL (onsite with Hybrid) Employment: Fulltime Job Description: Required Skills/Abilities: Excellent verbal and written communication skills. Excellent interpersonal and customer service skills. Excellent organizational skills and attention to detail. Ability to comprehend, interpret, and apply the appropriate sections of applicable laws, guidelines, regulations and policies. Ability to acquire a thorough understanding of the organization's hierarchy, jobs, qualifications, compensation practices, and the administrative practices related to those factors. Excellent time management skills with a proven ability to meet deadlines. Strong analytical and problem-solving skills. Proficient with Microsoft Office Suite or related software Education and Experience: Minimum of 5-7 years of experience in the Human Resources function, / IT Outsourcing space Working knowledge of multiple human...
HR Search Pros, Inc. HR (Human Resources) Executive Search Firm Irving, TX, USA
JOB POSTING #998 TITLE: HR Business Partner INDUSTRY: Manufacturing LOCATION: Irving, TX COMPENSATION: Depends on experience RELOCATION: Local candidates only GREAT, GROWING GLOBAL COMPANY NEEDS AN HR BUSINESS PARTNER TO SUPPORT THEIR CORPORATE FUNCTIONS. Summary Provide the full range of HR support to several of the company's corporate functions Work closely with business leaders coaching and advising them on all things HR (i.e., talent management and engagement, learning/OD, leadership development, etc.) Individual contributor role Requirements 7+ years of HR Business Partner experience Broad experience in all areas of HR Strong consultative/coaching skills Project management experience Strong analytical/problem-solving/decision-making skills Global experience is a plus Bachelor's degree To apply, please visit: https://hrsearchpros.com/jobs OR CONTACT: Ralph Chapman Email: Rchapman@HRSearchPros.com Please...
Vivir Healthcare Sydney NSW, Australia
Role: Permanent Full time Location: Sydney Remuneration up to $100k depending on experience About Vivir Healthcare Vivir Healthcare, Australia's leading Healthcare Provider within the Aged, Community and Disability. We make a difference to over 20,000 lives every week by providing Allied Health solutions in residential aged care facilities, community, as well as in the homes of elderly people, and retirement villages. Our team has been working to improve the lives of elderly Australians for over 20 years, blending expert aged care services with a genuine commitment to high-quality care. About The Role An exciting position has just become available for a HR Business Partner to join us in our Sydney office. As the HR Business Partner, you will provide effective and expert advice and solutions on all people issues, facilitating the optimum performance of the business. Your Responsibilities Would Include But Are Not Limited To Ensure compliance to Legal, IR,...
Senior people hr business partner
AnheuserBusch Jacksonville, FL, USA
Dreaming big is in our DNA. Brewing the world's most loved beers and creating meaningful experiences is what inspires us. We are owners, empowered to lead real change, deliver on tough challenges, and take accountability for the results. We are looking for talent that shares these values, that is ambitious, bold & resilient. We want talent that is looking for fast career growth, cross-functional experiences, global exposure and robust training & development. SALARY: $130-400 - $146,700 COMPANY: Michelob ULTRA. Cutwater Spirits. Budweiser. Kona Brewing Co. Stella Artois. Bud Light. That's right, over 100 of America's most loved brands, to be exact. But there's so much more to us than our top-notch portfolio of beers, seltzers, and more. We are powered by a 19,000-strong team that shares our passion to create a future with more cheers. We look for people with talent, curiosity, and commitment and provide the teammates, resources and opportunities to unleash their...
CornerStone Staffing Dallas, TX, USA
3.8 Quick Apply Full-time 1 day ago Full Job Description The City of Dallas is seeking an experienced HR Business Partner who has strong experience is employee relations and human resources consultation Position: HR Business Partner Location: Dallas, TX Pay: $30/hour Schedule: Mon-Fri, business hours Duties/Responsibilities: Provides human resources consultation and support to a designated business unit to define and execute HR strategies that enable accomplishment of business objectives. Contributes to the development of workforce plans and understands external customer trends and issues in the industry that could potentially impact business. Education and Experience: Bachelor's degree in human resources, business/public administration or social science field 2+ years of previous HRBP experience To Apply for this Job: Click the Apply Online button, then: If you are currently registered with CornerStone...
Flex Austin, TX, USA
3.9 Full-time 1 hour ago Full Job Description At Flex, we welcome people of all backgrounds. Our employees thrive here by living our values: we support each other as we strive to find a better way, we move fast with discipline and purpose, and we do the right thing always. Through a respectful, inclusive and collaborative culture, a career at Flex offers the opportunity to make a difference, invest in your career growth and join our purpose - to make great products that create value and improve people's lives. Job Summary To support our extraordinary teams who build great products and contribute to our growth, we're looking to add a HR Business Partner located in Austin, TX. Reporting to the Site HR Business Partner Director, the HR Business Partner role involves a self-starter person who demonstrates accountability for results. The ideal candidate will have 8-10 years of progressive experience in HR across all primary functions. The HRBP is...
TIAA Bank Jacksonville, FL, USA
HR Business Partner - TIAA Bank The HR Business Partner job oversees a specialized type of HR work focused on HR consulting to the business. While managing large projects or processes with minimal oversight, this job consults with senior business leaders to solve significant people and cultural issues, oversees the talent aspects of organization structure changes and communicates the business value of HR initiatives. This job is considered a subject matter expert in the Human Resources field. Key Responsibilities And Duties Acts as a HR solutions partner to business and functional leaders delivering consistent advice and coaching to support and enable local business strategy. Liaises with functional or operational managers to implement local human resources programs that are appropriate for their business needs, consistent with the organization's overall human resources strategy. Partners with the business and talent management/staffing and recruiting colleagues on key...
AIT Worldwide Logistics Dallas, TX, USA
Wake up each day to be part of a movement. A movement to make an impact. A movement to go for it. A movement to provide solutions. A movement to be empowered. A movement to do the right thing. Our people deliver.® Empowerment, active listening, and creative thinking are a few attributes AIT teammates exercise to deliver a world-class customer experience. Forbes names AIT Worldwide Logistics one of America's Best Employers in 2022! Come join us at AIT and see why our employees rank us among the best places to work. Position Overview: The Sr. HR Business Partner will play a critical role in delivering the HR strategy across their regions. The Sr. HR Business Partner will be responsible for strategic and operational support to managers across their regions, with responsibilities related to performance management, coaching/training of People Leaders, new hire on-boarding, employee relations, career development and...
Uniting Remote (Harris Park NSW 2150, Australia)
Human Resources Business Partner At Uniting, we believe in taking real steps to make the world a better place. We work to inspire people, enliven communities, and confront injustice. About the opportunity: We currently have an opportunity for an energetic, creative, and highly customer-centric HR (Human Resources) professional to join our People and Culture team. Reporting to the HR Business Partnering Lead you will partner with the Head of Operations and their management team to provide strategic and tactical HR advice that supports our people to deliver exceptional services to our clients. As a true HR generalist, you will have the opportunity to work on various leadership, capability, culture, engagement, and operational programs that deliver our people strategy. About the team: The People and Culture team at Uniting moves at pace, has fun, and consistently strive to build a better future for our people on the frontline. We...
HubSpot Australia
Our HubSpot People Operations team prides itself on our innovative and flexible approach to HR, and we are looking for an exceptional individual to join us in either Sydney, Singapore or Japan in Senior HRBP position. In this role you will partner with all business functions in the region. You will provide People Operations support across the company on a variety of HR-related matters including, but not limited to: manager coaching, team mentoring, international operations, talent management, performance management, and conflict resolution - in a way that may not be traditional to HR but embraces our unique culture and solves for our rapid business growth. It's an exciting time and place to be a part of People Operations! In This Role You Will Successfully operate as a HR and employment law consultant to the JAPAC business. Provide strategic business partnership and coaching to people managers to positively impact the motivation, development, and retention of talent....
Raytheon Intelligence & Space Sydney NSW, Australia
Build close relationships with your stakeholders Work for one of Australia's leading Defence companies Competitive salary + 12% Super Join a team passionate about the work we do! Raytheon Australia is the nation's leading provider of whole-of-life capabilities for the Australian Defence Force. Our team of 1,500 employees have ensured our long-term incumbency, positioning us to successfully deliver on a range of diverse programs including delivery of advanced combat-enabling technologies, integrated weapons systems, communications, cyber security, and capability lifecycle management solutions. We are seeking two HR Business Partner's to join our People and Culture team, partnering with Business Managers and Leadership teams to identify, develop and implement solutions across a range of HR activities including recruitment, workforce planning and performance management. This role will see you providing a consultancy and advisory service to managers and employees...
Raytheon Australia Macquarie Park NSW 2113, Australia
Build close relationships with your stakeholders Work for one of Australia's leading Defence companies Competitive salary + 12% Super Join a team passionate about the work we do! Raytheon Australia is the nation's leading provider of whole-of-life capabilities for the Australian Defence Force. Our team of 1,500 employees have ensured our long-term incumbency, positioning us to successfully deliver on a range of diverse programs including delivery of advanced combat-enabling technologies, integrated weapons systems, communications, cyber security, and capability lifecycle management solutions. We are seeking two HR Business Partner's to join our People and Culture team, partnering with Business Managers and Leadership teams to identify, develop and implement solutions across a range of HR activities including recruitment, workforce planning and performance management. This role will see you providing a consultancy and advisory service to managers and employees on...
Xoriant Corporation Austin, TX, USA
Roles and Responsibilities • Coach, advise and partner with management and employees on a variety of areas including performance management, employee engagement, and employee relations • Drive execution excellence and organizational change on initiatives including talent reviews, performance management, succession planning, diversity, equity and inclusion, rewards and recognition, learning & development, and overall process improvements • Listen to and conduct thorough and objective investigations with regard to employee concerns or complaints • Partner and collaborate with all functional HR groups including, Compensation, HRIS, Learning & Development, and Talent Acquisition • Work closely with management and employees to improve work relationships, build morale, and increase productivity and retention • Participate in and/or lead HR projects and initiatives • Work independently and exercise sound judgment in all activities • Analyze trends and metrics in...
Gartner Irving, TX, USA
About this Role: The HR Partner will collaborate with the HR team and the business to implement HR initiatives in line with the organization's strategic objectives. By developing strong internal client relationships and providing coaching, the HRP will help drive people management initiatives. Working as part of an influential, global HR team, the HR Partner (HRP) will work with a strong multi-disciplined group of colleagues. Key responsibilities include providing HR consultation and support in the areas of performance management, change management, recruiting life cycle and employee relations. What you'll do: Employee Relations – Serve as primary point of contact for managers and associates in actively responding to and addressing associate concerns in a timely manner. Conduct complete investigations and make recommendations based on findings/facts. Escalate issues to HR leadership and BU leadership as appropriate. Collaborate with HR leadership and inside counsel...
NextEra Energy, Inc. Houston, TX, USA
Requisition ID: 66977 NextEra Energy Resources is the world's largest generator of renewable energy from the wind and sun, and a world leader in battery storage. We provide energy-related products and services that grow our economy, protect the environment, support our communities and help customers meet their energy needs. We are leading the decarbonization of the U.S. economy with our goal to reach Real Zero carbon emissions from our operations by 2045 while improving customer affordability and reliability. Are you interested in creating a cleaner environment for future generations? Join our world-class, innovative team today. Position Specific Description The Sr Human Resources Business Partner (Sr HRBP) will report to the VP Human Resource for NextEra Energy Resources. The Sr HRBP provides strategic consulting on people and organization development in support of several high growth business units throughout NextEra Energy Resources. The Sr HRBP will use their...
NextEra Energy Resources Houston, TX, USA
Apex Group The Rocks NSW 2000, Australia
The Role & Key Responsibilities Skills Required Additional Information The role duties and responsibilities will include but not limited to the following: Work with business stakeholders to provide solutions on HR topics Support the Senior HRBP in delivering actions plans for HR policy and procedures Partner with Talent Acquisition and hiring managers to ensure the right talent is secured into the business including offering new employees Onboard and induct new employees into the business and work with leaders to ensure the success of new recruits Work closely with the region and local HR team to deliver on key KPIs and objectives Assist the Senior HRP to implement the people strategy and key employee lifecycle activities Provide HR data insights and HR reports for the local AU business Assist in the management of local payroll Work closely with the Head of HR to drive strategic HR projects across the region Ad hoc projects as required 3+ years HR...
© 2021/2022 Institute of Data
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| 9,596
|
{"url":"https:\/\/electronics.stackexchange.com\/questions\/601655\/i2c-slave-pulls-down-scl-while-doing-nothing","text":"# I2C slave pulls down SCL while doing nothing\n\nI have an MCU (Texas Instruments tm4c123gh6pm) in which I have configured an I2C slave device. I have taken 3.3V from the MCU pin that provides them and connect them to two 4.7 kOhm resistors (to make the pull-ups). I have tried with 2 1.5V batteries in series as well.\n\nWhen I connect one of the above two lines to the slave SDA, it remains high (fine) but, for some reason, as soon as I connect SCL to the other pull-up res., it goes low (to 0.8 V or so). For some reason, the slave's SCL pulls down the line while the it is doing nothing (I haven't even connected it to the master).\n\nWhy this happens?\n\nHere is how I initialize the I2C slave:\n\nstatic void I2C1_Init(void) {\nSysCtlPeripheralEnable(SYSCTL_PERIPH_I2C1);\nSysCtlPeripheralEnable(SYSCTL_PERIPH_GPIOA);\n\nGPIOPinConfigure(GPIO_PA6_I2C1SCL);\nGPIOPinConfigure(GPIO_PA7_I2C1SDA);\n\nGPIOPinTypeI2CSCL(GPIO_PORTA_BASE, GPIO_PIN_6);\nGPIOPinTypeI2C(GPIO_PORTA_BASE, GPIO_PIN_7);\n\nI2CSlaveEnable(I2C1_BASE);\n}\n\n\n## IMPORTANT EDIT\n\nI'm providing the pull-up resistors from an external power supply, whose ground I connect to the MCU GND pin. The MCU is powered by a laptop through its USB port. I have observed that the problem above doesn't arise if I don't connect these 2 grounds, but as far as I understand I do need to connect them, right?","date":"2022-08-16 19:34:59","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\": 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\": 0.5141334533691406, \"perplexity\": 2787.16029541168}, \"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-2022-33\/segments\/1659882572515.15\/warc\/CC-MAIN-20220816181215-20220816211215-00221.warc.gz\"}"}
| null | null |
package com.netease.nim.demo.chatroom.viewholder;
import android.text.TextUtils;
import android.view.View;
import android.widget.ImageView;
import android.widget.TextView;
import com.netease.nim.demo.R;
import com.netease.nim.demo.chatroom.widget.ChatRoomImageView;
import com.netease.nim.uikit.common.adapter.TViewHolder;
import com.netease.nimlib.sdk.chatroom.constant.MemberType;
import com.netease.nimlib.sdk.chatroom.model.ChatRoomMember;
/**
* Created by hzxuwen on 2015/12/18.
*/
public class OnlinePeopleViewHolder extends TViewHolder {
private static final String TAG = OnlinePeopleViewHolder.class.getSimpleName();
private ChatRoomImageView userHeadImage;
private TextView nameText;
private ChatRoomMember chatRoomMember;
private ImageView identityImage;
@Override
protected int getResId() {
return R.layout.online_people_item;
}
@Override
protected void inflate() {
identityImage = findView(R.id.identity_image);
userHeadImage = findView(R.id.user_head);
nameText = findView(R.id.user_name);
}
@Override
protected void refresh(Object item) {
chatRoomMember = (ChatRoomMember) item;
if (chatRoomMember.getMemberType() == MemberType.CREATOR) {
identityImage.setVisibility(View.VISIBLE);
identityImage.setImageDrawable(context.getResources().getDrawable(R.drawable.master_icon));
} else if (chatRoomMember.getMemberType() == MemberType.ADMIN) {
identityImage.setVisibility(View.VISIBLE);
identityImage.setImageDrawable(context.getResources().getDrawable(R.drawable.admin_icon));
} else {
identityImage.setVisibility(View.GONE);
}
userHeadImage.loadAvatarByUrl(chatRoomMember.getAvatar());
nameText.setText(TextUtils.isEmpty(chatRoomMember.getNick()) ? "" : chatRoomMember.getNick());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,299
|
The Oberlin Review is proud to announce "The Friends of The Oberlin Review Endowment"!
We have until August 31 to raise $25,000, the minimum needed to establish an endowed fund.
The five percent return on the fund will supplement the Review's general operating costs. This money will allow us to work on projects that were otherwise not financially feasible.
If we don't make the goal in a year, any money we do raise will be put into a current use fund designated for our general operating costs and we would target it for updating our equipment and educating our staff through speakers and conferences.
We're confident, though, that we'll be able to reach our goal, ensure the financial future of the Review and preserve our journalistic integrity.
A five-year donation pledge of $100 from 50 people will easily get us to our goal.
Make sure to put in the memo line 'Friends of The Oberlin Review Endowment'.
Remember that your company may have a gift-matching program!
If you can't make a dollar donation, contact the Long Term Manager, Ireta Kraal, '02, at endowment@oberlinreview.org about other ways you can help out.
Established in 2001 by current and former members of The Oberlin Review Staff, this fund is created to support the operating budget of The Oberlin Review. At such time as the balance in this fund reaches $25,000, this fund shall become a permanently endowed fund. The income from this fund, which shall be expended at the Oberlin College Board-approved payout rate, is to be used to provide general operating support for The Oberlin Review on an annual basis. If the fund does not reach $25,000 by August 31, 2002, this fund shall become a current-use fund to support The Oberlin Review during fiscal year 2002-03.
Read about the endowment in the November 9, 2001 edition of the review. Click here.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,691
|
\section{Introduction}
In recent years, Boolean SAT solving techniques have improved
dramatically. Today's SAT solvers are considerably faster and able to
manage larger instances than yesterday's. Moreover, encoding and
modeling techniques are better understood and increasingly
innovative. SAT is currently applied to solve a wide variety of hard
and practical combinatorial problems, often outperforming
dedicated algorithms.
The general idea is to encode a (typically, NP) hard problem instance,
$\mu$, to a Boolean formula, $\varphi_\mu$, such that the solutions of
$\mu$ correspond to the satisfying assignments of $\varphi_\mu$. Given
the encoding, a SAT solver is then applied to solve $\mu$.
Tailgating the success of SAT technology are a variety of tools which
can be applied to specify and then compile problem instances to
corresponding SAT instances.
For example, \citeN{npspec2005} introduce \textsf{NP-SPEC}, a
logic-based specification language which allows to specify
combinatorial problems in a declarative way. At the core of this
system is a compiler which translates specifications to CNF formula.
The general objective of such tools is to facilitate the process of
providing high-level descriptions of how the (constraint) problem at
hand is to be solved. Typically, a constraint based modeling language
is introduced and used to model instances. Drawing on the analogy
to
programming languages, given such a description, a
compiler then provides a low-level executable for the underlying
machine. Namely, in our context, a formula for the underlying SAT or
SMT solver.
One obstacle when seeking to optimize CNF encodings derived from
high-level descriptions, is that CNF encodings are
\textit{``bit-level''} representations and do not maintain
\textit{``word-level''} information. For example, from a CNF encoding
one cannot know that certain bits originate from the same integer
value in the original constraint. This limits the ability to apply
optimizations which rely on such word-level information.
We mention two relevant tools.
Sugar \cite{sugar2009}, is a SAT-based constraint solver. To solve a
finite domain linear constraint satisfaction problem it is first
encoded to a CNF formula by Sugar, and then solved using the MiniSat
solver \cite{minisat2003}. \bee\ is like Sugar, but applies
optimizations. Sugar is the first system which demonstrates the
advantage in adopting the, so-called, unary order-encoding to
represent integers. We follow suite, and introduce additional novel
encoding techniques that take advantage of, previously unobserved,
properties of the order-encoding.
MiniZinc~\cite{miniZinc2007}, is a constraint modeling language which
is compiled by a variety of solvers to the low-level target language
FlatZinc for which there exist many solvers. It creates a standard for
the source language (which we follow loosely). \bee\ is like FlatZinc,
but with a focus on a subset of the language relevant for finite
domain constraint problems.
We present a tool, \bee\ {\small (\textbf{\textsf{B}}en-Gurion
University \textbf{\textsf{E}}qui-propagation
\textbf{\textsf{E}}ncoder)} which translates models in a constraint
based modeling language, similar to Sugar and FlatZinc, to CNF.
Conceptually, \bee\ maintains two representations for each constraint
in a model so that each constraint is also viewed as a Boolean
function. Partial evaluation, and other word-level techniques, drive
simplification through the constraint part; whereas, equi-propagation
\cite{Metodi2011}, and other bit-level techniques, drive
simplification through the Boolean part.
Finally, an encoding technique is selected for a constraint, depending
on its context, to derive a CNF.
The name, ``\bee'' refers both to the constraint language as well
as to its compiler to CNF. \bee\ is not a constraint solver, but can
be applied in combination with a SAT solver to solve finite domain
constraint problems.
We report on our experience with applications which indicates that
using \bee, like any compiler, has two main advantages. On the one
hand, it facilitates the process of programming (or modeling). On the
other hand, given a program (a model), it simplifies the corresponding
CNF which, in many cases, is faster to solve than with other
approaches. The tool integrates with SWI Prolog and can be downloaded
from~\cite{bee2012}.
\section{Representing Integers}
A fundamental design choice when encoding finite domain constraints
concerns the representation of integer variables.
\citeN{DBLP:conf/cp/Gavanelli07} surveys several of the possible
choices (the \emph{direct-}, \emph{support-} and \emph{log-}
\emph{encodings}) and introduces the \emph{log-support encoding}.
We focus in this paper on the use of unary representations and
primarily on the, so-called, \emph{order-encoding} (see
e.g.~\cite{baker,BailleuxB03}) which has many nice properties when
applied to small finite domains. We describe the setting where all
integer variables are represented in the order-encoding except for
those involved in a global ``all-different'' constraint which take a
dual representation with channeling between the order-encoding and the
\emph{direct encoding}. This choice derives from the observation by
\citeN{direct4allDiff} that the direct-encoding is superior when
encoding the all-different constraint.
Let bit vector $X=[x_1,\ldots,x_n]$ represent a finite domain integer
variable. In the \emph{order-encoding}, $X$ constitutes a monotonic
non-increasing Boolean sequence. Bit $x_i$ is interpreted as $X\geq
i$. For example, the value 3 in the interval $[0,5]$ is represented
in 5 bits as $[1,1,1,0,0]$. In the \emph{direct-encoding}, $X$
constitutes a characteristic function (exactly one bit takes value 1)
and $x_i$ is interpreted as stating $X= i-1$. For example, the value
3 in the interval $[0,5]$ is represented in 6 bits as $[0,0,0,1,0,0]$.
An important property of a Boolean representation for finite domain
integers is the ability to represent changes in the set of values a
variable can take.
It is well-known that the order-encoding facilitates the propagation
of bounds. Consider an integer variable $X=[x_1,\ldots,x_n]$ with
values in the interval $[0,n]$. To restrict $X$ to take values in the
range $[a,b]$ (for $1\leq a\leq b\leq n$), it is sufficient to assign
$x_{a}=1$ and $x_{b+1}=0$ (if $b<n$). The variables $x_{a'}$ and
$x_{b'}$ for $0\geq a'> a$ and $b<b'\leq n$ are then determined true
and false, respectively, by \emph{unit propagation}. For example,
given $X=[x_1,\ldots,x_9]$, assigning $x_3=1$ and $x_6=0$ propagates
to give $X=[1,1,1,x_4,x_5,0,0,0,0]$, signifying that
$dom(X)=\{3,4,5\}$.
This property is exploited in Sugar \cite{sugar2009} which also
applies the order-encoding.
We observe, and apply in \bee, an additional property of the
order-encoding: its ability to specify that a variable cannot take a
specific value $0\leq v\leq n$ in its domain by equating two
variables: $x_{v}=x_{v+1}$.
This indicates that the order-encoding is well-suited not only to
propagate lower and upper bounds, but also to represent integer
variables with an arbitrary, finite set, domain.
For example, given $X=[x_1,\ldots,x_9]$, equating $x_2=x_3$ imposes
that $X\neq 2$. Likewise $x_5=x_6$ and $x_7=x_8$ impose that $X\neq 5$
and $X\neq 7$. Applying these equalities to $X$ gives,
$X=[x_1,\underline{x_2,x_2},x_4,\underline{x_5,x_5},\underline{x_7,x_7},x_9]$,
signifying that $dom(X)=\{0,1,3,4,6,8,9\}$.
The order-encoding has many additional nice features that are
exploited in \bee\ to simplify constraints and their encodings to
CNF. To illustrate one, consider a constraint of the form
$\mathtt{A+B=5}$ where \texttt{A} and \texttt{B} are integer values in
the range between 0 and 5 represented in the order-encoding. At the
bit level we have: $\mathtt{A=[a_1,\ldots,a_5]}$ and
$\mathtt{B=[b_1,\ldots,b_5]}$. The constraint is satisfied precisely
when $\mathtt{B=[\neg a_5,\ldots,\neg a_1]}$. Instead of encoding the
constraint to CNF, we substitute the bits $\mathtt{b_1,\ldots,b_5}$ by
the literals $\mathtt{\neg a_5,\ldots,\neg a_1}$, and remove the
constraint. In Prolog, this is implemented as a unification and does
not generate any clauses in the encoding.
\section{Constraints in \bee}
Boolean constants ``$\true$'' and ``$\false$'' are viewed as
(integer) values ``1'' and ``0''.
Constraints are represented as (a list of) Prolog terms. Boolean and
integer variables are represented as Prolog variables, which may be
instantiated when simplifying constraints.
Table~\ref{tab:beeStntax} introduces the syntax for (a simplified
subset of) \bee. In the table, $\mathtt{X}$ and $\mathtt{Xs}$ (possibly
with subscripts) denote a literal (a Boolean variable or
its negation) and a vector of literals,
$\mathtt{I}$ (possibly with subscript)
denotes an integer variable, and $\mathtt{c}$ (possibly with
subscript) denotes an integer constant.
\begin{table}[t]
\centering
\begin{tabular}{rlll}
\hline\hline
\multicolumn{4}{l}{\bf\small Declaring Variables}\\
\hline
(1) &$\mathtt{new\_bool(X)}$ &\qquad\qquad &declare Boolean \texttt{X}
\\
(2) &$\mathtt{new\_int(I,c_1,c_2)}$ & &
declare integer \texttt{I}, $\mathtt{c_1\leq I\leq c_2}$ \\
(3) & $\mathtt{ordered([X_1,\ldots,X_n])}$ &
&
$\mathtt{X_1\geq X_2\geq\cdots\geq X_n}$ (on Booleans)\\
\hline
\multicolumn{4}{l}{\bf\small Boolean (reified) Statements~
\hfill $\mathtt{op\in\{or, and, xor, iff\}}$}\\
\hline
(4) & $\mathtt{bool\_eq(X_1,X_2)}$ ~or~ $\mathtt{bool\_eq(X_1,-X_2)}$&
$\mathtt{}$&
$\mathtt{X_1 = X_2}$ ~or~ $\mathtt{X_1 = \neg X_2}$\\
(5) & $\mathtt{bool\_array\_op([X_1,\ldots,X_n])}$ &
$\mathtt{}$&
$\mathtt{X_1 ~op~ X_2 \cdots op~ X_n}$\\
(6) & $\mathtt{bool\_array\_op\_reif([X_1,\ldots,X_n],~X)}$ &
$\mathtt{}$&
$\mathtt{X_1 ~op~ X_2 \cdots op~ X_n\Leftrightarrow X}$\\
(7) & $\mathtt{bool\_op\_reif(X_1,X_2,~X)}$ &
$\mathtt{}$&
$\mathtt{X_1 ~op~ X_2\Leftrightarrow X}$\\
(8) & $\mathtt{bool\_array\_lex(Xs_1,Xs_2)}$ &$\mathtt{}$&
$\mathtt{Xs_1}$ precedes $\mathtt{Xs_2}$ in the lex order\\
\hline
\multicolumn{4}{l}{\bf\small Integer relations (reified)
\hfill $\mathtt{rel\in\{leq, geq, eq, lt, gt, neq\}}$}\\
\multicolumn{4}{l}{\bf\small and arithmetic \hfill
~$\mathtt{op\in\{plus, times, div, mod, max, min\}}$,
$\mathtt{op'\in\{plus, max, min\}}$ }\\
\hline
(9) & $\mathtt{int\_rel(I_1,I_2)}$ &
$\mathtt{}$&
$\mathtt{I_1 ~rel~ I_2}$\\
(10) & $\mathtt{int\_rel\_reif(I_1,I_2,~X)}$ &
$\mathtt{}$&
$\mathtt{I_1 ~rel~ I_2 \Leftrightarrow X}$\\
(11) &$\mathtt{int\_op(I_1,I_2,~I)}$ &
$\mathtt{}$&
$\mathtt{I_1 ~op~ I_2 = I}$\\
(12) &$\mathtt{int\_array\_op'([I_1,\ldots,I_n],~I)}$ &
$\mathtt{}$&
$\mathtt{I_1 ~op'\cdots op'~ I_n = I}$\\
\hline
\multicolumn{4}{l}{\bf\small All Different and cardinality~
\hfill $\mathtt{rel{\in}\{leq, geq, eq, lt, gt, neq\}}$}\\
\hline
(13) & $\mathtt{allDiff([I_1,\ldots,I_n])}$ &
&
$\mathtt{\bigwedge_{i<j}I_i \neq I_j}$\\
(14) & $\mathtt{bool\_array\_sum\_rel([X_1,\ldots,X_n],~I)}$ &
$\mathtt{}$&
$\mathtt{(\Sigma ~X_i)~ rel~ I}$\\
(15) & $\mathtt{comparator(X_1,X_2,X_3,X_4)}$ &
&
$\mathtt{sort([X_1,X_2])=[X_3,X_4]}$\\
\hline\hline
\end{tabular}
\caption{Syntax for a subset of \bee. }
\label{tab:beeStntax}
\end{table}
On the right column of the table are brief explanations regarding the
constraints. The table introduces 15 constraint templates.
Constraints (1-2) are about variable declarations: Booleans and
integers. Constraint (3) signifies that a bit sequence is monotonic
non-increasing, and is used to specify that an integer variable is in
the order-encoding.
Constraints (4-7) are about Boolean (and reified Boolean)
statements. The cases for $\mathtt{bool\_array\_or([X_1,\ldots,X_n])}$
and $\mathtt{bool\_array\_xor([X_1,\ldots,X_n])}$ facilitate the
specification of clauses and of \texttt{xor} clauses (supported in the
CryptoMiniSAT solver~\cite{Crypto}).
Constraint (8) specifies that two bit-vectors are ordered
lexicographically.
Constraints (9-12) are about integer relations and operations.
Constraints (13-14) are the all-different constraint on integers and
the cardinality constraint on Booleans. Constraint (15) specifies
that sorting a bit pair $\mathtt{[X_1,X_2]}$ (decreasing order)
results in the pair $\mathtt{[X_3,X_4]}$. This is a basic building
block for the construction of sorting networks \cite{Batcher68} used
to encode cardinality constraints during compilation as described
in~\cite{AsinNOR11} and in~\cite{DBLP:conf/lpar/CodishZ10}.
\section{An Example \bee\ Application: magic graph labeling}
\label{sec:magic}
We illustrate the application of \bee\ to solve a
graph labeling problem.
A typical \bee\ application has the form depicted as
Figure~\ref{fig:generic} where the predicate \texttt{solve/2} takes a
problem \texttt{Instance} and provides a \texttt{Solution}. The
specifics of the application are in the call to \texttt{encode/3}
which given the \texttt{Instance} generates the \texttt{Constraints}
that solve it together with a \texttt{Map} relating instance variables with
constraint variables. The calls to \texttt{compile/2} and
\texttt{sat/1} compile the constraints to a \texttt{CNF} and solve it
applying a SAT solver. If the instance has a solution, the SAT solver
binds the constraint variables accordingly. Then, the call to
\texttt{decode/2}, using the \texttt{Map}, provides a
\texttt{Solution} in terms of the instance variables.
The definitions of \texttt{encode/3} and \texttt{decode/3} are
application dependent and provided by the user. The predicates
\texttt{compile/2} and \texttt{sat/1} provide the interface to \bee\
and the underlying SAT solver.
\begin{figure}[t]
\begin{SProg}
:- use\_module(bee\_compiler, [compile/2]). \\
:- use\_module(sat\_solver, [sat/1]). \\
\vspace{-3mm}\\
solve(Instance, Solution) :- \\
\qquad encode(Instance, Map, Constraints),\\
\qquad compile(Constraints, CNF), \\
\qquad sat(CNF), \\
\qquad decode(Map, Solution).\\
\end{SProg}
\caption{A generic application of \bee.}
\label{fig:generic}
\end{figure}
Graph labeling is about finding an assignment of integers to the
vertices and edges of a graph subject to certain conditions. Graph
labelings were introduced in the 60's and hundreds of papers on a wide
variety of related problems have been published since then. See for
example the survey by \citeN{Gallian2011} with more than 1200
references. Graph labelings have many applications. For instance in
radars, xray crystallography, coding theory, etc.
We focus here on the vertex-magic total labeling (VMTL) problem where
one should find for the graph $G=(V,E)$ a labeling that is a
one-to-one map $V\cup E \rightarrow \{1,2,\ldots,|V|+|E|\}$ with the
property that the sum of the labels of a vertex and its incident edges
is a constant $K$ independent of the choice of vertex.
A problem instance takes the form $vmtl(G,K)$ specifying the graph $G$
and a constant $K$. The query $\mathtt{solve(vmtl(G,K), Solution)}$
poses the question: ``Does there exist a vmtl labeling for $G$ with
magic constant $K$?'' It binds $\mathtt{Solution}$ to indicate such a
labeling if one exists, or to ``unsat'' otherwise.
\begin{figure}[t]
\centering
\begin{tabular}{l}
\hline
\hspace{1cm} An Instance \hspace{2cm} The Graph \hspace{2cm} The Map\\
\hline
$\begin{array}{l}
\mathtt{Instance = vmtl(G,K),}\\
\mathtt{G=(V,E),}\\
\mathtt{V=[1,2,3,4],}\\
\mathtt{E=[(1,2),(1,3),}\\
\qquad \mathtt{(2,3),(3,4)], }\\
\mathtt{K=14}
\end{array}$\quad\qquad
$\begin{array}{l}
\xymatrix@C=9pt@R=15pt{
& 4\ar@{-}[d] & \\
& 3\ar@{-}[dl]\ar@{-}[dr] & \\
2\ar@{-}[rr]&&1 }
\end{array}$
\qquad\quad$\mathtt{M=} \left[\begin{array}{ll}
\mathtt{((1,2),~E_1),} & \mathtt{(1,~V_1),} \\
\mathtt{((1,3),~E_2),} & \mathtt{(2,~V_2),} \\
\mathtt{((2,3),~E_3),} & \mathtt{(3,~V_3),} \\
\mathtt{((3,4),~E_4),} & \mathtt{(4,~V_4)} \\
\end{array} \right]$\\
\hline
\hspace{1cm} The Constraints\\
\hline
\qquad$\mathtt{Cs=}
\left[\begin{array}{lll}
\mathtt{new\_int(V_{1}, 1, 8),} &
\mathtt{new\_int(E_{1}, 1, 8),} &
\mathtt{int\_array\_plus([V_1,E_1,E_2], K),}
\\
\mathtt{new\_int(V_{2}, 1, 8),} &
\mathtt{new\_int(E_{2}, 1, 8),} &
\mathtt{int\_array\_plus([V_2,E_1,E_3], K),}
\\
\mathtt{new\_int(V_{3}, 1, 8),} &
\mathtt{new\_int(E_{3}, 1, 8),} &
\mathtt{int\_array\_plus([V_3,E_2,E_3,E_4], K),}
\\
\mathtt{new\_int(V_{4}, 1, 8),} &
\mathtt{new\_int(E_{4}, 1, 8),} &
\mathtt{int\_array\_plus([V_4,E_4], K),}
\\
\mathtt{new\_int(K, 14, 14),} &
\multicolumn{2}{l}{
\mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4])}}
\end{array}\right]$\\
\hline
\end{tabular}
\caption{A VMTL instance with the constraints and map generated by
\texttt{encode/3}. }
\label{fig:vmtl-instance}
\end{figure}
Figure~\ref{fig:vmtl-instance} illustrates an example problem instance
together with the constraints, \texttt{Cs} and the map, \texttt{M},
generated by the \texttt{encode/3} predicate for this instance.
The constraints introduce integer variables for the vertices and
edges, specify that these variables take ``all different'' values,
and specify that the labels for each vertex with its incident edges
sum to $\mathtt{K}$.
Solving the constraints from Figure~\ref{fig:vmtl-instance} for the
example VMTL instance binds the Map, \texttt{M}, as follows,
indicating a solution:
\[\mathtt{M=}{\small \left[\begin{array}{ll}
\mathtt{((1,2),~[1,1,1,1,1,1,1,0]),} & \mathtt{(1,~[1,1,1,1,0,0,0,0]),} \\
\mathtt{((1,3),~[1,1,1,0,0,0,0,0]),} & \mathtt{(2,~[1,1,1,1,1,0,0,0]),} \\
\mathtt{((2,3),~[1,1,0,0,0,0,0,0]),} & \mathtt{(3,~[1,0,0,0,0,0,0,0]),} \\
\mathtt{((3,4),~[1,1,1,1,1,1,1,1]),} & \mathtt{(4,~[1,1,1,1,1,1,0,0])} \\
\end{array} \right]}
\]
In Section~\ref{results} we report that using \bee\ enables
us to solve interesting instances of the VMTL problem not previously
solvable by other techniques.
\section{Compiling \bee\ to CNF}
\label{sec:compiling}
The compilation of a constraint model to a CNF
using \bee\ goes through three phases.
In the first phase, (unary) bit blasting, integer variables (and
constants) are represented as bit vectors in the order-encoding. Now
all constraints are about Boolean variables.
The second phase, the main loop of the compiler, is about constraint
simplification. Three types of actions are applied: equi-propagation,
partial evaluation, and decomposition of constraints. These are
specified as a set of transitions which we write in the form
$c_1\overset{\theta}{\longmapsto}c_2$ to specify that constraint $c_1$
reduces to constraint $c_2$ generating the (possibly empty)
substitution $\theta$. Simplification is applied repeatedly until no
rule is applicable.
In the third, and final phase, simplified constraints are encoded to
CNF. We elaborate below.
To simplify the presentation, we assume that integer variables are
represented in a positive interval starting from~$0$. As later
detailed in Section~\ref{sec:implementation} there is no such
limitation in \bee.
\vspace{-3mm}
\paragraph{\underline{Bit-blasting:}}
Each integer variable declaration
$\mathtt{new\_int(I,c_1,c_2)}$ triggers a unification
$\mathtt{I=[1,\dots,1,X_{c_1+1},\ldots,X_{c_2}]}$ and introduces a
constraint $\mathtt{ordered(I)}$ to specify that the bits representing
$\mathtt{I}$ are in the order-encoding.
To illustrate bit-blasting, consider again the VMTL
example detailed in Figure~\ref{fig:vmtl-instance}. Each variable in
the \texttt{Map} occurs in a $\mathtt{new\_int}$ declaration. So the
following unifications are performed:
\[{\small\begin{array}{ll}
\mathtt{V_{1}=[1,V_{1,2},V_{1,3},V_{1,4},V_{1,5},V_{1,6},V_{1,7},V_{1,8}]}, &
\mathtt{E_{1}=[1,E_{1,2},E_{1,3},E_{1,4},E_{1,5},E_{1,6},E_{1,7},E_{1,8}]}, \\
\mathtt{V_{2}=[1,V_{2,2},V_{2,3},V_{2,4},V_{2,5},V_{2,6},V_{2,7},V_{2,8}]}, &
\mathtt{E_{2}=[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},E_{2,6},E_{2,7},E_{2,8}]}, \\
\mathtt{V_{3}=[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},V_{3,6},V_{3,7},V_{3,8}]}, &
\mathtt{E_{3}=[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},E_{3,6},E_{3,7},E_{3,8}]}, \\
\mathtt{V_{4}=[1,V_{4,2},V_{4,3},V_{4,4},V_{4,5},V_{4,6},V_{4,7},V_{4,8}]}, &
\mathtt{E_{4}=[1,E_{4,2},E_{4,3},E_{4,4},E_{4,5},E_{4,6},E_{4,7},E_{4,8}]},\\
\mathtt{K=~[1,1,1,1,1,1,1,1,1,1,1,1,1,1]}&
\end{array}}\]
Integer variables occurring in an \texttt{allDiff} constraint are
bit-blasted twice: first, in the order-encoding, when declared, as
explained above, and second, in the direct encoding, when processing
the \texttt{allDiff} constraint, as described below.
\vspace{-3mm}
\paragraph{\underline{Equi-propagation}} is about detecting
situations in which a small number of constraints imply an equality of
the form $X=L$ where $X$ is a Boolean variable and $L$ is a Boolean
literal or constant. In this case $X$ becomes redundant
and can be replaced by $L$ in all constraints.
In \bee\ we consider as candidates for equi-propagation, individual
constraints together with constraints specifying that their integer
variables are in the order-encoding. If $X=L$ is such an equality,
then equi-propagation is implemented by unifying $X$ and $L$. This
unification applies to all occurrences of $X$ and in this sense
``propagates'' to other constraints involving $X$. Once
equi-propagation detects such an equation, this may trigger further
equi-propagation from other constraints.
For example, consider the constraint $\mathtt{int\_neq(I_1,I_2)}$
where $\mathtt{I_1=[x_1,x_2,x_3,x_4}]$ and
$\mathtt{I_2=[1,1,0,0}]$. We propagate that $\mathtt{(x_2=x_3)}$ because
\[
\left(
\begin{array}{l}
\mathtt{I_1=[x_1,x_2,x_3,x_4] ~\wedge~ I_2=[1,1,0,0] ~\wedge}\\
\mathtt{int\_neq(I_1,I_2) ~\wedge~ ordered(I_1)}
\end{array}
\right)\models \mathtt{(x_2=x_3)}.
\]
To see why, consider that $\mathtt{ordered(I_1)}$ implies that
$\mathtt{x_2\geq x_3}$. Furthermore, also $\mathtt{x_2\leq x_3}$ as
otherwise $\mathtt{x_2=1}$ and $\mathtt{x_3=0}$ which implies that
$\mathtt{I_1=[1,1,0,0}]$, contradicting $\mathtt{int\_neq(I_1,I_2)}$.
In \bee, equi-propagation is implemented by a collection of ad-hoc
transition rules for each type of constraint. While this approach is
not complete --- there are equations implied by a constraint that
\bee\ will not detect --- the implementation is fast, and works well
in practice.
An alternative approach is to implement equi-propagation, using BDD's,
as described in~\cite{Metodi2011}. This approach, though complete, is
slower and not included in the current release of \bee.
The following are two of the simplification (equi-propagation) rules
of \bee\ that apply to $\mathtt{int\_neq}$ constraints:
\begin{description}
\item[$\mathtt{neq_1}:$] applies when one of the (order-encoding)
integers in the relation is a constant and $\theta=\{X_1=X_2\}$:
\[
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_2,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&1, \hspace{-3mm}& 0,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&1, \hspace{-3mm}& 0,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\]
\item[$\mathtt{neq_2}:$] applies when the integers share common
variables as in the rule template and $\theta=\{X_1=X_2\}$:
\[
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_2,\hspace{-3mm}&\ldots{]}, \\
{[}\ldots,\hspace{-3mm}&\neg X_2,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\ldots{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}&\ldots{]}, \\
{[}\ldots,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\neg X_1,\hspace{-3mm}&\ldots{]}
\end{array}
\right)
\]
\end{description}
For the rule $\mathtt{neq_1}$, observe that after applying this
rule the constraint obtained is a tautology. Hence it is subsequently
removed by one of the other ``partial evaluation'' rules.
For the rule $\mathtt{neq_2}$, to see why the equation
$X_1=X_2$ is implied by the constraint (on the left side of the rule),
consider all possible truth values for the variables $X_1$ and $X_2$:
(a) If $X_1=0$ and $X_2=1$ then both integers in the relation take the
form $[\ldots,0,1,\ldots]$ violating their specification as
\texttt{ordered}, so this is not possible. (b) If $X_1=1$ and $X_2=0$
then both numbers take the form $[1,\ldots,1,0,\ldots,0]$ and are
equal, violating the $\mathtt{neq}$ constraint. The only possible
bindings for $X_1$ and $X_2$ are those where $X_1=X_2$.
The template expressed in rule $\mathtt{neq_2}$ is not contrived. It
comes up frequently as a result of applying other equi-propagation
rules.
\vspace{-3mm}
\paragraph{\underline{Partial evaluation}} is about simplifying
constraints in view of variables that are (partially) instantiated,
either because of information from the constraint model or else due to
equi-propagation. Typical cases include constant elimination and
elimination of tautologies.
The following are some of \bee's partial evaluation rules that apply
to $\mathtt{int\_neq}$ constraints ($\epsilon$ denotes the empty
substitution).
\begin{description}
\item[$\mathtt{neq_3}:$] applies to remove replicated variables:
\[
\mathtt{int\_neq}\left(
\begin{array}{cccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&Y_1,\hspace{-3mm}&Y_1,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\overset{\epsilon}{\longmapsto}
\mathtt{int\_neq}\left(
\begin{array}{ccc}
{[}\ldots,\hspace{-3mm}&X_1,\hspace{-3mm}& \ldots{]} \\
{[}\ldots,\hspace{-3mm}&Y_1,\hspace{-3mm}& \ldots{]}
\end{array}
\right)
\]
\item[$\mathtt{neq_4}:$] applies to remove leading 1 bits (there is a
similar rule for trailing 0's):
\[
\mathtt{int\_neq}([1,1,X_3,\ldots], [Y_1,Y_2,Y_3,\ldots])
\overset{\epsilon}{\longmapsto}
\mathtt{int\_neq}([1,X_3,\ldots],[Y_2,Y_3,\ldots])
\]
\end{description}
We now detail three of the simplification rules (equi-propagation and
partial evaluation) that apply to a constraint of the form
\texttt{int\_plus(A,B,C)} where we assume for simplicity of
presentation (the tool supports the general case) that
$\mathtt{A=[A_1,\ldots,A_n]}$, ~$\mathtt{B=[B_1,\ldots,B_m]}$, and
$\mathtt{C=[C_1,\ldots,C_{n+m}]}$. We denote by $\mathtt{min(I)}$ (or
$\mathtt{max(I)}$) the minimal (or maximal) value that integer
variable \texttt{I} can take, determined by the number of leading ones
(or trailing zeros) in its bit representation.
Rule $\mathtt{plus_{1}}$ is standard propagation for interval
arithmetics. Rule $\mathtt{plus_2}$ removes redundant bits
(assigned values through $\mathtt{plus_{1}}$).
Rules $\mathtt{plus_{3(a)}}$ and $\mathtt{plus_{3(b)}}$ remove
constraints and may seem contrived: \texttt{{3(a)}} assumes that
$\mathtt{m=0}$ and \texttt{{3(b)}} assumes that $\mathtt{n=m}$ and
that $\mathtt{C}$ represents the (same) constant $\mathtt{n}$.
However, in the general case, when $\mathtt{n}$, $\mathtt{m}$ are
arbitrary and constant \texttt{C} is represented in $\mathtt{m+n}$
bits, then application of the other rules will reduce the constraint
to one of these special cases.
\begin{description}
\item[$\mathtt{plus_1}:$] applies to propagate bounds:
$
\mathtt{int\_plus(A,B,C)}
\overset{\theta}{\longmapsto}
\mathtt{int\_plus(A,B,C)}
$
where \\
\hspace*{-7mm}$\theta=\left\{
\begin{array}{ll}
C_{max\{min(C), min(A)+min(B)\}} = 1, & C_{min\{max(C), max(A)+max(B)\}+1} = 0,\\
A_{max\{min(A), min(C)-max(B)\}} = 1, & A_{min\{max(A), max(C)-min(B)\}+1} = 0, \\
B_{max\{min(B), min(C)-max(A)\}} = 1, & B_{min\{max(B), max(C)-min(A)\}+1} = 0
\end{array}\right\}
$\smallskip
\item[$\mathtt{plus_{2}}:$] applies to remove leading 1's
(there are similar rules for trailing 0's and for the case when
the 1's or 0's are on $\mathtt{[B_1,\ldots,B_m]}$):
\[
\mathtt{int\_plus}\left(\begin{array}{l}
\mathtt{[1,A_2,\ldots,A_{n}], }\\
\mathtt{[B_1,\ldots,B_m],}\\
\mathtt{{[}1,C_2,\ldots,C_{n+m}{]}}
\end{array}\right)
\overset{\epsilon}{\longmapsto}
\mathtt{int\_plus}\left(\begin{array}{l}
\mathtt{[A_2,\ldots,A_{n}], }\\
\mathtt{[B_1,\ldots,B_m],}\\
\mathtt{{[}C_2,\ldots,C_{n+m}{]}}
\end{array}\right)
\]
\item[$\mathtt{plus_{3(a)}}:$] applies when \texttt{A} or \texttt{B}
is the empty bit list and $\theta = \sset{C_{i}=A_i}{1 \leq i \leq
n}$
\[
\mathtt{int\_plus([A_1,\ldots,A_n], [~], [C_1,\ldots,C_n])}
\overset{\theta}{\longmapsto} none
\]
\item[$\mathtt{plus_{3(b)}}:$] applies when \texttt{C} is a constant
\texttt{n} and $\theta = \sset{A_{i}=\neg B_{n-i+1}}{1 \leq i
\leq n}$
\[
\mathtt{int\_plus([A_1,\ldots,A_n], [B_1,\ldots,B_n], [1,\ldots1,0,\ldots,0])}
\overset{\theta}{\longmapsto}
none
\]
\end{description}
We illustrate the simplification of a \texttt{int\_plus} constraint by
the following example.
\begin{example}[simplifying \texttt{int\_plus}: equi-propagation and
partial evaluation]\label{ex:plus14}
Consider constraint $\mathtt{int\_plus(A,B,C)}$ where $A$ and $B$
are integer variables with domain $[1..8]$ and $C$ is the constant
14 represented in 16 bits. Constraint simplification follows the
steps:
\medskip
\begin{tabular}{l}
\fbox{$\begin{array}{l}
\mathtt{int\_plus(}\\
~~\mathtt{[1,A_2,A_3,A_4,A_5,A_6,A_7,A_8],} \\
~~\mathtt{[1,B_2,B_3,B_4,B_5,B_6,B_7,B_8],}\\
~~\mathtt{[\underbrace{1,\qquad\ldots\ldots~\quad,1}_{14
~times},0,0]}\\
\mathtt{)}
\end{array}$}
$\xrightarrow{\mathtt{plus_1}}$
\fbox{$\begin{array}{l}
\mathtt{int\_plus(}\\
~~\mathtt{[1,1,1,1,1,1,A_7,A_8],} \\
~~\mathtt{[1,1,1,1,1,1,B_7,B_8],}\\
~~\mathtt{[\underbrace{1,\quad\ldots\ldots~~,1}_{14 ~times},0,0]}\\
\mathtt{)}
\end{array}$}
$\xrightarrow{\mathtt{plus_{2}}}$
\\ ~\\
$\xrightarrow{\mathtt{plus_{2}}}$
\fbox{\texttt{int\_plus}$\left(\begin{array}{l}
\mathtt{[A_7,A_8],~}
\mathtt{[B_7,B_8],}\\
\mathtt{[1,1,0,0]}
\end{array}\right)$}
$\xrightarrow{\mathtt{plus_{3(b)}}}$
\fbox{
$\begin{array}{l}
\mathtt{none}, ~binding: \\
\mathtt{B_7=\neg A_8,~B_8=\neg A_7})
\end{array}$
}
\end{tabular}
\medskip\noindent After constraint simplification
variables \texttt{A} and \texttt{B} take the form: $\mathtt{[1, 1, 1,
1, 1, 1, A_{7}, A_{8}]}$ and $\mathtt{[1, 1, 1, 1, 1,
1,\neg A_{8}, \neg A_{7}]}$ (and nothing is left to encode to CNF).
\end{example}
\vspace{-3mm}
\paragraph{\underline{Decomposition}} is about replacing complex
constraints (for example about arrays) with simpler constraints (for
example about array elements). Consider, for instance, the constraint
$\mathtt{int\_array\_plus(As,Sum)}$. It is decomposed to a list of
$\mathtt{int\_plus}$ constraints applying a straightforward divide and
conquer recursive definition. At the base case, if \texttt{As=[A]}
then the constraint is replaced by \texttt{int\_eq(A,Sum)}, or if
$\mathtt{As=[A_1,A_2]}$ then it is replaced by
$\mathtt{int\_plus(A_1,A_2,Sum)}$.
In the general case \texttt{As} is split into two halves, then
constraints are generated to sum these halves, and then an additional
$\mathtt{int\_plus}$ constraint is introduced to sum the two sums.
As another example, consider the $\mathtt{int\_plus(A_1,A_2,A)}$
constraint. One approach, supported by \bee, decomposes the
constraint as an odd-even merger (from the context of odd-even sorting
networks)~\cite{Batcher68}. Here, the sorted sequences of bits
$\mathtt{A_1}$ and $\mathtt{A_2}$ are merged to obtain their sum
$\mathtt{A}$.
This results in a model with $O(n\log n)$
\texttt{comparator} constraints (and later in an encoding with
$O(n\log n)$ clauses).
Another approach, also supported in \bee, does not decompose the
constraint but encodes it directly to a CNF of size $O(n^2)$,
as in the context of so-called totalizers~\cite{BailleuxB03}.
A hybrid approach, leaves the choice to \bee, depending on the size of
the domains of the variables involved.
Finally, we note that the user can configure \bee\ to fix the way it
compiles this constraint (and others).
\paragraph{\underline{CNF encoding}} is the last phase and applies to
all remaining simplified constraints. The encoding of constraints to
CNF is standard and similar to the encodings in
Sugar~\cite{sugar2009}.
\paragraph{\underline{Cardinality constraints}} are about the
cardinality of sets of Boolean variables and are specified by the
template $\mathtt{bool\_array\_sum\_rel([X_1,\ldots,X_n],~I)}$.
Cardinality constraints are normalized, see e.g., \cite{EenS06}, so we
only consider $\mathtt{rel\in\{leq,eq\}}$. Partial evaluation rules
for cardinality constraints are the obvious. For example, in the
special case when \texttt{I} is a constant:
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,1,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_1,X_2,X_4],~2)}$
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,0,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_1,X_2,X_4],~3)}$
$\mathtt{bool\_array\_sum\_leq([X_1,X_2,-X_1,X_4],~3)\mapsto
bool\_array\_sum\_leq([X_2,X_4],~2)}$
\medskip\noindent
The special case, when \texttt{I} is the constant~1 is called the
``at-most-one'' constraint and it has been studied extensively (for a
recent survey see \cite{Frisch+Giannaros/10/SAT}). In \bee, we support
two different encodings for this case (the user can choose). The first
is the standard ``pairwise'' encoding which specifies a clause $(\neg
x_i\vee\neg x_j)$ for each pair of Boolean variables $x_i$ and $x_j$.
This encoding introduces $O(n^2)$ clauses and is sometimes too large.
The second, is a more compact encoding which follows
the approach described in~\cite{NewAtMostOne}.
In the general case (when $\mathtt{I}>1$) the constraint is
decomposed, much the same as the $\mathtt{int\_array\_plus}$
constraint, to a network of $\mathtt{int\_plus}$ constraints.
\paragraph{\underline{The All-different constraint}}
specifies that a set of integer variables take all different
values. Although we adopt the order-encoding for integer variables, it
is well accepted that for these constraints the direct encoding is
superior \cite{direct4allDiff}. For this reason, in \bee, when
processing the constraint, a dual representation is chosen. When
integer variable $\mathtt{I}$, occurring in an \texttt{allDiff}
constraint, is declared, it was unified with its unary representation
in the order-encoding: $\mathtt{I=[x_1,\ldots,x_n]}$. In addition, we
associate \texttt{I} with a new bit-blast,
$\mathtt{[d_0,\ldots,d_{n}]}$, in the direct encoding. We introduce
for each such \texttt{I} a channeling formula to capture the relation
between its two representations.
\[\mathtt{channel([x_1,\ldots,x_n],[d_0,\ldots,d_{n}])=
\left(\begin{array}{r} d_0 = \neg x_1 \\
\wedge~ d_n = x_n
\end{array}\right) \wedge
\bigwedge_{i=1}^{n-1}(d_i\leftrightarrow x_{i}\wedge\neg x_{i+1})}
\]
During constraint simplification, the
$\mathtt{allDiff([I_1,\ldots,I_n])}$ constraint is viewed as a bit
matrix where each row consists of the bits
$\mathtt{[d_{i0},\ldots,d_{im}]}$ for $\mathtt{I_i}$ in the direct
encoding. The element $d_{ij}$ is true iff $I_i$ takes the value $j$.
The $j^{th}$ column specifies which of the $I_i$ take the value $j$
and hence, at most one variable in a column may take the value true.
\bee\ distinguishes the special case when $\mathtt{[I_1,\ldots,I_n]}$
must take precisely $n$ different values. In this case the constraint
is about ``permutation''. We denote this by a flag (*) as in
$\mathtt{allDiff^*([I_1,\ldots,I_n])}$. In this case, exactly one bit
in each column of the representation must take the value true.
To simplify an \texttt{allDiff} constraint, \bee\ applies
simplification rules to the implicit cardinality constraints on the
columns and also two specific \texttt{allDiff} rules. The first is
essentially the usual domain consistent propagator~\cite{regin}
focusing on Hall sets of size 2. The second rule applies only to an
$\mathtt{allDiff^*}$ constraint which is about permutation.
We denote the values that $\mathtt{I_i}$ can take as
$\mathtt{dom(I_i)}$.
\begin{description}
\item[$\mathtt{allDiff_1}:$]
when $\mathtt{dom(I_1) = dom(I_2)=\{v_1,v_2\}}$:
\[
\mathtt{allDiff([I_1,I_2,I_3,\ldots,I_n])}
\overset{\theta}{\longmapsto}
\mathtt{allDiff([I_3,\ldots,I_n])}
\]
where $\theta = \bigcup_{3\leq i\leq n}
\{d_{i,v_1}=0, d_{i,v_2}=0\} \cup
\{d_{1,v_1}= -d_{2,v_1}, d_{1,v_2}= -d_{2,v_2} \}$.
\item[$\mathtt{allDiff_2}:$] when
$\mathtt{\{v_1,v_2\}\subseteq dom(I_1)\cap dom(I_2)}$,
and for $i\geq 3$,
$\mathtt{\{v_1,v_2\}\cap dom(I_i)=\emptyset}$
\[
\mathtt{allDiff^*([I_1,\ldots,I_n])}
\overset{\theta}{\longmapsto}
\mathtt{allDiff^*([I_1,\ldots,I_n])}
\]
where $\theta = \bigcup_{j\neq v_1, j\neq v_2}\{d_{1j}=0, d_{2,j}=0\}$.
\end{description}
To illustrate the two rules for \texttt{allDiff} consider the
following.
\begin{example}
Consider an \texttt{allDiff} constraint on 5 integer variables
taking values in the interval $[0,7]$ where the first two can take
only values 0 and 1. So, they are a Hall set of size two and rule
$\mathtt{allDiff_1}$ applies. We present the simplification step on
the order encoding representation (though it is triggered through
the direct encoding representation): \vspace{-2mm}
\[\small
\mathtt{allDiff}\left(
\begin{array}{cccc}
{[}X_{1,1},\hspace{-3mm}&0,\hspace{-3mm}& \ldots,\hspace{-3mm}&0{]} \\
{[}X_{2,1},\hspace{-3mm}&0,\hspace{-3mm}& \ldots,\hspace{-3mm}&0{]} \\
{[}X_{3,1},\hspace{-3mm}&X_{3,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{3,7}{]} \\
{[}X_{4,1},\hspace{-3mm}&X_{4,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{4,7}{]} \\
{[}X_{5,1},\hspace{-3mm}&X_{5,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{5,7}{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{allDiff}\left(
\begin{array}{ccc}
{[}1,X_{3,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{3,7}{]} \\
{[}1,X_{4,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{4,7}{]} \\
{[}1,X_{5,2},\hspace{-3mm}& \ldots,\hspace{-3mm}&X_{5,7}{]}
\end{array}
\right)
\]
where $\theta = \{ X_{1,1}=\neg X_{2,1} , X_{3,1}=0 , X_{4,1}=0 ,
X_{5,1}=0\}$.
Now consider a setting where an \texttt{allDiff} constraint is about 5
variables that can take 5 values (permutation) and the first two are
the only two that can take values 0 and 1. So rule
$\mathtt{allDiff_2}$ applies. We present the simplification step on
the order encoding representation (though it is triggered through
the direct encoding representation):
\vspace{-2mm}
\[\small
\mathtt{allDiff^*}\left(
\begin{array}{lccr}
{[}X_{1,1},\hspace{-3mm}&X_{1,2},\hspace{-3mm}&X_{1,3},\hspace{-3mm}&X_{1,4}{]} \\
{[}X_{2,1},\hspace{-3mm}&X_{2,2},\hspace{-3mm}&X_{2,3},\hspace{-3mm}&X_{2,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{3,3},\hspace{-3mm}&X_{3,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{4,3},\hspace{-3mm}&X_{4,4}{]} \\
{[}1,\hspace{-3mm}&1,\hspace{-3mm}&X_{5,3},\hspace{-3mm}&X_{5,4}{]}
\end{array}
\right)
\overset{\theta}{\longmapsto}
\mathtt{allDiff^*}\left(
\begin{array}{lccr}
{[}X_{1,1},\hspace{-3mm}&0,\hspace{-3mm}&0,\hspace{-3mm}&0{]} \\
{[}X_{2,1},\hspace{-3mm}&0,\hspace{-3mm}&0,\hspace{-3mm}&0{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{3,3},\hspace{-3mm}&X_{3,4}{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{4,3},\hspace{-3mm}&X_{4,4}{]} \\
{[}1,\hspace{-0mm}&1,\hspace{-0mm}&X_{5,3},\hspace{-3mm}&X_{5,4}{]}
\end{array}
\right)
\]
where $\theta = \{ X_{1,2},\ldots,X_{1,4} = 0 , X_{2,2},\ldots,X_{2,4} = 0\}$.
\end{example}
When no further simplification rules apply the \texttt{allDiff}
constraint is decomposed to the corresponding cardinality constraints
on the columns of its bit matrix representation.
\section{Constraint simplification in the VMTL example}
Consider again the VMTL example and the constraints from
Figure~\ref{fig:vmtl-instance}. We focus on three
constraints and follow the steps made when compiling these (we write
``14'' as short for $\mathtt{[\underbrace{1,1,\ldots,1}_{14}]}$).
\smallskip
$\begin{array}{ll}
(1) & \mathtt{int\_array\_plus([V_4,E_4], 14)} \\
(2) & \mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4]),} \\
(3) & \mathtt{int\_array\_plus([V_3,E_2,E_3,E_4], 14),}
\end{array}$
\smallskip\noindent In the first steps, constraint (1) is decomposed to
an \texttt{int\_plus} constraint which has the same form as the
constraint in Example~\ref{ex:plus14}. So, we have the bindings
$\mathtt{V_4 = [1,1,1,1,1,1,V_{4,7}, V_{4,8}]}$ and
$\mathtt{E_4 = [1,1,1,1,1,1,\neg V_{4,8},\neg V_{4,7}]}$.
Now, consider the \texttt{allDiff} constraint (2). \bee\ determines
that this constraint is about permutation (8 integer variables with 8
different values in the range [1,8]). The simplification rules for
\texttt{allDiff} detect that $\mathtt{\{V_4,E_4\}}$ must take together
the two values 6 and 8 (using a simplification rule similar to
$\mathtt{neq_2}$) triggerring the substitution $\mathtt{\{V_{4,7} =
V_{4,8}\}}$. Now rule $\mathtt{allDiff_1}$ detects a Hall set
$\mathtt{\{V_4,E_4\}}$ of size two:
\[\small \mathtt{allDiff([V_1,V_2,V_3,V_4,E_1,E_2,E_3,E_4])}
\xrightarrow{\theta}
\mathtt{allDiff([V_1,V_2,V_3,E_1,E_2,E_3])}
\]
where $\theta$ is the unification that imposes
$\mathtt{V_1,V_2,V_3,E_1,E_2,E_3 \neq 6,8}$. So we have the following
bindings (where the impact of $\theta$ is underlined):
\[\small \begin{array}{ll}
\mathtt{V_{1}=
[1,V_{1,2},V_{1,3},V_{1,4},V_{1,5},\underline{V_{1,7},V_{1,7}},0]} \qquad&
\mathtt{E_{1}=
[1,E_{1,2},E_{1,3},E_{1,4},E_{1,5},\underline{E_{1,7},E_{1,7}},0]} \\
\mathtt{V_{2}=
[1,V_{2,2},V_{2,3},V_{2,4},V_{2,5},\underline{V_{2,7},V_{2,7}},0]} &
\mathtt{E_{2}=
[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},\underline{E_{2,7},E_{2,7}},0]} \\
\mathtt{V_{3}=
[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},\underline{V_{3,7},V_{3,7}},0]} &
\mathtt{E_{3}=
[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},\underline{E_{3,7},E_{3,7}},0]} \\
\mathtt{V_{4}=
[1,1,1,1,1,1,V_{4,7},V_{4,7}]} &
\mathtt{E_{4}=
[1,1,1,1,1,1,\neg V_{4,7},\neg V_{4,7}]}
\end{array}
\]
Consider now the constraint (3). Equi-propagation (because of bounds)
dictates that $\mathtt{max(V_1)=max(V_2)=max(V_3)=5}$, so this
constraint then simplifies as follows:
\[\small
\fbox{$\begin{array}{l}
\mathtt{int\_array\_plus([}\\
~~\mathtt{[1,V_{3,2},V_{3,3},V_{3,4},V_{3,5},0,0,0]},\\
~~\mathtt{[1,E_{2,2},E_{2,3},E_{2,4},E_{2,5},0,0,0]},\\
~~\mathtt{[1,E_{3,2},E_{3,3},E_{3,4},E_{3,5},0,0,0]},\\
~~\mathtt{[1,1,1,1,1,1,\neg V_{4,7},\neg V_{4,7}]},~14~
\mathtt{])}
\end{array}$}
\longmapsto
\fbox{$\begin{array}{l}
\mathtt{int\_array\_plus([}\\
~~\mathtt{[V_{3,2},V_{3,3},V_{3,4},V_{3,5}]},\\
~~\mathtt{[E_{2,2},E_{2,3},E_{2,4},E_{2,5}]},\\
~~\mathtt{[E_{3,2},E_{3,3},E_{3,4},E_{3,5}]},\\
~~\mathtt{[\neg V_{4,7},\neg V_{4,7}]},~5~
\mathtt{])}
\end{array}$}
\]
\medskip After applying simplification and decomposition rules on all
the constraints from Figure~\ref{fig:vmtl-instance} until no further
rules can be applyed, the constraints will be encoded to CNF. The
generated CNF contains 301 clauses and 48 Boolean variables.
Compiling the same set of constraints from
Figure~\ref{fig:vmtl-instance} without applying simplification rules
generates a larger CNF which contains 642 clauses and 97 Boolean
variables.
\section{Another Example \bee\ Application:
DNA word problem}
The DNA word problem (Problem \texttt{033} of CSPLib) seeks the
largest parameter $n$, such that there exists a set $S$ of $n$
eight-letter words over the alphabet $\Sigma=\{A,C,G,T\}$ with the
following properties:
\textbf{(1)} Each word in $S$ has exactly 4 symbols from $\{C,G\}$;
\textbf{(2)} Each pair of distinct words in $S$ differ in at least 4 positions;
and
\textbf{(3)} For every $x,y\in S$: $x^R$ (the reverse of $x$) and $y^C$ (the
word obtained by replacing each $A$ by $T$, each $C$ by $G$, and vice
versa) differ in at least 4 positions.
In~\cite{dnaWordPaper}, the authors present a strategy to solve this
problem where the four letters are modeled by bit-pairs
$\tuple{t,m}$. Each eight-letter word can then be viewed as the
combination of a \emph{``t-part''}, $\tuple{t_1,\ldots,t_8}$, which is
a bit-vector, and a \emph{``m-part''}, $\tuple{m_1,\ldots,m_8}$, also
a bit-vector. Building on the approach described
in~\cite{dnaWordPaper}, we pose conditions on sets of
\emph{``t-parts''} and \emph{``m-parts''}, $T$ and $M$, so that their
Cartesian product $S=T\times M$ will satisfy the requirements of the
original problem. From the three conditions below, $T$ is required to
satisfy (1$'$) and (2$'$), and $M$ is required to satisfy (2$'$) and
(3$'$).
For a set of bit-vectors $V$, the conditions are:
\textbf{(1$'$)} Each bit-vector in $V$ sums to 4;
\textbf{(2$'$)} Each pair of distinct bit-vectors in $V$ differ in at
least 4 positions; and
\textbf{(3$'$}) For each pair of bit-vectors (not necessarily distinct)
$u,v\in V$, $u^R$ (the reverse of $u$) and $v^C$ (the complement of
$v$) differ in at least 4 positions. This is equivalent to requiring
that $(u^r)^c$ differs from $v$ in at least 4 positions.
It is this strategy that we model in our \bee\ encoding.
An instance takes the form $\mathtt{dna(n_1,n_2)}$ signifying the
numbers of bit-vectors, $n_1$ and $n_2$ in the sets $T$ and $M$.
Without loss of generality, we impose, to remove symmetries, that
$T$ and $M$ are lexicographically ordered.
A solution is the Cartesian product $S=T\times M$.
In Section~\ref{results} we report that using \bee\ enables us to
solve interesting instances of the problem not previously solvable
by other techniques.
\section{Implementation}
\label{sec:implementation}
\bee\ is implemented in (SWI) Prolog and can be applied in conjunction
with the CryptoMiniSAT solver \cite{Crypto} through a Prolog interface
\cite{satPearl}. \bee\ can be downloaded from \cite{bee2012} where one
can find also the examples from this paper and others. The
distribution includes also a solver, which we call Bumble\bee, which
enables to specify a \bee\ model as an input file and solve it. The
output is a set of bindings to the declared variables in the model.
In \bee, Boolean variables are represented as Prolog variables. The
negation of \texttt{X} is represented as \texttt{-X}. The truth
values, $\true$ and $\false$, are denoted \texttt{1} and \texttt{-1}.
Integer variables (including negative range values) are represented in
the order-encoding. When processing (bit-blasting) a declaration
$\mathtt{new\_int(I,Min,Max)}$, Prolog variable \texttt{I} is unified
with the tuple \texttt{(Min,Max,Bits,LastBit)} where \texttt{Min} and
\texttt{Max} are constants indicating the interval domain of
\texttt{I}, \texttt{Bits} is a list of $\mathtt{(Max-Min)}$ variables,
and \texttt{LastBit} is the last variable of \texttt{Bits}. This
representation is more concise than the one assumed for simplicity in
the previous sections and it also supports negative
numbers. Maintaining direct access to the last bit in the
representation (we already can access the first bit through the list
\texttt{Bits}) facilitates a (constant time) check if the lower and
upper bound values of a variable has changed. This way we can more
efficiently determine when (certain) simplification rules apply.
We make a few notes: (1) Integer variables must be declared before
use; (2) \bee\ allows the use of constants in constraints instead of
declaring them as integer variables (for example
$\mathtt{int\_gt(I,5)}$ represents a declaration
$\mathtt{new\_int(I',5,5)}$ together with the constraint
$\mathtt{int\_gt(I,I')}$); (3) integer variables can be negated.
\bee\ maintains constraints as a Prolog list
(of terms). Each type of constraint is associated with corresponding
rules for simplification, decomposition, and encoding to CNF.
After bit-blasting, constraints are first simplified (equi-propagation
and partial evaluation) using these rules until no further rules
apply. During this process, if a pair of literals is equated (e.g.~ as
in \texttt{X=Y, X=-Y, X=1, X=-1}), then they are unified, thus
propagating the effect to other constraints.
After constraint simplification, some constraints are decomposed, and
this process repeats.
We end up with a set of ``basic'' constraints (which cannot be further
decomposed or simplified). These are then encoded to CNF.
\section{Experiments}\label{results}
We report on our experience in applying \bee. To appreciate the ease
in its use, and for further details, the reader is encouraged to view
the example encodings available with the tool \cite{bee2012}.
All experiments run on an Intel Core 2 Duo E8400 3.00GHz CPU with 4GB
memory under Linux (Ubuntu lucid, kernel 2.6.32-24-generic).
\bee\ is written in Prolog and run using SWI Prolog v6.0.2 64-bits.
Comparisons with Sugar (v1.15.0) are based on the use of identical
constraint models, apply the same SAT solver (CryptoMiniSat v2.5.1),
and run on the same machine.
For all of the tables describing
experiments, columns indicate:
\vspace{-3mm}
\qquad\begin{minipage}{0.45\linewidth}
\qquad\begin{itemize}
\item[\texttt{comp:}] compile time (seconds)
\item[\texttt{clauses:}] number of CNF clauses
\end{itemize}
\end{minipage}
\begin{minipage}{0.4\linewidth}
\qquad\begin{itemize}
\item[\texttt{vars:}] number of CNF variables
\item[\texttt{sat:}] SAT solving time (seconds)
\end{itemize}
\end{minipage}
\medskip
We first focus on the impact of the dual representation for
\allDifferent\ constraints. We report on the application of \bee\ to
Quasi-group completion problems (QCP), proposed by
\citeN{DBLP:conf/cp/GomesSC97} as a constraint satisfaction benchmark,
where the model is a conjunction of \allDifferent\ constraints.
\vspace{-3mm}
\paragraph{\underline{Quasi-group completion:}}
\begin{table}[t]
\scriptsize
\centering
\begin{oldtabular}{lc|rrrr|rrrr|rrr}
\hline
\multicolumn{2}{c|}{instance} & \multicolumn{4}{c|}{\bee\ (dual encoding)} &
\multicolumn{4}{c|}{\bee\ (order encoding)} & \multicolumn{3}{c}{Sugar} \\
\hline
& &comp &clauses & vars & sat & comp &clauses& vars & sat & clauses & vars & sat \\
\hline
25-264-0 & sat & 0.23 & 6509 & 1317 & 0.33 & 0.36 & 33224 & 887 & 8.95 & 126733 & 10770 & 34.20 \\
25-264-1 & sat & 0.20 & 7475 & 1508 & 3.29 & 0.30 & 34323 & 917 & 97.50 & 127222 & 10798 & 13.93 \\
25-264-2 & sat & 0.21 & 6531 & 1329 & 0.07 & 0.30 & 35238 & 905 & 2.46 & 127062 & 10787 & 8.06 \\
25-264-3 & sat & 0.21 & 6819 & 1374 & 0.83 & 0.29 & 32457 & 899 & 18.52 & 127757 & 10827 & 44.03 \\
25-264-4 & sat & 0.21 & 7082 & 1431 & 0.34 & 0.29 & 32825 & 897 & 19.08 & 126777 & 10779 & 85.92 \\
25-264-5 & sat & 0.21 & 7055 & 1431 & 3.12 & 0.30 & 33590 & 897 & 46.15 & 126973 & 10784 & 41.04 \\
25-264-6 & sat & 0.21 & 7712 & 1551 & 0.34 & 0.33 & 39015 & 932 & 69.81 & 128354 & 10850 & 12.67 \\
25-264-7 & sat & 0.21 & 7428 & 1496 & 0.13 & 0.30 & 36580 & 937 & 19.93 & 127106 & 10794 & 7.01 \\
25-264-8 & sat & 0.21 & 6603 & 1335 & 0.18 & 0.27 & 31561 & 896 & 10.32 & 124153 & 10687 & 9.69 \\
25-264-9 & sat & 0.21 & 6784 & 1350 & 0.19 & 0.27 & 35404 & 903 & 34.08 & 128423 & 10853 & 38.80 \\
25-264-10 & unsat & 0.21 & 6491 & 1296 & 0.04 & 0.30 & 33321 & 930 & 10.92 & 126999 & 10785 & 57.75 \\
25-264-11 & unsat & 0.12 & 1 & 0 & 0.00 & 0.28 & 37912 & 955 & 0.09 & 125373 & 10744 & 0.47 \\
25-264-12 & unsat & 0.16 & 1 & 0 & 0.00 & 0.29 & 39135 & 984 & 0.08 & 127539 & 10815 & 0.57 \\
25-264-13 & unsat & 0.12 & 1 & 0 & 0.00 & 0.29 & 35048 & 944 & 0.09 & 127026 & 10786 & 0.56 \\
25-264-14 & unsat & 0.23 & 5984 & 1210 & 0.07 & 0.28 & 31093 & 885 & 11.60 & 126628 & 10771 & 15.93 \\
\hline
Total & & & & & 8.93 & & & & 349.58 & & & 370.63 \\
\hline
\end{oldtabular}
\caption{QCP results for $25\times 25$ instances with 264 holes}
\label{tab:qcpExpr}
\end{table}
We consider 15 instances from the 2008 CSP
competition\footnote{\url{http://www.cril.univ-artois.fr/CPAI08/}}.
Table~\ref{tab:qcpExpr} considers three settings: \bee\ with its dual
encoding for \allDifferent\ constraints, \bee\ using only the order
encoding (equivalent to using $\mathtt{int\_neq}$ constraints instead of
\allDifferent), and Sugar. The results indicate that:
(1) Application of \bee\ using the dual representation for
\allDifferent\ is 38 times faster and produces 20 times less clauses
(in average) than when using the order-encoding alone (despite the
need to maintain two encodings);
(2) Without the dual representation, solving encodings generated by
\bee\ is only slightly faster but \bee\ generates CNF encodings 4
times smaller (on average) than those generated by Sugar.
Observe that 3 instances are found unsatisfiable by \bee\ (indicated
by a CNF with a single clause and no variables). We comment that Sugar
preprocessing times are higher than those of \bee\ and not indicated
in the table.
\bigskip To further appreciate the impact of the tool we describe
results for three additional applications which shift the
state-of-the-art with respect to what could previously be solved.
The experiments clearly illustrate that \bee\ decreases the size of
CNF encodings as well as the subsequent SAT solving time.
\vspace{-3mm}
\paragraph{\underline{Magic labels:}}
In \cite{MacDougall2002} the authors conjecture that the $n$ vertex
complete graph, $K_n$, for $n\geq 5$ has a vertex magic total labeling
with magic constants for specific range of values of $k$, determined
by $n$. This conjecture is proved correct for all odd $n$ and verified
by brute force for $n=6$.
We address the cases for $n=8$ and $n=10$ which involve 15 instances
(different values of $K$) for $n=8$, and 23 (different values of $K$)
for $n=10$. Starting from the simple constraint model (illustrated by
the example in Figure~\ref{fig:vmtl-instance}), we add additional
constraints to exploit that the graphs are symmetric:
(1) We assume that the edge with the smallest label is $e_{1,2}$;
(2) We assume that the labels of the edges incident to $v_1$ are
ordered and hence introduce constraints $e_{1,2} < e_{1,3} < \cdots <
e_{1,n}$;
(3) We assume that the label of edge $e_{1,3}$ is smaller than the
labels of the edges incident to $v_2$ (except $e_{1,2}$) and introduce
constraints accordingly.
In this setting \bee\ can solve all except 2 instances with a 4 hour
timeout and Sugar can solve all except~4.
Table~\ref{tab:k8} depicts results for the 10 hardest instances for
$K_8$ and the 20 hardest for $K_{10}$ with a 4 hour time-out. \bee\
compilation times are on the order of 0.5 sec/instance for $K_8$ and
2.5 sec/instance for $K_{10}$. Sugar encoding times are slightly
larger. The instances are indicated by the magic constant, $k$; the
columns for \bee\ and Sugar indicate SAT solving times (in seconds).
The bottom two lines indicate average encoding sizes (numbers of
clauses and variables).
\begin{table}[t]
\scriptsize \hspace{-10mm}
\begin{minipage}{0.37\linewidth}
\begin{oldtabular}{c|c|r|r}
\hline
$K_8$& $k$ & \multicolumn{1}{c}{\bee} & \multicolumn{1}{c}{Sugar} \\
\hline
&143 & 1.26 & 2.87\\
&142 & 10.14 & 1.62\\
&141 & 7.64 & 2.94\\
&140 & 14.68 & 6.46\\
&139 & 25.60 & 6.67\\
&138 & 12.99 & 2.80\\
&137 & 22.91 & 298.58\\
&136 & 14.46 & 251.82\\
&135 & 298.54 & 182.90\\
&134 & 331.80 & $\infty$\\
\hline
\multicolumn{2}{c}{\vspace{-1mm}Average}\\
\hline
\multicolumn{2}{r|}{clauses {$\mathbf{\times 1000}$}} &248& 402\\
\multicolumn{2}{r|}{vars} &5688&9370\\
\end{oldtabular}
\end{minipage}
\begin{minipage}{0.26\linewidth}
\begin{oldtabular}{c|c|r|r}
\hline
$K_{10}$&k & \multicolumn{1}{c}{\bee} & \multicolumn{1}{c}{Sugar} \\
\hline
&277 & 5.31 & 9.25\\
&276 & 7.11 & 9.91\\
&275 & 13.57 & 19.63\\
&274 & 4.93 & 9.24\\
&273 & 45.94 & 9.03\\
&272 & 22.74 & 86.45\\
&271 & 7.35 & 9.49\\
&270 & 6.03 & 55.94\\
&269 & 5.20 & 11.05\\
&268 & 94.44 & 424.89\\
\hline
\multicolumn{2}{c}{\vspace{-1mm}~}\\
\hline
\multicolumn{4}{r}{clauses {$\mathbf{\times 1000}$}}\\
\multicolumn{4}{r}{vars}\\
\end{oldtabular}
\end{minipage}
\begin{minipage}{0.25\linewidth}
\begin{oldtabular}{r|r|r}
\hline
k~~ & \multicolumn{1}{c}{\bee} & \multicolumn{1}{c}{Sugar} \\
\hline
267~ & 88.51 & 175.70\\
266~ & 229.80 & 247.56\\
265~ & 1335.31 & 259.45\\
264~ & 486.09 & 513.61\\
263~ & 236.68 & 648.43\\
262~ & 1843.70 & 6429.25\\
261~ & 2771.60 & 7872.76\\
260~ & 4873.99 & $\infty$\\
259~ & $\infty$ & $\infty$\\
258~ & $\infty$ & $\infty$\\
\hline
\multicolumn{2}{c}{\vspace{-1mm}Average}\\
\hline
\multicolumn{1}{r|}{} &1229&1966 \\
\multicolumn{1}{r|}{} &15529&25688 \\
\end{oldtabular}
\end{minipage}
\caption{VMTL results for $K_8$ and $K_{10}$ (times are in seconds)}
\label{tab:k8}
\end{table}
The results indicate that the Sugar encodings are (in average) about
60\% larger, while the average SAT solving time for the \bee\
encodings is about 2 times faster (average excluding instances
where Sugar times-out).
To address the two VMTL instances not solvable using the \bee\ models
described above ($K_{10}$ with magic labels 259 and 258), we partition
the problem fixing the values of $e_{1,2}$ and $e_{1,3}$ and
maintaining all of the other constraints. Analysis of the symmetry
breaking constraints indicates that this results in 198 new instances
for each of the two cases. The original VMTL instance is solved
if any one of of these 198 instances is solved. So, we solve them in
parallel. Fixing $e_{1,2}$ and $e_{1,3}$ ``fuels'' the compiler so the
encodings are considerably smaller.
The instance for $k= 259$ is solved in 1379.50 seconds where
$e_{1,2}=1$ and $e_{1,3}=6$. The compilation time is 2.09 seconds and
the encoding consists in 1056107 clauses and 14143 variables.
To the best of our knowledge, the hard instances from this suite are
beyond the reach of all previous approaches to program the search for
magic labels. The SAT based approach presented in \cite{Jaeger2010}
cannot handle these.\footnote{Personal communication (Gerold J\"ager),
March 2012.} The comparison with Sugar indicates the impact of the
compiler.
\vspace{-3mm}
\paragraph{\underline{DNA word problem:}}
\citeN{DBLP:journals/constraints/ManciniMPC08} provide a comparison of
several state-of-the-art solvers applied to the DNA word problem with
a variety of encoding techniques. Their best reported result is
a solution with 87 DNA words, obtained in 554 seconds, using an
OPL~\cite{opl} model with lexicographic order to break symmetry.
In~\cite{dnaWordPaper}, the authors report a solution composed from
two pairs of (t-part and m-part) sets $\tuple{T_1,M_1}$ and
$\tuple{T_2,M_2}$ where $|T_1|=6$, $|M_1|=16$, $|T_2|=2$,
$|M_2|=6$. This forms a set $S$ with $(6\times 16)+(2\times 6)=108$
DNA words.
Marc van Dongen reports a larger solution with 112
words.\footnote{See
\url{http://www.cs.st-andrews.ac.uk/~ianm/CSPLib/}.}
Using \bee, we find, in a fraction of a second, a template of size 14
and a map of size 8. This provides a solution of size $14\times 8=112$
to the DNA word problem. Running Comet (v2.0.1) we find a 112 word
solution in about 10 seconds using a model by H\aa kan
Kjellerstrand.\footnote{See
\url{http://www.hakank.org/comet/word_design_dna1.co}.}
We also prove that there does not exist a template of size 15 (0.15
seconds), nor a map of size 9 (4.47 seconds). These facts were unknown
prior to \bee.
Proving that there is no solution to the DNA word problem with
more than 112 words, not via the two part t-m strategy, is still an
open problem.
\vspace{-3mm}
\paragraph{\underline{Model Based Diagnostics}}
(MBD) is an artificial intelligence based
approach that aims to cope with the, so-called, diagnosis
problem~(e.g.~\cite{Reiter87}). In~\cite{MBD}, we (with other
researchers) focus on a notion of minimal cardinality MBD and apply
\bee\ to model and solve the instances of a standard MBD benchmark.
Experimental evidence (see \cite{MBD}), indicates that our approach is
superior to all existing algorithms for minimal cardinality MBD. We
determine, for the first time, minimal cardinality diagnoses for the
entire standard benchmark.
Prior attempts to apply SAT for MBD (for example, by \citeN{Smith05}
and \citeN{Feldman10DX} where a MaxSAT solver is used) indicate that
SAT solvers perform poorly on the standard benchmarks. So, \bee\
really makes the difference.
\section{Conclusion}
We introduce \bee, a compiler to encode finite domain constraints to
CNF. A key design point is to apply bit-level techniques, locally as
prescribed by the word-level constraints in a model.
Optimizations are based on equi-propagation and partial
evaluation. Implemented in Prolog, compilation times are
typically small (measured in seconds) even for instances which result
in several millions of CNF clauses. On the other hand, the reduction
in SAT solving time can be larger in orders of magnitude.
It is well-understood that making a CNF smaller is not the ultimate
goal: often smaller CNF's are harder to solve. Indeed, one often
introduces redundancies to improve SAT encodings: so removing them is
counter productive. Our experience is that \bee\ reduces the size of
an encoding in a way that is productive for the subsequent SAT
solving. In particular, by removing variables that can be determined
``at compile time'' to be definitely equal (or definitely different)
in any solution.
The simplification rules illustrated in Section~\ref{sec:compiling}
apply standard constraint programming techniques (i.e.~to reduce
variable domains). However, equi-propagation is more powerful. It
focuses, in general, in specializing the bit-level representation of
the constraints in view of equations implied by the constraints. In
this way it captures many of the well-known constraint programming
preprocessing techniques, and more.
Future work will investigate: how to strengthen the implementation of
equi-propagation using BDD's and SAT solving techniques, how to
improve the compiler implementation using better data-structures for
the constraint store (for example applying a CHR based approach for
the simplification rules), and how to enhance the underlying
constraint language.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Burberry and Marcus Rashford Launch New Campaign to Help Young People Around the World
British Celebrities
Burberry and Marcus Rashford Raise Funds For Youth Charities
2 November 2020 by Kara Kia
Burberry is partnering with English footballer Marcus Rashford MBE to support numerous youth organisations in London and Manchester and charities with a global reach. After campaigning heavily over the summer to ensure that the government continues to provide free school meals to young people affected by the coronavirus, the Manchester United and England striker was approached by Burberry to fund local and international youth organisations that hold a special place in Rashford's own journey.
In honour of Rashford, Burberry will continue working with London Youth to fund grants and programmes for 15 youth centres that will "make a positive impact in some of London's most deprived communities," Burberry said in a press statement. The heritage fashion house is also partnering with the Norbrook Youth Club and Woodhouse Park Lifestyle Centre in Manchester, which both played a central role in Rashford's upbringing. Volunteers from both organisations will be collaborating to uplift charities in the Wythenshawe area, where Rashford grew up.
To extend Rashford's global reach, Burberry is also making donations to New York City nonprofit Wide Rainbow, which will "provide art supplies, food deliveries, and music education to young people in these communities as well as fund the creation of art murals to invigorate schools and shelters in New York and Los Angeles." In Asia, Burberry is joining forces with the International Youth Foundation to contribute to the Global Youth Resiliency Fund. "This will enable young entrepreneurs and community leaders, especially in Asia, to develop solutions to challenges including closing nutrition gaps and unlocking access to livelihoods," Burberry said.
To celebrate Burberry's commitment to Rashford's cause, the off-the-field hero shared a letter to his 10-year-old self about believing in the unique value that you bring to the world, and despite where you come from, to consider where you are going. "I encourage you to dream, because sometimes dreams are all you will have," Rashford wrote in a letter to his younger self and all young people. "I encourage you to lay your head on the pillow and close your eyes tight and think of better days to come, because luckily for us, our dreams have come true. Would I be the Marcus Rashford you see stood in front of you today if it wasn't for the hardship and struggle? Simple answer? No. You should never be ashamed to ask for help. Take pride in knowing that your struggle will play the biggest role in your purpose. Never drop your head in shame . . . There have been many days that you have felt lesser than others, but no more. Your voice, your stance, your family, your community, and friends, all matter. Whenever you feel like you have very little, know that there are always people willing to give. Just look at those doors wide open welcoming you in for a snack or a chat. If I had to ask one thing of you it would be this. Please, never go to bed feeling like you don't have a role to play in this life because, believe me when I tell you, the possibilities are endless."
Ahead, take a closer look at Rashford's portraits for Burberry's charitable partnership, as well as a copy of his letter to his 10-year-old self in full.
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var uuid = require('node-uuid');
var logger = require('log4js').getLogger();
var FakeAjaxBear = function(bearDataSource) {
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bearDataSource = [{
bear: {
bearId: '4a82199e-7c30-4a66-b194-6d40127fbb89',
userId: '12345678-34567-67',
bearUsername: "jason",
socialId: '24567789',
avatarId: 23,
hasSignedIn: false
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var document = window.document;
var $ = window.jQuery;
this.getCurrentBear = function() {
var $deferred = $.Deferred();
setTimeout(function() {
$deferred.resolve(bearDataSource[0]);
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return $deferred.promise();
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this.signIn = function(cmd) {
var $deferred = $.Deferred();
setTimeout(function() {
var o = {
response: 'OK'
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$deferred.resolve(o);
}, 50);
return $deferred.promise();
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module.exports = FakeAjaxBear;
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{"url":"https:\/\/cran.csiro.au\/web\/packages\/meshed\/vignettes\/univariate_irregular.html","text":"# MGPs for univariate data at irregularly spaced locations\n\n#### 2021-10-06\n\nCompared to the univariate gridded Gaussian case, we now place the data irregularly and assume we observe counts rather than a Gaussian response.\n\nlibrary(magrittr)\nlibrary(dplyr)\nlibrary(ggplot2)\nlibrary(meshed)\nset.seed(2021)\n\nSS <- 30 # coord values for jth dimension\ndd <- 2 # spatial dimension\nn <- SS^2 # number of locations\np <- 3 # number of covariates\n\n# irregularly spaced\ncoords <- cbind(runif(n), runif(n))\ncolnames(coords) <- c(\"Var1\", \"Var2\")\n\nsigmasq <- 1.5\nnu <- 0.5\nphi <- 10\ntausq <- .1\nee <- rnorm(n) * tausq^.5\n\n# covariance at coordinates\nCC <- sigmasq * exp(-phi * as.matrix(dist(coords)))\n# cholesky of covariance matrix\nLC <- t(chol(CC))\n# spatial latent effects are a GP\nw <- LC %*% rnorm(n)\n\n# covariates and coefficients\nX <- matrix(rnorm(n*p), ncol=p)\nBeta <- matrix(rnorm(p), ncol=1)\n\n# univariate outcome, fully observed\ny_full <- 1 + X %*% Beta + w + ee\n\n# now introduce some NA values in the outcomes\ny <- y_full %>% matrix(ncol=1)\ny[sample(1:n, n\/5, replace=FALSE), ] <- NA\n\nsimdata <- coords %>%\ncbind(data.frame(Outcome_full= y_full,\nOutcome_obs = y,\nw = w))\n\nsimdata %>% ggplot(aes(Var1, Var2, color=w)) +\ngeom_point() +\nscale_color_viridis_c() +\ntheme_minimal() + ggtitle(\"w: Latent GP\")\n\nsimdata %>% ggplot(aes(Var1, Var2, color=y)) +\ngeom_point() +\nscale_color_viridis_c() +\ntheme_minimal() + ggtitle(\"y: Observed outcomes\")\n\nIn spmeshed we can choose settings$forced_grid=FALSE, in which case we are fitting the model $y = X \\beta + w + \\epsilon,$ where $$w \\sim MGP$$ are spatial random effects and $$\\epsilon \\sim N(0, \\tau^2 I_n)$$. If instead we choose settings$forced_grid=TRUE, then the model becomes $y = X \\beta + H w + \\epsilon',$ where $$w$$ are now sampled on a grid of knots, $$H$$ is a matrix that interpolates $$w$$ to the observed locations, and $$\\epsilon \\sim N(0, R + \\tau^2 I_n)$$ where $$R$$ is a diagonal matrix that corrects the measurement error variance. Refer to Peruzzi et al (2021)1 for details.\n\nRegardless of the model chosen, for spatial data, an exponential covariance function is used by default: $$C(h) = \\sigma^2 \\exp( -\\phi h )$$ where $$h$$ is the spatial distance.\n\nThe prior for the spatial decay $$\\phi$$ is $$U[l,u]$$ and the values of $$l$$ and $$u$$ must be specified. We choose $$l=1$$, $$u=30$$ for this dataset.2\n\nSetting verbose tells spmeshed how many intermediate messages to output while running MCMC. We set up MCMC to run for 3000 iterations, of which we discard the first third. We use the argument block_size=25 to specify the coarseness of domain partitioning. In this case, the same result can be achieved by setting axis_partition=c(10, 10).\n\nFinally, we note that NA values for the outcome are OK since the full dataset is on a grid. spmeshed will figure this out and use settings optimized for partly observed lattices, and automatically predict the outcomes at missing locations. On the other hand, X values are assumed to not be missing.\n\nmcmc_keep <- 200 # too small! this is just a vignette.\nmcmc_burn <- 400\nmcmc_thin <- 2\n\nmesh1_total_time <- system.time({\nmeshout1 <- spmeshed(y, X, coords,\nblock_size = 25,\nn_samples = mcmc_keep, n_burn = mcmc_burn, n_thin = mcmc_thin,\nverbose = 0,\nsettings=list(forced_grid=FALSE, cache=FALSE),\nprior=list(phi=c(1,30))\n)})\n#> Warning in spmeshed(y, X, coords, block_size = 25, n_samples = mcmc_keep, : Data\n#> look not gridded: force a grid with settings$forced_grid=T. And now with the gridded knots. We build a grid with as many knots as there are observations, but we could use a smaller or a bigger one. mesh2_total_time <- system.time({ meshout2 <- spmeshed(y, X, coords, grid_size = c(30, 30), block_size = 25, n_samples = mcmc_keep, n_burn = mcmc_burn, n_thin = mcmc_thin, n_threads = 2, verbose = 0, prior=list(phi=c(1,30)) )}) Post-processing proceeds the same way as when the data locations are on a grid. Let\u2019s do that for the second model. summary(meshout2$lambda_mcmc[1,1,]^2)\n#> Min. 1st Qu. Median Mean 3rd Qu. Max.\n#> 1.425 2.286 2.767 3.174 3.628 7.606\nsummary(meshout2$theta_mcmc[1,1,]) #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> 1.138 2.471 3.386 3.261 3.900 6.171 summary(meshout2$tausq_mcmc[1,])\n#> Min. 1st Qu. Median Mean 3rd Qu. Max.\n#> 0.07923 0.15122 0.16977 0.17221 0.19697 0.26921\nsummary(meshout2$beta_mcmc[1,1,]) #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> -1.403 -1.345 -1.327 -1.328 -1.312 -1.261 We proceed to plot predictions across the domain along with the recovered latent effects. We plot the predictions at the input locations, i.e. the points not on the grid of knots, which we find via forced_grid==0. We plot the recovered latent process on the grid, for a nicer picture. # process means wmesh <- data.frame(w_mgp = meshout2$w_mcmc %>% summary_list_mean())\n# predictions\nymesh <- data.frame(y_mgp = meshout2$yhat_mcmc %>% summary_list_mean()) outdf <- meshout2$coordsdata %>%\ncbind(wmesh, ymesh) %>%\nleft_join(simdata)\n\noutdf %>% filter(forced_grid==1) %>%\nggplot(aes(Var1, Var2, fill=w_mgp)) +\ngeom_raster() +\nscale_fill_viridis_c() +\ntheme_minimal() + ggtitle(\"w: recovered\")\n\n\noutdf %>% filter(forced_grid==0) %>%\nggplot(aes(Var1, Var2, color=y_mgp)) +\ngeom_point() +\nscale_color_viridis_c() +\ntheme_minimal() + ggtitle(\"y: predictions\")\n\n1. spmeshed implements a model which can be interpreted as assigning $$\\sigma^2$$ a folded-t via parameter expansion if settings$ps=TRUE, or an inverse Gamma with parameters $$a=2$$ and $$b=1$$ if settings$ps=FALSE, which cannot be changed at this time. $$\\tau^2$$ is assigned an exponential prior.\u21a9\ufe0e","date":"2021-12-05 13:58:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5812417268753052, \"perplexity\": 10022.954632096533}, \"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-2021-49\/segments\/1637964363189.92\/warc\/CC-MAIN-20211205130619-20211205160619-00050.warc.gz\"}"}
| null | null |
Q: Using global variables Cpp I am newbie in C++. Previously, I learnt the C language and decide to shift to learn C++ language. However, I find something hard for me to follow, particularly in using global variables. I wrote a program and I get wrong. When I tried to debug, my program show error in the line
item* list = ::array[index].head;
in which item is
class item
{
public:
int key, data;
item* next;
};
and array is a global variable which is define as
class arrayitem
{
public:
item* head;
item* tail;
};
arrayitem* array;
Edit: This is what I get when I debug my program:
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Fix those winter blues with some retail therapy! Just 25 minutes from Wroxton House Bicester Village has over 130 designer boutiques, enough that everyone will find a bargain.
Pass by Reception before you leave and ensure you pick up and voucher for an additional 10% off that already reduced prices.
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| 8,426
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Die Talsperre Nunes () liegt in der Region Nord Portugals im Distrikt Bragança. Sie staut den Tuela, den linken (östlichen) Quellfluss des Tua zu einem Stausee auf. Die Kleinstadt Vinhais liegt ungefähr vier Kilometer westlich der Talsperre.
Die Talsperre Nunes wurde 1995 fertiggestellt.
Absperrbauwerk
Das Absperrbauwerk ist eine Bogengewichtsmauer aus Beton mit einer Höhe von 21,5 m über der Gründungssohle (18,5 m über dem Flussbett). Die Mauerkrone liegt auf einer Höhe von 542,3 m über dem Meeresspiegel. Die Länge der Mauerkrone beträgt 65,5 m. Die Staumauer verfügt sowohl über einen Grundablass als auch über eine Hochwasserentlastung. Über den Grundablass können maximal 52,36 m³/s abgeleitet werden, über die Hochwasserentlastung maximal 1.350 m³/s.
Stausee
Beim normalen Stauziel von 535,5 m (maximal 541,3 m bei Hochwasser) fasst der Stausee rund 0,138 Mio. m³ Wasser – davon können 0,098 Mio. m³ genutzt werden. Das minimale Stauziel liegt bei 532 m.
Kraftwerk
Es sind zwei Francis-Turbinen mit einer maximalen Leistung von zusammen 9,9 MW installiert. Die durchschnittliche Jahreserzeugung liegt bei 41,56 Mio. kWh.
Das Kraftwerk ist im Besitz der Sociedade Produtora de Energia Mini-Hídrica, Lda.
Siehe auch
Liste von Wasserkraftwerken in Portugal
Liste von Talsperren der Welt (Portugal)
Einzelnachweise
Nunes
Nunes
Nunes
Nunes
Nunes
SNunes
Distrikt Bragança
Erbaut in den 1990er Jahren
|
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| 3,465
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Mark and Suvanna's wedding was set in the beautiful countryside of Concan, Texas back in March of 2014. This wedding was particularly important to us because Suvanna is Athan's sister. It was a beautiful ceremony that we will not forget and we were more than honored to have been able to capture such an amazing moment in this couple's lives. We love coming back to this video to enjoy all those memories again and absolutely wish Mark and Suvanna all the best.
I was first referred to Rob and Tam for their wedding by a friend at the AACHD. It was by pure chance that upon meeting Rob we realized we had actually gone to the same high school over a thousand miles away (and over a decade ago) in Philadelphia. Their beautiful ceremony was held at the Blanton Museum of Art on UT's campus on a sunny summer day. Sometimes, chance just brings people together.
Christina and Montana came to us seeking unique wedding invitation pictures. We found a hidden lake in the outskirts of Ft. Worth to use as a backdrop for their pictures. Needless to say, it was a lot of fun and we wish them the best of luck in their marriage!
Marcos and Nikki choose the beautiful setting of Austin's Hummingbird House for their special day. The moments leading up to ceremony were masked by a light misty precipitation, however as soon as the two were pronounced husband and wife, the sun peaked out from behind the clouds.
I was very happy when Luke and Heather came to me asking for a video for their wedding. The weather was great, the setting was beautiful and of course everyone their looked their absolute best.
I wish Luke and Heather the best and thank them so much for allowing me to capture their amazing day.
|
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"redpajama_set_name": "RedPajamaC4"
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| 8,187
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Q: Plotly: How to add extra information to marker of 3D scatter plot? What I'd like to do is adding each string info into only each balloon like below capture. But without showing them next to each plot because it looks messy when plots are many.
My data frame is below, I added info to each row as string.
import plotly.express as px
import numpy as np
import pandas as pd
from sklearn import datasets
from sklearn.manifold import TSNE
digits = datasets.load_digits()
X_reduced = TSNE(n_components=3, random_state=0).fit_transform(digits.data)
myinfo = ['ID' + str(i+1) for i in range(X_reduced.shape[0])]
df = pd.DataFrame(data=X_reduced, columns=columns, dtype='float')
df['info'] = myinfo ## <- This is what i want to show in a balloon.
print(df)
a b c info
0 18.150545 -1.938995 1.192057 ID1
1 -8.773806 2.475938 -10.885373 ID2
2 4.305762 5.132519 -7.347726 ID3
3 5.327038 7.438313 8.786077 ID4
4 -15.810744 -9.374986 0.310686 ID5
... ... ... ... ...
1792 1.192152 -0.850245 12.781646 ID1793
1793 14.183956 -3.544634 -1.009673 ID1794
1794 -0.907307 2.596647 -4.307703 ID1795
1795 1.608326 0.700724 9.267643 ID1796
1796 1.809730 5.428222 -1.771497 ID1797
What I tried is that I assigned the string info as text='info' of px.scatter_3d() as following code.
fig = px.scatter_3d(df, x='a', y='b', z='c', color='a', size_max=6 , text='info', opacity=0.7)
fig.show()
Does someone know how to get rid of these info beside plots?
Thank you in advance.
A: Using your setup, all you have to do is run:
fig.update_traces(mode = 'markers')
...or:
fig.for_each_trace(lambda t: t.update(mode = 'markers'))
This removes the strings associated with the markers, but retains the added information in the hoverlabel:
This approach lets you keep your call to px.scatter_3d(..., text='info') as it stands.
Another approach would be to edit the hoverinfo directly with a custom_value as described in the post Plotly 3D plot annotations
|
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| 8,789
|
\section{Introduction}
\subsection{Context and purpose}
If nuclear matter is compressed to sufficiently large densities it turns into quark matter, which is weakly interacting at asymptotically large densities. The nature of this transition is unknown because first-principle calculations
within Quantum Chromodynamics (QCD) are currently inaccessible for matter at low temperature and large baryon density
due to the strong-coupling nature of the problem and the so-called sign problem of lattice gauge theory \cite{Aarts:2015tyj}. In a simple picture, extreme compression makes the nucleons overlap, and at some point it is the quarks, no longer the nucleons,
which are the relevant degrees of freedom. Since there is no strict order parameter for confinement and chiral symmetry breaking, it appears that there is no qualitative difference between low-density nuclear matter and ultra-dense quark matter, such that this transition is allowed to be continuous. This scenario is realized at zero baryon density, where we do know from first principles that there is a crossover from the hadronic phase to the quark-gluon plasma. At large baryon densities, the situation is more complicated due to the presence of Cooper pairing. Since asymptotically dense and sufficiently cold matter is in the color-flavor locked (CFL) phase \cite{Alford:1998mk,Alford:2007xm}, the actual question is whether one can go continuously from nuclear matter to CFL rather than to unpaired quark matter \cite{Schafer:1998ef,Alford:1999pa,Hatsuda:2006ps,Schmitt:2010pf,Alford:2018mqj,Chatterjee:2018nxe,Cherman:2018jir}.
Here we will ignore Cooper pairing and address the question of a possible continuity between unpaired, isospin-symmetric nuclear matter and unpaired quark matter, as for instance envisioned
within a percolation picture \cite{Baym:1979etb,Celik:1980td}.
Combining nuclear matter and quark matter in the same calculation on a microscopic level is very challenging, even if the rigor of a first-principle calculation is given up. However, besides the theoretical motivation just discussed, such studies are highly desired in the context of neutron stars, where for instance mass and radius, but also gravitational wave signals from neutron star mergers, depend on thermodynamic properties of the matter in the core of the star and on the possible existence of a first-order phase transition between nuclear matter and quark matter \cite{Alford:2013aca,Bastian:2018wfl,Christian:2018jyd,Most:2018eaw}. Therefore, various studies exist which combine two distinct models, one with nucleons as degrees of freedom, one with quarks as degree of freedom. They are then either glued together at a first-order phase transition, or a smooth interpolation is introduced between the two phases \cite{Masuda:2012ed,PhysRevD.88.085001,Masuda2016}.
In either case, these approaches have no predictive power for the nature and the location of the quark-hadron phase transition. Other studies attempt to find a sufficiently general parameterization of the equation of state without relying on a microscopic theory, using microscopic and/or astrophysical input as constraints for the parameterization \cite{Alford:2013aca,Kurkela:2014vha,Tews:2018kmu}. Another strategy is to use a single
model that has either quark degrees of freedom or nucleonic degrees of freedom, but not both. In that case, potential chiral and
deconfinement phase transitions are determined consistently within the given model, but at least one side of the phase transition is then only a very rough approximation to the real world \cite{Schaefer:2007pw,Palhares:2010be,BRATOVIC2013131,Fraga:2018cvr}. An example is the Nambu-Jona-Lasinio (NJL) model \cite{Buballa:2003qv,Ruester:2005jc,Buballa:2016fjh}, which usually does not contain baryons, although it is conceivable to include them \cite{Ishii:1995bu,Alkofer:1994ph,CHRISTOV199691}.
Our present approach, based on the gauge/gravity duality, is at best a distorted
version of QCD, in some sense similar to an NJL model, which however {\it does} have a well-defined concept of baryons in a chirally broken phase and of
chirally symmetric quark matter. Our main results are qualitative and thus at this point we do not attempt to make quantitative predictions for the physics of neutron stars. We do make a step in this direction, however, by fitting the parameters of our holographic model to properties of nuclear matter at saturation
and by calculating the speed of sound for all densities. The results indicate that -- together with future improvements --
our approach can be a useful alternative to the models currently employed for dense matter in neutron stars.
There exist other recent studies that have connected results from the gauge/gravity duality to neutron star physics \cite{Hoyos:2016zke,Annala:2017tqz,Jokela:2018ers}. However, these studies have combined a holographic equation of state for quark matter with a traditional approach for nuclear matter and thus fall in the above category of combining two distinct theories.
\subsection{Method}
We employ the gauge/gravity correspondence ("holography") which allows for a non-pertur\-bative calculation
of strong-coupling physics via the classical gravity approximation of certain string theories \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}. Specifically, we work with the model developed by Sakai and Sugimoto \cite{Sakai:2004cn,Sakai:2005yt}, who introduced fundamental chiral fermions through $N_f$ D8 and $\overline{\rm D8}$ branes ("flavor branes") into Witten's original model for low-energy QCD based on a background of $N_c$ D4 branes \cite{Witten:1998zw}, see Refs.\ \cite{Peeters:2007ab,Gubser:2009md,Rebhan:2014rxa} for reviews. Here, $N_f$ and $N_c$ are the numbers of flavors and colors, respectively, and the model is usually
evaluated
in the limit $N_f\ll N_c$, which is a necessary condition to neglect effects of the flavor branes on the background geometry. We shall work in the so-called decompactified
limit of the model, where the background is given by the deconfined geometry.
In this limit, the gluon dynamics are decoupled and the dual field theory is expected to resemble the NJL model in certain aspects \cite{Antonyan:2006vw,Davis:2007ka,Preis:2012fh}. The price we have to pay for employing this limit is
that the rigorous connection to large-$N_c$ QCD is lost -- which in any case only exists in the limit of
small 't Hooft coupling, which is inaccessible within the gravity approximation. But, in the decompactified limit,
the chiral phase transition depends nontrivially on the baryon chemical potential even if changes of the background geometry are neglected. This results in a richer
phase structure which may well be closer to
real-world $N_c=3$ QCD, at least with regard to the chiral phase transition.
Baryons are introduced following the general recipe of the gauge/gravity correspondence, i.e., through branes that
wrap around an internal subspace of the ten-dimensional geometry and that have $N_c$ string endpoints. In the
non-supersymmetric Sakai-Sugimoto model, this picture becomes equivalent to instanton solutions of the gauge theory on the flavor branes \cite{Hata:2007mb}.
Here, "instanton" refers to an object localized in position space and the holographic direction, i.e., they are similar to
ordinary Yang-Mills instantons with the time direction replaced by the holographic coordinate. The instantons are introduced in the chirally broken phase, where the D8 and $\overline{\rm D8}$ branes are connected, and they carry topological baryon number due to the presence of a Chern-Simons term in the action. This
is our holographic version of "nuclear matter". In the
chirally restored phase, the flavor branes are disconnected and baryon number is carried by quarks, represented by strings attached with one end at one of the $N_f$ flavor branes (either D8 or $\overline{\rm D8}$, depending on the chirality of the quarks) and the other end at one of the $N_c$ D4 branes that are responsible for the curved background geometry. This is our holographic version of "quark matter". One of our main results is the observation that the two embeddings of the flavor branes (connected and disconnected)
can be continuously
transformed into each other in the presence of instantons.
Throughout the paper we shall work in the chiral limit. Since the D8 and $\overline{\rm D8}$ branes intercept the D4 branes, quarks are massless in the chirally restored phase.
For discussions about nonzero quark masses in the Sakai-Sugimoto model see
Refs.\ \cite{Evans:2007jr,Bergman:2007pm,Dhar:2007bz,Aharony:2008an,Hashimoto:2008sr}. In the massless limit,
chiral symmetry is exact, which suggests a true phase transition between (chirally broken) nuclear matter and (chirally restored) quark matter, either of first or second order. It thus appears
contradictory to address the question of the continuity between nuclear and quark matter in the chiral limit.
However, our present goal is more modest: we seek to connect chirally broken and chirally restored phases
in the "order parameter space", along stationary, but not necessarily stable, points of the free energy. This is one step towards an effective potential defined in the entire "order parameter space", which depends on chemical potential and temperature and where nuclear and quark matter correspond to local minima that can be continuously connected whenever they exist. Whether the model allows for a quark-hadron continuity between stable phases can be addressed more comprehensively only after including quark masses.
The main difficulty of our approach is the treatment of the baryonic phase. Single baryons in the
Sakai-Sugimoto model have been studied extensively in the
literature, from analytical flat-space approximations to purely numerical studies \cite{Sakai:2004cn,Sakai:2005yt,Hata:2007mb,Seki:2008mu,Cherman:2011ve,Rozali:2013fna}. Here we are interested in a many-baryon system, which requires various approximations. We follow the approach developed in Refs.\ \cite{Ghoroku:2012am,Li:2015uea,Preis:2016fsp,Preis:2016gvf}, which is based on the pioneering work \cite{Bergman:2007wp}, where
pointlike instantons were considered. Our work is a direct improvement of Ref.\ \cite{Preis:2016fsp}, whose approach we extend by accounting for the
interactions of the instantons in the bulk.
This is done by using the exact two-instanton solution in flat space, which is a special case of the general Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction
\cite{Atiyah:1978ri}, and which has been discussed previously in the context of the Sakai-Sugimoto model to study
the nucleon-nucleon interaction \cite{Kim:2008iy,Hashimoto:2009ys}.
As we shall see, the instanton
interactions are crucial for our main observation.
There are various other, in some aspects complementary, approximations to many-baryon phases in the Sakai-Sugimoto model,
see for instance Refs.\ \cite{Kim:2007vd,Rho:2009ym,Kaplunovsky:2010eh,Kaplunovsky:2012gb,deBoer:2012ij,Bolognesi:2013nja,Kaplunovsky:2015zsa}. One of them is based on a homogeneous ansatz for the gauge fields in the bulk \cite{Rozali:2007rx,Li:2015uea,Elliot-Ripley:2016uwb}, which is
expected to yield a better approximation for large densities, but which is less transparent from a physical point of view because it is not built from single instantons.
Our paper is organized as follows. The main point of Sec.\ \ref{sec:setup} is to introduce our ansatz for the non-abelian gauge fields on the flavor branes, based on the single instanton and two-instanton solutions in flat space. This is discussed in Sec.\ \ref{sec:ansatz}
with the technical background deferred to appendix \ref{app:flat}. In Secs.\ \ref{sec:eom} and \ref{sec:mini} we solve the equations of motion and set up the calculation of the stationary points of the free energy. Sec.\ \ref{sec:results} contains our main results. In particular, we discuss the continuity between the chirally broken and chirally symmetric phases in Sec.\ \ref{sec:continuity},
and we compute the speed of sound in Sec.\ \ref{sec:sound}. We give our conclusions in Sec.\ \ref{sec:conclusions}.
\section{Setup}
\label{sec:setup}
\subsection{Action}
\label{sec:action}
The general setup of the model in the given limit has been used in various previous works. The study most relevant to the present paper is Ref.\ \cite{Preis:2016fsp}, whose notation we shall employ. Here we will not go into details regarding the construction of the model itself and refer the reader to the reviews and original works quoted in
the introduction and, more specifically, to Refs.\ \cite{Li:2015uea,Preis:2016fsp}. The most important difference to
Ref.\ \cite{Preis:2016fsp} is our ansatz for the non-abelian field strength, which we will explain in detail.
Our starting
point is the action for the U($N_f$) gauge fields on the D8 and $\overline{\rm D8}$ branes in the
deconfined geometry. In our treatment of baryons we shall restrict ourselves to $N_f=2$, i.e., we shall not include hyperons, and thus the non-abelian field strengths will be approximated with the help of
SU(2) instantons. For consistency we shall use $N_f=2$ also for the quark matter phase, although for our purpose it would be trivial to add a third massless flavor in the chirally symmetric phase. We shall briefly comment on the three-flavor case in Sec.\ \ref{sec:solution}. The action has a Dirac-Born-Infeld (DBI) and a Chern-Simons (CS) contribution,
\begin{equation}
S= S_{\rm DBI} + S_{\rm CS} \, ,
\end{equation}
with
\begin{subequations} \label{SDBICS}
\begin{eqnarray}
S_{\rm DBI} &=& \frac{{\cal N}}{M_{\rm KK}^3T} \int d^3 x \int_{u_c}^\infty du\, u^{5/2}\nonumber\\ [2ex]
&&\times {\rm str} \sqrt{(1+u^3f_Tx_4'^2+\hat{F}_{u0}^2)\left(1+\frac{F_{ij}^2}{2u^3\lambda_0^2}\right)+\frac{f_TF_{iu}^2}{\lambda_0^2}+\frac{f_T(F_{ij}F_{ku}\epsilon_{ijk})^2}{4u^3\lambda_0^4}} \, , \label{DBI}\\[2ex]
S_{\rm CS} &=& \frac{3{\cal N}}{2\lambda_0^2M_{\rm KK}^3T} \int d^3 x \int_{u_c}^\infty du\,\hat{a}_0{\rm Tr}[F_{ij}F_{ku}]\epsilon_{ijk} \, , \label{CS}
\end{eqnarray}
\end{subequations}
where we have abbreviated
\begin{equation}
{\cal N} \equiv \frac{N_c M_{\rm KK}^4\lambda_0^3}{6\pi^2} \, , \qquad \lambda_0\equiv \frac{\lambda}{4\pi} \, ,
\end{equation}
with the 't Hooft coupling $\lambda$.
For convenience, we have started with a mostly dimensionless formulation, the relation to the corresponding dimensionful quantities
can be found in Refs.\ \cite{Li:2015uea,Preis:2016fsp}. The only dimensionful quantities in the present formulation are in the
prefactor of the action, the temperature $T$ and the Kaluza-Klein mass $M_{\rm KK}$, which is the inverse radius of the compactified extra dimension with coordinate $x_4$. In the dimensionless coordinates used here, $x_4 \equiv x_4 + 2\pi$.
The integration over imaginary time has already been performed, yielding the prefactor $1/T$. The remaining integrals
are taken over position space $\mathbb{R}^3$ and the holographic coordinate $u\in [u_c,\infty]$, where $u_c$ is the
location of the tip of the connected flavor branes\footnote{The notation "$u_c$" originates from Ref.\ \cite{Bergman:2007wp}, where
pointlike instantons were used as an approximation, which cause a cusp at the tip of the connected flavor branes, hence the subscript $c$. We keep using this notation even though the embedding is smooth in the presence of instantons with nonzero width.} and the holographic boundary is located at $u=\infty$. The embedding of the flavor branes, given by the function $x_4(u)$ and
by $u_c$ itself will later be determined dynamically, with the boundary condition $x_4(\infty)-x_4(u_c)=\ell/2$, where $\ell$
is the dimensionless asymptotic separation of the D8 and $\overline{\rm D8}$ branes, its dimensionful version given by $L=\ell/M_{\rm KK}$.
We have denoted derivatives with respect to $u$ by a prime. The DBI action has been written in terms of the symmetrized
trace "str", to be taken over the 2-dimensional flavor space. For simplicity, we shall later work with the ordinary trace.
This is a
tremendous simplification, and it has been shown for a very similar calculation that the use of the symmetrized
trace prescription does not make a large quantitative difference in the results \cite{Preis:2016fsp}. The expression in the
square root is obtained from ${\rm det}(g+2\pi \alpha'{\cal F})$, where $g$ is the induced metric on the flavor branes,
${\cal F}$ is the field strength tensor, and $\alpha'=\ell_s^2$ with the string length $\ell_s$ (which is absorbed in the definition of the dimensionless field strengths). For the explicit form of the metric see for instance Ref.\ \cite{Preis:2016fsp}. It contains the temperature-dependent function
\begin{equation}
f_T(u)\equiv 1-\frac{u_T^3}{u^3} \, , \qquad u_T = \left(\frac{4\pi}{3}\right)^2 \frac{t^2}{\ell^2} \, ,
\end{equation}
with the dimensionless temperature $t=T/M_{\rm KK}$. The field strengths are
decomposed according to ${\rm U(2)}\cong {\rm U(1)}\times {\rm SU(2)}$. Within our ansatz, the only
nonzero abelian field strength is $\hat{F}_{u0}=i\hat{a}_0'$ with the temporal component of the abelian part of the gauge field
$\hat{a}_0(u)$ (the factor $i$ is needed in the imaginary time formalism). The quark chemical potential enters the action as
a boundary condition for this abelian gauge field, $\hat{a}_0(\infty) = \mu$. The only nonzero non-abelian field strengths are
$F_{ij}$ and $F_{iu}$, $i=1,2,3$. They are responsible for the nonzero baryon number through the topological charge provided by
the CS contribution, and their form plays the main role in our calculation. We do not attempt to
solve the full equations of motion for all gauge fields plus the embedding function in $\vec{x}$ and $u$. (In the action (\ref{SDBICS}) we have already neglected the spatial derivatives of $\hat{a}_0$ and $x_4$.) We shall rather insert an ansatz
for the square of the non-abelian field strengths, which depends on $\vec{x}$ and $u$, and then average over position space
such that the spatial integration in the action becomes trivial. Then, we will proceed without further approximation and solve the equations of motion for $\hat{a}_0(u)$ and $x_4(u)$ fully dynamically.
We will exclusively work with the deconfined geometry, whose metric has been used to derive the DBI action (\ref{DBI}). Since we are interested in the chiral phase transition, we need to work in a regime where the critical temperature of the chiral phase transition is larger than that of the deconfinement transition. Since $T_c^{\rm deconf} = M_{\rm KK}/(2\pi)$ (for all $\mu$ unless backreactions are taken into account) and $T_c^{\rm chiral} \simeq 0.15/L$ (at $\mu=0$) \cite{Aharony:2006da}, this requires $\ell/\pi \lesssim 0.31$ ($\ell=\pi$ is the antipodal limit used in the original works by Sakai and Sugimoto). We shall also assume that the deconfined geometry is preferred down to arbitrarily small temperatures, which is justified in the decompactified limit, where the radius of the $x_4$ circle is taken to infinity (at fixed $L$) such that $M_{\rm KK}$ and thus $T_c^{\rm deconf}$ goes to zero. In this limit, the chiral phase transition is entirely determined by the dynamics on the flavor branes, not by the background geometry.
\subsection{Ansatz for non-abelian field strengths}
\label{sec:ansatz}
With our ansatz for the non-abelian field strengths we attempt to capture as many effects as possible from a many-instanton
system and at the same time try to keep the expressions as simple as possible to make feasible the full evaluation of the
equations of motion for the abelian gauge field and the embedding function of the flavor branes. We will construct our
ansatz from the 1-instanton and the 2-instanton solutions in flat space, which we now discuss.
\subsubsection{Single instanton}
For large, but not infinite, 't Hooft coupling,
the leading-order expression for a single instanton has been discussed for the confined geometry in Ref.\ \cite{Hata:2007mb} for
maximally separated flavor branes (= on opposite ends of the $x_4$ circle, $\ell = \pi$) and for non-maximal separation
in Ref.\ \cite{Seki:2008mu}. Here we need the analogous result for the deconfined geometry from Ref.\ \cite{Preis:2016fsp}, which reads
\begin{equation} \label{Fiz0}
F_{iz}^{(1)}F_{iz}^{(1)} = \frac{12(\rho/\gamma)^4}{\gamma^2[x^2+(z/\gamma)^2+(\rho/\gamma)^2]^4} \, ,
\end{equation}
where the superscript "$(1)$" is added to indicate the single-instanton solution, where $x^2 = x_1^2+x_2^2+x_3^2$, and where $z\in [-\infty,\infty]$ is the holographic coordinate along the connected flavor branes, defined by
\begin{equation} \label{uz}
u = (u_c^3+u_c z^2)^{1/3} \, ,
\end{equation}
such that $z=0$ corresponds to the tip of the connected branes, $u=u_c$. The shape of
the instanton is given by
\begin{equation} \label{rhogam}
\rho = \frac{\rho_0u_c^{3/4}}{\lambda^{1/2}}\left[\frac{f_T(u_c)}{\beta_T(u_c)}\right]^{1/4} \sqrt{\alpha_T(u_c)}\, , \qquad \gamma = \frac{3\gamma_0 u_c^{3/2}}{2}\sqrt{\alpha_T(u_c)} \, ,
\end{equation}
where, as Eq.\ (\ref{Fiz0}) shows, $\rho$ can be interpreted as the width in the holographic direction, while $\rho/\gamma$ corresponds to the spatial width. Therefore, $\gamma$ is a
"deformation parameter", which characterizes the deviation of the instanton from SO(4) symmetry. This deviation has also been
observed in a purely numerical evaluation of a single instanton \cite{Rozali:2013fna}.
In the large-$\lambda$ expansion, $\rho_0=6\sqrt{2\pi/\sqrt{5}}\simeq 10.06$, $\gamma_0=2\sqrt{2/3}\simeq 1.633$ \cite{Preis:2016fsp}, and we have abbreviated
\begin{equation}
\alpha_T(u_c) \equiv 1-\frac{5u_T^3}{8u_c^3} \, , \qquad \beta_T(u_c) \equiv 1- \frac{u_T^3}{8u_c^3}-\frac{5u_T^6}{16u_c^6} \, ,
\end{equation}
such that at zero temperature $f_T(u_c)=\alpha_T(u_c)=\beta_T(u_c)=1$.
The solution (\ref{Fiz0}) is the flat-space Belavin-Polyakov-Schwarz-Tyupkin (BPST) instanton \cite{1975PhLB...59...85B}, which is exact for a single instanton in the limit of large $\lambda$ because the width goes to zero for infinite $\lambda$
and thus the effect of curved space is negligible. For our many-baryon system this is no longer true, and in principle we would have to construct a (numerical) solution in curved space. For simplicity, however, we shall keep the form (\ref{Fiz0})
as an ansatz. In particular, we shall keep the dependence of $\rho$ and
$\gamma$ on $u_c$ from Eq.\ (\ref{rhogam}), but treat $\rho_0$ and $\gamma_0$ as free parameters, which can be adjusted to capture effects that
are known from the numerical solution of a single instanton in curved space and can also be adjusted to reproduce known properties of realistic nuclear matter\footnote{This approach was already suggested
and explored in Ref.\ \cite{Preis:2016fsp}. However, in this reference, $\rho\propto u_c$, $\gamma\propto u_c^{3/2}$ was used, i.e., the dependence
of $\rho$ on $u_c$ was chosen differently for simplicity.
Here we work with the "correct" dependence suggested by
the lowest-order, single-instanton solution, which leads to a somewhat more difficult calculation, but which turns out to be crucial
(together with taking into account instanton interactions) for our main observations.}. Since $u_c$ will be determined dynamically for each $\mu$ and $T$, the instanton width in the spatial and holographic directions will depend on density and temperature.
Self-duality of the instanton solution
implies
\begin{equation} \label{selfdual}
F_{iz}F_{iz} = \frac{F_{ij}F_{ij}}{2\gamma^2} = -\frac{F_{iz}F_{jk}\epsilon_{ijk}}{2\gamma} \, .
\end{equation}
Note that all field strengths squared are proportional to the $2\times 2$ unit matrix in flavor space. At this point, $\gamma$ appears as a mere rescaling of the holographic coordinate $z$. However, when we insert our ansatz into the DBI action $\gamma$ becomes a nontrivial parameter
that cannot be eliminated by rescaling $z$.
The single-instanton solution has topological charge 1, as it should be,
\begin{equation} \label{NB}
-\frac{1}{8\pi^2}\int d^3x \int_{-\infty}^\infty dz \, {\rm Tr}[F_{ij}^{(1)}F_{kz}^{(1)}]\epsilon_{ijk} = \int_{-\infty}^\infty dz \, q_0(z) = 1
\, ,
\end{equation}
where, for later convenience, we have denoted\footnote{In Ref.\ \cite{Preis:2016fsp}, $q_0$ was denoted by $D$.}
\begin{equation}
q_0(z) \equiv \frac{3\rho^4}{4(\rho^2+z^2)^{5/2}} \, .
\end{equation}
\subsubsection{Two instantons}
The second ingredient for our ansatz is the flat-space two-instanton solution. If the separation of the two instantons
is sufficiently large, this solution is well approximated by two single instantons. Let us for now consider SO(4) symmetric instantons, denote the width of both instantons by $\rho$, and let us place one instanton at the origin of the four dimensional $(\vec{x},z)$ space and the other one separated by a distance $\delta$ in the $x_1$ direction.
Then, the exact flat-space
solution that we will use is parameterized by $\rho$ and $\delta$ and reads
\begin{eqnarray} \label{calFsq}
F_{iz}^{(2)}F_{iz}^{(2)}&=& \frac{4096 \rho^4}{[(\delta^2 + 4 |x|^2) (\delta^2 + 8 \rho^2 + 4 |x|^2) - 16 \delta^2 x_1^2]^4} \Big\{3 \delta^8
+16 \delta^6 (3 \rho^2 + |x|^2 + 8 x_1^2) \nonumber\\[2ex]
&& +
32 \delta^4 \Big[6 \rho^4 + |x|^4 + 8 x_1^4 + 48 |x|^2 x_1^2 + 4 \rho^2 (|x|^2 + 8 x_1^2)\Big] \nonumber\\[2ex]
&&+ 256 \delta^2 |x|^2 \Big[|x|^4 + 8 |x|^2 x_1^2 - \rho^2 (|x|^2 - 4 x_1^2) + 768 |x|^8 \Big] \Big\} \, ,
\end{eqnarray}
with $|x|^2=x_1^2+x_2^2+x_3^2+z^2$. The parameters $\rho$ and $\delta$ lose their interpretation of width and separation as $\delta$ is decreased and the instantons start to overlap. The derivation of Eq.\ (\ref{calFsq}) is explained in appendix \ref{app:flat}. It is based on the general ADHM construction \cite{Atiyah:1978ri} and its application to the Sakai-Sugimoto model \cite{Kim:2008iy,Hashimoto:2009ys}. For simplicity, we have assumed the instantons to have the same orientation, termed "defensive skyrmions" in Ref.\ \cite{Kim:2008iy}. This renders the subsequent calculation much simpler, but we should keep in mind that in principle the relative orientation of the instantons in the many-instanton system has to be determined by minimizing the free energy. In this sense, the free energy we calculate has to be understood as an upper limit, likely to be reduced by choosing a nontrivial relative orientation. We introduce the deformation parameter $\gamma$ by the same rescaling as in the single-instanton solution (\ref{Fiz0}): we replace $\rho\to \rho/\gamma$, $z\to z/\gamma$ and divide $F_{iz}^{(2)}F_{iz}^{(2)}$ by $\gamma^2$. Then, we can apply the self-duality (\ref{selfdual}) for the two-instanton solution, and we verify that the topological charge is 2,
\begin{equation} \label{NB2}
-\frac{1}{8\pi^2}\int d^3x \int_{-\infty}^\infty dz \, {\rm Tr}[F_{ij}^{(2)}F_{kz}^{(2)}]\epsilon_{ijk} = \int_{-\infty}^\infty dz \, q_{\rm int}(d,z) = 2 \, .
\end{equation}
Here we have introduced the function
\begin{eqnarray} \label{Phi}
q_{\rm int}(d,z) &=& \frac{3\sqrt{2}\rho^8}{4}\frac{h_1{\cal S}_1+h_2{\cal S}_2}{(a^2+b)^{5/2}b^2} \, ,
\end{eqnarray}
with the polynomials
\begin{subequations} \label{p1p2}
\begin{eqnarray}
h_1&\equiv& 2\Big[\rho^{10}d^6(4d^4+7d^2-2)^2+\rho^{8}z^2d^2(4d^2-1)^2(4d^6+12d^4+7d^2-2)\nonumber\\[2ex]
&&+\rho^6z^4(96d^{10}+116d^8-91d^6-14d^4+11d^2-1)+2\rho^4z^6d^2(26d^6-8d^4-23d^2+5)\nonumber\\[2ex]
&&+4\rho^2z^8d^4(2d^2-3)+4z^{10}d^4\Big] \, , \\[2ex]
h_2 &\equiv& \rho^8d^4(4d^4+7d^2-2)^2+2\rho^6z^2d^2(32d^8+76d^6+15d^4-17d^2+2)\nonumber\\[2ex]
&&+\rho^4z^4(92d^8+144d^6+5d^4-18d^2+2)+8\rho^2z^6d^2(9d^4+8d^2-2)+28z^8d^4\, ,
\end{eqnarray}
\end{subequations}
and
\begin{subequations}\label{SSab}
\begin{eqnarray}
a&\equiv& z^2-\rho^2(d^2-1) \, , \qquad b \equiv \rho^2[4d^2(\rho^2+z^2)-\rho^2] \, , \\[2ex]
{\cal S}_1 &\equiv& \sqrt{\frac{-a+\sqrt{a^2+b}}{b}} \, , \qquad
{\cal S}_2 \equiv \sqrt{a+\sqrt{a^2+b}} \, .
\end{eqnarray}
\end{subequations}
The particular way (\ref{Phi}) of writing the function $q_{\rm int}$ is not only motivated by compactness, but also
turns out to be useful for integrating and differentiating $q_{\rm int}$. We have also introduced the overlap parameter
\begin{equation} \label{d}
d\equiv \frac{\delta}{2\rho/\gamma} \, ,
\end{equation}
which is the distance between the instantons normalized by twice their spatial width, such that, if they were spheres with
radius $\rho/\gamma$, they would overlap for $d<1$.
\subsubsection{Many-instanton approximation}
The ADHM construction in principle provides exact many-instanton solutions, which
however become increasingly complicated and are much too unwieldy for our purpose. Elements of these
general solutions have been used in the Sakai-Sugimoto model in Refs.\ \cite{Hashimoto:2009as,Hashimoto:2010ue}, see
also a brief discussion in Ref.\ \cite{Kaplunovsky:2015zsa}.
Here we construct a simple approximation for a many-instanton system, taking into account only 2-body interactions.
To this end, we first define an interaction energy density by subtracting the two individual single-instanton contributions
from the full two-instanton solution,
\begin{equation} \label{Int}
{\cal I}(\vec{x}_1,z_1,\vec{x}_2,z_2) \equiv (F_{iz}^{(2)})^2(\vec{x}_1,z_1,\vec{x}_2,z_2)- (F_{iz}^{(1)})^2(\vec{x}_1,z_1)-(F_{iz}^{(1)})^2(\vec{x}_2,z_2) \, .
\end{equation}
Via the self-duality (\ref{selfdual}), analogous relations hold for the other two relevant quadratic forms $F_{ij}F_{ij}$ and
$F_{ij}F_{kz}\epsilon_{ijk}$. The two instantons are centered at the points $(\vec{x}_1,z_1)$ and $(\vec{x}_2,z_2)$.
Each term in Eq.\ (\ref{Int}) also depends on $\vec{x}$ and $z$, which we have not indicated explicitly in the arguments.
The field strength squared for a system of $N_I$ instantons is then approximately constructed as
\begin{equation}\label{FsqN}
F_{iz}^2 \simeq \sum_n^{N_I} (F_{iz}^{(1)})^2(\vec{x}_n,z_n)+\frac{1}{2}\sum_n^{N_I}\sum_{m\neq n}^{N_I}{\cal I}(\vec{x}_n,z_n,\vec{x}_m,z_m) \, .
\end{equation}
Let us first comment on the holographic location of the instantons. Previous studies have shown that at low baryon density the instantons are located at $z=0$, the tip of the connected flavor branes, while at large density additional
instanton layers at $|z|>0$ are expected to appear \cite{Kaplunovsky:2012gb,Elliot-Ripley:2015cma,Preis:2016fsp,Elliot-Ripley:2016uwb}. Allowing for
multiple layers makes our calculation somewhat more complicated, both the numerical evaluation and the following
equations. Therefore, and since our emphasis is on the new interaction terms, we discuss all modifications that arise
due to multiple layers in appendix \ref{app:layers}. With the help of this appendix, we have checked that allowing for
more layers does not change our main conclusion regarding the continuous connection between nuclear and quark matter.
Here we continue with a single layer, i.e., we center all instantons around the point $z=0$ in the holographic direction. Consequently, $z_n=z_m=0$ in Eq.\ (\ref{FsqN}), and we shall
omit the arguments referring to the holographic coordinate from now on. Next, we put the instantons on a regular lattice in position
space and apply the nearest-neighbor approximation. We denote the number of nearest neighbors by $p$ and the lattice vectors
that connect each instanton with its nearest neighbors by $\vec{\delta}_m$, with the nearest neighbor distance $\delta$, i.e., $|\vec{\delta}_1|=\ldots =|\vec{\delta}_p|\equiv \delta$. Then, Eq.\ (\ref{FsqN}) becomes
\begin{eqnarray} \label{Fsum}
F_{iz}^2 &\simeq& \sum_n^{N_I} (F_{iz}^{(1)})^2(\vec{x}_n)+\frac{1}{2}\sum_m^p\sum_{n}^{N_I}{\cal I}(\vec{x}_n,\vec{x}_n+\vec{\delta}_m) \nonumber\\[2ex]
&\to& N_I\left[(1-p)\left\langle(F_{iz}^{(1)})^2(0)\right\rangle +\frac{p}{2}\left\langle(F_{iz}^{(2)})^2(0,\vec{\delta})\right\rangle\right] \nonumber\\[2ex]
&=& \frac{2\lambda_0^2 n_I}{3\gamma}\left[(1-p)q_0(z)+\frac{p}{2}q_{\rm int}(d,z)\right] \, .
\end{eqnarray}
In the second line we have employed a further approximation by averaging the field strengths squared over position space, denoted by angular brackets, and used Eq.\ (\ref{Int}).
This has allowed us to shift every single term of the sum to the origin, $\vec{x}_n\to 0$, with $\vec{\delta}$ connecting an instanton in the origin to one if its nearest neighbors, say along the $x_1$ direction.
We have used the spatial integrals
for the one-instanton and two-instanton solutions from Eqs.\ (\ref{NB}) and (\ref{NB2}), respectively. Moreover, we have
introduced the dimensionless instanton density (per flavor)
\begin{equation}
n_I = \frac{48\pi^4 N_I}{\lambda^2 v} \, ,
\end{equation}
where $v$ is the dimensionless three-volume of the system, related to the
dimensionful volume $V$ by $v=VM_{\rm KK}^3$. The instanton distance $d$ in the last line of Eq.\ (\ref{Fsum}) is of course related to the instanton density. This relation
depends on the lattice structure. We write the volume of a Wigner-Seitz cell as
\begin{equation} \label{s}
\frac{v}{N_{I}}= \frac{\delta^3}{r} \, ,
\end{equation}
such that $r$ is the inverse Wigner-Seitz volume in units of the nearest-neighbor distance $\delta$.
Consequently,
with Eq.\ (\ref{d}) we can express $d$ as
\begin{equation} \label{d1}
d = \frac{\gamma}{\rho}\left(\frac{6\pi^4 r}{\lambda^2 n_I}\right)^{1/3} \, .
\end{equation}
The parameters $p$ and $r$ carry all information about the lattice structure in our approximation. For a cubic, a body centered cubic, and a face centered cubic crystal we have $(p,r) = (6,1), (8,3\sqrt{3}/4), (12,\sqrt{2})$, respectively. By setting the number of nearest neighbors $p$
to zero the result becomes independent of $r$ and we recover the non-interacting
approximation of Ref.\ \cite{Preis:2016fsp}. Replacing the field strengths squared
by their spatial average is an enormous simplification since now the equations of motion
will be ordinary differential equations, only depending on the holographic coordinate $u$. A more refined approximation would be to average not over the squared field strengths, but over the entire square root of the DBI action (\ref{DBI}) before solving the equations of motion. Although still resulting in ordinary differential equations, this procedure would be much more complicated since
there is no simple analytic form for the spatial integral over the DBI Lagrangian, even if the
above analytic ansatz for the non-abelian field strengths is used.
\subsection{Solving the equations of motion}
\label{sec:eom}
We now insert our ansatz (\ref{Fsum}) together with the corresponding expressions for $F_{ij}^2$, $F_{ij}F_{kz}\epsilon_{ijk}$ into
the action (\ref{SDBICS}). As already mentioned, instead of the symmetrized trace we take the standard trace for simplicity.
Since all field strengths squared are diagonal in flavor space, this simply gives an overall factor $N_f$, and we can write
\begin{equation} \label{S1}
S = {\cal N} N_f \frac{V}{T} \int_{u_c}^\infty du\, {\cal L} \, ,
\end{equation}
with the Lagrangian
\begin{equation} \label{Lag}
{\cal L} = u^{5/2}\sqrt{(1+u^3f_Tx_4'^2-\hat{a}_0'^2+g_1)(1+g_2)}-\hat{a}_0 n_I q(u) \, ,
\end{equation}
where
\begin{equation}
g_1 \equiv \frac{f_T n_I}{3\gamma}\frac{\partial z}{\partial u} q(u) \, , \qquad g_2 \equiv \frac{\gamma n_I}{3u^3}\frac{\partial u}{\partial z} q(u) \, ,
\end{equation}
with
\begin{equation}
q(u) \equiv 2\frac{\partial z}{\partial u} \left[(1-p)q_0(z)+\frac{p}{2}q_{\rm int}(d,z)\right] \, , \qquad \int_{u_c}^\infty du\, q(u) = 1 \, .
\end{equation}
(Here we are switching back and forth between the coordinates $u$ and $z$, whose relation is given in Eq.\ (\ref{uz}).)
We have brought the Lagrangian into the same form as in Ref.\ \cite{Preis:2016fsp}, with all effects of the instanton interaction absorbed in the function $q(u)$. Therefore, for the solutions of the equations
of motion $\hat{a}_0(u)$ and $x_4(u)$, we can simply follow this reference. With the integration constant $k$ we
obtain
\begin{eqnarray} \label{a0x4}
\hat{a}_0'&=& \frac{n_IQ}{u^{5/2}} \zeta\, , \qquad x_4'=\frac{k}{u^{11/2}f_T}\zeta \, ,
\end{eqnarray}
where we have abbreviated
\begin{equation} \label{zeta}
\zeta\equiv \frac{\sqrt{1+g_1}}{\sqrt{1+g_2-\frac{k^2}{u^8f_T}+\frac{(n_IQ)^2}{u^5}}} = \frac{\sqrt{1+u^3f_Tx_4'^2-\hat{a}_0'^2+g_1}}{\sqrt{1+g_2}}\, ,
\end{equation}
and
\begin{eqnarray} \label{Q}
Q(u) &\equiv& \int_{u_c}^u du'\, q(u') = (1-p)Q_0(z)+ \frac{p}{2}Q_{\rm int}(d,z) \, ,
\end{eqnarray}
where
\begin{subequations}
\begin{eqnarray}
Q_0(z) &\equiv& \int_{-z}^z dz'\,q_0(z') = \frac{(3\rho^2 + 2z^2)|z|}{2(z^2+\rho^2)^{3/2}} \, , \\[2ex]
Q_{\rm int}(d,z) &\equiv& \int_{-z}^z dz'\,q_{\rm int}(z') = \frac{\sqrt{2}\rho^2}{2}\frac{H_1 {\cal S}_1+H_2{\cal S}_2}{(a^2+b)^{3/2}b} \, , \label{Psi}
\end{eqnarray}
\end{subequations}
with the polynomials
\begin{subequations} \label{q1q2}
\begin{eqnarray}
H_1&=&\rho^2 z \Big[6 \rho^6 d^4 (4 d^4 +7 d^2 -2 ) + \rho^4 z^2 (16 d^8 + 80 d^6 + 36 d^4- 16 d^2+1) \nonumber\\[2ex]
&& + 2 \rho^2 z^4 d^2 (16 d^4+ 28 d^2 -5 ) + 16 z^6 d^4\Big] \, , \\[2ex]
H_2&=& z\Big[3 \rho^6 d^2 (4 d^4 + 7 d^2-2 ) +\rho^4 z^2 (8 d^6+ 40 d^4 + 17 d^2-5 ) \nonumber\\[2ex]
&&+ 2\rho^2 z^4 (8 d^4 + 14 d^2 -1) + 8 z^6 d^2 \Big] \, ,
\end{eqnarray}
\end{subequations}
and ${\cal S}_1$, ${\cal S}_2$, $a$, $b$ defined in Eq.\ (\ref{SSab}). At first sight,
finding an analytic expression for the integral in Eq.\ (\ref{Psi}) seems hopeless, due to the
complicated form of $q_{\rm int}$. As explained in appendix \ref{app:deriv}, however, an analytical form can be found
from the observation that taking successive derivatives does not change the structure of the function, only the exponents in the denominator and the explicit form of the polynomials that multiply ${\cal S}_1$ and ${\cal S}_2$.
The solution (\ref{a0x4}) includes the case of a trivial embedding function, $x_4'=k=0$, which corresponds to the chirally symmetric quark matter phase. It also includes the mesonic phase, which has vanishing baryon number, $n_I=0$, and thus a constant field $\hat{a}_0(u)=\mu$. In the baryonic phase both $k$ and $n_I$ are nonvanishing. For all cases, the asymptotic behavior at the holographic boundary $u\to \infty$ can be written as
\begin{equation}
\hat{a}_0'(u) = \frac{n_I}{u^{5/2}} +\ldots \, , \qquad x_4'(u) = \frac{k}{u^{11/2}} + \ldots \, .
\end{equation}
This expansion follows from Eq.\ (\ref{a0x4}) and $Q=1+{\cal O}(u^{-6})$, $\zeta = 1+{\cal O}(u^{-5})$ and confirms the interpretation of $n_I$ as the baryon number density.
\subsection{Stationary points of the free energy}
\label{sec:mini}
According to the on-shell action (\ref{S1}) the free energy is ${\cal N}N_f\Omega$ with
\begin{equation} \label{Omega}
\Omega = \int_{u_c}^\infty du\, {\cal L} \, .
\end{equation}
Here, the Lagrangian is obtained by re-inserting the solutions (\ref{a0x4}) into Eq.\ (\ref{Lag}).
As written, the free energy
is divergent. We can easily deal with this divergence by introducing a cutoff
for the upper boundary of the $u$ integral. This introduces a cutoff dependent term that turns out to be independent of
chemical potential and temperature. We can therefore simply subtract this vacuum term since it does not affect the physics.
The free energy depends on various parameters. Firstly, there are the model parameters
$\lambda$, $M_{\rm KK}$, and $\ell$.
Secondly, there are the quantities $k$, $n_I$, $u_c$, $\rho$, $\gamma$, which, in principle, have to be determined dynamically. However, as shown in Ref.\ \cite{Preis:2016fsp}, this leads to pointlike instantons at the
baryon onset (which, unphysically, is of second order), while we know that at finite $\lambda$
the instantons should have a nonzero width. We have checked that this observation remains unchanged in the present approach and thus we shall use the form of $\rho$ and $\gamma$ from Eq.\ (\ref{rhogam}) and treat $\rho_0$ and $\gamma_0$ as parameters, increasing the number of model parameters from 3 to 5 \cite{Preis:2016fsp}. This can be understood as an attempt to capture nontrivial effects, for instance known from other approximations and numerical studies, for which our original ansatz is too simplistic. It can also be understood as
a phenomenological approach, since it gives us additional freedom to reproduce properties of real-world nuclear matter.
Within this approach, it remains to find the stationary points of $\Omega$ with respect to $k$, $n_I$, and $u_c$. The three stationarity equations
\begin{equation}
\frac{\partial \Omega}{\partial k} = \frac{\partial \Omega}{\partial n_I} = \frac{\partial \Omega}{\partial u_c} = 0
\end{equation}
can be written more explicitly as
\begin{subequations} \allowdisplaybreaks \label{minimOm}
\begin{eqnarray}
\frac{\ell}{2} &=& \int_{u_c}^\infty du\, x_4' \, , \label{dk} \\[2ex]
\mu n_I &=&\int_{u_c}^\infty du \,u^{5/2}\left[\frac{g_1\zeta^{-1}+g_2\zeta}{2q}\left(q-\frac{d}{3}\frac{\partial q}{\partial d}\right) + \frac{\zeta n_I^2 Q}{u^5}\left(Q -\frac{d}{3}\frac{\partial Q}{\partial d}\right)\right]
\, , \label{dnI} \\[2ex]
st &=& 2u_c^{7/2}+\int_{u_c}^\infty du \, u^{5/2} \left\{7 - \zeta\left[7(1+g_2)+2\frac{(n_I Q)^2}{u^5}+\frac{k^2}{u^8f_T}\right]\right.
\nonumber\\[2ex]
&&\left.+\frac{g_1\zeta^{-1}+g_2\zeta}{2q}\left(5q-\frac{3d}{2}\frac{\partial q}{\partial d}
+\frac{\rho}{2}\frac{\partial q}{\partial \rho}\right)-\frac{\zeta n_I^2Q}{u^5}\left(\frac{3d}{2}\frac{\partial Q}{\partial d}-\frac{\rho}{2} \frac{\partial Q}{\partial \rho}\right)\right\} \, , \label{duc0}
\end{eqnarray}
\end{subequations}
where the derivatives with respect to $d$ are taken at fixed $\rho$ and vice versa.
The first two equations, the derivatives with respect to $k$ and $n_I$, are straightforwardly derived.
The derivative with respect to $u_c$ is more complicated, and we explain the derivation in appendix \ref{app:mini}. We have denoted the dimensionless entropy density by $s$,
\begin{equation} \label{entropy}
s = -\frac{\partial \Omega}{\partial t} = \frac{3u_T^3}{t}\int_{u_c}^\infty \frac{du}{u^{1/2}f_T}\left(g_1\zeta^{-1}+\frac{k^2}{u^8f_T}\zeta \right) \, .
\end{equation}
Note that $s$ is computed solely from the derivative with respect to the explicit dependence on the dimensionless temperature $t$ because the implicit dependence through $k$, $n_I$, $u_c$ vanishes at the stationary point.
We see from Eq.\ (\ref{dk}) that stationarity with respect to $k$ is equivalent to the boundary condition
$x_4(\infty)-x_4(u_c)=\ell/2$, given by the fixed asymptotic separation of the flavor branes. Therefore,
Eq.\ (\ref{dk}) should be viewed as a constraint, it is not a minimization of the free energy.
This constraint restricts the parameter space to a two-dimensional surface in $(k,n_I,u_c)$ space.
Local minima of the potential are points that fulfill Eqs.\ (\ref{minimOm}) and in whose vicinity the potential increases in all directions on this two-dimensional surface (and not necessarily in all directions in the three-dimensional $(k,n_I,u_c)$ space).
\subsection{Choice of parameters}
\label{sec:parameters}
The stationarity equations (\ref{minimOm}) have to be solved simultaneously for $k$, $n_I$, $u_c$ for given chemical potential $\mu$ and temperature $t$. In the remainder of the paper we shall set $t=0$. Moreover, we shall work with the cubic lattice, $p=6$, $r=1$. We have checked that other lattice structures slightly change our results
quantitatively, but not qualitatively. It remains to fix the model parameters $\lambda$, $M_{\rm KK}$, $\ell$, $\rho_0$, $\gamma_0$. Here we do not attempt to study this parameter space systematically, as it
was done in Ref.\ \cite{Preis:2016fsp} within a simpler version of the model.
We shall rather work with a single parameter set, which we determine by requiring the model to reproduce the basic properties of nuclear matter
at saturation (a first step in this direction was made in Ref.\ \cite{Preis:2016gvf}). This is, firstly, a useful preparation
for future extensions and applications of this model. For instance, if the equation of state is calculated and used to describe dense matter inside neutron stars, it is important to anchor the approach to known low-density properties. And, secondly,
the fact that it is possible to reproduce properties of real-world nuclear matter can be viewed as a phenomenological
validation of the model. We will reproduce 4 properties with 5 free parameters, and thus the fact that the matching works might not seem very impressive. But, due to the complicated, highly nonlinear structure of the stationarity equations it is a priori not obvious that the equations set by the constraints do have a solution.
The physical constraints we impose are as follows. We require the vacuum mass of the nucleon to be $m_N = 939\, {\rm MeV}$. The remaining conditions are properties of infinite, isospin-symmetric, zero-temperature nuclear matter at saturation: the binding energy $E_B=-16.3\, {\rm MeV}$, the saturation density $n_0 = 0.153\, {\rm fm}^{-3}$, and the incompressibility, for which only a certain range is known, $K\simeq(200-300)\, {\rm MeV}$. Relating these quantities to our model gives the following four conditions,
\begin{subequations} \label{4conds}
\begin{eqnarray}
m_N+E_B &=& \frac{\lambda N_cM_{\rm KK}}{4\pi} \,\mu_0 \, , \label{binding}\\[2ex]
m_N &=& \frac{\lambda N_cM_{\rm KK}}{4\pi} \, m_0 \, , \label{mN} \\[2ex]
n_0&=&\frac{\lambda^2 M_{\rm KK}^3}{48\pi^4} \,n_I^0 \, , \label{n0} \\[2ex]
K&=&\frac{\lambda N_c M_{\rm KK}}{4\pi}\,\kappa \label{comp}\, .
\end{eqnarray}
\end{subequations}
The right-hand sides depend on the parameters of the model, through the dimensionless quantities $\mu_0$, $m_0$, $n_I^0$, $\kappa$, which are functions of $\rho_0$, $\gamma_0$, $\lambda$, and $\ell$. Here, $\mu_0$ is the quark chemical potential at the baryon onset, and $n_I^0$ is the baryon density just above the onset. To relate $\mu_0$ and $n_I^0$ to their dimensionful counterparts, we have used Eqs.\ (2.44) and (2.45) of Ref.\ \cite{Preis:2016fsp}. Moreover, $m_0$ is the constituent quark mass in the vacuum, such that $N_cm_0$ is the vacuum
mass of the baryon. In our approach, it can be computed from the chemical potential in the limit of vanishing baryon density, $n_I\to 0$. Since in this limit the instantons are infinitely far away from each other we can obviously use the non-interacting limit to compute the vacuum mass. We find that the solution of Eqs.\ (\ref{minimOm}) for $n_I\to 0$
with $\mu$ approaching a nonzero value is (for zero temperature)
\begin{subequations}\allowdisplaybreaks
\begin{eqnarray}
u_c &=& \frac{16\pi}{\ell^2}\left[\frac{\Gamma\left(\frac{9}{16}\right)}{\Gamma\left(\frac{1}{16}\right)}\right]^2 \, ,\label{ucMes} \\[2ex]
k&=&u_c^4 \, , \\[2ex]
m_0&\equiv & \mu = \frac{3u_c^3\rho_0^4}{8\gamma_0\lambda^2}\int_{u_c}^\infty du\,u^{5/2} \frac{u^8-u_c^8+\gamma_0^2 u_c^4u(u^3-u_c^3)}{(u^3-u_c^3)\sqrt{u^8-u_c^8}\left(u^3-u_c^3+\frac{\rho_0^2u_c^{5/2}}{\lambda}\right)^{5/2}}
\, .\label{M0}
\end{eqnarray}
\end{subequations}
We mention already here that there is a second solution for $n_I\to 0$, where $\mu$ does go to zero as well. This
solution will play crucial role later, see Eq.\ (\ref{smallMu}). Finally, $\kappa$ is the dimensionless version of the incompressibility at saturation,
\begin{equation} \label{kappa}
\kappa = - 9 \left.\frac{\partial \Omega}{\partial n_I}\right|_{n_I=n_I^0} \, .
\end{equation}
This derivative with respect to the baryon density is taken
at fixed temperature,
but with $\mu$, $k$, $u_c$ being functions of
$n_I$. In contrast, the same derivative, used above for minimizing the free energy, was taken at fixed temperature {\it and} fixed $\mu$, $k$, $u_c$. We compute the derivative in (\ref{kappa}) purely numerically.
The 4 conditions (\ref{4conds}) are used to fix $\rho_0$, $\gamma_0$, $\lambda/\ell$, and $\ell/M_{\rm KK}=L$. This matching is not unique because $K$ is not known exactly and because it is conceivable that different, disconnected regions in parameter space satisfy the physical constraints. It is beyond the scope of this paper to present a systematic analysis of the multi-dimensional parameter space. The parameter set that will be used throughout the
rest of the paper is
\begin{equation} \label{parachoice}
\rho_0 = 4.3497 \, , \qquad \gamma_0 = 3.7569 \, , \qquad \frac{\lambda}{\ell}=15.061 \, , \qquad \frac{\ell}{\pi}=\frac{M_{\rm KK}}{3185\, {\rm MeV}} \, .
\end{equation}
It can be checked numerically that these values satisfy Eqs.\ (\ref{4conds}): when solving
the stationarity equations (\ref{minimOm}) it is useful to eliminate $\ell$ completely from the equations by an appropriate rescaling of all
quantities. After that rescaling, the stationarity equations yield, using the values (\ref{parachoice}), $\mu_0\simeq0.2531\,\ell^{-2}$, $m_0\simeq 0.2576\, \ell^{-2}$, $n_I^0 \simeq 0.02325\, \ell^{-5}$, $\kappa\simeq 0.06\, \ell^{-2}$. It is then easy to check that these values satisfy the physical conditions at saturation; in particular $K\simeq 220\, {\rm MeV}$, which is within the experimentally known range \cite{Blaizot:1995zz,Youngblood:2004fe}, possibly somewhat smaller \cite{Stone:2014wza}. The predicted value for $L$ from Eq.\ (\ref{parachoice}) can
be used to compute the critical temperature for the chiral transition at zero chemical potential.
Equating the free energies of the mesonic phase and the quark matter phase -- for instance using
appendix B of Ref.\ \cite{Li:2015uea} -- yields a first-order phase transition with $T_c^{\rm chiral}\simeq 0.15384/L$, from which we obtain $T_c^{\rm chiral}\simeq 156\, {\rm MeV}$. This is very close to the transition temperature for the chiral crossover in real-world QCD. We should keep in mind, however, that we work in the chiral limit with two quark flavors. Since lattice simulations are not available in this limit, the exact value and even the order of the transition is not rigorously known. It is nevertheless worth pointing out that from fixing our parameters to cold nuclear matter, we obtain a prediction for the finite-temperature phase transition which can in principle be compared to QCD.
Besides the absolute value of $L$ let us also briefly comment on the relation
between $\ell$ and $M_{\rm KK}$ in Eq.\ (\ref{parachoice}). This relation suggests that by choosing $M_{\rm KK}$
arbitrarily any value of $\ell$ can be obtained. However, $\ell$ is constrained because the asymptotic separation of the flavor branes can obviously not be larger than half of the circumference of the $x_4$ circle, and hence $\ell<\pi$ is an absolute limit. As discussed at the end of Sec.\ \ref{sec:action}, a more restrictive condition comes from the observation that there is a chirally broken phase in the deconfined geometry only if $\ell/\pi<0.30768$. Interestingly, at this upper limit, we have $\lambda\simeq 14.6$ and $M_{\rm KK}\simeq 980\, {\rm MeV}$, which is close to the original fits in the confined geometry, which were obtained using completely different constraints based on the rho meson mass and the pion decay constant \cite{Sakai:2004cn}.
The decompactified limit, which allows us to apply the deconfined geometry to arbitrarily small temperatures, implies $\ell\ll \pi$. It is a matter of interpretation whether this stronger constraint should be obeyed since one might interpret the decompactified limit as a separate model, allowing for extrapolations of $\ell/\pi$ beyond the originally allowed regime. Nevertheless, it is reassuring that $\ell\ll\pi$ does not contradict our physical constraints.
\section{Results}
\label{sec:results}
\subsection{Solution of the stationarity equations}
\label{sec:solution}
Having fixed all parameters, we can solve the stationarity equations and compute the free energy. This has to be done numerically in general, but in one instance we have found an analytical expansion. This is the limit of small $\mu$, where the solution to Eqs.\ (\ref{minimOm}) is
\begin{equation} \label{smallMu}
u_c = \frac{ 2^4\cdot3^2}{\phi^2}\,\mu^2 + \ldots \, , \qquad k = \frac{2^{16}\cdot3^{11} \ell}{\phi^9}\, \mu^9 + \ldots\, , \qquad n_I = \frac{2^{15}\cdot3^7 }{\phi^7} \mu^6 + \ldots\, ,
\end{equation}
where we have abbreviated
\begin{equation}
\phi\equiv \frac{\rho_0^2}{\lambda\gamma_0} \, .
\end{equation}
With Eq.\ (\ref{d}), this solution yields in particular $d\propto \mu^{-1/2}$,
i.e., the ratio of the distance over the width of the instantons becomes infinitely large as $\mu$ goes to zero. As a consequence, the instantons are essentially non-interacting in this limit, and Eq.\ (\ref{smallMu}) does not depend on the interaction terms\footnote{The solution (\ref{smallMu}) does not exist if $\rho\propto u_c$ instead of $\rho\propto u_c^{3/4}$, which explains why this solution was not found in
Ref.\ \cite{Preis:2016fsp}.}.
We show the full numerical solution in Fig.\ \ref{fig:nIuc}. In this figure, we have plotted $n_I$ (upper panels) and
$u_c$ (lower panels)\footnote{Here and in all following figures, the axes labels have to be understood to include appropriate powers of $\ell$, i.e., $n_I$ stands for $n_I\ell^5$, $\mu$ for $\mu \ell^2$, $u_c$ for $u_c \ell^2$, and $\Omega$ for $\Omega\ell^7$; the overlap parameter $d$ does not scale with $\ell$.}. The third variable $k$ is not plotted because
it shows a qualitatively identical behavior as $u_c$, although there is no simple analytical relation between them. Each quantity is shown in a linear plot (left) and a double-logarithmic plot (right) to make all important features of the solution visible. In the left plots, solid lines corresponds to the stable solution, dashed lines indicate
metastable and unstable branches of the solution.
Fig.\ \ref{fig:nIuc} also includes the result from the mesonic phase, which has zero baryon number, $n_I=0$ and where
$u_c$ is constant in $\mu$ and assumes the value (\ref{ucMes}).
We see that there is a first-order baryon onset, which occurs by construction due to our
matching of the parameters to real-world nuclear matter at saturation. The double-logarithmic plots show that the two branches in the left plots are continuously connected. As a check, we have also included the analytical result (\ref{smallMu}) to confirm that it is in exact agreement with the full
result at (very) small $\mu$. Moreover, we see that at small, but not too small, chemical potentials, $n_I$ and $u_c$ become three-valued in the baryonic phase. This multivaluedness is not visible in the linear plots. We do not have any particular interpretation for this behavior. It does not play any role for the ground state since in this regime the mesonic phase has a smaller free energy than any of the three baryonic solutions. We shall thus not further discuss this feature, although it is prominently visible in the double-logarithmic plots.
\begin{figure}[tbp]
\centering
\hbox{\includegraphics[width=.5\textwidth]{nImu}\includegraphics[width=.5\textwidth]{nImulog}}
\hbox{\includegraphics[width=.5\textwidth]{ucmu}\includegraphics[width=.5\textwidth]{ucmulog}}
\caption{\label{fig:nIuc} Dimensionless baryon density $n_I$ (upper panels) and location of the tip of the connected flavor branes $u_c$ (lower panels) as a function of the dimensionless chemical potential $\mu$ at zero temperature. In the left panels, solid lines are the stable branches, dashed lines are metastable or unstable. The right panels show the same quantities in double-logarithmic plots, now all branches shown as solid lines, together with the analytical result for small $\mu$
(blue dashed).
}
\end{figure}
\begin{figure}[tbp]
\centering
\hbox{\includegraphics[width=.5\textwidth]{Pmulog}\includegraphics[width=.5\textwidth]{dmulog}}
\caption{\label{fig:dP} {\it Left panel:} Pressure of the baryonic phase (black solid), mesonic phase (straight red solid line), and chirally symmetric phase (blue dashed horizontal line), all normalized to the pressure of the chirally symmetric phase $P_{||}$. {\it Right panel:} instanton distance over instanton width $d$ as a function of $\mu$ for the baryonic phase. The (black) solid
segment corresponds to the stable phase, while the dashed segments are unstable or metastable. The thin (red) curves
are the results with non-interacting instantons.
}
\end{figure}
In the left panel of Fig.\ \ref{fig:dP} we show the pressure $P=-\Omega$ for the mesonic phase (straight red line) and the baryonic phase (solid black curve), both divided by the pressure of the quark phase $P_{||}$, which is represented by the (blue) dashed horizontal line. Here, the zero-temperature results for the mesonic and quark phases are (see for instance Ref.\ \cite{Li:2015uea}),
\begin{subequations}
\begin{eqnarray}
P_{\cup} &=& \frac{2^{15}\pi^4}{7\ell^7}\frac{\Gamma\left(\frac{15}{16}\right)\tan\frac{\pi}{16}}{\Gamma\left(\frac{7}{16}\right)}\left[\frac{\Gamma\left(\frac{9}{16}\right)}{\Gamma\left(\frac{1}{16}\right)}\right]^7 \, , \label{OmM}\\[2ex]
P_{||} &=& \frac{2}{7}\left[\frac{\sqrt{\pi}}{\Gamma\left(\frac{3}{10}\right)\Gamma\left(\frac{6}{5}\right)}\right]^{5/2}\mu^{7/2} \, , \label{OmQ}
\end{eqnarray}
\end{subequations}
where the subscripts indicate the shape of the flavor branes of the two phases.
The favored state for a given $\mu$ is the one with the largest value of $P/P_{||}$. We see that for small $\mu$ the mesonic phase (the vacuum) is preferred,
until at $\mu_0\simeq 0.253/\ell^2$ there is a first-order phase transition to the baryonic phase. This first-order
phase transition is weak in the sense that the chemical potential at the transition is not much smaller than the vacuum mass of the nucleon, as required from the matching to QCD. Therefore, in the double-logarithmic plot, the multivaluedness of the baryonic pressure around this first-order phase transition is not visible. At $\mu\simeq 42.7/\ell^2$ there is a
first-order phase transition to the chirally symmetric phase. In physical units, this
corresponds to a critical chemical potential of about 160 GeV or, using the value for the dimensionless baryon density at
the transition $n_I\simeq 3780/\ell^5$, to more than $10^5$ times saturation density. This is an enormously large value, for instance compared to the chiral transition suggested by phenomenological models, but also because at these ultra-high
densities we are approaching the regime where perturbative QCD is expected to be valid. If
two-layer or multi-layer instanton solutions are taken into account, which decrease the free energy of the baryonic phase \cite{Preis:2016fsp}, the chiral phase transition will be even further shifted to
larger densities. On the other hand, it has been shown, at least without instanton interactions \cite{Preis:2016fsp}, that the chiral phase transition can be arbitrarily close to the baryon onset
for certain choices of the parameters $\gamma_0$ and $\rho_0$, which then, however, lead to unrealistically large binding energies of nuclear matter \cite{Preis:2016gvf}. In any case, we should take the large value of the critical chemical potential with care for various reasons.
Firstly, we have neglected backreactions on the background geometry, which can be expected to become large at large baryon densities. Even if our approximation is taken seriously, the large value may be
a consequence of the large-$N_c$ limit that is inherent in our calculation and/or the lack of asymptotic freedom of the model. We should also recall that for consistency we work with two flavors also in the chirally symmetric phase. In reality, however, {\it strange} quark matter is the ground state at high densities. (Even
more quark flavors appear if the chemical potential is sufficiently large, but, eventually having in mind neutron star applications, let us ignore them.) In our approximation of massless quarks, an additional quark flavor in the chirally symmetric phase simply changes the prefactor of the free energy from $N_f=2$ to $N_f=3$. Since the free energy is negative, this reduces the free energy, i.e., makes strange quark matter more favorable, as it should be, and we find a critical chemical potential of about 30 GeV, still large, but reduced by about a factor 5 compared to two-flavor quark matter. Since here we are mostly interested in qualitative results, we leave a more systematic discussion of the
chiral phase transition for the future.
In the right panel of Fig.\ \ref{fig:dP} we show the instanton overlap parameter $d$. The (black) solid curve, bounded by the two dots, is the stable baryonic segment, from the baryon onset (large $d$) to the chiral transition (smaller $d$), while the dashed
segments are unstable or metastable. The result shows that the instantons "refuse" to overlap strongly. Even at the highest densities we have $d>1$, because, as the instantons are squeezed, they become smaller. As a consequence, the non-interacting solution is a very good approximation for large parts of the curve. This solution, obtained in our approximation
by setting the number of nearest neighbors to zero, $p=0$, is shown by the two thin (red) curves. The right branch of the non-interacting result ends at large $\mu$, we have not found any solutions beyond this point. Since the numerics get increasingly difficult at these large chemical potentials we cannot exclude that this branch continues beyond the point where we have stopped. On the other hand, we were able to continue the left branch to much smaller values of $\mu$ than shown in the plot. Therefore, if instanton interactions are neglected these two branches appear to be disconnected.
Only in the presence of interactions all baryonic solutions we have found are continuously connected. And, they
are also continuously connected to the chirally symmetric phase, as we shall discuss now.
\subsection{Connecting nuclear matter with quark matter}
\label{sec:continuity}
\begin{figure}[tbp]
\centering
\includegraphics[width=\textwidth]{continuity}
\caption{\label{fig:schematic} Schematic plot of the free energy $\Omega$ as a function of the chemical potential $\mu$,
showing the continuous path that connects nuclear matter with quark matter. The various branches are labelled by the geometry of the flavor branes in the $(u,x_4)$ subspace, with the dot on the connected flavor branes indicating the instantons.
}
\end{figure}
Our results show a continuity between nuclear and quark matter in the following sense: the curve $[k(\mu),u_c(\mu),n_I(\mu)]$
describing the stationary points of the free energy is continuous and is a path that
can be traced from the baryonic phase into the quark matter phase (along which $\mu$ is non-monotonic). This continuous
path manifests itself also in the free energy, which is shown schematically in Fig.\ \ref{fig:schematic}. We have chosen
a schematic representation for convenience, which is a distorted, but not disrupted, version of the path obtained from the
actual calculation. We have also indicated the geometries of the flavor branes in the two-dimensional $(u,x_4)$ subspace of the model, together with a "legend" that explains these geometries. If we follow the state with the lowest free energy, the transition from nuclear to quark matter is {\it not} continuous. For instance $n_I$, which is the (negative) derivative of
$\Omega$ with respect to $\mu$, as well as $u_c$ are discontinuous. Therefore, the continuous path just described passes through metastable and unstable states. At zero chemical potential, the baryonic phase connects continuously to the chirally symmetric phase. With the help of the analytical solutions
(\ref{smallMu}) this statement is exact and does not rely on a numerical calculation: the small-$\mu$ expansions show for instance that $u_c$ indeed vanishes as $\mu$ goes to zero. We can investigate this small-$\mu$ regime
in more detail for instance by studying the behavior of the instantons. In Fig.\ \ref{fig:series} we show the
instanton profiles at certain locations of the continuous path.
\begin{figure}[tbp]
\centering
$z$ vs. $x_1$\hspace{2.1cm} $x_2$ vs. $x_1$\hspace{2.2cm} $u$ vs. $x_4$\hspace{2.3cm}$d$ vs. $\mu$ \hspace{0.2cm}
\includegraphics[width=\textwidth]{series}
\caption{\label{fig:series} {\it First column:} instanton profiles cut along the $z$-$x_1$ plane at $x_2=x_3=0$, with $z\in[-2,2]$ and an $x_1$ range of length 2. {\it Second column:} cubic instanton lattice in the $x_1$-$x_2$ plane at $z=x_3=0$, with an $x_1$ range of length
2 and $x_2$ range of length 2/3. {\it Third column:} embedding function of the flavor branes, $x_4\in[-0.6,0.6]$ and $u\in [0,10]$, showing the continuous transition from the chirally broken to the chirally restored phase. The branes are asymptotically fixed at $x_4(u=\infty)=\pm 1/2$. {\it Fourth column:} same double-logarithmic plot as in the left panel of Fig.\ \ref{fig:dP}, with the large (red) dot indicating the $(d,\mu)$ values for each row and the line style distinguishing between stable (solid) and metastable or unstable (dashed) phases.}
\end{figure}
This figure shows the instanton profiles via cuts in the $(z,x_1)$ and $(x_1,x_2)$ subspaces in the first two columns.
The profiles are given by the field strengths squared from the first line of Eq.\ (\ref{Fsum}), and in each row the color
scale is adjusted to the maximal value of the profile. For simplicity, we have
calculated the profiles from the non-interacting limit, which makes almost no difference since, as we have seen and as this figure further illustrates, the instantons never overlap significantly.
The figure
also shows the corresponding embedding of the flavor branes (third column) and indicates the value of $\mu$ and $d$
for each row (fourth column). The first three rows correspond to stable solutions, the first row being located at the baryon onset and the third row at the chiral phase transition. As we move towards the chirally symmetric phase the instantons become infinitesimally thin in the holographic direction, while they spread out in the spatial direction and become infinitely wide\footnote{From a top-down string-theoretical point of view one might argue that the resulting large derivatives of the field strengths require derivative corrections of higher order in $\alpha'$ to the DBI action. Here we do not include such terms for simplicity.}. This is a consequence of the scaling of the holographic width $\rho\propto u_c^{3/4}$ and the spatial width $\rho/\gamma \propto u_c^{-3/4}$. Since $u_c$ goes to zero according to Eq.\ (\ref{smallMu}) we have $\rho\propto \mu^{3/2}$ and $\rho/\gamma \propto \mu^{-3/2}$. Note that nonzero-temperature effects will slightly change this picture. The geometry dictates that
$u_c\ge u_T$ and thus $u_c>0$ for $T>0$. Hence the instantons will always retain a nonzero width in the holographic direction and a finite spatial width within the present ansatz.
We leave a more thorough study of nonzero temperatures to the future.
Having emphasized the continuous connection between nuclear and quark matter, let us discuss and interpret this result. It is striking that the connection relies on the nuclear and quark matter branches to "meet" at zero chemical potential. The reason why the quark matter solution exists all the way down to $\mu=0$ is that we work in the chiral limit and thus quarks can be placed into the system at infinitesimally small chemical potential. Of course, at small chemical potential
the quark matter phase is energetically disfavored compared to the mesonic phase (in real-world QCD one would expect an infinite energy cost due to confinement). Nucleons, on the other hand, do have a mass and are placed into the system at a nonzero chemical potential. Therefore, the uppermost branch in Fig.\ \ref{fig:schematic} should not be confused with ordinary nuclear matter where the density is reduced as we approach the origin.
Ordinary low-density nuclear matter sits on the lower branch and is superseded by the mesonic phase as the chemical potential is decreased. We thus emphasize that we are not simply decreasing the density of two phases until all matter is gone and then claim a continuous connection between them. We rather observe that all branches, from the mesonic phase through the nuclear matter phase up to the quark matter phase are continuously connected if we include all solutions of the equations of motion and all stationary points of the free energy, which are not necessarily stable. In fact, the uppermost branch in Fig.\ \ref{fig:schematic} can be expected to be unstable
because it has the largest free energy of all solutions we have found. It should thus
be a local maximum, not a local minimum of the free energy. By plotting the free energy for a fixed $\mu$ as a function of $n_I$ and $u_c$, with $k$ determined such that the constraint (\ref{dk}) is fulfilled,
we have confirmed this expectation.
Despite being unstable, the existence of this branch is interesting for the following reason. Ultimately, we are interested in a single continuous potential, defined in the entire "order parameter space". If there is a first-order
phase transition between nuclear and quark matter, one should be able to use this potential to connect the two phases continuously.
The knowledge of the full potential is for instance needed for the calculation of the surface tension of an interface between nuclear and quark matter \cite{Palhares:2010be,Fraga:2018cvr}. Our approximation does not yield such a potential. It is easy
to check that there are regions in parameter space where the potential becomes complex. However, compared to previous approximations within the same model (and compared to the vast majority of field-theoretical models) we have made a step forward by at least finding a path in parameter space along stationary points of the potential that connects nuclear and quark matter.
Since we are working with massless quarks, the transition from the chirally broken to the chirally restored branch at zero chemical potential can be expected to involve a discontinuity in second derivatives of the free energy, just like an ordinary second-order phase transition. We shall see in the next section that this is indeed the case, by computing the speed of sound. In other words, since chiral symmetry is an exact symmetry, instantons are topologically protected and a transition from a state with nonzero instanton number to a state without instantons must involve some kind of discontinuity. These arguments suggest an obvious improvement for the future, namely to include nonzero quark masses, which breaks chiral symmetry explicitly. In this case, firstly, the quark and nuclear branches may connect without reaching back to zero chemical potential (since
massive quarks require a finite energy and thus the quark matter branch cannot start at zero chemical potential). And, secondly, the transition between the chirally broken and (approximately) symmetric phases is allowed to be smooth, including the second derivatives of the free energy. It is then even conceivable that the multivalued curve in Fig.\ \ref{fig:schematic} "straightens out" at large densities, such that an actual quark-hadron continuity occurs in the phase diagram. Therefore, and despite the various approximations we have made and despite the various differences to real-world QCD, we believe that our approach can be very useful in understanding the
transition between nuclear and quark matter from a physical point of view.
\subsection{Speed of sound}
\label{sec:sound}
The general form of the
speed of sound is
\begin{eqnarray} \label{sound}
c_s^2 &=& \frac{\partial P}{\partial \epsilon} =\frac{n^2\frac{\partial s}{\partial T}+s^2\frac{\partial n}{\partial\mu}-ns\left(\frac{\partial n}{\partial T}+\frac{\partial s}{\partial \mu}\right)}{(\mu n + sT)\left(\frac{\partial n}{\partial \mu}\frac{\partial s}{\partial T}-
\frac{\partial n}{\partial T}\frac{\partial s}{\partial \mu}\right)} \, ,
\end{eqnarray}
where the derivative of the pressure $P$ with respect to the energy density $\epsilon=-P+\mu n + sT$ is taken at fixed entropy per particle $s/n$, and
where the derivatives of entropy density $s$ and baryon number density $n$ with respect to the baryon chemical potential $\mu$ are taken at fixed temperature $T$ and vice versa. We derive
Eq.\ (\ref{sound}) in appendix \ref{app:soundgeneral}, see also Refs.\ \cite{Herzog:2008he,Alford:2012vn} for similar derivations in the context of superfluids and appendix A of Ref.\ \cite{Floerchinger:2015efa}.
As above, we restrict ourselves to zero temperature. In this case
the speed of sound in the baryonic phase and the chirally restored phase becomes
\begin{equation}\label{nmu}
c_s^2 = \frac{n}{\mu}\left(\frac{\partial n}{\partial \mu}\right)^{-1} \, ,
\end{equation}
which is a useful relation to keep in mind for the following results.
Since a phase can only be thermodynamically stable if its density increases monotonically with the corresponding chemical potential, this relation also shows that a negative speed of sound squared indicates an instability (assuming that $n$ and $\mu$ themselves are both positive).
We compute the speed of sound numerically for the
baryonic solution discussed in the previous subsection, i.e., with the parameters from Eq.\ (\ref{parachoice}).
The results for the mesonic phase and the quark matter phase are derived in appendix \ref{app:sound} and read
\begin{subequations}
\begin{eqnarray}
\mbox{mesonic:} \qquad c_s^2(\mu,T\to 0) &=& \frac{1}{5} \, , \\[2ex]
\mbox{chirally symmetric:} \qquad c_s^2(\mu,T) &=& \frac{2}{5}\frac{u_T\sqrt{n_I^2+u_T^5}(n_I^2+5u_T^5)+\mu n_I(n_I^2+6u_T^5)}{(n_I^2+6u_T^5)(\mu n_I+2u_T\sqrt{n_I^2+u_T^5})} \nonumber\\[2ex]
&=& \left\{\begin{array}{cc} \displaystyle{\frac{1}{6}} & \;\;\mbox{for}\; \mu=0
\\[2ex] \displaystyle{\frac{2}{5}} & \;\;\mbox{for}\; T=0 \end{array}\right. \, . \label{cs2}
\end{eqnarray}
\end{subequations}
The result for the mesonic phase away from the zero-temperature limit can only be computed numerically, but is not needed here. The general result for the chirally symmetric phase is written in terms of the baryon density $n_I$, which depends on $\mu$ and $T$ in a non-analytical way. (Although there are no instantons in the chirally symmetric phase we have kept denoting the dimensionless baryon density by $n_I$ for consistency.)
\begin{figure}[tbp]
\centering
\includegraphics[width=0.7\textwidth]{sound}
\caption{\label{fig:sound} Speed of sound squared $c_s^2$ in units of the speed of light as a function of the dimensionless quark chemical potential $\mu$, computed with the parameters (\ref{parachoice}). The thick (red) curve shows the result for the stable phase, from the mesonic phase at low $\mu$ through the nuclear matter phase at intermediate $\mu$ to the quark matter phase at large $\mu$. The thin (black) curve corresponds to metastable and unstable phases. }
\end{figure}
In Fig.\ \ref{fig:sound} we show the speed of sound squared
as a function of chemical potential in the baryonic phase and, in the regimes where they are favored, in
the mesonic and quark matter phases. Close to the baryon onset, barely visible on the plot, $c_s^2$ is negative. The solution is unstable for densities smaller than about 66\% of the saturation density, then becomes metastable and finally stable at saturation.
Besides this instance, there are three other turning points of the curve $n_I(\mu)$ close to which its derivative is negative. In contrast to the turning point close to the baryon onset, in these three instances the curve $n_I(\mu)$ goes through a point where its derivative is zero such that, according to Eq.\ (\ref{nmu}), the speed of sound diverges. Apart from small vicinities of the turning points the speed of sound is real for the entire
nuclear matter solution. We should keep in mind, however, that $c_s^2>0$ is a necessary, but not sufficient,
stability criterion. As we know from the previous subsection, certain segments of the thin (black) curve in
Fig.\ \ref{fig:sound} correspond to maxima of the free energy and thus are unstable.
At very small $\mu$, i.e., in the regime of the continuous transition from the connected to the disconnected flavor branes,
the speed of sound squared approaches 1/6. This can be seen from the analytical result (\ref{smallMu}), which shows
$n_I\propto \mu^6$, from which
$c_s^2=1/6$ follows with the help of Eq.\ (\ref{nmu}). This value is different from the speed of sound in the
chirally restored phase, which is $c_s^2=2/5$, as shown for large chemical potentials in the figure. Consequently, as we go from the chirally broken to the chirally restored phase at zero chemical potential, the speed of sound is discontinuous. As explained above, it is expected that second derivatives of the free energy show discontinuities due to the
different symmetries of the phases. Since the speed of sound (\ref{nmu}) contains a second derivative, its discontinuity at zero chemical potential is no surprise.
Interestingly, 1/6 is the value of the speed of sound in the chirally restored phase if we set $\mu=0$ and then take the limit $T\to0$ (note the different values of $c_s$ at $T=\mu=0$ depending on the order in which the limits of the function in Eq.\ (\ref{cs2}) are taken).
Let us now comment on the stable (red) branch. At the baryon onset, the speed of sound jumps from its mesonic value
down to $c_s^2\simeq 0.025$, then increases to a maximum of about $c_s^2\simeq 0.47$, then decreases and finally jumps down to
the constant value of the quark matter phase at the chiral transition. In QCD, we know that for asymptotically large $\mu$ the speed of sound assumes the conformal value $c_s^2=1/3$ and perturbative corrections decrease this value. We also know
that nuclear matter at saturation has a much smaller -- non-relativistic -- speed of sound, close to the value we have found here. It is unknown how the low-density and high-density regimes are connected. It has been pointed out that the simplest scenario,
connecting the two regimes with a monotonic function, creates tension with astrophysical observations \cite{Bedaque:2014sqa}. More precisely, a sufficiently stiff equation of state (= a sufficiently large speed of sound) is required to obtain observed neutron star masses of twice the solar mass \cite{Demorest:2010bx,Antoniadis:2013pzd}, possibly larger
\cite{Linares:2018ppq}. Hence,
in the intermediate density regime the speed of sound most likely has to exceed the conformal value.
There have been suggestions in the literature that the conformal limit may be an absolute upper bound, but counterexamples have been pointed out within the gauge/string duality \cite{Hoyos:2016cob,Ecker:2017fyh}. Values larger than $c_s^2>1/3$ are routinely reached in
phenomenological models of nuclear matter or in extrapolations of low-density effective theories, but in these calculations the speed of sound typically behaves monotonically and cannot be traced into the chirally symmetric phase. Therefore, phenomenologically motivated interpolations between the low-density and high-density QCD results have been studied and the consequences for masses and radii of neutron stars have been tested \cite{Tews:2018kmu}. It is intriguing that the nuclear matter of our holographic calculation does show a non-monotonic behavior, as required from putting together constraints from QCD and neutron stars. Of course, our result is different from QCD at asymptotically large $\mu$. The reason is that
the mass scale $M_{\rm KK}$ plays a role even in this asymptotic regime, allowing
the speed of sound to be different from $1/3$. This might be interpreted as a non-perturbative strong-coupling effect, having in mind that we need to stop trusting the model for asymptotically large densities, where QCD exhibits asymptotic freedom. We also mention again that the chiral transition occurs at an extremely large density, a fact easy to be overlooked due to the use of the logarithmic scale in Fig.\ \ref{fig:sound}, and, as explained above, that one should treat the quantitative interpretation of the model with some care, especially at large densities. Nevertheless, our approach allows for a consistent microscopic calculation of nuclear and quark matter with a qualitative prediction for the speed of sound that is very interesting in view of recent astrophysical constraints.
\section{Summary and outlook}
\label{sec:conclusions}
We have shown that chirally broken and chirally symmetric phases in the holographic Sakai-Sugimoto model can be continuously connected in the presence of instantons on the flavor branes. Geometrically, this continuity is realized by transforming the curved, connected flavor branes for left- and right-handed fermions into straight, disconnected branes. The transformation appears dynamically by solving
the equations of motion and computing the embedding of the flavor branes in the deconfined geometry of the model and in the presence of an interacting many-instanton
system. We have approximated the instanton interactions with the help of the exact flat-space two-instanton solution.
We have found
that, as the connected flavor branes become straight and approach the disconnected embedding, the instantons become
infinitesimally thin in the holographic direction, but spread out to become infinitely wide in the spatial direction.
Nevertheless, they barely overlap since in this limit also their density goes to zero. Most previous related studies had
included instantons only in the confined geometry or did not include interactions between the instantons, i.e.,
to the best of our knowledge this continuity has been observed for the first time in the present paper.
Our observation is closely related to the transition between nuclear matter (= instanton system with connected flavor branes)
and chirally symmetric quark matter (= disconnected flavor branes) in dense QCD. While our continuity does not directly connect stable nuclear matter at the lowest densities with stable ultra-dense quark matter, it does so on a path that goes through metastable and unstable
stationary points of the potential. In particular, the actual phase transition between nuclear and quark matter turns out to be of first order. Since we have worked in the chiral limit, it is obvious that the actual transition cannot be continuous, since chiral symmetry is exact. It would thus be very interesting to include quark masses into our approach and
see whether and how the continuity is affected. This can be done with worldsheet instantons \cite{Hashimoto:2008sr}, which have been used to study the effect of quark masses on the hadron spectrum \cite{Aharony:2008an,Hashimoto:2009hj,Hashimoto:2009st,Bigazzi:2018cpg}, or with a tachyonic effective action \cite{Bergman:2007pm,Dhar:2007bz}.
Besides this main theoretical result we have also pointed out phenomenological applications of our approach. We have shown that it is possible to fit the parameters of the model to reproduce the basic properties of isospin-symmetric nuclear matter at saturation. In doing so, it was crucial to introduce two additional parameters that characterize the instanton shape, increasing the
number of model parameters from 3 to 5. We have worked with a single parameter set, and future studies are required for a more
systematic investigation of the parameter space. For instance, we have seen that for the chosen parameters the chiral transition occurs at an extremely large baryon density -- probably unrealistically large, at least compared to predictions of other models.
It would be interesting to see whether different parameter sets can be found that are also in agreement with nuclear physics and yield different
values for the chiral transition. We have also calculated the speed of sound to further connect our
results to the phenomenology of dense matter. Our result shows a non-monotonic behavior of the speed of sound in nuclear matter that
reaches a maximum larger than that of the quark matter phase, before it jumps to the quark matter value at the chiral transition. Interestingly, this non-monotonic behavior is suggested by putting together constraints from
QCD and astrophysical data, most notably the largest observed neutron star mass of about two solar masses. It is therefore a natural extension of our work to compute the equation of state and the
resulting maximum mass of a star that can be reached with it.
Let us conclude with mentioning some further extensions that may improve our calculation. Obvious technical improvements concern our
many-instanton approximation. We have constructed the interaction terms solely from 2-instanton contributions, and those, in turn, are based on the flat-space solutions. Moreover, we have simply averaged over position space before solving the equations of motion and the embedding of the flavor branes. Also, we have kept the background geometry fixed although the gauge fields
on the flavor branes become very large at large densities and backreactions may be non-negligible. Improvements on all these points are difficult, but since we are working within a top-down approach it is in principle known how to proceed. More direct extensions are for instance the study of two- or multi-layer solutions, which we have briefly discussed but not included into our numerical evaluation; isospin-breaking terms, which are needed for a more realistic study of neutron star matter;
and nonzero-temperature effects, which we have included in almost all our equations, but not in the
numerical results.
\acknowledgments
We would like to thank N.\ Evans, A.\ Rebhan, and S.\ Reddy for valuable comments.
A.S.\ is supported by the Science \& Technology Facilities Council (STFC) in the form of an Ernest Rutherford Fellowship. KBF thanks the University of Southampton for their hospitality during his sabbatical.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 103
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Hartford Accident & Indemnity Co., Defendants.
through doctrine of equitable subrogation.
extrinsic evidence and properly resolved matter without trial.
relative to remainder interest. 26 U.S.C.A. § 7403.
lien. 26 U.S.C.A. § 6323(i)(2).
Kirschenbaum, on the brief), for defendants-appellants.
Gabriel W. Gorenstein, Asst. U.S. Atty., New York City (Roger S. Hayes, U.S.
Atty. and Ping C. Moy, Asst. U.S. Atty., on the brief), for plaintiff- appellee.
Before: NEWMAN and MAHONEY, Circuit Judges, and EGINTON, [FN*] District Judge.
District of Connecticut, sitting by designation.
further consideration of equitable subrogation.
Chase Manhattan mortgage was senior to the federal tax lien.
FN1. The first name is spelled "Marie" in some of the papers.
favor of the Government in the amount of $259,601.84.
in a deed. No additional instrument is required. See, e.g., Winick v.
236 (1983); United States v. Kocher, 468 F.2d 503, 506-07 (2d Cir.1972), cert.
the entire property through use of "a standard statutory or commercial table."
764 F.2d 1126, 1130-31 (5th Cir.1985).
not be prejudiced by remaining junior to the Government.
758, 229 N.E.2d 435, 439 (1967). See, e.g., The Thrift v. Michaelis, 259 N.Y.
A.D. 1042, 145 N.Y.S.2d 335 (2d Dep't 1955); Union Savings Bank of Patchogue v.
Association v. Skow, 25 A.D.2d 880, 881, 270 N.Y.S.2d 234, 236 (2d Dep't 1966).
F.2d 909, 914-15 (2d Cir.1960).
the lender to offer an excuse for his failure to discover the intervening lien.
of retaining a first mortgage on the property.
A more subtle question is the effect of equitable subrogation in this case.
remaining $25,000 after the Government.
Miller v. Federal Land Bank of Spokane, 587 F.2d 415, 420 (9th Cir.1978), cert.
denied, 441 U.S. 962, 99 S.Ct. 2407, 60 L.Ed.2d 1067 (1979); Malinoski v.
prejudice the senior creditor. See Walther v. Bank of New York, 772 F.Supp.
to satisfy the subrogated Chase Manhattan lien from the remainder interest.
mortgage and hypothetical interest payments on prior mortgage).
UNITED STATES of America, Plaintiff-Appellee,v.Zenowia BARAN, Ostap Baran, and Self Reliant (NY) Federal Credit Union,Defendants-Appellants,Joseph F. Farano, Maria Farano, Joseph M. Farano, Phyllis Marie Farano, andHartford Accident & Indemnity Co., Defendants.
Argued April 26, 1993.Decided June 14, 1993.
United States sought to foreclose federal tax lien on life estate in property which had been sold to purchasers unaware of tax lien. The United States District Court for the Southern District of New York, Miriam Goldman Cedarbaum, J., foreclosed lien, and owners and their mortgagee appealed. The Court of Appeals, Jon O. Newman, Circuit Judge, held that remand was required to determine whether tax lien should be subordinated to junior lien of mortgagee through doctrine of equitable subrogation.
Under New York law, life estate may be created by reservation in deed; no additional instrument is required.
Phrase "life estate" has well-established meaning: it is estate in land giving life tenant full and exclusive possession of property for duration of tenant's life.
In construing unambiguous deed, district court properly declined to consider extrinsic evidence and properly resolved matter without trial.
It was within district court's discretion to accept government's method for determining value of life estate in real property subject to federal tax lien, even though appraiser stated that life estate was probably unmarketable, where property owners made no claim that standard Internal Revenue Service (IRS) table for valuing life estates erroneously overstated economic value of life estate relative to remainder interest. 26 U.S.C.A. § 7403.
Remand was required to determine whether federal tax lien should be subordinated to junior lien of mortgagee through doctrine of equitable subrogation, even though district court held that mortgagee was not entitled to be subrogated to senior mortgage, where record was unclear as to whether court recognized that it had discretion, under New York law, to subrogate mortgagee to discharged senior lien. 26 U.S.C.A. § 6323(i)(2).
Under New York law, mortgagee is subrogated to senior lien, where funds of mortgagee are used to satisfy lien of existing, known incumbrance and, unbeknown to mortgagee, another lien on property exists which is senior to mortgage but junior to lien satisfied with mortgage funds, regardless of whether mortgagee has excuse for failure to discover intervening lien; purpose of subrogation is to prevent intervening lienor from converting mistake of mortgagee into magical gift for himself.
Court could infer that mortgagee discharged senior lien on property under agreement with mortgagor, as required, under New York law, for mortgagee to be equitably subrogated to senior lien as against undiscovered intervening lien, where mortgagee made payment to discharge earlier mortgage as part of closing at which mortgagee supplied financing with intent of retaining first mortgage on property.
Assuming doctrine of equitable subrogation applied, under New York law, mortgagee's lien would be senior to federal tax lien on life interest, even though, as a result, mortgage would be secured to greater extent than funds actually paid to discharge senior mortgage; since tax lien was not on entire property, but only on life estate, it could be satisfied out of proceeds of sale only to extent of value of life estate, while, in contrast, senior mortgage was in entire property. *26 Burton Aronson, Garden City, NY (Samuel Kirschenbaum, Kirschenbaum & Kirschenbaum, on the brief), for defendants-appellants.
Gabriel W. Gorenstein, Asst. U.S. Atty., New York City (Roger S. Hayes, U.S. Atty. and Ping C. Moy, Asst. U.S. Atty., on the brief), for plaintiff- appellee.
FN* The Honorable Warren W. Eginton of the United States District Court for the District of Connecticut, sitting by designation.
In 1980, Maria Farano, [FN1] the owner of a house located at 52 Stoneleigh Road, Yonkers, New York (the "property"), deeded the property to her children, Joseph M. Farano and Phyllis Marie Farano (the "Farano children"), "subject to a life estate in the lives of" herself and her husband Joseph F. Farano (the "Farano parents"). In 1982, the IRS filed a federal tax lien against the Farano parents for an unpaid assessed balance of $82,221.86. Pursuant to 26 U.S.C. § 6321 (1988), the tax lien attached to all property owned by the Farano parents, and has continued to accrue interest. There appears to be agreement that Maria Farano's deed was ineffective to create a life estate for her husband. The tax lien therefore applied, at most, to Maria's life estate. In 1986, the Farano children and the Farano parents together conveyed the property to Zenowia and Ostap Baran for $440,000. The purchase was financed by Self Reliant. Apparently unaware of the federal tax lien, Self Reliant, in addition to loaning the Barans money to pay the Faranos, provided a check at the closing for $68,293.90 to discharge a prior mortgage held by Chase Manhattan Bank. The Chase Manhattan mortgage was senior to the federal tax lien.
1. Creation of a life estate. The Barans and Self Reliant make several related arguments concerning the 1980 deed that conveyed the property to the Farano children "subject to" a life estate in favor of Maria Farano. They contend that a deed cannot create a life estate, that the life estate did not include the entire premises, that the life estate terminated upon an attempt to attach Maria Farano's assets, and that summary judgment was inappropriate because the deed was ambiguous. None of these arguments is convincing. New York authorities recognize that a life estate can be created *28 by reservation in a deed. No additional instrument is required. See, e.g., Winick v. Winick, 26 A.D.2d 663, 272 N.Y.S.2d 869 (2d Dep't 1966), app. denied, 19 N.Y.2d 581, 279 N.Y.S.2d 1026, 226 N.E.2d 707 (1967). The phrase "life estate" has a well-established meaning. It is an estate in land giving the life tenant full and exclusive possession of the property for the duration of the life tenant's life. See In re Hinman's Will, 22 Misc.2d 655, 657, 200 N.Y.S.2d 170, 172 (Surr.Ct.1960). Because there was no ambiguity in the deed, the District Court properly declined to consider extrinsic evidence, see Uihlein v. Matthews, 172 N.Y. 154, 158-60, 64 N.E. 792, 794 (1902), and properly resolved the matter without a trial, see Hurd v. Lis, 92 A.D.2d 653, 654, 460 N.Y.S.2d 173, 174 (3d Dep't 1983).
The District Court acted within its discretion in accepting the Government's valuation method. 26 U.S.C. § 7403 (1988) allows the Government to seek the sale of the whole property in order to maximize the value of its interest. See United States v. Rodgers, 461 U.S. 677, 691, 103 S.Ct. 2132, 2142, 76 L.Ed.2d 236 (1983); United States v. Kocher, 468 F.2d 503, 506-07 (2d Cir.1972), cert. denied, 411 U.S. 931, 93 S.Ct. 1897, 36 L.Ed.2d 390 (1973). Often, as in this case, "interests in property, when sold separately, may be worth ... significantly less than the sum of their parts." Rodgers, 461 U.S. at 694, 103 S.Ct. at 2143. Apparently wishing to retain ownership of the property in the Barans, the defendants requested an opportunity to satisfy the Government's claim prior to a sale of the property. The District Court was free to grant that request while still valuing the life estate as a percentage of the value of the entire property through use of "a standard statutory or commercial table." See id. at 698-99, 103 S.Ct. at 2145. Defendants make no claim that the standard IRS table for valuing life estates erroneously overstates the economic value of a life estate relative to the remainder interest. See Harris v. United States, 764 F.2d 1126, 1130-31 (5th Cir.1985).
Under 26 U.S.C. § 6323(i)(2) (1988), we look to state law to determine whether equitable subrogation is available. New York courts have routinely applied subrogation "where the funds of a mortgagee are used to satisfy the lien of an existing, known incumbrance when, unbeknown to the mortgagee, another lien on the property exists which is senior to his but junior to the one satisfied with his funds." King v. Pelkofski, 20 N.Y.2d 326, 333-34, 282 N.Y.S.2d 753, 758, 229 N.E.2d 435, 439 (1967). See, e.g., The Thrift v. Michaelis, 259 N.Y. 302, 181 N.E. 580 (1932); Whitestone Savings & Loan Association v. Moring, 286 A.D. 1042, 145 N.Y.S.2d 335 (2d Dep't 1955); Union Savings Bank of Patchogue v. Dudine, 40 Misc.2d 155, 242 N.Y.S.2d 692 (Sup.Ct.1963). *29 The purpose of subrogation is to prevent a junior lienor from converting the mistake of the lender "into a magical gift for himself." Long Island City Savings & Loan Association v. Skow, 25 A.D.2d 880, 881, 270 N.Y.S.2d 234, 236 (2d Dep't 1966). In effect, subrogation erases the lender's mistake in failing to discover intervening liens, and grants him the benefit of having obtained an assignment of the senior lien that he caused to be discharged. See Pipola v. Chicco, 274 F.2d 909, 914-15 (2d Cir.1960).
A more subtle question is the effect of equitable subrogation in this case. The Government appears to suggest that subrogation is inappropriate because it cannot increase the extent to which Self Reliant's interest is secured. The Government's theory is that because the discharged Chase Manhattan mortgage (worth approximately $68,000) was for less than the difference between the value of the entire property (approximately $353,000) and the value of the life interest subject to the tax lien (approximately $260,000), it makes no difference whether Self Reliant has priority to the extent of the Chase Manhattan mortgage. In other words, the Government contends, Self Reliant would be secured to the extent of $93,000 ($353,000- $260,000) regardless of whether it took that amount from the proceeds of a sale after satisfaction of the tax lien or whether it took $68,000 ahead of the Government and took the remaining $25,000 after the Government.
FN2. The Government contends that under the doctrine of marshaling of assets, Self Reliant should be required to satisfy its lien solely from the remainder interest. Marshaling of assets cannot be required, however, when it will prejudice the senior creditor. See Walther v. Bank of New York, 772 F.Supp. 754, 767 (S.D.N.Y.1991). Self Reliant would be prejudiced if it were required to satisfy the subrogated Chase Manhattan lien from the remainder interest.
Although the decision to apply equitable subrogation is committed to the discretion of the trial court, the record is unclear as to whether the District Court recognized that it had discretion under New York law to subrogate Self Reliant to the discharged Chase Manhattan lien. Accordingly, we will vacate the judgment and remand the matter to the District Court for further consideration of the equities, as well as for any adjustments in the relative interests of the parties required by the payment or non-payment of interest on the portion of the Self Reliant mortgage corresponding to the prior Chase Manhattan mortgage. See Pipola v. Chicco, 274 F.2d at 915 (mortgagee and mortgagor entitled to priority over tax lien to extent of *30 discharged prior mortgage and hypothetical interest payments on prior mortgage).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,443
|
\section{\label{sec:Introduction}Introduction}
One of the most surprising experimental results of the last decades has been the
discovery of tiny neutrino masses and relatively large neutrino mixings.
Although non-vanishing neutrino masses are a clear indication of physics beyond
the Standard Model (SM), the mechanism and the scales responsible for the neutrino mass generation remain a total mystery.
It seems unlikely that the very small neutrino masses are generated by the
same Higgs mechanism responsible for the masses of the other SM fermions, since extremely small Yukawa couplings, of the order of $\lesssim 10^{-12}$, must be invoked.
A more `natural' way to generate neutrino masses involve the addition of new states that, once integrated out, generate the dimension five Weinberg operator
\begin{equation}
\mathcal{O}_5=\frac{c}{\Lambda}LLHH.
\end{equation}
This is embodied by the so-called seesaw mechanisms~\cite{seesawI,seesawII,seesawIII, Mohapatra:1986bd}.
The smallness of neutrino masses relative to the weak scale implies either that the scale of new physics $\Lambda$ is very large (making it impossible to experimentally discriminate the different seesaw mechanisms), or that the Wilson coefficient $c$ is extremely small (for instance, coming from loop effects involving singly or doubly charged scalars~\cite{radiativeseesaw}).
A different approach is given by neutrinophilic Two-Higgs-Doublet Models~\cite{Gabriel:2006ns, Davidson:2009ha}. In this framework, a symmetry ($U(1)$ or $Z_2$) compels one of the doublets to couple to
all SM fermions but neutrinos, hence being responsible for their masses, while the other Higgs
couples to the lepton doublets and right-handed neutrinos. If the second doublet
acquires a vacuum expectation value (vev) around the eV scale, this leads to
small neutrino masses. These models, however, are either ruled out or severely constrained by electroweak precision data and low energy flavor physics~\cite{Machado:2015sha,Bertuzzo:2015ada}.
A variation of this idea, in which the symmetry is taken
to be a local $U(1)$ and leads to the typical Lagrangian of the inverse seesaw
scenario, suffers from an accidental lepton number symmetry that has to be
explicitly broken to avoid the presence of a massless Nambu-Goldstone boson in the spectrum~\cite{Bertuzzo:2017sbj}.
All aforementioned models have one of the two following features: (i) The model is realized at very high scales, or (ii) the model is based on explicit breaking of lepton number or other symmetries that protect neutrino masses (e.g. in TeV scale type II or inverse seesaw models).
Neutrinos, however, are the {\em darkest} between the SM particles, in
the sense that they can couple through the renormalizable {\em
neutrino portal} $LH$ operator with generic dark
sectors~\cite{Schmaltz:2017oov}. This fact has been used in connection
to thermal Dark Matter with mass in the sub-GeV region (see for
instance Refs.~\cite{Falkowski:2011xh,Batell:2017cmf}). In this letter
we propose to use such a portal to explicitly connect a new light dark
sector with the generation of neutrino masses. In this way, we are
able to lower the scale of neutrino mass generation below the
electroweak one by resorting to a dynamical gauge symmetry breaking of
this new sector. The dark sector is mostly secluded from experimental
scrutiny, as it only communicates with the SM by mixing among scalars,
among neutrinos and dark fermions, and through kinetic and mass mixing
between the gauge bosons. This scheme has several phenomenological
consequences at lower energies, and in particular it offers a natural
explanation for the long-standing excess of electron-like events
reported by the MiniBooNE collaboration~\cite{Aguilar-Arevalo:2018gpe,
Bertuzzo:2018itn}.
\section{\label{sec:model} The Model}
To avoid any neutrino mass contribution from the Higgs mechanism, we introduce
a new dark gauge symmetry $U(1)_{\cal D}$, under which the SM particles are uncharged,
but the new sector is charged.
To build a Dirac neutrino mass term we need a $SU(2)_L$ singlet right-handed dark neutrino $N$, and a dark scalar doublet $\phi$, both having the same $U(1)_{\cal D}$ charge $+1$.
The absence of chiral anomalies require a second
right-handed neutrino, $N'$, with an opposite $U(1)_{\cal D}$ charge, thus restoring lepton number symmetry.
We add to the particle content a dark scalar $SU(2)_L$ singlet $S_{2}$, with dark charge $+2$, whose vev spontaneously breaks lepton number, giving rise to a Majorana mass component for the dark neutrinos. As we will see shortly, this setup leads to an inverse seesaw structure in which the lepton number breaking parameter is promoted to a dynamical quantity. Finally, this scalar content enjoys an accidental global symmetry which is spontaneously broken. To avoid a massless Goldstone boson, an extra dark scalar $SU(2)_L$ singlet $S_{1}$, with dark charge $+1$, is included in the spectrum. Its vev breaks all accidental global symmetries.
This field will allow for mixing among all the scalar fields, including the SM Higgs.
The dark scalar $S_{1}$ will spontaneously break $U(1)_{\cal D}$ by acquiring a vev, while $\phi$ and $S_2$
will only develop an induced vev after the breaking of the electroweak and dark symmetries.
By making a well motivated choice for the hierarchy of the vevs, our model allows a dynamical generation of the light neutrino masses and mixings at very low scale.
Our model predicts masses for the dark scalars within the reach of current experiments as well as a light dark vector boson, $Z_{\cal D}$, that has small kinetic mixing with the photon and mass mixing with the SM $Z$ boson.
The dark particles communicate with the SM ones via mixing: flavor
mixing (neutrinos), mass mixing (scalars) and mass mixing and kinetic
mixing ($Z_{\cal D}$), giving rise to a simple yet rich phenomenology.
\vspace{0.3cm}
\subsection{\label{subsec:scalars} The Dark Scalar Sector}
Let us start discussing the scalar sector of the model. This will motivate the region of parameter space on which we will focus throughout the paper. The most general $SU(2)_L\times U(1)_{Y} \times U(1)_{\cal D}$
invariant scalar potential that can be constructed out of the fields and charges outlined above is \begin{widetext}
\begin{align}\begin{aligned}\label{eq:potential}
V = &-m_H^2(H^\dagger H)+m^2_{\phi}(\phi^\dagger\phi)
-m_{1}^2S_{1}^*S_{1} + m_{2}^2 S_{2}^* S_{2} -\left[\frac{\mu}{2} S_{1}(\phi^\dagger H)+\frac{\mu'}{2}S_{1}^2 S_{2}^* + \frac{\alpha}{2}(H^\dagger \phi)S_{1} S_{2}^*+{\rm h.c.} \right] \\
& ~~~~~+ \lambda_{H\phi}' \phi^\dagger H H^\dagger \phi + \sum_\varphi^{\{H,\phi,S_1,S_{2}\}}\lambda_\varphi(\varphi^\dagger \varphi)^2 + \sum_{\varphi<\varphi'}^{\{H,\phi,S_1,S_2\}}\lambda_{\varphi\varphi'}(\varphi^\dagger \varphi)(\varphi^{\prime\dagger} \varphi')\, .
\end{aligned}\end{align}
\end{widetext}
(In the last sum, the notation $\varphi<\varphi'$ is to avoid double counting.)
We denote the vevs of the scalar fields as
$(H , \phi , S_1 , S_2)|_{\rm vev}\equiv\left(v,v_\phi,\omega_1,\omega_2\right)/\sqrt{2}$.
We stress that we are supposing the bare mass terms of $H$ and $S_1$ to be negative, while we take the corresponding ones for $\phi$ and $S_2$ to be positive. This ensures that, as long as $\mu=\mu'=\alpha \equiv 0$ ({\it i.e.} if there is no mixing among the scalar fields), the latter fields do not develop a vev, while the former do. In turn, this implies that the vevs $v_\phi$ and $\omega_2$ must be induced by $\mu$, $\mu'$, and/or $\alpha$.
We now observe that $\mu$, $\mu'$, and $\alpha$ explicitly break two
accidental $U(1)$ global symmetries, making these parameters
technically natural~\footnote{ One of the symmetries is lepton number,
the other is a symmetry under which only $\phi$ and $L$ are charged,
with opposite charge. Since there are only two global symmetries for
3 parameters, having two of them non-zero necessarily generates the
third by renormalization group running. }. For our purposes, this
means that $\mu$, $\mu'$ and $\alpha$ can be taken small in a natural
way, and justifies our working hypothesis $v_\phi,\omega_2\ll
v,\omega_1$. As we will see later, this hierarchy of vevs will
provide a low scale realization of the inverse seesaw mechanism with
low scale dynamics associated to it. Explicitly, we obtain
\begin{align}\label{eq:vevs}
&v_\phi\simeq \frac{1}{8\sqrt{2}}\left(\frac{\alpha\mu' \, v \omega_1^3}{M_{S'_{\cal D}}^2M_{H_{\cal D}}^2}+4\frac{\mu \, \omega_1 v}{M_{H_{\cal D}}^2}\right)\, ,~~~~~{\rm and}\\ %
&\omega_2\simeq \frac{1}{8\sqrt{2}}\left(\frac{\alpha\mu \, v^2\omega_1^2}{M_{S'_{\cal D}}^2 M_{H_{\cal D}}^2}+4\frac{\mu' \, \omega_1^2}{M_{S'_{\cal D}}^2}\right) \, ,
\end{align}
with $M_{H_{\cal D}}^2$ and $M_{S'_{\cal D}}^2$ approximately being the physical masses of the respective scalars (to be defined below).
In order to avoid large mixing between $H$ and $\phi$, we will always make the choice $\omega_1 \ll v$.
The scalar spectrum contains, in addition to the SM-like scalar $h_{\rm SM}$ with mass $m_{h_{\rm SM}}\simeq $ 125 GeV, three CP-even dark scalars $H_{\cal D}$, $S_{\cal D}$ and
$S'_{\cal D}$, with masses $M_{H_{\cal D}}$, $M_{S_{\cal D}}$ and
$M_{S'_{\cal D}}$, two CP-odd dark scalars $A_{\cal D}$ and $a_{\cal D}$
with masses $M_{A_{\cal D}}$ and $M_{a_{\cal D}}$, and a charged dark scalar $H^\pm_{\cal D}$ with mass $M_{H^\pm_{\cal D}}$.
Explicitly, the masses of the CP-even scalars are~\footnote{Radiative
corrections will naturally contribute to the masses of these
scalars. There are potentially several contributions according to
Eq.~(\ref{eq:potential}), the quartic couplings being the most
dangerous ones. In order to avoid fine-tuning{, we will always
demand the masses of the lightest scalars to satisfy $M_{\rm lightest}
\gtrsim \sqrt{\lambda} M_{\rm heavy}/8\pi$, where $M_{\rm heavy}$ denotes
any of the heavy scalar masses.} By the same argument we expect $\mu, \mu'$
and $\alpha v$ to be below $M_{\rm lightest}$. { Our computation
ignores the threshold at the Planck scale, which must be
stabilized by other means (for instance, supersymmetrizing the
theory).}}
\begin{align}\begin{aligned}\label{eq:scalar_masses}
m_{h_{\rm SM}}^2 &\simeq 2 \, \lambda_H v^2\, , \\
M_{S_{\cal D}}^2 &\simeq 2 \, \lambda_{S_{1}} \omega_1^2\, , \\
M_{H_{\cal D}}^2 &\simeq m_{\phi}^2 + \frac{\lambda_{H\phi}+\lambda_{H\phi}'}{2} v^2 \, ,\\
M_{S'_{\cal D}}^2 & \simeq m_{2}^2 + \frac{\lambda_{H S_2}}{2} \, v^2 \, ,\\
\end{aligned}\end{align}
while the masses of the CP-odd and charged scalars are given by
\begin{align}
M_{A_{\cal D}} & \simeq M_{H_{\cal D}}\, , \\
M_{a_{\cal D}} & \simeq M_{S'_{\cal D}}\, , \\
M_{H_{\cal D}^\pm}^2 & \simeq M_{H_{\cal D}}^2 - \frac{\lambda_{H\phi }' v^2}{2}\, .
\end{align}
As for the composition of the physical states, since the mixing in the scalar sector is typically small, we can generically define
\begin{equation}
\varphi_{\rm physical} = \varphi - \sum_{\varphi'\neq \varphi}\theta_{\varphi \varphi'} \varphi'\,,
\end{equation}
where $\varphi_{\rm physical}$ denotes the physical scalar field that has the largest $\varphi$ admixture.
Then, the mixing in the CP-even scalar sector is given by
\begin{align}\label{eq:scalar_mixing}
\theta_{H \phi } &\simeq \, \left[(\lambda_{H\phi}+\lambda_{H\phi }')\, v_\phi v - \mu\omega_1/2\sqrt{2}\right]/\Delta M^2_{h_{SM}H_{\cal D}}\, , \nonumber \\
\theta_{H {S_1} } &\simeq \, \lambda_{H S_1}\, \omega_1 v/\Delta M^2_{h_{SM}S_{\cal D}}\, , \nonumber \\
\theta_{H {S_2} } &\simeq \lambda_{HS_2}\, \omega_2 v/\Delta M^2_{h_{SM} S'_{\cal D}} \, , \nonumber\\
\theta_{\phi {S_1} } &\simeq \mu v /2\sqrt{2}\Delta M^2_{H_{\cal D}S_{\cal D}} \, ,\\
\theta_{\phi {S_2} } &\simeq \alpha \, \omega_1 v/4\Delta M^2_{H_{\cal D}S'_{\cal D}} \, \nonumber ,\\
\theta_{{S_1} {S_2} } & \simeq \mu'\omega_1/\sqrt{2}\Delta M^2_{S_{\cal D}S'_{\cal D}} \, , \nonumber
\end{align}
where $\Delta M^2_{\varphi \varphi'}\equiv M^2_\varphi-M^2_{\varphi'}$,
while the Nambu-Goldstone bosons associated with the $W^\pm$, $Z$ and $Z_{\cal D}$ bosons are defined as
\begin{align}
G_W^\pm &\simeq H^\pm - \frac{v_\phi}{v}\phi^\pm \,, \nonumber\\
G_Z & \simeq {\rm Im}(H^0) + \frac{v_\phi}{v} {\rm Im}(\phi^0)\,,\\
G_{Z_{\cal D}}&\simeq{\rm Im}(S_1)+\frac{2\omega_2}{\omega_1}{\rm Im}(S_2)+\frac{v_\phi}{\omega_1}{\rm Im}(\phi^0)-\frac{v_\phi^2}{\omega_1 v}{\rm Im}(H^0).\nonumber
\end{align}
We see that our hypothesis $v_\phi, \omega_2 \ll \omega_1 \ll v$ prevents any relevant modification to the Higgs-like couplings, and $h_{\rm SM}$ ends up being basically like the SM Higgs boson.
Moreover, due to the mixing with the Higgs field, the dark scalars and the longitudinal mode of the $Z_{\cal D}$ will also couple to SM fermions via SM Yukawa couplings. Nevertheless, such couplings to light fermions are quite small as they are suppressed by the hierarchy of vevs. If the spectrum enjoys light degrees of freedom (below the $100$ MeV scale), an interesting phenomenology may be associated with this sector. A dedicated study will be pursued in a future work.
\subsection{\label{subsec:neutrinos} Neutrino Masses and Mixings}
Let us now discuss the generation of neutrino masses and mixings, and how the dynamics of the dark sector outlined so far ensures light neutrinos. The most general Lagrangian in the neutrino sector, compatible with our charge assignment, is
\begin{align}
\mathcal{L}_\nu=& -y_\nu \, \overline{L} \widetilde{\phi} N + y_N \, S_2 \,\overline{N}N^c + y_{N'}\, S_{2}^* \, \overline{N'}N'^c \nonumber\\
&+ m\, \overline{N'}N^c+{\rm h.c.}\, ,
\label{eq:Lnu}
\end{align}
where $y_\nu$, $m$, $y_N$ and $y_{N'}$ are matrices in flavor space. After the two-steps spontaneous breaking $SU(2)_L \times U(1)_Y \times U(1)_{\cal D}\xrightarrow[\quad ]{v} U(1)_{\rm em} \times U(1)_{\cal D} \xrightarrow[\quad ]{\omega_1} U(1)_{\rm em}$, the neutrino mass matrix
in the $(\nu,\,N,\,N')$ basis is
\begin{equation}
\mathcal{M}_\nu=\frac{1}{\sqrt{2}}\left(
\begin{array}{c c c}
0 & y_\nu \, v_\phi & 0\\
y_\nu^T \, v_\phi & y_N \, \omega_2 & \sqrt{2}\,m\\
0 & \sqrt{2}\,m^T &y_{N'} \,\omega_2
\end{array}\right)\, .
\end{equation}
As already stressed, $v_\phi$ generates a Dirac mass term, while $\omega_2$ plays the key role to generate a naturally small term $y_{N^{\prime}}\omega_2$, which can be identified as the tiny mass term of the inverse seesaw $\mu_{\rm ISS}$ (the dimensionful parameter of inverse seesaw that breaks lepton number by two units), and we obtain a dynamically generated inverse seesaw neutrino mass matrix. The mass matrix $m$ can always be made diagonal, and in principle take any value, but given the smallness of the Dirac and $\mu_{\rm ISS} $-terms, it is clear that we can generate light neutrino masses even with values of $m$ smaller than that in the usual inverse seesaw scenario.
More precisely, the light neutrino mass matrix is given at leading order by
\begin{equation}
m_\nu \simeq (y_\nu^T v_\phi) \frac{1}{m^T} (y_{N^{\prime}} \omega_2) \frac{1}{m} (y_\nu v_\phi)\, .
\end{equation}
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Numass.png}
\caption{\label{fig:massdiagram}
Diagram for the dynamically induced light neutrino masses in our model.}
\end{figure}
Inspection of Eq.~(\ref{eq:Lnu}) makes clear why we can substantially lower the
scale of neutrino mass generation, since in our construction the light neutrino masses are generated effectively as a dimension nine operator (see Fig.~\ref{fig:massdiagram}). Schematically, we start with
\begin{equation}
\label{eq:dsix}
{\cal L}_\nu^{\rm eff} \sim y_\nu^2 \, y_{N^{\prime}}\frac{(\overline{L^c} \phi)(\phi^T L)}{m^2} S_{2}^* \, .
\end{equation}
Remembering that the vevs of $\phi$ and $S_2$ are induced by the dynamics of the scalar sector, we can rewrite the previous operator in terms of $H$ and $S_1$, the fields whose vev's are present even in the limit $\left\{ \mu, \mu', \alpha\right\}\to 0$. We obtain
\begin{equation}
\label{eq:dnine}
{\cal L}_\nu^{\rm d=9} \sim y_\nu^2 \, y_{N^{\prime}} \frac{\mu^2}{M^4_{H_{\cal D}}} \frac{\mu'}{M^2_{S'_{\cal D}}}\frac{(\overline{L^c} H)(H^T L)}{m^2} (S_{1}^* S_{1})^2 \, ,
\end{equation}
from which it is clear that, ultimately, neutrinos masses are generated by a dimension 9 operator (see, e.g., Refs.~\cite{higherdim} for generation of neutrino masses from higher dimensional effective operators). In addition, we have a further suppression due to the fact that $\mu$ and $\mu'$ can be taken small in a technically natural way.
The mixing between active and dark neutrinos can be explicitly written as
\begin{equation}\label{eq:mix}
\nu_\alpha = \sum_{i=1}^{3} U_{\alpha i} \, \nu_i + U_{\alpha \mathcal{D}} \, N_{\cal D}\, ,
\end{equation}
$\alpha=e,\mu,\tau,{\cal D}$, where $\nu_i$ and $\nu_{\alpha}$ are the neutrinos mass and flavor eigenstates, respectively (we denote by $\alpha = {\cal D}$ the 6 dark neutrinos flavor states, while $U_{\alpha \mathcal{D}}$ is a $9\times 6$ matrix). Schematically, we have that the mixing between light and heavy neutrinos is $y_\nu v_\phi/m$. Note that the dark neutrino can be made very light, without introducing too large mixing, even for $y_\nu\sim\mathcal{O}(1)$ since $v_\phi\ll v$.
\vspace{0.3cm}
\subsection{\label{subsec:gauge} $Z_{\cal D}$ and the Gauge Sector}
The new vector boson will, in general, communicate with the SM sector via
either mass mixing or kinetic mixing. The relevant part of the dark Lagrangian is
\begin{widetext}
\begin{equation}
{\cal L}_{\cal D} \supset \frac{m^2_{Z_{\cal D }}}{2} \,
Z_{{\cal D}\mu} Z_{\cal D}^{\mu} + g_{\cal D} Z_{\cal D}^\mu \, J_{\mathcal{D}\mu} + e \epsilon \, Z_{\cal D}^\mu \,
J_\mu^{\rm em}
+ \frac{g}{c_W} \epsilon' \, Z_{\cal D}^\mu \,
J_\mu^{\rm Z} \, ,
\label{eq:kmix}
\end{equation}
\end{widetext}
where $m_{Z_{\cal D}}$ is the mass of $Z_{\cal D}$, $g_{\cal D}$ is the $U(1)_{\cal D}$ gauge coupling, $e$ is the electromagnetic coupling, $g/c_W$ is the $Z$ coupling in the SM, while $\epsilon$ and $\epsilon'$ parametrize the kinetic and mass mixings, respectively. The electromagnetic and $Z$ currents are denoted by $\
J^{\rm em}_\mu$ and $J^{Z}_\mu$, while $J_{\mathcal{D}\mu}$ denotes the dark current.
In the limits we are considering, the $Z$ and $W^\pm$ masses are essentially unchanged with respect to the SM values, while the new gauge boson mass reads
\begin{equation}
m_{Z_{\cal D}}^2\simeq g_{\cal D}^2 \left(\omega_1^2 + v_\phi^2 + 4 \, \omega_2^2\right)\simeq g_{\cal D}^2 \, \omega_1^2\, ,
\end{equation}
with mass mixing between $Z$ and $Z_{\cal D}$ given by
\begin{align}
\epsilon' \simeq \frac{2 g_{\cal D} }{g/c_W} \frac{v_\phi^2}{v^2}\, .
\end{align}
Of course, a non-vanishing mass mixing $\epsilon'$ implies that the $Z$ boson inherits a coupling to the dark current
\begin{align}
{\cal L}_{Z} = \frac{m_Z^2}{2} Z_\mu Z^\mu + \frac{g}{c_W} Z^\mu J_\mu^Z - g_{\cal D} \epsilon' Z^\mu J_{\mathcal{D}\mu}\, .
\end{align}
While the new coupling allows for the possibility of new invisible $Z$ decays, the large hierarchy $v_\phi \ll v$ guarantees that the new contributions to the invisible decay width are well inside the experimentally allowed region. The vev hierarchy also protects the model from dangerous $K$, $B$ and $\Upsilon$ decays with an on-shell $Z_{\cal D}$ in the final state~\cite{Davoudiasl:2012ag, Babu:2017olk}.
The kinetic mixing parameter $\epsilon$ is allowed { at tree-level} by all symmetries of the model. Moreover, it is radiatively generated (see e.g. Ref.~\cite{Holdom:1985ag}) by a loop of the $H^\pm_{\cal D}$ scalar which magnitude is
\begin{align}
\epsilon_{\rm LOOP} \sim \frac{e g_{\cal D}}{480 \pi^2} \frac{m_{Z_{\cal D}}^2}{m_{H^\pm_{\cal D}}^2}.
\end{align}
{ In what follows, we will take $\epsilon$ as generated at tree-level, with $\epsilon_{\rm TREE} \gg \epsilon_{\rm LOOP}$ to guarantee the radiative stability of the theory.}
The kinetic mixing will lead to interactions of the $Z_{\cal D}$ to charged fermions, as well as decays if kinematically allowed (see e.g. Ref.~\cite{Ilten:2018crw} for constraints).
\section{\label{sec:pheno}Phenomenological Consequences}
We would like at this point to make some comments about the possible
phenomenological consequences of our model. To illustrate the discussion
let us consider a benchmark point consistent with our working
hypothesis $v_{\phi}, \omega_2 \ll \omega_1 \ll v$. This point is defined by the
input values given in Tab.~\ref{tab1}, producing the physical observables
in Tab.~\ref{tab2}.
We see that for this point the changes in the masses of the SM gauge
bosons as well as the mixings of the Higgs with the new scalars are
negligible, so we do not foresee any major problems to pass the
constraints imposed to the SM observables by the Tevatron, LEP or the
LHC data. { Moreover, our model is endowed with all the
features needed to explain the excess of electron-like events
observed by the MiniBooNE experiment: a new dark vector boson,
$Z_{\cal D}$, that couples to the SM fermions by kinetic mixing and
also directly to a dark neutrino, $\nu_{\cal D}$, which mixes with
the active ones. As shown in \cite{Bertuzzo:2018itn}, the dark
neutrino can be produced via neutrino-nucleus scattering in the
MiniBooNE detector and, if $m_{N_{\cal D}}>m_{Z_{\cal D}}$,
subsequently decay as $N_{\cal D} \to Z_{\cal D} + \nu_i$. The
$Z_{\cal D}$ can then be made to decay primarily to $e^+ e^-$ pairs
with a rate that results in an excellent fit to MiniBooNE energy
spectra and angular distributions.}
In general, this model may in principle also give contributions to the muon $g-2$ \footnote{Since additional electrically charged/neutral scalar ($H^\pm_{\cal D}, H_{\cal D}, A_{\cal D}$) fields and a light dark gauge boson($Z_{\cal D}$) field are present in our model, they will induce a shift in the leptonic magnetic moments and mediate LFV decays via the interactions as shown in Eq.~\ref{eq:Lnu} and Eq.~\ref{eq:kmix}. The contribution to muon magnetic moment from neutral dark Higgs fields ($H_{\cal D}, A_{\cal D}$) with flavor-changing couplings is negligible in our framework. The dominant contribution will arise from singly charged scalar ($H^\pm_{\cal D}$) via the interaction term $y_\nu \, \overline{L} \widetilde{\phi} N$. But, the singly charged scalar correction to muon $g-2$ is negative and rather destructive to the other contributions. Whereas, the one loop contribution of the dark gauge boson ($Z_{\cal D}$) to muon $g-2$ is quite promising and a dedicated study will be pursued further on that. It is worth mentioning that there will be another small contribution to muon $g-2$ from the $W$ boson exchange via mixing between active and dark neutrinos.}, to atomic parity violation, polarized electron scattering, neutrinoless double $\beta$ decay, rare meson decays as well as to other low energy observables such as
the running of the weak mixing angle $\sin^2\theta_{W}$.
There might be consequences to neutrino experiments too. It can, for instance,
modify neutrino scattering, such as coherent neutrino-nucleus scattering, or
impact neutrino oscillations experimental results
as this model may give rise to non-standard neutrino interactions in matter.
Furthermore, data from accelerator neutrino experiments, such as MINOS, NO$\nu$A, T2K, and MINER$\nu$A, may be used to probe $Z_{\cal D}$ decays to charged leptons, in particular, if the channel $\mu^+\mu^-$ is kinematically allowed.
We anticipate new rare Higgs decays, such as $h_{\rm SM} \to Z Z_{\cal D}$, or
$H^\pm_{\cal D} \to W^\pm Z_{\cal D}$, that depending on $m_{Z_{\cal D}}$ may affect LHC
physics. Finally, it may be interesting to examine the apparent anomaly seen in $^8$Be decays~\cite{Kozaczuk:2016nma} in the light of this new dark sector.
The investigation of these effects is currently under way but beyond the scope of this letter and shall be presented in a future work.
\begin{table}[htb]
\begin{tabular}{||c|c|c|c||}
\hline\hline
\multicolumn{4}{c}{\bf Vacuum Expectation Values}\\
\hline \hline
$v$ (GeV) & $\omega_1$ (MeV) & $v_{\phi}$ (MeV) & $\omega_2$ (MeV)\\
\hline
$246$ & $136$ & $0.176$ & $0.65$ \\
\hline \hline
\multicolumn{4}{c}{\bf Coupling Constants}\\
\hline\hline
$\lambda_H$ & $\lambda_{H\phi}=\lambda_{H\phi}'$ & $\lambda_{H S_1}$ & $\lambda_{H S_2}$ \\
\hline
$0.129$ & $10^{-3}$ & $10^{-3}$ & $-10^{-3}$ \\
\hline
$\lambda_{\phi S_1}$ & $\lambda_{\phi S_2}$ & $\lambda_{S_1}$ & $\lambda_{S_1 S_2}$ \\
\hline
$10^{-2}$ & $10^{-2}$ & $2$ & $0.01$ \\
\hline
$\mu$ (GeV) & $\mu'$ (GeV) & $\alpha$ & $g_{\cal D}$\\
\hline
$0.15$ & $0.01$ &$10^{-3}$ & 0.22 \\
\hline\hline
\multicolumn{4}{c}{\bf Bare Masses}\\
\hline\hline
\multicolumn{2}{||c|}{$m_{\phi}$ (GeV)} & \multicolumn{2}{c||}{$m_2$ (GeV)} \\
\hline \hline
\multicolumn{2}{||c|}{100} & \multicolumn{2}{c||}{5.51} \\
\hline \hline
\end{tabular}
\caption{\label{tab1} Input values for a benchmark point in our model that can provide an explanation of the low energy MiniBooNE excess~\cite{Aguilar-Arevalo:2018gpe,Bertuzzo:2018itn}. See Tab.~\ref{tab2} for the respective physical masses and mixings.}
\end{table}
\begin{table*}[htb]
\begin{tabular}{||c|c|c|c|c|c|c|c|c||}
\hline\hline
\multicolumn{9}{c}{\bf Masses of the Physical Fields}\\
\hline\hline
$m_{h_{\rm SM}}$ (GeV) & $m_{H_{\cal D}}$ (GeV) & $m_{S_{\cal D}}$ (MeV) & $m_{S'_{\cal D}}$ (MeV) & $m_{H^\pm_{\cal D}}$ (GeV) & $m_{A_{\cal D}}$ (GeV) & $ m_{a_{\cal D}}$ (MeV) & $m_{Z_{\cal D}}$ (MeV) & $m_{N_{\cal D}}$ (MeV)\\
\hline
$125$ & $100$ & $272$ & $320$ & 100 & 100 & 272 & 30 & 150 \\
\hline \hline
\multicolumn{9}{c}{\bf Mixing between the Fields }\\
\hline\hline
$\theta_{H\phi} $ & $\theta_{HS_1}$ & $\theta_{HS_2}$ & $\theta_{\phi S_1}$ & $\theta_{\phi S_2}$ & $\theta_{S_1 S_2} $ & $e\epsilon$ & $\epsilon' $ & $|U_{\alpha N}|^2$\\
\hline
$1.3\times 10^{-6}$ & $2.1 \times 10^{-6}$ & $10^{-8}$ & $1.2 \times 10^{-3}$ & $8.3 \times 10^{-7}$ & $3.4\times 10^{-2}$ & $2\times 10^{-4}$ & $3.6 \times 10^{-14}$ & $
\mathcal{O}(10^{-6})$ \\
\hline \hline
\end{tabular}
\caption{\label{tab2} Physical masses and mixings for the benchmark point of our model that can provide an explanation of the low energy MiniBooNE excess~\cite{Aguilar-Arevalo:2018gpe,Bertuzzo:2018itn}. The light-heavy neutrino mixing is schematically denoted by $|U_{\alpha N}|^2$, and $m_{N_{\cal D}}$ denotes the order of magnitude of the diagonal entries of the dark neutrino mass matrix.}
\end{table*}
\section{\label{sec:Conclusion}Final Conclusions and Remarks}
The main purpose of this letter has been to explicitly connect the generation of neutrino masses to the existence of a new light dark sector. Doing so, we are able to lower the scale of neutrino mass generation substantially below the electroweak one by resorting to a dynamical breaking of a new $U(1)_{\cal D}$ dark gauge symmetry under which SM particles are neutral.
Our secluded sector consists of the minimal dark field content able to ensure anomaly cancellation, as well as the spontaneous breaking of the dark gauge symmetry without the appearance of a Nambu-Goldstone boson. It consists of a new scalar doublet, two scalar singlets and a set of six new fermion singlets, all charged under the dark symmetry. A judicious choice of dark charges allows to generate neutrino masses by a dynamical inverse seesaw mechanism, but unlike the usual inverse seesaw scenario, the so-called $\mu_{\rm ISS}$-term is here dynamically generated, and can be small in a technically natural way. Interestingly, neutrino masses effectively emerge only at the level of dimension 9 operators, and we can have a new light dark gauge boson in the spectrum.
The dark sector is mostly secluded from experimental scrutiny, as it only communicates with the SM by mixing: the SM Higgs mixing with dark scalars, neutrinos
mixing with dark fermions, and through kinetic and mass mixing with the dark gauge boson.
The low scale phenomenology of the model is simple yet rich. It is possible that
our model gives sizable contributions to several experimental observables
such as the value of the muon $g-2$, the Majorana mass in neutrinoless double
$\beta$ decay or influence atomic parity violation, polarized electron scattering,
or rare meson decays, among others.
Moreover, the mechanism we propose in this letter could provide an novel explanation to the MiniBooNE low energy excess of electron-like events~\cite{Bertuzzo:2018itn}.
As a final remark, let us stress that we presented here only the low scale realization of the model,
imposed by the hierarchy of vevs we have selected. Nevertheless, we could have chosen a different one, for instance, $\omega_1\gtrsim v$. In that case we would have a high scale realization of the model, with unique phenomenological consequences at the LHC, for instance displaced vertex or prompt multi-lepton signatures.
\section*{Acknowledgments}
\noindent
We thank Kaladi Babu for useful discussions and Oscar \'{E}boli for careful reading of the manuscript. This work was supported by Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo (FAPESP) under contracts 12/10995-7 and 2015/25884-4, and by Conselho Nacional de Ci\^encia e Tecnologia (CNPq).
R.Z.F. is grateful for the hospitality of the Fermilab Theoretical Physics Department
during the completion of this work.
The work of S.J. is supported in part by the US Department of Energy Grant
(DE-SC0016013) and the Fermilab Distinguished Scholars Program. Fermilab is
operated by the Fermi Research Alliance, LLC under contract No. DE-AC02-07CH11359 with the United States Department of Energy. This project has also received support from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement N$^\circ$ 690575 (InvisiblesPlus) and
N$^\circ$ 674896 (Elusives).
|
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"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Check if file is image or pdf I need to check if a file is a image or pdf in php/laravel.
This is what I have now:
return $file['content-type'] == 'image/*';
In addition to the 'image/*' I need to add 'application/pdf'
How could this be added?
Update
To be clearer, is there a way to add more allowable types without having to do a OR-condition. I got the answer now with in_array!
A: I like this approach it saves a bit of typing
return (in_array($file['content-type'], ['image/jpg', 'application/pdf']));
A: You can simply use an OR statement, i.e.
return ($file['content-type'] == 'image/*' || $file['content-type'] == 'application/pdf');
This is assuming you still just want to return true/false. So the caller will know that the file was either a PDF or an image.
Or do you mean that the return statement must produce a value which distinguishes between the two types? It's not clear.
If it's the latter than you might want something more like
$type = null;
switch ($file['content-type']) {
case "image/*":
$type = "image";
break;
case "application/pdf":
$type = "pdf";
break;
}
return $type;
A: You can check the content-type of the file using OR condition to add additional check condition.
return ($file['content-type'] == 'image/*' || $file['content-type'] == 'application/pdf')
But it would be better if you put all conditional values in array and check existence of them using in_array.
return (in_array(['image/jpg', 'application/pdf'], $file['content-type']));
A: return $file['content-type'] == 'image/*' || return $file['content-type'] == 'application/pdf';
"||" meaning OR
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"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
An isomorphism between two graphs is a bijection between their vertex
sets that preserves adjacency. Checking if two given graphs are
isomorphic is a problem that has been extensively studied since the
1970s (so much that it was even called a ``disease'' among algorithm
designers) and since then many efficient algorithms have been proposed
and practically used
(e.g.,~\cite{babai,mckay_practical,mckay_piperno,conauto}). Most
state-of-the-art isomorphism checking algorithms and tools
(e.g.,~\verb|nauty|, \verb|bliss|, \verb|saucy|, and \verb|Traces|)
are based on \emph{canonical labellings} \cite{mckay_piperno}. They
assign a unique vertex labelling to each graph, so that two graphs are
isomorphic if and only if their canonical labellings are the same. Such
algorithms can also compute graph automorphism groups. Graph
isomorphism checking and computing canonical labellings and
automorphism groups are useful in many applications. For example,
they are used in many algorithms that enumerate and count
isomorph-free families of some combinatorial objects
\cite{mckay_isomorph_free}. Traditional graph isomorphism applications
are found in mathematical chemistry (e.g.~identification of chemical
compounds within chemical databases) and electronic design automation
(e.g., verification whether the electronic circuits represented by a
circuit schematic and an integrated circuit layout are the same), and
there are more recent applications in the field of machine learning
(e.g.,~\cite{pmlr-v48-niepert16}).
In previous decades we have seen the rise of automated and interactive
theorem proving and their application in formalizing many areas of
mathematics and computer science \cite{ams-formal,itp_milestones}.
Some of the most famous interactive theorem proving results had
significant combinatorial components. For example, the proof of four
color theorem \cite{four-colour} required analyzing several hundred
non-isomorphic graph configurations, some requiring to analyze more 20
million different cases. Formal proof of Kepler's conjecture on
optimal sphere packing in 3d Euclidean space \cite{flyspeck} required
identifying, enumerating and analyzing several thousand non-isomorphic
graphs \cite{tame-graphs}, that was done using a custom formally
verified algorithm. First steps in verifying generic algorithms for
isomorph-free enumeration of combinatorial objects have been made
(e.g., Farad\v zev-Read enumeration has been formally verified within
Isabelle/HOL \cite{faradzev-read}). Several other such algorithms
(e.g., \cite{mckay_isomorph_free}) would benefit from having a trusted
graph isomorphism checking, automorphism group computation, and
canonical labelling.
In order to apply isomorphism checking in theorem proving
applications, it needs to be verified. The easy case is when an
isomorphism between two graphs is explicitly constructed, since it is
then easy to independently verify that the graphs are
isomorphic. However, when an algorithm claims that the given graphs
are not isomorphic, that claim is not easily independently verifiable.
One approach would be to have the isomorphism checking algorithm and
its implementation fully formally verified within an interactive
theorem prover. That would be a very hard and tedious task. The other
approach, which we investigate in this paper, is to extend the
isomorphism checking algorithm, so that it can generate
\emph{certificates} for non-isomorphism. Such certificates would then
be verified by an independent checker, that is much simpler than the
isomorphism checking algorithm itself. Similar approach has been
successfully applied in SAT and SMT solving
\cite{drat-trim,largest-proof,lammich_unsat}. In particular, we shall
extend a canonical labelling algorithm, so that it generates a
certificate certifying that the canonical labelling is correctly
computed.
In the rest of the paper we will investigate the scheme for canonical
labelling and graph isomorphism checking described by McKay and
Piperno \cite{mckay_piperno} and formulate it in form of a proof
system (in spirit of proof systems for SAT and SMT
\cite{KrsticSMT,DPLLT,MJSAT}). We prove the soundness and completeness
of the proof system and implement a proof checker for it. We build our
custom implementation of McKay and Piperno's algorithm that exports
canonical labelling certificates (proofs in our proof-system) that are
independently checked by our proof checker and report experimental
results.
\section{Background}
Notation used in this text is mostly borrowed from McKay and Piperno
\cite{mckay_piperno}.
Let $G = (V, E)$ be an undirected graph with the set of vertices
$V = \{ 1, 2,\ldots, n \}$ and the set of edges
$E \subseteq V \times V$. Let $\pi$ be a \emph{coloring} of the
graph's vertices, i.e.~a surjective function from $V$ to
$\{ 1, \ldots, m \}$. The set $\pi^{-1}(k)$ ($1 \le k \le m$) is a
\emph{cell} of a coloring $\pi$ corresponding to the color $k$
(i.e.~the set of vertices from $G$ colored with the color $k$). We
will also call this cell \emph{the $k$-th cell of $\pi$}, and the
coloring $\pi$ can be represented by the sequence
$\pi^{-1}(1),\pi^{-1}(2),\ldots,\pi^{-1}(m)$ of its cells. The pair
$(G, \pi)$ is called a \emph{colored graph}.
A coloring $\pi'$ is \emph{finer} than $\pi$ (denoted by
$\pi' \preceq \pi$) if $\pi(u) < \pi(v)$ implies $\pi'(u) < \pi'(v)$
for all $u,v \in V$ (note that this implies that each cell of $\pi'$
is a subset of some cell of $\pi$). A coloring $\pi'$ is
\emph{strictly finer} than $\pi$ (denoted by $\pi' \prec \pi$) if
$\pi' \preceq \pi$ and $\pi' \ne \pi$. A coloring $\pi$ is
\emph{discrete} if all its cells are singleton. In that case, the
coloring $\pi$ is a permutation of $V$ (i.e.~the colors of $\pi$ can
be considered as unique labels of the graph's vertices).
A permutation $\sigma \in S_n$ acts on a graph $G = (V, E)$ by
permuting the labels of vertices, i.e.~the resulting graph is
$G^\sigma = (V^\sigma, E^\sigma)$, where $V^\sigma = V$, and
$E^\sigma = \{ (u^\sigma, v^\sigma)\ |\ (u,v) \in E\}$. The
permutation $\sigma$ acts on a coloring $\pi$ such that for the
resulting coloring $\pi^\sigma$ it holds that
$\pi^\sigma(v^\sigma) = \pi(v)$ (that is, the $\sigma$-images of
vertices are colored by $\pi^\sigma$ in the same way as their
originals were colored by $\pi$). For a colored graph $(G,\pi)$, we
define $(G, \pi)^\sigma = (G^\sigma, \pi^\sigma)$.
The composition of permutations $\sigma_1$ and $\sigma_2$ is denoted
by $\sigma_1\sigma_2$. It holds
$(G, \pi)^{\sigma_1\sigma_2} = ((G,\pi)^{\sigma_1})^{\sigma_2} = (G^{\sigma_1},
\pi^{\sigma_1})^{\sigma_2} = ((G^{\sigma_1})^{\sigma_2},
(\pi^{\sigma_1})^{\sigma_2})$. It can be noticed that, if a discrete
coloring $\pi$ is seen as a permutation, it holds
$\pi^\sigma = \sigma^{-1}\pi$, for each $\sigma \in S_n$.
Two graphs $G_1$ and $G_2$ (or colored graphs $(G_1, \pi_1)$ and
$(G_2,\pi_2)$) are \emph{isomorphic} if there exists a permutation
$\sigma \in S_n$, such that $G_1^\sigma = G_2$ (or
$(G_1^\sigma, \pi_1^\sigma) = (G_2,\pi_2)$, respectively).
The problem of \emph{graph isomorphism} is the problem of determining
whether two given (colored) graphs are isomorphic.
A permutation $\sigma \in S_n$ is an \emph{automorphism} of a graph
$G$ (or a colored graph $(G,\pi)$), if $G^\sigma = G$ (or
$(G^\sigma, \pi^\sigma) = (G, \pi)$, respectively). The set of all
automorphisms of a colored graph $(G,\pi)$ is denoted by
$Aut(G,\pi)$. It is a subgroup of $S_n$.
The \emph{orbit} of a vertex $v \in V$ with respect to $Aut(G,\pi)$
is the set $\Omega(v,G,\pi) = \{ v^\sigma \ |\ \sigma \in Aut(G,\pi) \}$.
\section{McKay and Piperno's scheme}
\label{sec:mckay}
The graph isomorphism algorithm considered in this text is described
by McKay and Piperno in \cite{mckay_piperno}. The central idea is to
find a \emph{canonical form} of a graph $(G,\pi)$ denoted by
$C(G,\pi)$ which is some distinguished colored graph obtained by
relabelling the vertices of $G$ (i.e.~there exists a permutation
$\sigma \in S_n$ such that $C(G,\pi)=(G,\pi)^\sigma$), so that the
canonical forms of two graphs are identical if and only if the graphs
are isomorphic. That way, the isomorphism of two given graphs can be
checked by calculating canonical forms of both graphs and then
comparing them for equality.
McKay and Piperno describe an abstract framework for constructing
canonical forms. It is parameterized by several abstract functions
that, when fixed, give a concrete, deterministic algorithm for
constructing canonical forms. Any choice for those functions that
satisfies certain properties (given as axioms) gives rise to a correct
algorithm.
In the abstract framework, the canonical form of a graph is calculated
by constructing the \emph{search tree} of a graph $(G,\pi_0)$, where
$\pi_0$ is the initial coloring of the graph $G$. This tree is denoted
by $\mathcal{T}(G, \pi_0)$. Each node of this tree is labeled with a
unique sequence $\nu = [v_1,v_2,\ldots,v_k]$ of distinct vertices of
$G$. The root of the tree is assigned the empty sequence $[\ ]$. The
length of a sequence $\nu$ is denoted by $|\nu|$, and the prefix of a
length $i$ of a sequence $\nu$ is denoted by $[\nu]_i$. A subtree of
$\mathcal{T}(G,\pi_0)$ rooted in the node $\nu$ is denoted by
$\mathcal{T}(G,\pi_0,\nu)$. A permutation $\sigma \in S_n$ acts on
search (sub)trees by acting on each of the sequences assigned to its
nodes.
Each node $\nu$ of the search tree is also assigned a coloring of $G$,
denoted by $R(G,\pi_0,\nu)$. The \emph{refinement function} $R$ is one
of the parameters that must satisfy several properties for the
algorithm to be correct.
\begin{itemize}
\item The coloring $R(G, \pi_0, \nu)$ must be finer than $\pi_0$ (the axiom R1)
\item It must individualize the vertices from $\nu$, i.e.~each vertex
from $\nu$ must be in a singleton cell of $R(G,\pi_0,\nu)$ (the
axiom R2).
\item The function $R$ must be \emph{label invariant}, that is,
it must hold that
$R(G,\pi_0,\nu)^\sigma = R(G^\sigma, \pi_0^\sigma, \nu^\sigma)$ (the
axiom R3).
\end{itemize}
If the coloring assigned to the node $\nu$ is discrete, then this node
is a leaf of the search tree. Otherwise, we choose some of the
non-singleton cells of the coloring $R(G,\pi_0,\nu)$, called \emph{a
target cell}, and denoted by $T(G, \pi_0, \nu)$. The function $T$ is
called \emph{a target cell selector}, and it is another parameter that
must satisfy several properties for the algorithm to be correct.
\begin{itemize}
\item For search tree leaves, $T$ returns an empty cell (the axiom T1).
\item For search tree inner nodes it returns a non-singleton cell of
$R(G,\pi_0,\nu)$ as the target cell (the axiom T2).
\item The function $T$
also has to be label invariant, i.e.~it must hold
$T(G,\pi_0,\nu)^\sigma = T(G^\sigma, \pi_0^\sigma, \nu^\sigma)$ (the
axiom T3).
\end{itemize}
If $T(G,\pi_0,\nu) = \{ w_1, \ldots, w_m \}$, then the children of the
node $\nu = [v_1,\ldots,v_k]$ in $\mathcal{T}(G,\pi_0)$ are the nodes
$[\nu,w_1]$, $[\nu,w_2]$,\ldots, $[\nu,w_m]$ in that order, where
$[\nu,w_i]$ denotes the sequence $[v_1,\ldots,v_k,w_i]$. Note that the
coloring of each child of the node $\nu$ will individualize at least
one vertex more, which guarantees that the tree $\mathcal{T}(G,\pi_0)$
will be finite. Also note that if the order of child nodes is fixed
(and this can be achieved by assuming that the cell elements $w_1$ to
$w_m$ are given in increasing order), once the functions $R$ and $T$
are fixed, the tree $\mathcal{T}(G, \pi_0)$ is uniquely determined
from the initial colored graph $(G, \pi_0)$.
Since the coloring $\pi = R(G,\pi_0,\nu)$ is discrete if and only if
the node $\nu$ is a leaf, and since discrete colorings can be
considered as permutations of the graph's vertices, a graph
$G_\nu = G^\pi$ can be assigned to each leaf $\nu$.
\begin{exa}
In the example given in Figure \ref{fig:tree}, the target cell is
chosen to be the first non-singleton cell of $R(G, \pi_0, \nu)$ (in
each internal search tree node there is only one non-singleton cell,
so the target cell selector $T$ is trivial). The coloring
$R(G, \pi_0, [\ ])$ is equal to $\pi_0$, and
$\pi = R(G, \pi_0, [\nu, w])$ introduces new cells with respect to
$R(G, \pi_0, \nu)$ only by individualizing $w$. The resulting
colorings are depicted below each node in Figure \ref{fig:tree} (as
sequences of cells, which are sets of vertices).
\begin{figure}[!ht]
\begin{center}
\input{tree.tikz}
\end{center}
\caption{A search tree that generates all permutations of the vertex
set. In each search tree node, the colored graph is drawn (each
graph node contains the number of its color), and the sequence
$\nu$ and the coloring $R(G, \pi_0, \nu)$ given as a sequence of its
cells are printed.}
\label{fig:tree}
\end{figure}
\end{exa}
Each node $\nu$ of $\mathcal{T}(G,\pi_0)$ is also assigned a
\emph{node invariant}, denoted by $\phi(G,\pi_0,\nu)$. Node invariants
map tree nodes to elements of some totally ordered set (such as
numbers, or sequences of numbers, ordered lexicographically). The node
invariant function $\phi$ is another parameter of the algorithm, and
must satisfy the following conditions for the algorithm to be correct:
\begin{itemize}
\item If $|\nu'| = |\nu''|$ and $\phi(G,\pi_0,\nu') < \phi(G,\pi_0,\nu'')$,
then
$\phi(G,\pi_0,[\nu', w_1]) <
\phi(G,\pi_0,[\nu'',w_2])$ for each
$w_1 \in \mathcal{T}(G,\pi_0,\nu')$,
$w_2 \in \mathcal{T}(G,\pi_0,\nu'')$ (the axiom $\phi$1).
\item If $\phi(G,\pi_0,\nu_1) \ne \phi(G,\pi_0,\nu_2)$ for two
distinct leaves $\nu_1$ and $\nu_2$, then $G_{\nu_1} \ne G_{\nu_2}$
(the axiom $\phi$2). However, note that two leaves with equal
invariants may have distinct corresponding graphs.
\item $\phi(G^\sigma, \pi_0^\sigma, \nu^\sigma) = \phi(G,\pi_0,\nu)$
for each node $\nu$ and for each permutation $\sigma$ (that is, $\phi$ is
label invariant) (the axiom $\phi$3).
\end{itemize}
Note that if $\phi(G, \pi_0, \nu) = [f(G, \pi_0, [\nu]_0), f(G, \pi_0,
[\nu]_1), \dots, f(G, \pi_0, [\nu]_{|\nu|})]$ for any label invariant
function $f$ mapping to a totally ordered set, then $\phi$ trivially
satisfies all of the aforementioned conditions.
Let
$\phi_{max} = \max \{ \phi(G,\pi_0,\nu)\ |\ \nu\ \text{is a leaf}
\}$. A \emph{maximal leaf} is any leaf $\nu$ such that
$\phi(G,\pi_0,\nu) = \phi_{max}$. Notice that there can be several
maximal leaves, and recall that their corresponding graphs may be
distinct. Let
$G_{max} = \max\{ G_\nu\ |\ \nu\ \text{is a maximal leaf} \}$ (that
is, $G_{max}$ is a maximal graph assigned to some maximal
leaf)\footnote{We assume that there is a total ordering defined on
graphs (for instance, we can compare their adjacency matrices
lexicographically).}. Again, there may be more than one maximal
leaves corresponding to $G_{max}$. The one labeled with the
lexicographically smallest sequence will be called the \emph{canonical
leaf} of the search tree $\mathcal{T}(G,\pi_0)$, and will be denoted
by $\nu^*$. Let $\pi^* = R(G,\pi_0,\nu^*)$ be the coloring of the leaf
$\nu^*$. The graph $G^{\pi^*} = G_{max}$ that corresponds to the
canonical leaf $\nu^*$ is the canonical form of $(G,\pi_0)$, that is,
we define $C(G,\pi_0) = (G^{\pi^*}, \pi_0^{\pi^*})$. So, in order to
find the canonical form of the graph, we must find the canonical leaf
of the search tree.
Note that once the node invariant function $\phi$, and the graph
ordering are fixed, the canonical leaf $\nu^*$ and the canonical form
$C(G, \pi_0)$ are uniquely
defined based only on the initial colored graph $(G, \pi_0)$.
\begin{exa}
Continuing the previous example, let $f(G, \pi_0, \nu)$ be the maximal
vertex degree of the cell $R(G, \pi_0, \nu)^{-1}(1)$ and let $\phi(G, \pi_0,
\nu)$ be defined as the sequence $[f(G, \pi_0, [\nu]_0), f(G, \pi_0, [\nu]_1), \dots, f(G, \pi_0,
[\nu]_{|\nu|})]$. Note that $f$ is label invariant, implying that
$\phi$ is a valid node invariant. The node invariants for the nodes of
the search tree from Figure \ref{fig:tree}, the
adjacency matrices of the graphs corresponding to the leaves of the tree,
and the canonical leaf of the tree are shown in Figure \ref{fig:invariant}.
\begin{figure}[h!]
\begin{center}
\input{invariant.tikz}
\end{center}
\caption{Node invariants and the canonical leaf for the search tree
from Figure \ref{fig:tree}. For all nodes the coloring
$R(G, \pi_0, \nu)$, and the invariant $\phi(G, \pi_0, \nu)$ are
printed. Colored graphs that correspond to leaves are drawn. }
\label{fig:invariant}
\end{figure}
\end{exa}
McKay and Piperno prove the following central result (Lemma~4 in
\cite{mckay_piperno}).
\begin{thm}
\label{thm:abstract_correctness}
If $R$, $T$, and $\phi$ satisfy all axioms, then for the function $C$
defined as above it holds that the colored graphs
$(G, \pi_0)$ and $(G', \pi_0')$ are isomorphic if and only if
$C(G, \pi_0) = C(G', \pi_0')$. \qed
\end{thm}
\subsection{Finding the canonical leaf}
Assume that the search tree $\mathcal{T}(G,\pi_0)$ is constructed. In
order to find its canonical leaf efficiently, the search tree is
\emph{pruned} by removing the subtrees for which it is somehow deduced
that they cannot contain the canonical leaf. There are two main types
of pruning mentioned in \cite{mckay_piperno} that are of our interest
here. The first type of pruning is based on comparing node invariants
-- if for two nodes $\nu'$ and $\nu''$ such that $|\nu'| = |\nu''|$ it holds
$\phi(G,\pi_0,\nu') > \phi(G,\pi_0,\nu'')$,
then the subtree $\mathcal{T}(G,\pi_0,\nu'')$ can be pruned
(operation $P_A$ in \cite{mckay_piperno}). The other type of pruning
is based on automorphisms -- if there are two nodes $\nu'$
and $\nu''$ such that $|\nu'|=|\nu''|$ and
$\nu' <_{lex} \nu''$ (where $<_{lex}$ denotes
the lexicographical ordering relation), and there is an automorphism
$\sigma \in Aut(G,\pi_0)$ such that
$\nu'^\sigma = \nu''$, then the subtree
$\mathcal{T}(G,\pi_0,\nu'')$ can be pruned (operation $P_C$
in \cite{mckay_piperno}).\footnote{In \cite{mckay_piperno}, the
operation $P_B$ is also considered, which removes a node whose
invariant is distinct from the invariant of the node at the same
level on the path to some leaf $\nu_0$ fixed in advance. This type
of pruning is used only for discovering the automorphism group of
the graph, and not for finding its canonical form, so it is not
considered in this work.}
In practical implementations, the search tree is not constructed
explicitly, and the pruning is done during its traversal, as soon as
possible. If a subtree is marked for pruning, its traversal will be
completely avoided, which is the main source of the efficiency
gains. The algorithm keeps track of the current canonical leaf
candidate, which is updated whenever a ``better'' leaf is discovered.
\begin{exa}
The search tree in Figure \ref{fig:invariant} may be pruned using
either of the two types of pruning operations. For example, nodes with
the node invariant $[2, 1]$ may be pruned, because a node with the node invariant
$[2, 2]$ is present in the tree and $[2, 1] < [2, 2]$. Some nodes may
also be pruned due to the automorphism $(a\ c)$, as illustrated
in Figure \ref{fig:autprune}.
\begin{figure}[h!]
\begin{center}
\input{autprune.tikz}
\end{center}
\caption{The search tree from Figure \ref{fig:invariant} pruned using
the automorphism $(a\ c)$}
\label{fig:autprune}
\end{figure}
\end{exa}
\section{Proving canonical labelling and non-isomorphism}
In order to verify that two graphs are indeed non-isomorphic, we can
either unconditionally trust the algorithm that calculated the
canonical forms (which turned out to be different), or extend the
algorithm to produce us \emph{a certificate (a proof)} for each
canonical form it calculates, which can be independently verified by
an external tool. A certificate is a sequence of easily verifiable
steps that proves the fact that the graph constructed by the algorithm
is equal to the canonical form $C(G, \pi_0)$, as defined by McKay and
Piperno's scheme. The proof system presented in this paper assumes
that the scheme is instantiated with a concrete refinement function,
target cell selector, and node invariant function, denoted
respectively by $\bar{R}$, $\bar{T}$, and $\bar{\phi}$. These
functions will be specified in the following text, and are similar to
the ones used in real-world implementations.
In this approach the correctness of abstract McKay and Piperno's
scheme is assumed. This meta-theory has been well-studied in the
literature \cite{mckay_practical, mckay_piperno}. It is also assumed
that our concrete functions $\bar{R}$, $\bar{T}$ and $\bar{\phi}$
satisfy the appropriate axioms (and that is going to be explicitly
proved in the following text). Therefore, soundness of the approach is
based on Theorem \ref{thm:abstract_correctness}, instantiated for this
specific choice of $R$, $T$ and $\phi$.
On the other hand, the correctness of the implementation of the algorithm
that calculates the canonical form of a given graph is neither assumed
nor verified. Instead,
for the given colored graph $(G, \pi_0)$ it produces a colored graph
$(G', \pi')$ and a proof $P$ that certifies that
$(G', \pi')$ is the canonical form of the graph $(G, \pi_0)$ (i.e.~$C(G,\pi_0) = (G',\pi')$).
The certificate is checked by an
independent proof-checker, whose correctness is assumed (or,
mechanically checked within a proof assistant). Note that the
implementation of the proof-checker is much simpler than the
implementation of the algorithm for calculating canonical form, so it
can be much more easily trusted or verified.
\subsection{Refinement function \texorpdfstring{$\bar{R}$}{}}
In this section we shall define a specific refinement function
$\bar{R}$ and prove its correctness i.e., prove that it satisfies
axioms R1-R3.
We say that the coloring $\pi$ is \emph{equitable} if for each two
vertices $v_1$ and $v_2$ of the same color, and each color $k$ of
$\pi$, the numbers of vertices of color $k$ adjacent to $v_1$ and
$v_2$ are equal. As in \cite{mckay_piperno}, we define the coloring
$\bar{R}(G,\pi_0,\nu)$ to be the coarsest equitable coloring finer
than $\pi_0$ that individualizes the vertices from the sequence
$\nu$. Such a coloring is obtained by individualizing the vertices from
$\nu$, and then making the coloring equitable in an iterative fashion
\cite{mckay_piperno}. In each iteration we choose a cell $W$ from the
current coloring $\pi$ (called a \emph{splitting cell}), and then
partition all the cells of $\pi$ with respect to the number of
adjacent vertices from $W$. This process is repeated until an
equitable coloring is obtained. It is well known \cite{mckay_piperno}
that such a coloring is unique up to the order of its cells. Thus, to
fully determine the coloring $\bar{R}(G,\pi_0,\nu)$, we must fix this
order in some way. The order of the cells depends on the order in
which the splitting cells are chosen, as well as on the order in which
the obtained partitions of a split cell are inserted into the
resulting coloring. In our setting, we use the following strategy:
\begin{itemize}
\item We always choose the splitting cell that appears first in the
current coloring $\pi$ (i.e.~the cell that corresponds to the
smallest color of $\pi$).
\item The obtained fragments of a split cell $X$ are ordered with
respect to the number of adjacent vertices from the current
splitting cell $W$ (ascending). After the fragments are ordered in
that fashion, the first fragment with the maximal possible size is
moved to the end of the sequence, preserving the order of other
fragments.
\end{itemize}
The procedure that computes $\bar{R}(G, \pi_0, \nu)$ is given in
Algorithm~\ref{algo:refinement}. This procedure first constructs the
initial equitable coloring, and then individualizes vertices from $\nu$
one by one, making the coloring equitable after each
step. Individualization of a vertex $v$ is done by replacing the cell
$W$ containing $v$ in the current coloring $\pi$ with two cells
$\{ v \}$ and $W \setminus \{ v \}$ in that order. In order to obtain
an equitable coloring finer than the current coloring $\pi$, the
procedure $\mathtt{make\_equitable}(G, \pi, \alpha)$ is invoked
(Algorithm~\ref{algo:calculate_coloring}), where $\alpha$ is the list
of splitting cells to use (a subset of the set of cells of $\pi$). In
case of initial equitable coloring (when $\nu$ is the empty sequence),
$\alpha$ contains all the cells of $\pi_0$. Otherwise, $\alpha$
contains only the cell $\{ v \}$, where $v$ is the vertex that is
individualized last. The procedure
$\mathtt{make\_equitable}(G, \pi, \alpha)$ implements the iterative
algorithm given in \cite{mckay_piperno}, respecting the order of cells
that is described in the previous paragraph.
\begin{algorithm}[!h]
\caption{$\mathtt{make\_equitable}(G, \pi, \alpha)$}
\label{algo:calculate_coloring}
\begin{algorithmic}
\REQUIRE $(G, \pi)$ is a colored graph
\REQUIRE $\alpha$ is a subset of cells from $\pi$
\ENSURE $\pi'$ is the coarsest equitable coloring of $G$ finer than $\pi$ obtained from $\pi$ by partitioning its cells
with respect to the cells from $\alpha$
\STATE{\textbf{begin}}
\STATE $\pi' = \pi$
\WHILE {$\pi'$ is not discrete and $\alpha$ is not empty}
\STATE\COMMENT{let $W$ be the first cell of $\pi'$ that belongs to $\alpha$}
\STATE remove $W$ from $\alpha$
\FOR{ each non-singleton cell $X$ of $\pi'$ }
\STATE partition $X$ into $X_1,X_2,\ldots X_k$, according to the number of adjacent vertices from $W$
\STATE\COMMENT{ assume that the sequence $X_1,X_2,\ldots, X_k$ is sorted by the number of adjacent vertices from $W$ (ascending)}
\STATE\COMMENT{ let $j$ be the smallest index such that $|X_j| = max\{ |X_i|\ |\ 1 \le i \le k \}$}
\STATE replace $X$ in $\pi'$ by $X_1,\ldots X_{j-1},X_{j+1},\ldots,X_k,X_j$ \COMMENT{in that order (\emph{in situ})}
\FOR{ each $X_i$ in $\{ X_1, \ldots, X_{j-1}, X_{j+1}, \ldots, X_k\}$ }
\STATE add $X_i$ to $\alpha$
\ENDFOR
\IF{ $X \in \alpha$ }
\STATE replace $X$ by $X_j$ in $\alpha$
\ENDIF
\ENDFOR
\ENDWHILE
\STATE{ $\mathbf{return}\ \pi'$ }
\STATE{\textbf{end}}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[!h]
\caption{$\bar{R}(G, \pi_0, \nu)$}
\label{algo:refinement}
\begin{algorithmic}
\REQUIRE $(G, \pi_0)$ is a colored graph
\REQUIRE $\nu$ is a sequence of vertices of $G$
\ENSURE $\pi = \bar{R}(G, \pi_0, \nu)$
is the coarsest equitable coloring finer than $\pi_0$ that
individualizes vertices from $\nu$ \STATE{\textbf{begin}}
\IF{$\nu = [\ ]$}
\STATE\COMMENT{let $\alpha$ be the list of all cells of $\pi_0$}
\STATE $\pi = \mathtt{make\_equitable}(G, \pi_0, \alpha)$
\ELSE
\STATE\COMMENT{let $\nu = [\nu', v]$}
\STATE $\pi' = \bar{R}(G, \pi_0, \nu')$
\STATE\COMMENT{let $W$ be the cell containing $v$ in $\pi'$}
\STATE\COMMENT{let $\pi''$ is obtained by replacing $W$ in $\pi'$ with two cells $\{ v \}, W\setminus \{ v \}$ in that order (\emph{in situ})}
\STATE\COMMENT{let $\alpha$ contain only the cell $\{ v \}$}
\STATE $\pi = \mathtt{make\_equitable}(G, \pi'', \alpha))$
\ENDIF
\STATE{$\mathbf{return}\ \pi$}
\STATE{\textbf{end}}
\end{algorithmic}
\end{algorithm}
The next two lemmas state some properties of the procedure
$\mathtt{make\_equitable}(G, \pi, \alpha)$, needed for proving the correctness
of the refinement function itself.
\begin{lem}
\label{lemma:make_equitable_1}
For each $\sigma \in S_n$,
$\mathtt{make\_equitable}(G^\sigma, \pi^\sigma, \alpha^\sigma) =
\mathtt{make\_equitable}(G,\pi,\alpha)^\sigma$.
\end{lem}
\begin{lem}
\label{lemma:make_equitable_2}
If $\pi' = \mathtt{make\_equitable}(G,\pi,\alpha)$, then $\pi' \preceq \pi$.
\end{lem}
Using the previous two lemmas, we can prove the following lemma which
states that the refinement function $\bar{R}$ satisfies the properties
required by the McKay and Piperno's scheme.
\begin{lem}
\label{lemma:R_correctness}
If $\pi = \bar{R}(G,\pi_0,\nu)$, then:
\begin{enumerate}
\item $\pi \preceq \pi_0$
\item $\pi$ individualizes vertices from $\nu$
\item for each $\sigma \in S_n$,
$\bar{R}(G^\sigma,\pi_0^\sigma,\nu^\sigma) =
\bar{R}(G,\pi_0,\nu)^\sigma$
\end{enumerate}
\end{lem}
The next lemma proves an interesting property of the procedure
$\mathtt{make\_equitable}(G,\pi,\alpha)$ that is of a great
importance for the consistency with our proof system.
\begin{lem}
\label{lemma:make_equitable}
Let $W$ be the cell of $\pi'$ that is chosen as a splitting cell in some arbitrary iteration of the
\texttt{while} loop of the procedure $\mathtt{make\_equitable}(G, \pi, \alpha)$. Let $W'$ be any cell of $\pi'$ that precedes
$W$ in $\pi'$ (that is, it corresponds to a smaller color).
Then for any two vertices of a same color in
$\pi'$ the numbers of their adjacent vertices from $W'$ are equal.\footnote{Informally speaking, the cell $W$ chosen as a
splitting cell in an arbitrary iteration of the \texttt{while} loop is the first cell of $\pi'$ for which the partitioning of $\pi'$ with respect to
$W$ would possibly have some effect on $\pi'$.}
\end{lem}
\subsection{Target cell selector \texorpdfstring{$\bar{T}$}{}}
We assume that the target cell $\bar{T}(G,\pi_0,\nu)$ that corresponds
to the node $\nu$ of the search tree is the first non-singleton cell
of the coloring $\bar{R}(G,\pi_0,\nu)$, if such a cell exists, or the
empty set, otherwise. It trivially satisfies the axioms T1 and T2,
while the following lemma states that the target cell chosen this way
is label invariant, and therefore satisfies the axiom T3.
\begin{lem}
\label{lemma:T_label_invariant}
For each $\sigma \in S_n$ it holds
$\bar{T}(G^\sigma, \pi_0^\sigma, \nu^\sigma) = \bar{T}(G,\pi_0,\nu)^\sigma$.
\end{lem}
\subsection{Node invariant \texorpdfstring{$\bar{\phi}$}{}}
Assume that a suitable \emph{hash function} (denoted by $\operatorname{hash}(G,\pi)$)
is defined over the colored graphs, such that
$\operatorname{hash}(G^\sigma,\pi^\sigma) = \operatorname{hash}(G,\pi)$ for each permutation
$\sigma \in S_n$.\footnote{One approach to construct such hash
function is based on quotient graphs \cite{mckay_piperno}. The
\emph{quotient graph} that corresponds to a colored graph $(G,\pi)$
is the graph whose vertices are the cells of $\pi$ labeled by the
cell number and size, and the edge that connects any two cells $W_1$
and $W_2$ is labeled by the number of edges between the vertices of
$W_1$ and $W_2$ in $G$. It is easy to argue that quotient graphs are
label invariant, so any hash function that depends only on the
quotient graph will also be label invariant.} In our setting, the
node invariant $\bar{\phi}(G,\pi_0,\nu)$ that corresponds to the node
$\nu = [v_1,v_2,\ldots,v_k]$ is a vector of numbers
$[h_1,h_2,\ldots,h_k]$, such that for each $i$ ($1 \le i \le k$) it holds
$h_i = \operatorname{hash}(G,R(G,\pi_0,[v_1,v_2,\ldots,v_i]))$. The node invariants
are ordered lexicographically.
The following lemma proves that the axioms $\phi$1-$\phi$3 are
satisfied for such defined node invariant function.
\begin{lem}
\label{lemma:phi_correct}
The following properties of the function $\bar{\phi}$ hold:
\begin{enumerate}
\item If $|\nu'| = |\nu''|$ and
$\bar{\phi}(G,\pi_0,\nu') <
\bar{\phi}(G,\pi_0,\nu'')$, then
$\bar{\phi}(G,\pi_0,[\nu', w_1]) <
\bar{\phi}(G,\pi_0,[\nu'',w_2])$, for each
$w_1 \in T(G,\pi_0,\nu')$,
$w_2 \in T(G,\pi_0,\nu'')$
\item If $\bar{\phi}(G,\pi_0,\nu_1) \ne \bar{\phi}(G,\pi_0,\nu_2)$ for
two distinct leaves $\nu_1$ and $\nu_2$, then
$G_{\nu_1} \ne G_{\nu_2}$
\item
$\bar{\phi}(G^\sigma, \pi_0^\sigma, \nu^\sigma) =
\bar{\phi}(G,\pi_0,\nu)$ for each node $\nu$ and for each permutation
$\sigma \in S_n$
\end{enumerate}
\end{lem}
\subsection{The proof system}
\emph{A proof} is a sequence of \emph{rule applications}, each
deriving a \emph{fact} from the rule's \emph{premises}, which are the
facts already derived by the preceding rules in the sequence. The
derived facts describe search tree nodes and their properties. We
define the following types of facts:
\begin{itemize}
\item $\bar{R}(G,\pi_0,\nu) = \pi$: the coloring that
corresponds to the node $\nu$ of the tree is equal
to some given coloring $\pi$ of $G$
\item $\bar{R}(G,\pi_0,\nu) \preceq \pi$: the
coloring that corresponds to the node $\nu$ is
finer than or equal to $\pi$.
\item $\bar{T}(G,\pi_0,\nu) = W$: the target cell
that corresponds to the node $\nu$ is $W$ (which is
a set of vertices from $G$).
\item $\Omega \vartriangleleft \operatorname{orbits}(G, \pi_0, \nu)$:
the set of graph vertices $\Omega$ is a subset of some orbit with respect to
$Aut(G,\pi)$, where $\pi = R(G,\pi_0,\nu)$.
\item
$\bar{\phi}(G,\pi_0, \nu') =
\bar{\phi}(G,\pi_0,\nu'')$: the node invariants that
correspond to the nodes $\nu'$ and $\nu''$ ($|\nu'|=|\nu''|$) are equal.
\item $\operatorname{pruned}(G,\pi_0, \nu)$: the node
$\nu$ is pruned from the tree.
\item $\operatorname{on\_path}(G,\pi_0,\nu)$: the node
$\nu$ is an ancestor of the canonical leaf.
\item $C(G,\pi_0) = (G',\pi')$: the colored graph $(G',\pi')$
is the canonical form of the colored graph $(G,\pi_0)$.
\end{itemize}
While some rules may be applied unconditionally, whenever their
premises are already derived, there are also rules that require some
additional conditions to be fulfilled in order to be applied. These
additional conditions should be easily checkable during the proof
verification. Also, proof rules use some functions that should be easy
to calculate during the proof verification:
\begin{itemize}
\item $\operatorname{ind}(\pi,v)$ -- this function returns the coloring that is
obtained from $\pi$ by individualizing the graph vertex $v$, i.e.~by
replacing the cell $W$ of $\pi$ containing $v$ by two cells
$\{ v \}$ and $W \setminus \{ v \}$ in that order (\emph{in situ}).
\item $\operatorname{split}(G,\pi,i)$ -- this function returns the coloring that is
obtained by partitioning the cells of $\pi$ with respect to the
number of adjacent vertices from the $i$-th cell of $\pi$. Each cell
$X$ of $\pi$ is replaced in $\operatorname{split}(G,\pi,i)$ by the cells
$X_1,\ldots,X_k$ obtained by its partitioning, ordered in the same
way as in Algorithm~\ref{algo:calculate_coloring}.
\item $\operatorname{cell}_j(\pi)$ is the $j$-th cell of $\pi$, and $|\operatorname{cell}_j(\pi)|$
denotes its size.
\item $\operatorname{discrete}(\pi)$ -- this predicate is true if the coloring $\pi$
is discrete.
\item $\operatorname{hash}(G,\pi)$ -- this function returns the hash value that
corresponds to the colored graph $(G,\pi)$.
We assume that the proof checker implements the same hash function
as the algorithm that has exported the proof, and that function is
label invariant.
\end{itemize}
The goal of the proving process is to
derive the fact $C(G,\pi_0) = (G',\pi')$ (i.e.~the final rule in the proof
sequence should derive such a fact).
The proof can be built during the search tree
traversal by exporting rule applications determined based on the
operations performed during the algorithm execution. However, it is
possible to optimize this approach and emit proofs after the search
tree has been completely processed -- such proofs can be much shorter,
avoiding derivation of many facts that are not relevant for the final
result. This will be
discussed in more details in Section~\ref{subsec:impl}.
\subsubsection{Refinement rules}
The rules given in Figure~\ref{rules:refinement} are used to verify the
correctness of the colorings constructed by the refinement function
$\bar{R}$. The rule $\mathtt{ColoringAxiom}$ states that the equitable
coloring assigned to the root of the search tree is finer than the
initial coloring $\pi_0$. This rule is exported once, at the very
beginning of the proof construction. The rule $\mathtt{Individualize}$
formalizes the individualization of vertices (an instance of this rule
can be exported after each individualization in Algorithm~\ref{algo:refinement}), and the rule $\mathtt{SplitColoring}$
formalizes the partitioning with respect to a chosen cell (an instance
of this rule can be exported in each iteration of the $while$ loop in
Algorithm~\ref{algo:calculate_coloring}). The rule
$\mathtt{Equitable}$ states that an equitable coloring is reached, and
can be exported at the end of Algorithm~\ref{algo:calculate_coloring}.
\begin{figure}[!h]
{\footnotesize
$$
\begin{array}{l}
\mathtt{ColoringAxiom:} \\
\begin{array}{c}
\\
\hline
\bar{R}(G, \pi_0, [\ ]) \preceq \pi_0
\end{array} \\
\\
\mathtt{Individualize:} \\
\begin{array}{c}
\bar{R}(G, \pi_0, \nu) = \pi \\
\hline
\bar{R}(G, \pi_0, [\nu, v]) \preceq \operatorname{ind}(\pi, v)
\end{array} \\
\\
\mathtt{SplitColoring:}\\
\begin{array}{c}
\bar{R}(G, \pi_0, \nu) \preceq \pi \\
\hline
\bar{R}(G, \pi_0, \nu) \preceq \operatorname{split}(G, \pi, i)
\end{array} \\
\left\{\begin{array}{l}
\textbf{where: } i = \min\{ j\ |\ \operatorname{split}(G, \pi, j) \prec \pi \} \\
\textbf{provided: }\text{such $i$ exists} \\
\end{array}\right. \\
\\
\mathtt{Equitable:} \\
\begin{array}{c}
\bar{R}(G, \pi_0, \nu) \preceq \pi \\
\hline
\bar{R}(G, \pi_0, \nu) = \pi
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: } \forall i.\ \operatorname{split}(G, \pi, i) = \pi \\
\end{array}\right. \\
\end{array}
$$
}
\caption{Rules for refinement}
\label{rules:refinement}
\end{figure}
\subsubsection{Rule for the target cell selection}
The target cell selection is formalized by a single rule given in
Figure~\ref{rules:target}. An instance of this rule can be exported in
the algorithm whenever the target cell selector is applied.
\begin{figure}[!h]
{\footnotesize
$$
\begin{array}{l}
\mathtt{TargetCell:} \\
\begin{array}{c}
\bar{R}(G, \pi_0, \nu) = \pi \\
\hline
\bar{T}(G, \pi_0, \nu) = \operatorname{cell}_i(\pi)
\end{array} \\
\left\{\begin{array}{l}
\textbf{where: } i = \min \{ j\ |\ |\operatorname{cell}_j(\pi)| > 1 \} \\
\textbf{provided: }\text{such $i$ exists} \\
\end{array}\right.
\\
\end{array}
$$}
\caption{Rule for target cell}
\label{rules:target}
\end{figure}
\subsubsection{Rules for node invariant equality detection}
The rules given in Figure~\ref{rules:invariant} are used for
formalizing the knowledge about the nodes at the same level of the
search tree with equal node invariants. Such knowledge is important
for formalizing the lexicographical order over the node
invariants. The rule $\mathtt{InvariantAxiom}$ formalizes the
reflexivity of the equality. It may be exported once for each node of
the tree. The rule $\mathtt{InvariantsEqual}$ formalizes when the node
invariants of two nodes at the same level are lexicographically equal
-- their parents should have equal node invariants and the hash values
assigned to the graph $G$ colored by corresponding colorings of the
two nodes should be equal. This rule may be exported whenever two
nodes at the same level in the tree have equal node invariants. The
rule $\mathtt{InvariantsEqualSym}$ formalizes the symmetricity of the
equality. It may be exported for any two nodes at the same level in
the tree for which the invariant equality fact has been already
derived.
\begin{figure}[!h]
{\footnotesize $$
\begin{array}{l}
\mathtt{InvariantAxiom:} \\
\begin{array}{c}
\\
\hline
\bar{\phi}(G,\pi_0, \nu) = \bar{\phi}(G,\pi_0, \nu)
\end{array}
\\
\\
\mathtt{InvariantsEqual:} \\
\begin{array}{c}
\bar{\phi}(G,\pi_0, \nu') = \bar{\phi}(G,\pi_0,\nu''), \\
\bar{R}(G, \pi_0, [\nu',v']) = \pi_1 \\
\bar{R}(G, \pi_0, [\nu'', v'']) = \pi_2\\
\hline
\bar{\phi}(G,\pi_0, [\nu',v']) = \bar{\phi}(G,\pi_0, [\nu'', v''])
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: }\operatorname{hash}(G,\pi_1) = \operatorname{hash}(G,\pi_2) \\
\end{array}
\right.\\
\\
\mathtt{InvariantsEqualSym:} \\
\begin{array}{c}
\bar{\phi}(G,\pi_0,\nu') = \bar{\phi}(G,\pi_0,\nu'') \\
\hline
\bar{\phi}(G,\pi_0,\nu'') = \bar{\phi}(G,\pi_0,\nu') \\
\end{array}
\\
\end{array}
$$}
\caption{Node invariant equality rules}
\label{rules:invariant}
\end{figure}
\subsubsection{Rules for orbits calculation}
Orbit calculation is formalized by rules given in Figure
\ref{rules:orbits}. The rule $\mathtt{OrbitsAxiom}$ states that any
singleton set of vertices is a subset of some of the orbits with
respect to $Aut(G,\pi)$ (where $\pi = R(G,\pi_0,\nu)$).
It may be exported once for each graph vertex and for each tree node.
The rule $\mathtt{MergeOrbits}$ is used for merging the orbits
$\Omega_1$ and $\Omega_2$ whenever two vertices $w_1 \in \Omega_1$ and
$w_2\in \Omega_2$ correspond to each other with respect to some
automorphism $\sigma \in Aut(G,\pi)$. It can be exported whenever two
orbits are merged at some tree node, after a new automorphism $\sigma$
had been discovered.
\begin{figure}[!h]
{\footnotesize $$
\begin{array}{l}
\mathtt{OrbitsAxiom:} \\
\begin{array}{c}
\\
\hline
\{ v \} \vartriangleleft \operatorname{orbits}(G,\pi_0,\nu) \\
\end{array} \\
\\
\mathtt{MergeOrbits:} \\
\begin{array}{c}
\Omega_1 \vartriangleleft \operatorname{orbits}(G,\pi_0, \nu),\\
\Omega_2 \vartriangleleft \operatorname{orbits}(G,\pi_0, \nu),\\
\hline
\Omega_1 \cup \Omega_2 \vartriangleleft \operatorname{orbits}(G,\pi_0,\nu) \\
\end{array} \\
\left\{\begin{array}{ll}
\textbf{provided: } & \exists \sigma \in Aut(G,\pi_0).\ \nu^\sigma = \nu\ \wedge\\
& \exists w_1\in \Omega_1.\ \exists w_2\in\Omega_2.\ w_1^\sigma = w_2\\
\end{array}\right. \\
\end{array}
$$}
\caption{Orbits calculation rules}
\label{rules:orbits}
\end{figure}
\subsubsection{Rules for pruning}
The rules given in Figure~\ref{rules:pruning} are used for
formalization of pruning. Note that the pruning within the proof
(i.e.~application of the pruning rules) has quite different purpose,
compared to the pruning during the search. Namely, the sole purpose of
pruning during the search (as described in \cite{mckay_piperno}) is to
make the tree traversal more efficient, by avoiding the traversal of
unpromising branches. In that sense, there is no use of pruning a
subtree that has been already traversed, or a leaf that has been
already visited. In other words, we can think of pruning during the
search as a synonym for \emph{not traversing}. On the other hand,
pruning within the proof (which will be referred to as \emph{formal
pruning}) has the purpose of \emph{proving} that some node is not an
ancestor of the canonical leaf. This must be done for all such nodes
(whether they are traversed during the search or not), in order to be
able to prove that the remaining nodes \emph{are} the ancestors of the
canonical leaf, i.e.~that they belong to the path from the root of the
tree to the canonical leaf. This enables deriving the canonical form,
which is done by the rules described in the next section.
As a consequence, applications of the pruning rules within the proof
may or may not correspond to effective pruning operations during the
search. Some formal pruning derivations will be performed
\emph{retroactively}, i.e.~after the corresponding subtree has been
already traversed. Also, it will be necessary to prune (non-canonical)
leaves when they are visited.
\begin{figure}[!h]
{\footnotesize $$
\begin{array}{l}
\mathtt{PruneInvariant:} \\
\begin{array}{c}
\bar{\phi}(G,\pi_0, \nu') = \bar{\phi}(G,\pi_0,\nu''), \\
\bar{R}(G, \pi_0, [\nu',v']) = \pi_1 \\
\bar{R}(G, \pi_0, [\nu'', v'']) = \pi_2\\
\hline
\operatorname{pruned}(G,\pi_0,[\nu'',v''])
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: } \operatorname{hash}(G,\pi_1) > \operatorname{hash}(G,\pi_2) \\
\end{array}
\right.
\\
\\
\mathtt{PruneLeaf:} \\
\begin{array}{c}
\bar{R}(G,\pi_0,\nu') = \pi_1 \\
\bar{R}(G,\pi_0,\nu'') = \pi_2,\\
\bar{\phi}(G,\pi_0, \nu') = \bar{\phi}(G,\pi_0,\nu'') \\
\hline
\operatorname{pruned}(G,\pi_0, \nu'')
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: }\operatorname{discrete}(\pi_2) \wedge (\neg \operatorname{discrete}(\pi_1) \vee G^{\pi_1} > G^{\pi_2}) \\
\end{array}
\right.\\
\\
\mathtt{PruneAutomorphism:} \\
\begin{array}{c}
\\
\hline
\operatorname{pruned}(G,\pi_0,\nu'')
\end{array} \\
\left\{\begin{array}{ll}
\textbf{provided: }& \exists \nu'.\ \nu' <_{lex} \nu''\ \wedge\\
& \exists \sigma \in Aut(G,\pi_0).\ \nu'^\sigma = \nu'' \\
\end{array}
\right.\\
\\
\mathtt{PruneOrbits:} \\
\begin{array}{c}
\Omega \vartriangleleft \operatorname{orbits}(G, \pi_0, \nu) \\
\hline
\operatorname{pruned}(G,\pi_0,[\nu, w_2])
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: } w_2\in \Omega \wedge \exists w_1 \in \Omega.\ w_1 < w_2 \\
\end{array}
\right.\\
\\
\mathtt{PruneParent:} \\
\begin{array}{c}
\bar{T}(G,\pi_0,\nu) = W,\\
\forall w \in W.\ \operatorname{pruned}(G,\pi_0,[\nu,w])\\
\hline
\operatorname{pruned}(G,\pi_0,\nu) \\
\end{array}
\end{array}
$$}
\caption{Pruning rules}
\label{rules:pruning}
\end{figure}
The rule $\mathtt{PruneInvariant}$ formalizes the invariant based
pruning (operation $P_A$ in \cite{mckay_piperno}) and may be applied
when such a pruning operation is done during the search. However, this
rule may also perform the retroactive pruning, in case when the node
$\nu''$ being pruned belongs to the path from the root of
the tree to the current canonical node candidate (which has been
already traversed), and the node $\nu'$ is a node which is
visited later, but it has a greater node invariant (that is the moment
when the algorithm performing the search updates the current canonical
node candidate). Note that the rule $\mathtt{PruneInvariant}$ may also
prune a leaf, when two leaves are found with distinct node invariants
(and their parents have equal node invariants).
The rule $\mathtt{PruneLeaf}$ formalizes the pruning of a leaf, and it
does not correspond to any type of pruning described in
\cite{mckay_piperno} as such (since no leaves are pruned during the
search). This rule treats some special cases not covered by
$\mathtt{PruneInvariant}$ rule. The first case is when a leaf
$\nu''$ has equal node invariant as some non-leaf node
$\nu'$ such that $|\nu'|=|\nu''|$. In that case, it is obvious that all the leaves
belonging to the subtree $\mathcal{T}(G,\pi_0,\nu')$ have
greater node invariants than the leaf $\nu''$, since node
invariants are compared lexicographically. The second case is when the
node $\nu'$ is also a leaf with an equal node invariant as
the leaf $\nu''$, but its corresponding graph is greater
than the graph that corresponds to $\nu''$. In both cases,
we can prune the leaf $\nu''$.
The rules $\mathtt{PruneAutomorphism}$ and $\mathtt{PruneOrbits}$
formalize the automorphism based pruning (operation $P_C$ in
\cite{mckay_piperno}). While this is obvious for the rule
$\mathtt{PruneAutomorphism}$, the connection of the rule
$\mathtt{PruneOrbits}$ to the automorphisms is more subtle. Namely, it
can be shown (Lemma~\ref{lemma:orbits}) that the fact
$\Omega \vartriangleleft \operatorname{orbits}(G,\pi_0,\nu)$ implies
that there is an automorphism $\sigma \in Aut(G,\pi_0)$ such that
$[\nu,w_1]^\sigma = [\nu, w_2]$. Therefore, this
kind of pruning is indeed an instance of $P_C$ operation described in
\cite{mckay_piperno}.
Finally, the rule $\mathtt{PruneParent}$ formalizes the fact that a
node cannot be an ancestor of the canonical leaf if none of its
children is an ancestor of the canonical leaf. This rule performs
retroactive pruning and plays an essential role in pruning already
traversed branches which turned out to be unfruitful.
\subsubsection{Rules for discovering the canonical leaf}
The rules given in Figure~\ref{rules:canonical} are used to formalize
the traversal of the remaining path of the search tree in order to
reach the canonical leaf, after all the pruning is done. The rule
$\mathtt{PathAxiom}$ states that the root node of the tree belongs to
the path leading to the canonical leaf. It is exported once, when all
the pruning is finished. The rule $\mathtt{ExtendPath}$ is then
exported for each node on the path leading to the canonical leaf, and
it states that if the node $\nu$ is on that path, and all
its children except $[\nu,w]$ are pruned, then
$[\nu,w]$ is also on that path. Finally, the rule
$\mathtt{CanonicalLeaf}$ is exported at the very end of the proof
construction, and it states that any leaf that belongs to the path
that leads to the canonical leaf must be the canonical leaf itself
(and, therefore, it corresponds to the canonical graph).
\begin{figure}[!h]
{\footnotesize $$
\begin{array}{l}
\mathtt{PathAxiom:} \\
\begin{array}{c}
\\
\hline
\operatorname{on\_path}(G,\pi_0,[\ ])
\end{array} \\
\\
\mathtt{ExtendPath:} \\
\begin{array}{c}
\operatorname{on\_path}(G,\pi_0,\nu) \\
\bar{T}(G,\pi_0,\nu) = W,\\
\forall w' \in W \setminus \{ w \}.\ \operatorname{pruned}(G,\pi_0,[\nu,w']) \\
\hline
\operatorname{on\_path}(G,\pi_0,[\nu,w])
\end{array} \\
\\
\mathtt{CanonicalLeaf:} \\
\begin{array}{c}
\operatorname{on\_path}(G,\pi_0,\nu) \\
\bar{R}(G,\pi_0,\nu) = \pi \\
\hline
C(G,\pi_0) = (G^\pi,\pi_0^\pi)
\end{array} \\
\left\{\begin{array}{l}
\textbf{provided: }\operatorname{discrete}(\pi)\\
\end{array}\right.\\
\end{array}
$$}
\caption{Canonical leaf rules}
\label{rules:canonical}
\end{figure}
\subsection{The correctness of the proof system}
In this section, we prove the correctness of the presented proof
system. More precisely, we want to prove the \emph{soundness}
(i.e.~that we can only derive correct canonical forms), and the
\emph{completeness} (i.e.~that we can derive the canonical form of any
colored graph). As already noted, we still rely on the correctness of
McKay and Piperno's abstract algorithm itself, that is, that the
canonical forms returned by the algorithm are equal if and only if the
two graphs are isomorphic. In other words, the verified proofs of
canonical forms only confirm that the derived canonical forms are
correct, without any conclusions about the isomorphism of the given
graphs.
\subsubsection{Soundness}
In case of soundness, we want to prove that if our proof derives the
fact $C(G,\pi_0) = (G',\pi')$, then $(G',\pi')$ is indeed the
canonical form of $(G,\pi_0)$ that should be returned by the instance
of McKay's algorithm described in the previous sections. We prove the
soundness of the proof system by proving the soundness of all its
rules, which is the subject of the following lemmas.
\begin{lem}
\label{lemma:refinement}
If the fact $\bar{R}(G,\pi_0, \nu) = \pi$ is derived,
then $\pi$ is the coloring assigned to the node $\nu$ of
the search tree. If the fact
$\bar{R}(G,\pi_0,\nu) \preceq \pi$ is derived, then
the coloring assigned to the node $\nu$ is finer than
or equal to $\pi$.
\end{lem}
\begin{lem}
\label{lemma:target}
If the fact $\bar{T}(G,\pi_0, \nu) = W$ is derived,
then the set of vertices $W$ is the target cell that corresponds to
the node $\nu$.
\end{lem}
\begin{lem}
\label{lemma:orbits}
If the fact $\Omega \vartriangleleft \operatorname{orbits}(G,\pi_0,\nu)$ is derived,
then $\Omega$ is a subset of an orbit with respect to $Aut(G,\pi)$,
where $\pi = \bar{R}(G,\pi_0,\nu)$. In other words, for each
$u_1,u_2 \in \Omega$, there exists an automorphism
$\sigma \in Aut(G,\pi)$, such that $u_1^\sigma=u_2$.
\end{lem}
\begin{lem}
\label{lemma:invariant}
If the fact
$\bar{\phi}(G,\pi_0, \nu_2) =
\bar{\phi}(G,\pi_0,\nu_2)$ is derived, then the nodes
$\nu_1$ and $\nu_2$ have equal node
invariants.
\end{lem}
\begin{lem}
\label{lemma:pruning}
If the fact $\operatorname{pruned}(G,\pi_0,\nu)$ is derived for some search tree
node $\nu$, then this node is not an ancestor of the canonical leaf.
\end{lem}
\begin{lem}
\label{lemma:max_prefix}
If the fact $\operatorname{on\_path}(G,\pi_0,\nu)$ is derived,
then $\nu$ is an ancestor of the canonical
leaf.
\end{lem}
\begin{lem}
\label{lemma:canonical}
If the fact $C(G,\pi_0)=(G',\pi')$ is derived, then
$(G',\pi')$ is the canonical form of the colored graph $(G,\pi_0)$.
\end{lem}
Together, these lemmas prove the main theorem that follows.
\begin{thm}
\label{thm:soundness}
Let $(G,\pi_0)$ be a colored graph. Assume a proof that corresponds
to $(G, \pi_0)$ such that it derives the fact
$C(G,\pi_0)=(G',\pi')$. Then $(G', \pi')$ is the canonical
form of the graph $(G,\pi_0)$.\qed
\end{thm}
\subsubsection{Completeness}
The completeness of the proof system means that for any colored graph
there is a proof that derives its canonical form. First, we prove the
following lemmas which claim the completeness of the proof system with
respect to particular types of facts.
\begin{lem}
\label{lemma:comp_R}
Let $\pi$ be the coloring assigned to a node
$\nu \in \mathcal{T}(G,\pi_0)$. Then there is a proof that derives the
fact $\bar{R}(G,\pi_0,\nu) = \pi$.
\end{lem}
\begin{lem}
\label{lemma:comp_T}
Let $W$ be the target cell assigned to a non-leaf node
$\nu \in \mathcal{T}(G,\pi_0)$. Then there is a proof that derives the
fact $\bar{T}(G,\pi_0,\nu) = W$.
\end{lem}
\begin{lem}
\label{lemma:comp_phi}
Let $\nu_1$ and $\nu_2$ be two nodes such that $|\nu_1|=|\nu_2|$, and
that have equal node invariants. Then there is a proof that derives
the fact $\bar{\phi}(G,\pi_0,\nu_1)=\bar{\phi}(G,\pi_0,\nu_2)$.
\end{lem}
\begin{lem}
\label{lemma:comp_leaf}
If $\nu$ is a non-canonical leaf of
$\mathcal{T}(G,\pi_0)$, then there is a proof that derives the fact
$\operatorname{pruned}(G,\pi_0,[\nu]_i)$ for some $i \le |\nu|$.
\end{lem}
\begin{lem}
\label{lemma:comp_tree}
If a subtree $\mathcal{T}(G,\pi_0,\nu)$ does not contain
the canonical leaf, then there is a proof that derives the fact
$\operatorname{pruned}(G,\pi_0,[\nu]_i)$ for some $i \le |\nu|$.
\end{lem}
\begin{lem}
\label{lemma:comp_path}
If a node $\nu$ is an ancestor of the canonical leaf of the tree
$\mathcal{T}(G,\pi_0)$, then there is a proof that derives the fact
$\operatorname{on\_path}(G,\pi_0,\nu)$.
\end{lem}
\begin{lem}
\label{lemma:comp_canon}
If $\pi^*$ is the coloring that corresponds to the canonical leaf
$\nu^*$ of the tree $\mathcal{T}(G,\pi_0)$, then there is a proof that
derives the fact $C(G,\pi_0) = (G^{\pi^*},\pi_0^{\pi^*})$.
\end{lem}
Together, these lemmas imply the main completeness result given by the
following theorem.
\begin{thm}
\label{thm:completeness}
Let $(G,\pi_0)$ be a colored graph, and let $(G',\pi')$ be its
canonical form. Then there is a proof that corresponds to $(G,\pi_0)$
deriving the fact $C(G,\pi_0)=(G',\pi')$.\qed
\end{thm}
\section{Implementation and evaluation}
\subsection{Proof format}
The proof is generated by writing the applied rules into a file.
Each rule is encoded as a sequence of numbers that contains the parameters
of the rule sufficient to reconstruct the rule within the
proof-checker. The exact sequences for each of the rules are given in
Table~\ref{table:proof_encoding}, using the notation consistent with the one used in the rules' definitions.
\begin{table}[h!]
\begin{center}
{\footnotesize
\begin{tabular}{|c|p{6.2cm}|}
\hline
\textbf{Rule} & \textbf{Sequence} \\
\hline\hline
\texttt{ColoringAxiom} & $0$ \\
\hline
\texttt{Individualize} & $1,|\nu|,\nu,v,\pi$ \\
\hline
\texttt{SplitColoring} & $2,|\nu|,\nu,\pi$ \\
\hline
\texttt{Equitable} & $3,|\nu|,\nu,\pi$ \\
\hline
\texttt{TargetCell} & $4,|\nu|,\nu,\pi$ \\
\hline
\texttt{InvariantAxiom} & $5,|\nu|,\nu$ \\
\hline
\texttt{InvariantsEqual} & $6,|\nu'|+1,[\nu',v'],\pi_1,|\nu''|+1,[\nu'',v''],\pi_2$ \\
\hline
\texttt{InvariantsEqualSym} & $7,|\nu'|,\nu',|\nu''|,\nu''$ \\
\hline
\texttt{OrbitsAxiom} & $8,v,|\nu|,\nu$ \\
\hline
\texttt{MergeOrbits} & $9,|\Omega_1|,\Omega_1,|\Omega_2|,\Omega_2,|\nu|,\nu,\sigma,w_1,w_2$ \\
\hline
\texttt{PruneInvariant} & $10,|\nu'|+1,[\nu',v'],\pi_1,|\nu''|+1,[\nu'',v''],\pi_2$ \\
\hline
\texttt{PruneLeaf} & $11,|\nu'|,\nu',\pi_1,|\nu''|,\nu'',\pi_2$ \\
\hline
\texttt{PruneAutomorphism} & $12,|\nu'|,\nu',|\nu''|,\nu'',\sigma$ \\
\hline
\texttt{PruneParent} & $13,|\nu|,\nu,|W|,W$ \\
\hline
\texttt{PruneOrbits} & $14,|\Omega|,\Omega,|\nu|,\nu,w_1,w_2$ \\
\hline
\texttt{PathAxiom} & $15$ \\
\hline
\texttt{ExtendPath} & $16,|\nu|,\nu,|W|,W,w$ \\
\hline
\texttt{CanonicalLeaf} & $17,|\nu|,\nu,\pi$ \\
\hline
\end{tabular}
}
\end{center}
\caption{Number sequences used to encode the rules}
\label{table:proof_encoding}
\end{table}
Note that each sequence starts with a \emph{rule code}, that is, a
number that uniquely determines the type of the encoded rule. Subsequent numbers
in the sequence encode the parameters specific for the rule. Vertex
sequences are encoded such that the length of the sequence precedes
the members of the sequence. Vertices are encoded in a zero-based
fashion, i.e.~the vertex $i$ is encoded with the number $i-1$. Sets of
vertices (orbits and cells) are encoded in the same way as vertex
sequences, but is additionally required that the vertices are listed
in the increasing order. Colorings and permutations are encoded as
sequences of values assigned to vertices $1,2,\ldots,n$ in that order
(the colors are, like vertices, encoded in a zero-based fashion).
The whole proof is, therefore, represented by a sequence of numbers,
where the first number is the number of vertices of the graph, followed by the sequences of numbers encoding the applied rules, in the order of their application. When such a sequence
is written to the output file,
each number is considered as a 32-bit unsigned integer, and encoded as
a six-byte UTF-8
sequence\footnote{\url{https://datatracker.ietf.org/doc/html/rfc2279}}
for compactness. Thus, the proof format is not human-readable.
\subsection{Implementation details}
\label{subsec:impl}
For the purpose of evaluation, we have implemented a prototypical
proof checker in the \texttt{C++} programming language. The
implementation contains about 1600 lines of code (excluding the code
for printing debug messages), but most of the code is quite simple and
can be easily verified and trusted. The main challenge was to provide
an efficient implementation of the derived facts database, since the
checker must know which facts are already derived in order to check
whether a rule application is justified. The facts are, just like
rules, encoded and stored in the memory as sequences of numbers. The
checker offers support for two different implementations of the facts
database. The first uses \texttt{C++} standard \texttt{unordered\_set}
container to store the sequences encoding the facts. This
implementation is easier to trust, but is less memory
efficient. Another implementation is based on \emph{radix tries}. This
implementation uses significantly less memory, since the sequences
that encode facts tend to have common prefixes. On the other hand,
since it is an \emph{in-house} solution, it may be considered harder
to trust, especially if we take into account that it is the most
complex part of the checker. However, our intensive code inspecting
and testing have not shown any bugs so far. The implementation of
radix tries contains about 150 additional lines of code.
We have also implemented \texttt{morphi} --- a prototypical
\texttt{C++} implementation of the canonical form search algorithm,
based on McKay and Piperno's scheme, instantiated with the concrete
functions $\bar{R}$, $\bar{T}$ and $\bar{\phi}$, as described in this
paper. It is extended with the ability to produce proofs in the
previously described format. The algorithm supports two strategies of
proof generation.
In the first strategy, the proof is generated \emph{during the
search}, with rules being exported whenever the respective steps of
the algorithm are executed. The proof generated in this way
essentially represents a formalized step-by-step reproduction of the
execution of the search algorithm.
In the second strategy, called the \emph{post-search} strategy, the
rules are exported after the search algorithm has finished and
produced a canonical leaf and a set of generators of the automorphism
group. The proof is then generated by initiating the search tree
traversal once more, this time being able to utilize the pruning
operations more extensively since the knowledge of the canonical leaf
and discovered automorphisms is available from the start.
The proof checker implementation is much simpler than the
implementation of the canonical form search algorithm. Namely, the
canonical search implementation uses many highly optimized
data-structures and algorithms, while almost all proof checker data
structures and algorithms are quite straightforward (usually
brute-force). For example, our canonical search implementation
includes an efficient data structure for representing colorings, a
union-find data-structure for representing orbits, an efficient
data-structure for representing sets of permutations (and finding
permutations that stabilize a given vertex set). It adapts to graph
size by using different integer types (8-bit, 16-bit or 32-bit). It
uses a specialized memory allocation system (a stack based
memory-pool). The refinement algorithm is specialized in cases where
cells have 1 or 2 elements. The invariant is calculated incrementally
(based on its value in the previous node). All those (and many more)
implementation techniques are absent in the proof checker. If added,
proof checking would become much more complex and harder to implement
and verify. Since our experiments (see Subsection
\ref{sec:experimental}) show that even with the simplest
implementation the proof checker is not much slower than the canonical
form search, we opted to keep the proof checker implementation as
simple as possible (and much simpler than the original algorithm).
\subsection{Experimental evaluation}
\label{sec:experimental}
The graph instances used for evaluation of the approach are taken
from the benchmark library provided by McKay and
Piperno\footnote{\url{https://pallini.di.uniroma1.it/Graphs.html}},
given in DIMACS format. We included all the instances from the
library in our evaluation, except those instances whose initial
coloring is not trivial (this is because our implementation currently
does not support colored graphs as inputs). In total, our benchmark
set consists of 1284 instances. The experiments were run on a computer
with four AMD Opteron 6168 1.6GHz 12-core processors (that is, 48
cores in total), and with 94GB of RAM.
The evaluation is done using \texttt{morphi}. The first goal of the
evaluation was to estimate how our implementation compares to the
state-of-the-art isomorphism checking implementations. For this
reason, we compared it to the state-of-the-art solver \texttt{nauty}
(also based on McKay's general scheme for canonical labelling) on the
same set of instances. Both solvers were executed on all 1284
instances, with 60 seconds time limit per instance. For the sake of
fair comparison, the proof generating capabilities were disabled in
our solver during this stage of the evaluation.
\begin{table}[!h]
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Solver} & \textbf{\# solved} & \textbf{Avg.~time} & \textbf{Avg.~time on solved} \\
\hline\hline
\texttt{nauty} & 1170 & 15.48 & 5.3 \\
\hline
\texttt{morphi} & 709 & 58.05 & 7.81 \\
\hline
\end{tabular}
\caption{Results of evaluation of \texttt{nauty} and \texttt{morphi}
on the entire benchmark set. The total number of instances in the
set was 1284. Times are given in seconds. Time limit per instance
was 60s. When average time was calculated, 120s was assumed for
unsolved instances.}
\label{table:nauty}
\end{table}
The results are given in Table \ref{table:nauty}. Note that when
average solving time was calculated, twice the time limit was used for
unsolved instances (that is, 120 seconds). This is the PAR-2 score,
that is often used in SAT competitions. The results show that
\texttt{nauty} is significantly faster on average, and it managed to
solve 461 more instances in the given time limit. However, on solved
instances, the average solving times are much closer to each other,
which suggests that our solver is comparable to \texttt{nauty} on a
large portion of the benchmark set (at least on those 709 instances
that our solver managed to solve in the given time limit). A more
detailed, per-instance based comparison is given in Figure
\ref{fig:plot_nauty}. The figure shows that, although most of the
instances are solved faster by \texttt{nauty}, on a significant
portion of them our solver \texttt{morphi} is still comparable (less
than an order of magnitude slower), and there are several instances on
which \texttt{morphi} performed better than \texttt{nauty}. Overall,
we may say that \texttt{morphi} is comparable to the state-of-the-art
solver \texttt{nauty} on average. This is very important, since it
confirms that our prototypical implementation did not diverge too much
from the modern efficient implementations of the McKay's algorithm,
making the approach presented in this paper relevant.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=180pt]{morphi_nauty}
\end{center}
\caption{A comparison of behaviour of \texttt{nauty} and \texttt{morphi} on particular instances. Times are given in seconds}
\label{fig:plot_nauty}
\end{figure}
In the rest of this section, we provide the evaluation of the proof
generation and certification, using our solver \texttt{morphi} (with
the proving capabilities enabled), and our prototypical checker, also
described in the previous section\footnote{The implementation of the
search algorithm and the checker, together with the per-instance
evaluation details, is available at:
\url{https://github.com/milanbankovic/isocert}}. We consider two
different versions of \texttt{morphi}, implementing two different
proof generating strategies, explained in the previous section:
\begin{itemize}
\item \texttt{morphi-d}: the \emph{during-search} strategy
\item \texttt{morphi-p}: the \emph{post-search} strategy
\end{itemize}
The instances used in this stage of evaluation are exactly those
instances that our solver \texttt{morphi} without proof generating
capabilities managed to solve in the time limit of 60 seconds
(i.e.~the instances selected in the previous stage of evaluation, 709
instances in total). Both versions of the solver \texttt{morphi} were
run on all these 709 instances, this time without a time limit, with
the proof generation enabled. In both cases, the generated proofs were
verified using our prototypical checker.
The main result of the evaluation is that the proof verification was
successful for all tested instances, i.e.~for all generated proofs our
checker confirmed their correctness.
For each instance we also measure several parameters that are
important for estimating the impact of proof generation and checking
on the overall performance, as well as for comparing the two variants
of the proof generating strategy. The \emph{solve time} is the time
spent in calculating the canonical form. The \emph{prove time} is the
time spent in proof generation. Notice that in case of
\texttt{morphi-d}, we can only measure the sum of the solve time and
the prove time, since the two phases are intermixed. The \emph{check
time} is the time spent in verifying the proof. We also measure the
\emph{proof size}.
Some interesting relations between the measured parameters on
particular instances are depicted by scatter plots given in
Figure~\ref{fig:plots}, and corresponding minimal, maximal and average
ratios are given in Table~\ref{table:avgs}. We discuss these
relations in more details in the following paragraphs.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=430pt]{plot_new}
\end{center}
\caption{Behavior of \texttt{morphi} and the checker on particular instances. Times are given in seconds,
and proof sizes are given in
kilobytes}
\label{fig:plots}
\end{figure}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{Ratio} & \textbf{Min} & \textbf{Max} & \textbf{Average} \\
\hline\hline
solve/prove (\texttt{morphi-p}) & 0.053 & 4.090 & 0.886 \\
\hline
(prove+solve)/check (\texttt{morphi-p}) & 0.003 & 3.547
& 0.558 \\
\hline
(prove+solve)/check (\texttt{morphi-d}) & 0.001 & 1.009
& 0.219 \\
\hline
\texttt{morphi-p}/\texttt{morphi-d} (solve+prove) & 0.169 & 2.529
& 1.070 \\
\hline
\texttt{morphi-p}/\texttt{morphi-d} (check) & 0.048 & 1.192
& 0.555 \\
\hline
\texttt{morphi-p}/\texttt{morphi-d} (proof size) & 0.166
& 1.000 & 0.674 \\
\hline
\end{tabular}
\end{center}
\caption{Minimum, maximum and average ratios between the observed parameters on all tested instances}
\label{table:avgs}
\end{table}
\paragraph{Solve time vs. prove time.} Top-left plot in Figure
\ref{fig:plots} shows the relation between the solve time and the
prove time for \texttt{morphi-p}. The solve time was about 89\% of the
prove time on average, which means that the proof generation tends to
take slightly more time than the search. In other words, the
invocation of the algorithm with proof generation enabled on average
takes over twice as much time as the canonical form search alone. This
was somewhat expected, since the proof is generated by performing the
search once more, using the information gathered during the first tree
traversal.
\paragraph{Solve+prove time vs. check time.} Top-middle and top-right
plots in Figure~\ref{fig:plots} show the relation between solve+prove
time and check time, for \texttt{morphi-p} and \texttt{morphi-d},
respectively. From the plots it is clear that the proof verification
tends to take significantly more time than the search and the proof
generation together, and this is especially noticeable on ''harder''
instances. The average (solve+prove)/check ratio was 0.558 for
\texttt{morphi-p}, with minimal ratio on all instances being
0.003. The phenomenon is even more prominent for \texttt{morphi-d}
(0.219 average, 0.001 minimal). Such behavior can be explained by the
fact that the checker is implemented with the simplicity of the code
in mind (in order to make it more reliable), and not its
efficiency. Most of the conditions in the rule applications are
checked by brute force. On the other hand, the search algorithm's
implementation is heavily optimized for best performance.
\paragraph{Comparing the proof generating strategies.} Bottom-left,
bottom-middle and bottom-right plots in Figure~\ref{fig:plots} compare
the \texttt{morphi-p} and \texttt{morphi-d} with respect to the
solve+prove times, check times and proof sizes, respectively. The
solve+prove times were similar on average (\texttt{morphi-d} being
slightly faster, with the average solve+prove time equal to 17.7s,
while for \texttt{morphi-p} it was 18.9s). Most of the points on the
bottom-left plot are very close to the diagonal, almost evenly
distributed on both sides of it. The average solve+prove time ratio
was very close to 1 (1.07). This suggests that, when the time
consumed by the algorithm is concerned, it does not matter which proof
generation strategy is employed. On the other hand, the size of a
\texttt{morphi-p} proof was about 67\% of the size of the
corresponding \texttt{morphi-d} proof on average, and a
\texttt{morphi-p} proof was never greater than the corresponding
\texttt{morphi-d} proof. Finally, the check time for proofs generated
by \texttt{morphi-p} was about 55.5\% of the check time for proofs
generated by \texttt{morphi-d} on average. This means that the proof
checking will be more efficient if we employ the post-search proof
generation strategy. One of the main reasons for that is significantly
smaller average proof size, but this might not be the only
reason. Namely, the check time does not depend only on the proof size,
but also on the structure of the proof, since some rules are harder to
check than others. The most expensive check is required by
\texttt{Equitable}, \texttt{InvariantsEqual}, \texttt{PruneInvariant}
and \texttt{PruneLeaf} rules, since it includes time consuming
operations such as verifying the equitability of the coloring,
calculating the hash function for colored graphs, or comparing the
graphs. These operations tend to be more expensive as we move towards
the leaves of the search tree, i.e.~as the colorings become closer to
discrete. In post-search proof generating strategy, the prunings tend
to happen higher in the search tree during the second tree traversal
(when the proof is actually generated), lowering the proportion of
these expensive rules in the proof.
\subsection{False proofs}
It is important to stress that the checker does not only accept
correct proofs, but that it indeed rejects false proofs as well. Some
issues with the solver were found and resolved during development as a
consequence of false proofs that had been rejected by the checker. For
example, there were issues with the invariant calculation and the
coloring refinement that led to incorrect canonical forms and proofs
being produced for some instances. There was also an issue that caused
emission of false facts into the proof in some occasions, specifically
during the traversal of a pruned subtree in search for automorphisms.
As a part of further testing, several artificial bugs were introduced
into the solver which were successfully caught by the checker.
\footnote{Some of the artificial bugs that we introduced for testing
are provided along with the source code, so the interested reader may
apply them to see how the checker responds to such false proofs.}
\section{Related work}
\paragraph{Certifying algorithms in general.}
There are many applications in the industry or science where a user
may benefit of having an algorithm which produces certificates that
can confirm the correctness of its outputs. McConnell et
al.~\cite{mcconnell} go one step further, suggesting that \emph{every}
algorithm should be \emph{certifying}, i.e.~be able to produce
certificates. They establish a general theoretical framework for
studying and developing certifying algorithms, and provide a number of
examples of practical algorithms that could be extended to produce
certificates. The authors stress two important properties of a
certifying algorithm that must be fulfilled: \emph{simplicity}, which
means that it is easy to prove that the (correct) certificates really
certify the correctness of the obtained outputs, and
\emph{checkability}, which means that it is easy to check the
correctness of a certificate. In both cases, the property of
\emph{being easy} is not strictly defined.
In case of simplicity, it is expected that the fact that a certificate
really proves the correctness of the corresponding output should be
obvious and easily understandable. This may be considered as a
potential drawback of our approach, since the soundness of our proof
system is not obvious and requires a non-trivial proof. However, we
did our best to provide an as precise as possible soundness proof in
the appendix of this paper, with hope that it is convincing enough to
the reader (we also prove the completeness, i.e.~the existence of a
certificate for each input, for which McConnell et
al.~\cite{mcconnell} do not insist to be simple to prove). Our goal is
to formalize the soundness proof within Isabelle/HOL theorem prover,
and we believe that the existence of a formal soundness proof is a
good substitution for the simplicity property stressed by the authors.
When checkability is concerned, the authors suggest that an algorithm
for certificate checking should either have a simple logical
structure, such that its correctness can be easily established, or it
should be formally verified \cite{mcconnell}. We believe that our
checker indeed has a simple logical structure, since its most
complicated parts such as calculating $\operatorname{split}(G,\pi,i)$
function are implemented using a brute force, without any sofisticated
data structures or programming techniques. Moreover, we have already
argued (see Subsection \ref{subsec:impl}) that its implementation is much
simpler than the implementation of the canonical labelling algorithm
itself, and can be more easily verified (which we plan to do in the
future).
McConnell et al.~\cite{mcconnell} also suggest that the complexity of
a certificate checking algorithm should preferably be linear on its
input, and its input is composed of an input/output pair of a
certifying algorithm, and of the corresponding certificate. In our
case, each rule from the certificate is read and checked exactly once,
but checking of some rules may require traversing the graph's
adjacency matrix, so the complexity is not exactly linear. However,
since the size of the proof is usually much greater than the size of
the graph, the complexity may be considered to be almost linear.
\paragraph{Certifying algorithms in SAT solving.}
Certificate checking has been successfully used, for example, in
checking unsatisfiability results given by SAT or SMT solvers
\cite{drat-trim,largest-proof,lammich_unsat}, and this enabled
application of external SAT and SMT solvers from within interactive
theorem provers \cite{sat_isabelle,bohme_phd,sledgehammer-smt}.
A proof system used in case of SAT solvers is very simple and is based
on the resolution. While early approaches considered simple resolution
proofs \cite{zhang2003validating} which were very easy to check, but
much harder to produce, modern approaches are mostly based on
so-called \emph{reversed unit-propagation} (RUP) proofs
\cite{Gelder2008}, which are easy to generate, but harder to
check. While this fact violates the checkability property to some
extent, it is widely adopted by the SAT solving community, and the
proof formats such as DRAT \cite{drat-trim} became an industry
standard in the field. Our proof system has a similar trait, since we
retained some complex computations within the checker, in order to
make our proofs smaller and easier to generate. As with RUP proofs, we
consider this as a good tradeoff.
\paragraph{Certifying pseudo-boolean reasoning.}
The SAT problem is naturally generalized to the \emph{pseudo-boolean}
(PB) satisfiability problem \cite{roussel2021pseudo}, where it is also
useful to have certifying capabilities. Recently, a simple but
expressive proof format for proving the PB unsatisfiability was
proposed by Elffers et al.~\cite{elffers2020justifying}, based on
\emph{cutting planes} \cite{cook1987complexity}. That proof format was
then employed for expressing unsatisfiability proofs for different
combinatorial search and optimization problems, such as constraint
satisfaction problems involving the alldifferent constraint
\cite{elffers2020justifying}, the subgraph isomorphism problem
\cite{gocht2021subgraph}, and the maximum clique and the maximum
common subgraph problem \cite{gocht2020certifying}. The main idea is
to express some combinatorial problem $P$ as a pseudo-boolean problem
$PB(P)$, and then to express the reasonings of a dedicated solver for
$P$ in terms of PB constraints implied by the constraints of $PB(P)$
(such PB constraints are exported by the dedicated solver during the
solving process). The obtained PB proof is then checked by an
independent checker which relies on a small number of simple rules
(such as addition of two PB constraints or multiplying/dividing a PB
constraint with a positive constant), as well as on generalized RUP
derivations \cite{elffers2020justifying}. The authors advocate the
generality of their approach, enabling it to express the reasoning
employed in solving many different combinatorial problems, using a
simple and general language which does not know anything about the
concepts and specific semantics of considered problems.
In contrast, our proof system is tied to a particular problem -- the
problem of constructing canonical graph labellings, and a particular
class of algorithms -- those that could be instantiated from the
McKay/Piperno scheme \cite{mckay_piperno}.\footnote{Actually, our
proof system is tied to a particular instance of McKay/Piperno's
scheme, fixed by a choice of $R$, $T$ and $\phi$ functions, but it
should be relatively easy to adapt it for other such
instantiations.} Citing McKay, the canonical forms returned by the
algorithm considered in this paper are ``designed to be efficiently
computed, rather than easily described'' \cite{mckay_isomorph_free}.
Since the canonical labelling specification is given in terms of the
algorithm, it seems that certification must also be tied to it. This
is the main reason that led to our decision to develop an
algorithm-specific proof format which is powerful enough to capture
all the reasonings that appear in the context of this particular class
of labelling algorithms.
The question remains whether the canonical labelling problem, like
some other graph problems previously mentioned, may be encoded in
terms of PB constraints in a way simple enough to be trusted (as noted
by Gocht et al.~\cite{gocht2020certifying}, an encoding should be
simple -- since it is not formally verified, any errors in the
encoding may lead to "proving the wrong thing"). Moreover, the
encoding should permit expressing the operations made by the canonical
labelling algorithm in terms of PB constraints implied by the
encoding. Up to the authors' knowledge, it is not obvious whether such
encoding exists. The main issue that we see in such an approach is
that the canonical form of the graph, as defined by McKay and Piperno
\cite{mckay_piperno}, is not simple to describe in advance using PB
constraints, since it is defined implicitly by the canonical
labelling algorithm itself. For instance, one should capture the
notion of "coarsest equitable coloring finer than a given coloring
$\pi$", which is especially hard if we know that such a coloring is
not unique, so we must fix one particular ordering of the cells (and
that ordering must be also captured using the PB language). The
similar case is with describing hash functions used in calculating
node invariants.
\section{Conclusions and further work}
We have formulated a proof-system for certifying that one graph is a
canonical form of another, which is a central step in checking graph
isomorphism. The proof system is based on the state-of-the-art scheme
of McKay and Piperno and we have proved its soundness and
completeness. Our algorithm implementation produces proofs that can be
checked by an independent proof checker. The proof checker
implementation is significantly simpler than the core algorithm
implementation, so it can be more easily trusted.
One of the main problems of our approach is that our proof format is
very tightly connected to the canonical labelling search algorithm
itself. Therefore, it is very hard to use our proof format and proof
checker for different isomorphism checking approaches. On the other
hand, McKay/Piperno scheme is broad enough to cover several
state-of-the art algorithms and we believe that our proofs and
checkers can be easily adapted to certify all of them. Our
prototypical implementation \texttt{morphi} is slower compared to to
state-of-the-art implementations like \texttt{nauty}, but still
comparable on many instances. We plan to make further optimizations to
\texttt{morphi}. We are also considering a possibility to adapt
\texttt{nauty} source-code to emit proofs in our format, which is
non-trivial, since \texttt{nauty} has been developed for more than 30
years and is a very complex and highly optimized software system.
The main direction of our further work is to go one step further and
to integrate isomorphism checking into an interactive theorem
prover. We have already made first steps in that direction: we have
formally defined abstract scheme of McKay and Piperno in Isabelle/HOL
and formally proved its correctness (Theorem
\ref{thm:abstract_correctness})\footnote{Formalization is available at
\url{https://github.com/milanbankovic/isocert}}. The remaining step
is to formalize our proof system, formally prove its soundness and
implement a verified proof-checker within Isabelle/HOL.
\section*{Acknowledgment}
\noindent This work was partially supported by the Serbian Ministry of
Science grant 174021. We are very grateful to the anonymous reviewers
whose insightful comments and remarks helped us to make this text much
better.
\bibliographystyle{alpha}
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<!DOCTYPE html>
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https://www.littlethings.com/gender-reveal-wedding-surprise/
Mom Thinks Baby's Gender Is The Secret, But Sees Unexpected Message On Baby Bump And Loses It
by Barbara Diamond
Barbara is a passionate writer and animal lover who has been professionally blogging for over 10 years and counting.
So much love is wrapped up in this one video.
On October 3, 2015, a traditional gender reveal party turned into so much more.
While Dad was Team Pink, the expecting mother and their son were Team Blue all the way.
Since the day was so jam-packed with activities and events, the camerawoman knew she needed to capture each and every moment. It all kicked off with a scavenger hunt that led them to the location of the gender reveal party.
First up was a trip to Big Lots and then it was off to Grandma's house. As you'll see, it was a down-home and festive family affair with body paint, contests, and pink and blue balloons.
The plan was to blindfold Mom and paint her baby bump with either pink or blue. Then, she would be ushered outside where she'd lift her shirt and reveal the gender to herself and her loved ones.
When she was finally able to look down at her bump, Mom saw she was pregnant with the boy she'd been hoping for… but there was also a diamond ring painted around the blue cartoon baby.
Yes, a wedding was afoot, and it was going to happen much sooner than she could have ever expected.
As it turned out, she had already bought her own wedding dress without even knowing it. Her friends had told her they were going to be models in a bridal show — but it was all just a ruse.
Please enjoy this emotional and intimate surprise, and SHARE it with your friends on Facebook!
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| 1,825
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Штефан Кунц (; Нојнкирхен, 30. октобар 1962) бивши је немачки професионални фудбалер, играо је на позицији нападач. Наступао је за репрезентацију Немачке у периоду између 1993—1997. године, стигавши до четвртфинала Светског првенства 1994. и победивши на Европском првенству 1996. Постигао је шест голова за репрезентацију, укључујући у полуфиналној победи против репрезентације Енглеске.
Од 2006. до 2008. године био је спортски директор Бохума. Између 2008. и 2016. био је председник одбора Кајзерслаутерна. Био је тренер јуниорске репрезентације Немачке од 2016.
Каријера
Клуб
Сениорску каријеру је започео 1980. у Борусији Нојнкирхену. Године 1983. је играо у Бохуму, 1986. у Ирдингену 05, а 1989. у Кајзерслаутерну. У Бундеслиги Немачке је наступио у 449 утакмица и постигао је 179 голова. Године 1986. и 1994. изабран је за најбољег стрелца.
Године 1995. је потписао уговор са турским Бешикташом, на захтев тренера Кристофа Даума. Постигао је први гол 24. септембра 1995. у петом колу првенства.
Следеће године је играо за Арминију из Билефелда, од 1998. за Бохум, 2002. за Фурпач и 2004. потписао је уговор са Лимбахом.
Репрезентација
Био је члан јуниорске репрезентације Немачке 1983—1985. године.
Као члан репрезентације Немачке од 1993, играо је на Европском првенству 1996, где је имао кључну улогу у утакмици против репрезентације Енглеске у полуфиналу, постигавши изједначујући гол за 1:1 убрзо након што је Енглеска повела. Укупно је за репрезентацију постигао шест голова.
Тренер
Тренерску каријеру је започео 1999. у Борусији Нојнкирхен. Следеће године је био тренер Карлсруеа, 2003. Ваљдофа и Ахлена. Као тренер јуниорске репрезентације Немачке освојио је 2017. два Европска првенства до 21 године. У финалу су победили репрезентацију Шпаније са 1:0, а 2021. репрезентацију Португала са истим резултатом.
Успеси
Клуб
Кајзерслаутерн
Куп Немачке: 1989/90.
Бундеслига Немачке: 1990/91.
Суперкуп Немачке: 1991.
Репрезентација
Немачка
Европско првенство: 1996.
Тренер
Немачка
Европско првенство до 21 године
Шампиони: 2017, 2021.
Другопласирани: 2019.
Индивидуални
Фудбалер године: 1991.
Најуспешнији стрелац Бундеслиге: 1986.
Најуспешнији стрелац купа Немачке: 1988, 1990.
Референце
Спољашње везе
Штефан Кунц на fussballdaten.de
Штефан Кунц на WorldFootball.net
Штефан Кунц на National-Football-Teams.com
Рођени 1962.
Биографије живих особа
Немачки фудбалери
Немачки фудбалски тренери
Фудбалери европски прваци
Фудбалери на Европском првенству 1996.
Фудбалери на Светском првенству 1994.
Фудбалери Бешикташа
Фудбалери Арминије Билефелд
Фудбалери Бохума
Фудбалери Кајзерслаутерна
Фудбалери Бундеслиге
Нападачи у фудбалу
Немачки фудбалски репрезентативци
|
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\section*{Acknowledgments}
The authors gratefully acknowledge valuable discussions with R.~Alkofer
and N.~Ishii. We are indebted to H.~Witala for bringing Ref.~\cite{Car98} to
our attention and to B.~Blankleider for helping to clarify the
relation of our study to the general considerations of Ref.~\cite{Bla99a}.
The work of M.O. was supported by the Deutsche For\-schungsge\-mein\-schaft
under contract DFG We 1254/4-1. He thanks H.~Weigel and H.~Reinhardt
for their continuing support.
The work of M.A.P. was supported by the U.S. Department of Energy
under contracts DE-FG02-87ER40365,
DE-FG02-86ER40273 and DE-\-FG\-05-\-92ER40750,
the National Science Foundation under contract PHY9722076 and
the Florida State University Supercomputer Computations Research Institute
which is partially funded by the Department of Energy under
contract DE-FC05-85ER25000.
The work of L.v.S. was supported by the U.S. Department of Energy, Nuclear
Physics Division, under contract number W-31-109-ENG-38 and by the BMBF under
contract number 06-ER-809. Parts of the calculations were performed on the
IBM SP3 Quad Machine of the Center for Computational Science and Technology
at Argonne National Laboratory.
\section{Supplements on Nucleon Charge Conservation} \label{supplCC}
\newcommand{\widetilde{\psi}_M=S^{-1}_M D^{-1}_M \psi_M}{\widetilde{\psi}_M=S^{-1}_M D^{-1}_M \psi_M}
\newcommand{\widetilde{\psi}_E\not =S^{-1}_E D^{-1}_E \psi_E}{\widetilde{\psi}_E\not =S^{-1}_E D^{-1}_E \psi_E}
\begin{figure*}[t]
\centerline{\parbox{0.6\linewidth}{
\begin{equation*}
\begin{CD}
\int \bar\psi_M D_M^{-1}\Gamma^\mu_{q,M}\psi_M @>\text{Wick rotation}>>
\int\bar\psi_E D_E^{-1}\Gamma^\mu_{q,E}\psi_E \\
@V{\widetilde{\psi}_M=S^{-1}_M D^{-1}_M \psi_M}VV @VV{\widetilde{\psi}_E\not =S^{-1}_E D^{-1}_E \psi_E}V \\
\int \widetilde{\bar\psi}_M S_M\Gamma^\mu_{q,M}S_M D_M\widetilde{\psi}_M
@>\text{Wick rotation}>>
\begin{matrix}
\int \widetilde{\bar\psi}_E S_E\Gamma^\mu_{q,E}S_E D_E\widetilde{\psi}_E
\\[2pt]
\hskip 1.5cm +\hskip .5cm \text{residue terms}
\end{matrix}
\end{CD}
\end{equation*}}}
\stepcounter{section}
\refstepcounter{figure}
\centerline{\parbox{0.8\linewidth}{\small\textbf{Fig.~\thefigure.}
Interrelation of matrix elements in Minkowski and Euclidean space.
The integral sign is shorthand for the four-dimensional integration over
the relative momentum $k$, see Eq. (\ref{quark}).\label{box}}}
\end{figure*}
\addtocounter{section}{-1}
The missing step in the explicit verification of the correct charges $Q_P =
1$ for the proton and $Q_N = 0$ for the neutron is to prove that
\begin{eqnarray}
N_q - N_D \, = \, 2 N_X + 3 N_P \label{A1}
\end{eqnarray}
which is equivalent to $Q_N = 0$, see Eqs.~(\ref{chargeconds})
to~(\ref{chargesum}) in Sec.~\ref{ElmFFs}. To this end, note that from
Eqs.~(\ref{NqDef}) and~(\ref{NDDef}) the l.h.s above can be written as,
\begin{eqnarray}
N_q - N_D &=& \frac{P^\mu}{2M_n} i \int \frac{d^4k}{(2\pi)^4} \\
&& \hskip -1cm
\hbox{tr} \left[ \bar\psi (-k,P) \, \left(\frac{\partial}{\partial
k^\mu} S^{-1}(k_q) D^{-1}(k_s) \right) \, \psi(k,P) \right] \nn \\
&=& \frac{P^\mu}{2M_n} i \int
\frac{d^4k}{(2\pi)^4} \nn\\
&& \hskip -1cm
\hbox{tr} \Bigg[ \bar\psi (-k,P) \,
\left(\frac{\partial}{\partial k^\mu} S^{-1}(k_q) D^{-1}(k_s) \, \psi(k,P)
\, \right) \nn\\
&& \hskip -.4cm + \, \left( \frac{\partial}{\partial k^\mu} \bar\psi
(-k,P) \, S^{-1}(k_q) D^{-1}(k_s) \right) \, \psi(k,P) \; \Bigg] \nn\\
&& \hskip 3.5cm - \; \hbox{total derivative} \; ,\nn
\end{eqnarray}
with $k_q = \eta P +k $ and $k_s = \hat\eta P - k$ as in the previous
sections. A surface term which vanishes for normalizable BS wave functions
$\psi$ and $\bar\psi$ was not given explicitly. Using the BSEs for
$\widetilde\psi = S^{-1} D^{-1} \psi$ and $\widetilde{\bar\psi} = \bar\psi
S^{-1} D^{-1} $, {\it c.f.}, Eqs. (\ref{BSEpsi}) and (\ref{BSEpsibar}),
in analogy to the proof of current conservation in Sec.~\ref{TECO}, one
obtains,
\begin{eqnarray}
N_q - N_D &=& \frac{P^\mu}{2M_n} i \int \frac{d^4p}{(2\pi)^4}
\,\frac{d^4k}{(2\pi)^4} \label{A3} \\
&& \hskip -1.9cm \hbox{tr} \left[ \bar\psi (-p,P) \left( \frac{\partial}{\partial
p^\mu} K(p,k,P) + \frac{\partial}{\partial
k^\mu} K(p,k,P) \right) \psi(k,P) \right] . \nn
\end{eqnarray}
From Eq.~(\ref{x_kern_par}) for the explicit form of the exchange kernel it
follows that
\begin{eqnarray}
\left( \frac{\partial}{\partial
p^\mu} \, + \, \frac{\partial}{\partial k^\mu} \right) \, K(p,k,P) &=& -
\frac{1}{2N_s^2} \label{A4} \\
&& \hskip -4cm \Bigg\{ \, 2 \left( \frac{\partial}{\partial q^\mu} S(q)
\right) P(-p_1^2) P(-p_2^2)
+ S(q) \nn\\
&& \hskip -3.7cm
\Big( \, 3 p_{1\mu} \, P_1'(-p_1^2) P(-p_2^2)
- \, 3 p_{2\mu} \, P_1(-p_1^2) P'(-p_2^2) \, \Big) \, \Bigg\} \; ,\nn
\end{eqnarray}
since $q = (1\!-\!2\eta )P -p-k$, $p_1 = -(1\!-\!3\eta )P/2 + p +k/2$
and $p_2 = (1\!-\!3\eta )P/2 - p/2 -k$. Comparing to the definitions of
$N_X$ and $N_P$, Eqs.~(\ref{NXDef}) and~(\ref{NPDef}) in Sec.~\ref{SecNucNorm}
respectively, we see that Eq.~(\ref{A4}) inserted in Eq.~(\ref{A3}) gives
(\ref{A1}) as required.
\section{Calculation of the Impulse Approximation Diagrams}
\label{residue}
\stepcounter{figure}
In this appendix we discuss the difficulties in the formal
transition from the Minkowski to the Euclidean metric which are encountered
in the connection between Bethe-Salpeter amplitudes $\widetilde{\psi}$ and
Bethe-Salpeter wave functions $\psi$ in a general (boosted) frame of
reference of the nucleon bound state.
As the generic example for this discussion, we choose the first diagram in
Figure \ref{IAD} which describes the impulse-approximate contribution arising
from the coupling of the photon to the quark within the nucleon,
$\langle \widehat J^\mu_{q}\rangle$ according to Eq.~(\ref{Eq:6.10}) with
Eq.~(\ref{jq}). Please refer to Fig.~ \ref{IAD} and Table~\ref{mom_table}
for the momentum definitions employed herein.
In the Mandelstam formalism, such matrix elements between bound states are
related to the corresponding BS wave functions in the in Minkowski
space, here to the nucleon BS wave functions $\psi_M$.\footnote{In the
following the subscript $_M$ stands for definitions in Minkowski space and
$_E$ for the corresponding ones in Euclidean space.}
Upon the transition to the Euclidean metric, the corresponding
contribution to the observable, here to the nucleon form factors,
is determined by the ``Euclidean BS wave function'' $\psi_E$. In the rest
frame of the nucleon bound state this transfer from $M \to E$ of the BS wave
functions commutes with the replacement of the wave functions by the
truncated BS amplitudes; that is, unique results are obtained from
the Euclidean contributions based on either employing the Minkowski space wave
functions or the BS amplitudes which are related by the truncation of the
propagators of the constituent legs, here
$\widetilde{\psi}_M=S_M^{-1}D_M^{-1}\psi_M$, or, {\em vice versa},
$\psi_M=S_M D_M\widetilde{\psi}_M$.
At finite momentum transfer $Q^2$ one needs to employ BS wave functions in a
more general frame of reference, here we use the Breit frame in which neither
the incoming nor the outgoing nucleon are at rest.
As described in Sec. \ref{ElmFFs}, the ``Euclidean wave function'' $\psi_E$
in this frame is obtained from the solution to the BSE in the rest frame by
analytic continuation, in particular, by inserting complex values for the
argument of the Chebyshev polynomials, see Eqs. (\ref{compl_zy}).
This corresponds to the transition from left to right indicated by the arrow
of the upper line in Figure~\ref{box}.
In the analogous transition on the other hand, when the truncated BS
amplitudes are employed, the possible presence of singularities in the
propagators of legs has to be taken into account explicitly. In the present
example, these are the simple particle poles of the propagators of the
constituent quark and diquark that might be encircled by the closed path in
the $k_0$--integration. The corresponding residues have to be included in the
transition to the Euclidean metric in this case, which is indicated in the
lower line of Fig.~\ref{box}.
The conclusion is therefore that the naive relation between BS amplitudes and
wave functions can not be maintained in the Chebyshev expansion of the
Euclidean spherical momentum coordinates when singularities are encountered
in the truncation of the legs. Resorting to the Min\-kows\-ki space
definitions of BS amplitudes vs. wave functions, however, unique results are
obtained from either employing the domain of holomorphy of the BS wave
functions in the continuation to the Euclidean metric (with complex momenta)
or, alternatively and technically more involved, from keeping track of the
singularities that can occur in the Wick rotation when the truncated
amplitudes and explicit constituent propagators are employed.
The rest of this section concerns the description of how to account for
these singularities which, for our present calculations employing
constituent poles for quark and diquark, give rise to residue
terms as indicated in the lower right corner of Fig. \ref{box}.
To this end consider the quark contribution to the matrix elements of the
electromagnetic current which, from Eq.~(\ref{Eq:6.10}) with Eq.~(\ref{jq}),
is given by
\begin{eqnarray} \label{quark}
\langle \widehat J^\mu_{q} \rangle &=& q_q \, \fourint{k} \\
&& \hskip -.1cm
\widetilde{\bar\psi}(-k-\hat\eta Q,P')
\, D(k_s)S(p_{q}) \Gamma^\mu_{quark} S(k_{q}) \widetilde\psi(k,P) \, .\nn
\end{eqnarray}
We are interested in the location of the propagator poles herein.
For these poles, solving the corresponding quadratic equation for the zeroth
component of the relative momentum $k^0$, from Table~\ref{mom_table} and
Eqs.~(\ref{BF_def}), yields
\begin{eqnarray}
k_q^2-m_q^2-i\epsilon=0 \;\; \Leftrightarrow && \\
&&\hskip -2cm k^0_{pole,1} \, =\,
-\eta \, \omega_Q \, \pm \, W(m_{q},(\vec k -\eta/2 \,\vec
Q)^2 )
\nn\\
p_q^2-m_q^2-i\epsilon=0 \; \; \Leftrightarrow && \label{k2} \\
&&\hskip -2cm k^0_{pole,2} \,=\, -\eta\, \omega_Q
\, \pm\, W(m_{q}, (\vec k -(\eta/2-1) \vec Q)^2) \nn\\
k_s^2-m_s^2-i\epsilon=0 \; \; \Leftrightarrow && \label{k3} \\
&&\hskip -2cm k^0_{pole,3} \, = \,
\hat\eta\, \omega_Q \, \pm\, W(m_s,(\vec k+ \hat\eta/2 \, \vec Q)^2)
\nn
\end{eqnarray}
with $W(m,\vec k^2) = \sqrt{\vec k^2+m^2-i\epsilon}$ (and $\eta+\hat\eta =
1$).
\begin{figure}
\vspace{.6cm}
\hskip .2cm \epsfig{file=plot/residue.eps,width=0.95\linewidth}
\vspace{.1cm}
\caption{Location of the relevant singularities in the impulse-approximate
quark contribution to the form factors.
The relative momentum $k$ is the integration variable in the loop diagram
corresponding to Eq. (\ref{quark}).}
\label{cuts}
\end{figure}
For $Q=0$, {\it i.e.}, in the rest frame of the nucleon in which the
Bethe-Salpeter equation was solved, the naive Wick rotation is justified
for $1- \frac{m_{s}}{M} < \eta < \frac{m_q}{M}$, since there always is a
finite gap between the cuts contained in the hypersurface Re$k^0$=0 of the
Re$k^0$ -- Im$k^0$ -- $\vec k$ space.
As $Q$ increases, these cuts begin are shifted along both,
the $k^3$ and the $k^0$-axis, as sketched in Figure \ref{cuts}. This
eventually amounts to the effect that one of the two cuts arising from
each propagator crosses the Im $k^0$-axis. As indicated in the figure,
Wick rotation $k^0 \rightarrow ik^4$ is no longer possible for arbitrary
values of $k^3$ without encircling singularities. The corresponding residues
thus lead to
\begin{eqnarray} \label{residueint}
\langle\widehat J^\mu_{q}\rangle&\rightarrow& q_q \, \int_{E} \,
\frac{d^4k}{(2\pi)^4} \\
&& \hskip +.2cm
\widetilde{\bar\psi}(-k-\hat\eta Q,P')
\, D(k_{s})S(p_{q}) \Gamma^\mu_{q} S(k_{q})
\widetilde\psi(k,P)\nonumber \\
&& \hskip -.9cm + i \int\frac{d^3\vec k}{(2\pi)^3} \; \theta_{\vec k} \,
\widetilde{\bar\psi}(-k^4_{pole,1},-\vec k-\hat\eta \vec Q,P')
D(k_s) S(p_{q}) \times \nonumber \\
&& \hskip -.1cm \Gamma^\mu_{q}
\text{Res}(S(k_{q})) \psi(k^4_{pole,1},\vec k,P) \nn \\[2pt]
&& + \quad\text{analogous terms for $S(p_{q})$ and $D(k_s)$} \nonumber
\end{eqnarray}
upon transforming Eq.~(\ref{quark}) to the Euclidean metric.
Here, the residue integral is evaluated
at the position of the pole in the incoming quark propagator $S(k_q)$
on the Euclidean $k^4$-axis
\begin{eqnarray}
k_{pole,1}^4= -i\eta \sqrt{M^2+Q^2/4}+iW(m_q,(\vec k-\eta/2\, \vec Q)^2) \, ,
\nn \end{eqnarray}
where $\text{Res}(S(k_{q}))$ denotes the corresponding residue, and the
abbreviation
\begin{eqnarray}
\theta_{\vec k} \equiv \theta\left(\eta \,\omega_Q -
W(m_q,(\vec k- \eta/2 \,
\vec Q )^2)\right) \nn
\end{eqnarray}
was adopted to determine the integration domain for which the encircled
singularities of Figure \ref{cuts} contribute.
Analogous integrals over the spatial components of the relative momentum
$\vec k$ arise from the residues corresponding to the poles in the outgoing
quark propagator $S(p_q)$ and the diquark propagator $D(k_s)$ as given in
Eqs. (\ref{k2}) and (\ref{k3}).
\begin{figure}[t]
\vspace{.6cm}
\centerline{\epsfig{file=plot/vw.eps,width=2\figwidth}}
\caption{The impulse-approxiamte contribution, corresponding to the diagrams
of Fig.~\protect\ref{IAD}, to the electric form factor of the proton
(employing the dipole diquark amplitude). Results of the BS wave function
calculations for the $\eta$-values $1/3$ and $0.4$ are compared to the
respective amplitude plus residue calculations. The results are normalized to
the latter with $\eta$=1/3.}\label{vw}
\end{figure}
One verifies that these cuts (as represented by the shaded areas
in Fig.~\ref{cuts}) never overlap. Pinching of the deformed contour does not
occur, since there are no anomalous thresholds for spacelike momentum
transfer $Q^2$ in these diagrams.
In Figure \ref{vw} we compare results for the electric form factor of the
proton employing the nucleon BS amplitudes, together with the procedure to
account for the singularities as outlined above,
with the corresponding wave function calculations. The $n=2$ dipole diquark
amplitude of Sec.~\ref{QDBSE} is employed herein once more.
For $\eta = 1/3$ the Chebyshev expansion of the BS wave function to 9 orders
still turns out insufficient to provide for stable numerical results. This is
due to being too close to the range in $\eta$ that requires proper treatment of
the diquark pole contribution to the nucleon BSE which, with the present
value of\\\vfill\pagebreak\noindent
the mass $m_s/M=0.7$ is the case for $\eta \le 1 - m_s/M = 0.3$.
The considerably weaker suppression of higher orders in the Chebyshev
expansion of the BS wave function as compared to the expansion of the
BS amplitude enhances the residual $\eta$-dependence of the observables
obtained from the former expansion at a given order, in particular, when it
has to reproduce close-by pole contributions in the constituent propagators.
The impulse-approximate contributions to $G_E$ deviate substantially from
those employing the BS amplitude and residue calculations in this case.
On the other hand, for values of the momentum partitioning variable which are
a little larger than $1/3 $ such as $\eta=0.4$ used in the other results of
Fig.~\ref{vw}, unique results are obtained from both procedures.
Both, the BS wave function and amplitude calculation are in perfect
agreement for values of the momentum partitioning that are closer to
the middle of the range allowed to $\eta$.
For the seagull and exchange quark contributions corresponding to the
diagrams of Fig.~\ref{momrout} an analogous analysis of the singularity
structure is considerably more complicated. The explicit inclusion of their
contributions which allowed the calculation based on the BS amplitude
expansion also of these diagrams is numerically too involved. For these
contributions to the form factors we have to resort to the BS wave function
calculations. Unlike the impulse-approximate contributions we find, however,
that the deviations in the results for the exchange quark and seagull
contributions for $\eta= 1/3$ and $\eta = 0.4$ are smaller than the numerical
accuracy of the calculations and thus negligible. We attribute this to the
fact that the exchange quark and seagull contributions to the form
factors tend to fall off considerably faster with increasing $Q^2$ than those
of the impulse approximation. This can be seen in Figure~\ref{gedipole}.
Small residual $\eta$-dependences are observed for the momentum
transfers above 3GeV$^2$ also in the otherwise stable calculations. These give
rise to deviations in the results for the form factors of at most 4\% which
decrease rapidly and become negligible at lower $Q^2$.
\end{appendix}
\section{Conventions for Diquark Amplitudes}
\label{AppDqDetails}
The BS wave functions $\chi(p,P)$ and $\bar\chi(p,P)$ of the (scalar) diquark
bound-state are defined by the matrix elements,
\begin{eqnarray}
\chi_{\alpha\beta}(x,y;\vec P) &:=& \langle q_\alpha(x) q_\beta(y) |P_+
\rangle \\
\bar \chi_{\alpha\beta}(x,y;\vec P) &:=& \langle P_+ | \bar q_\alpha(x) \bar
q_\beta(y) \rangle \; , \nn \\
&=& \left(\gamma_0 \chi^\dagger(y,x;\vec P) \gamma_0 \right)_{\alpha\beta}
\label{conj_amp}\end{eqnarray}
Note that there is no need for time ordering here in contrast to
quark-antiquark bound states. The following normalization of the states is
used,
\begin{eqnarray}
\langle P'_\pm | P_\pm \rangle \, = \, 2 \omega_P \, (2\pi)^3 \delta^3 (\vec P'
- \vec P) \, , \; \omega_P^2 = \vec P^2 + m_s^2 \; ,
\end{eqnarray}
and the charge conjugate bound state being $ |P_- \rangle \, =\, C | P_+
\rangle $. The contribution of the charge conjugate bound state is included in
Eq.~(\ref{dq_pole_ms}) for $P_0 = - \omega_P $.
From invariance under space-time translations, the BS wave function has
the general form,
\begin{eqnarray}
\chi_{\alpha\beta}(x,y;\vec P) = e^{-iPX} \int
\frac{d^4p}{(2\pi)^4} \, e^{-ip(x-y)} \chi_{\alpha\beta}(p,P) \, ,
\end{eqnarray}
with $X = (1\!-\!\sigma) x + \sigma y $, $p := \sigma p_\alpha -
(1\!-\!\sigma) p_\beta $, and $ \, P = p_\alpha + p_\beta $, where
$p_\alpha$, $p_\beta$ denote the momenta of the outgoing quarks in the
Fourier transform $ \chi_{\alpha\beta}(p_\alpha ,p_\beta;\vec P) $ of $
\chi_{\alpha\beta}(x,y;\vec P) $, see Fig.~\ref{dqamps}. One thus has the
relation,
\begin{eqnarray}
\chi_{\alpha\beta}(p,P) := \chi_{\alpha\beta}(p+ (1\!-\!\sigma) P
,-p+\sigma P ;\vec P)\big\vert_{P_0 = \omega_{\scriptscriptstyle P}} \; .
\end{eqnarray}
In the definition of the conjugate amplitude, the convention
\begin{eqnarray}
\bar\chi_{\alpha\beta}(x,y;\vec P) \, = \, e^{iP\bar X} \int
\frac{d^4p}{(2\pi)^4} \, e^{-ip(x-y)} \, \bar\chi_{\alpha\beta}(p,P) \; ,
\end{eqnarray}
with $\bar X = \sigma x + (1\!-\!\sigma) y $, ensures that hermitian
conjugation from Eq.~(\ref{conj_amp}) yields,
\begin{eqnarray}
\bar \chi_{\alpha\beta} (p,P)\, = \, \left( \gamma_0 \chi^\dagger(p,P)
\gamma_0 \right)_{\alpha\beta} \; . \label{hcra}
\end{eqnarray}
In the conjugate amplitude $\bar\chi_{\alpha\beta}
(p,P) $, the definition of relative and total momenta corresponds to $p =
(1\!-\!\sigma) p'_\alpha - \sigma p'_\beta $ and $P = - p'_\alpha -
p'_\beta $ for the outgoing quark momenta $p'_\alpha , p'_\beta$ in
\begin{eqnarray}
\bar\chi_{\alpha\beta}(p'_\alpha ,p'_\beta ;\vec P) \, =\,
\left(\gamma_0 \chi^\dagger (-p'_\beta , -p'_\alpha ;\vec P) \gamma_0
\right)_{\alpha\beta} \; ,
\end{eqnarray}
{\it c.f.}, Fig.~\ref{dqamps}. Note here that hermitian conjugation implies
for the momenta of the two respective quark legs, $ p_\alpha \to - p_\beta'
$, and $ p_\beta \to - p_\alpha' $, which is equivalent
to $ \sigma \leftrightarrow (1\!-\!\sigma) $ and $P\to - P$. Besides the
hermitian conjugation of Eq.~(\ref{hcra}), one has from the antisymmetry of
the wave function, $\chi_{\alpha\beta}(x,y;\vec P) =
-\chi_{\beta\alpha}(y,x;\vec P)$. For the corresponding functions of the
relative coordinates/momenta, this entails that $\sigma$ and $(1\!-\!\sigma)$
have to be interchanged in exchanging the quark fields,
\begin{eqnarray}
\chi(x,P) &=& \left. - \chi^T(-x,P)\right|_{\sigma
\leftrightarrow (1-\sigma)} \; , \\
\quad \chi(p, P) &=& \left. -
\chi^T(-p,P)\right|_{\sigma \leftrightarrow (1-\sigma)} \;
. \nn
\end{eqnarray}
This interchange of the momentum partitioning can be undone by a charge
conjugation, from which the following identity is obtained,
\begin{eqnarray}
\chi^T(p,P) = - C \bar \chi(-p,-P) C^{-1} \; . \end{eqnarray}
This last identity is useful for relating $\bar\chi $ to $ \chi$ in
Euclidean space. In particular, this avoids the somewhat ambiguous definition
of the conjugation following from Eq~(\ref{conj_amp}) in Euclidean space with
complex bound-state momenta.
\input pictex/Fig2
One last definition for diquark amplitudes concerns the truncation of the
propagators $S$ attached to the quark legs, thus defining the amputated
amplitudes $\widetilde\chi$ and $\widetilde{\bar\chi}$ by
\begin{eqnarray}
\chi_{\alpha\beta}(p,P) &=& \left( S(p_\alpha) \widetilde
\chi(p,P) S^T(p_\beta)\right)_{\alpha\beta} \; , \\
\bar\chi_{\alpha\beta}(p,P) &=& \left( S^T(-p'_\alpha) \widetilde{
\bar\chi}(p,P) S(-p'_\beta)\right)_{\alpha\beta} \; . \end{eqnarray}
With the definitions above, the same relations hold for the amputated
amplitudes, in particular,
\begin{eqnarray}
\widetilde{\bar\chi}(p,P) &=& \gamma_0
\widetilde\chi^\dagger(p,P) \gamma_0 \; , \\
\widetilde\chi(p, P) &=& \left. - \widetilde\chi^T(-p,P)\right|_{\sigma
\leftrightarrow (1-\sigma)} \! . \nn
\end{eqnarray}
In Sec.~\ref{dq_corrs} the antisymmetric Green function $G^{(0)}$ for the
disconnected propagation of identical quarks with propagator $S(p)$,
\begin{eqnarray}
G^{(0)}_{\alpha\gamma , \beta\delta}(p,q,P) \, &=& \, (2\pi)^4 \delta^4(p-q)
\\
&& \hskip -2cm
S_{\alpha\beta}(\sigma P -p) \, S_{\gamma\delta}((1\!-\!\sigma)P+p)
\quad - \quad \mbox{crossed term}\, , \nn
\end{eqnarray}
was used to derive the normalization condition for the diquark
amplitudes. This notation is somewhat sloppy. In particular in
the second term proportional to $\delta^4(p+q)$, representing the crossed
propagation with exchange of the external quark lines, one may use either
$p$ or $-q$ in the arguments of the propagators. Exchanging one for the other
is possible only with, at the same time, exchanging $\sigma \leftrightarrow
1\!-\!\sigma $ as well, however. Momentum conservation entails that incoming
and outgoing quark-pairs in successive correlation functions can only be
connected if, besides the relative momenta $p = \pm q$, also the momentum
partitioning variables of the pairs match. We can include this condition
explicitly by introducing temporarily the dimensionless $\sigma' $ and
$\sigma $ (both in $[0,1]$) for the incoming and outgoing pair respectively,
thus writing,
\begin{eqnarray}
G^{(0)}_{\alpha\gamma , \beta\delta}(p,\sigma,q,\sigma',P) \, &=& \,
(2\pi)^4 \delta^4(p-q) \, \delta(\sigma - \sigma')
\label{A.15} \\
&&
S_{\alpha\beta}(\sigma P -p) \, S_{\gamma\delta}((1\!-\!\sigma)P+p) \,
\nn \\
&& \hskip -1.5cm - \, (2\pi)^4 \delta^4(p+q) \, \delta(\sigma
+\sigma' -1 ) \nn \\
&& S_{\alpha\delta}(\sigma P -p) \, S_{\gamma\beta}((1\!-\!\sigma)P+p) \,\;
. \nn
\end{eqnarray}
The inverse ${G^{(0)}}^{-1}$ can then be defined as,
\begin{eqnarray}
{G^{(0)}}^{-1}\hskip -.8cm _{\alpha\gamma , \beta\delta}(p,\sigma,q,\sigma',P)
\, &=& \frac{1}{4} \, \Big( \,
(2\pi)^4 \delta^4(p-q) \, \delta(\sigma - \sigma')
\label{A.16} \\
&&
S_{\alpha\beta}^{-1}(\sigma P -p) \, S_{\gamma\delta}^{-1}((1\!-\!\sigma)P+p)
\nn \\
&& \hskip -1.5cm - \, (2\pi)^4 \delta^4(p+q) \,
\delta(\sigma +\sigma' -1 ) \nn\\
&& S_{\alpha\delta}^{-1}(\sigma P -p) \,
S_{\gamma\beta}^{-1}((1\!-\!\sigma)P+p) \Big) \; , \nn
\end{eqnarray}
giving the exchange antisymmetric unity in the space of (identical) 2-quark
correlations upon (left or right) multiplication with $G^{(0)}$,
\begin{eqnarray}
&& \hskip -.2cm \int \frac{d^4k}{(2\pi)^4} \int_0^1 d\tilde\sigma \,
G^{(0)}_{\alpha\gamma , \rho\omega}(p,\sigma,k,\tilde\sigma,P) \,
{G^{(0)}}^{-1}\hskip -.8cm _{\rho\omega ,
\beta\delta}(k,\tilde\sigma,q,\sigma',P) = \nn \\
&& \hskip .2cm
\frac{1}{2} \bigg( \delta_{\alpha\beta}\, \delta_{\gamma\delta} \,
(2\pi)^4 \delta^4(p-q) \, \delta(\sigma - \sigma') \\
&& \hskip 2cm - \, \delta_{\alpha\delta} \,
\delta_{\gamma\beta}\, (2\pi)^4 \delta^4(p+q) \, \delta(\sigma
+\sigma' -1 ) \bigg) \; .\nn
\end{eqnarray}
This is the way the multiplication of 2-quark correlation functions
for identical quarks used in Sec.~\ref{dq_corrs} is understood properly.
Either Eq.~(\ref{A.15}) or Eq.~(\ref{A.16}) can be used in the derivation of
the normalization condition for the diquark amplitude $\chi$ for identical
quarks, Eq.~(\ref{dq_norm}).
\section{Supplements on the Nucleon BSE}
\label{SuppNBSE}
In this appendix we would first like to explore the possibility of having
exchange-symmetric arguments in the diquark amplitudes of the
quark-exchange kernel of the nucleon BSE. Consider the
invariant $x_1$ for the relative momentum in the incoming diquark in the
kernel~(\ref{x_kern_par}) corresponding to the definition of the
symmetric argument of $P(x)$ given in Eq.~(\ref{x_def}). From
Eqs.~(\ref{p_1}) and~(\ref{x_1}) one obtains (with $\sigma +\hat\sigma = 1$
and $\eta +\hat\eta = 1$ determining the momentum partitioning within the
diquark and nucleon respectively),
\begin{eqnarray}
x_1 & = & - p^2 - \sigma\hat\sigma k^2 - pk + (1\!-\!3\eta)Pp +
(2\sigma\hat\sigma\hat\eta\!-\!\eta) Pk \nn \\
&& \hskip 2cm + (\eta (1\!-\!2\eta)-\sigma\hat\sigma\hat\eta^2) P^2 \; .
\end{eqnarray}
One verifies readily that for $\sigma = (1\!-\!2\eta)/(1\!-\!\eta) $,
according to~(\ref{dq_mom_part}), the prefactor of the term $\propto P^2$
vanishes, and that of the term $\propto Pk$ becomes $\hat\sigma
(1\!-\!3\eta) $ leaving only the choice $\eta = 1/3$ for a $P$-independent
$x_1$. One might now argue that, from the antisymmetry considerations alone,
we should be free to add an arbitrary term proportional to the square of the
total diquark momentum in the definition of $x$. We will show now that such a
redefinition cannot lead to a $P$-independent $x$ either (for values of
$\eta$ different from $1/3$). Here, the diquark momentum
is $P_D = q+p_\alpha = \hat\eta P - k $ and with
\begin{eqnarray}
\hat x_1 & := & x_1 + C P_D^2 \nn \\
P_D^2 & = & \hat\eta^2 P^2 - 2\hat\eta Pk + k^2 \; ,
\end{eqnarray}
one finds that in order to have no terms $\propto P^2$ in $\hat x_1 $,
\begin{eqnarray}
C = - \hat\eta^{-2}\, (\eta (1\!-\!2\eta)-\sigma\hat\sigma\hat\eta^2) \; .
\end{eqnarray}
With this $C$, however, one has
\begin{eqnarray}
\hat x_1 = - p^2 - \eta \hat\eta^{-2}\!(1\!-\!2\eta) k^2 - pk +
(1\!-\!3\eta) ( Pp + \eta\hat\eta^{-1} Pk ) \, , \nn
\end{eqnarray}
independent of $\sigma$. This shows that the symmetric arguments of the
diquark amplitudes $\chi$ and, analogously, $\bar\chi$ can quite generally be
independent of the total nucleon momentum $P$ only for $\eta = 1/3$.
The remainder of this appendix describes the structure of the nucleon BSE for
the bound-state of scalar diquark and quark in some more detail. The form of
the bound-state pole in the scalar-fermion 4-point function,
Eq.~(\ref{nuc_pole_cont}), implies that the corresponding bound-state
amplitudes obey,
\begin{eqnarray}
\widetilde\psi(p ,P_n) \Lambda^+(P_n) &=& \widetilde\psi(p ,P_n)\; ,\\
\Lambda^+(P_n) \widetilde{\bar\psi}(p,P_n) &=&
\widetilde{\bar\psi}(p,P_n) \; , \nn
\end{eqnarray}
with $\Lambda^+(P_n) \, =\, (\fslash P_n + M_n)/2M_n $. Therefore, the
amplitudes can be decomposed as follows:
\begin{eqnarray}
\widetilde\psi(p ,P_n)\, &=& \label{psiDec.A} \\
&& \hskip -1.5cm S_1(p,P_n)\, \Lambda^+(P_n) \, +\,
S_2(p,P_n) \, \Xi(p,P_n)\, \Lambda^+(P_n) \; , \nn\\
\widetilde{\bar\psi}(p,P_n) \, &=& \label{psibarDec.A} \\
&& \hskip -1.5cm \, S_1(-p,P_n)\, \Lambda^+(P_n) \, +\,
S_2(-p,P_n)\, \Lambda^+(P_n)\, \Xi(-p,P_n) \; , \nn
\end{eqnarray}
with $\Xi(p,P_n) = (\fslash p - pP_n/M_n)/M_n $. This simply separates positive
from negative energy components of the amplitudes,
\begin{eqnarray}
&& \hskip -.2cm
\fslash P_n \, \Lambda^+(P_n) \,=\, M_n \, \Lambda^+(P_n) \; , \\
&& \hskip -.2cm
\fslash P_n \, \Xi(p,P_n)\, \Lambda^+(P_n) \,=\, - M_n \, \Xi(p,P_n)\,
\Lambda^+(P_n) \; , \nn \\
&& \hskip -.2cm
\hbox{and thus,} \quad \Lambda^+(P_n)\, \Xi(p,P_n)\,
\Lambda^+(P_n) \, = \, 0 \; . \nn
\end{eqnarray}
One furthermore has,
\begin{eqnarray}
\Lambda^+(P_n)\, \Xi(p,P_n) \Xi(p,P_n)\, \Lambda^+(P_n)
&=& \\
&& \hskip -1cm
\left(\frac{p^2}{M_n^2} - \frac{(pP_n)^2}{M_n^4} \right) \,
\Lambda^+(P) \; , \nn
\end{eqnarray}
which allows to rewrite the homogeneous BSE (\ref{hom_nuc_BSE}) in terms of
2-vectors $S^T(p,P_n) := (S_1(p,P_n),\, S_2(p,P_n))$, using the kernel
(\ref{x_kern_par}),
\begin{eqnarray}
S(p,P_n) &=& \\
&& \hskip -1.1cm
\frac{1}{2N_s^2} \int \frac{d^4k}{(2\pi)^4} \; P(x_1) P(x_2)
\, D(k_{s}) \, T(p,k,P_n) \, S(k,P_n) \; ,\nn
\end{eqnarray}
with $k_s = (1\!-\!\eta)P_n-k\, $, $\, k_q = k + \eta P_n$ and
\begin{eqnarray}
T(p,k,P_n) \, &=& \, \frac{1}{2}
\left( \begin{array}{cc}
1 & 0 \\
0 & ~~~~\left(\frac{p^2}{M_n^2} - \frac{(pP_n)^2}{M_n^4} \right)^{-1}
\end{array} \right) \times \\[+12pt]
&& \hskip -2.1cm \left( \begin{array}{l}
\hbox{tr} \left\{ S(q) S(k_q) \Lambda^+(P_n)\right\} \\
\hbox{tr} \left\{ S(q) S(k_q) \Lambda^+(P_n) \Xi(p,P_n) \right\} \end{array}
\right. \nn \\
&& \hskip -.1cm \left. \begin{array}{r}
\hbox{tr} \left\{ S(q) S(k_q) \Xi(k,P_n) \Lambda^+(P_n)\right\} \\
\hbox{tr} \left\{ S(q) S(k_q) \Xi(k,P_n) \Lambda^+(P_n) \Xi(p,P_n) \right\}
\end{array} \right) \, . \nn
\end{eqnarray}
After performing these traces
the transfer to Euclidean metric introducing 4-dimensional polar variables
is done according to the prescriptions,
\begin{eqnarray}
&&p^2,\, k^2 \, \to \, -p^2, \, -k^2 \; , \quad P_n^2 \to M_n^2 \; , \nn \\
&& pP_n \, \to \, i M_n p\, y \; , \quad kP_n \, \to \, i M_n k \, z \; ,
\label{WickRot.A} \\
&& pk \, \to \, - k \, p \, u(x,y,z) \; , \nn\\
&& u(x,y,z) \, = \, yz + x \, \sqrt{1-y^2}\sqrt{1-z^2} \; . \nn
\end{eqnarray}
In these variables, the nucleon BS-amplitudes are functions of the
modulus of the relative momentum and its azimuthal angle with the total
momentum. The matrix $T$ in the kernel, in addition to the moduli $p,k$ and
azimuthal angles $y,z$ of both relative momenta, also depends on the angle
$u$ between them,
\begin{eqnarray}
&&S(p,P_n) \, \to \, S(p,y) \; ,\nn\\
&&T(p,k,P_n) \, \to\, T(p,y,k,z,u(x,y,z))\; .
\end{eqnarray}
The azimuthal dependence of the amplitudes is taken into account by means of
a Chebyshev expansion to order $N$, see, {\it e.g.}, Ref.~\cite{Pre94},
\begin{eqnarray}
&& S(p,y) \, \simeq \, \sum_{n=0}^{N-1} (-i)^n \, S_n(p) T_n(y) \; ,
\label{ChebyS.A} \\
&& S_n(p) \, = \, i^n \, \frac{2}{N} \sum_{k=1}^N S(p,y_k) T_n(y_k) \; ,
\label{ChebyM.A} \\
&& \hskip -.2cm \hbox{where the} \;\;
y_k \, = \, \cos\left( \frac{\pi (k- 1/2)}{N} \right)
\nn \end{eqnarray}
are the zeros of the Chebyshev polynomial of degree $N$. Here,
Chebyshev polynomials of the 1st kind are used with, for later convenience, a
somewhat non-standard normalization $T_0 := 1/\sqrt{2}$. An explicit
factor $(-i)^n$ was introduced in order to obtain real Chebyshev moments
$S_n(p)$ for all $n$. Analogous formulae are obtained for expansions in
Chebyshev polynomials of the 2nd kind, which are used in
Refs.~\cite{Hel97b,Oet98}. The nucleon BSE now reads,
\begin{eqnarray}
S_m(p) &=& \\
&& \ - \frac{1}{2N_s^2} \, \int \frac{k^3 dk}{(4\pi)^2} \,
\sum_{n=0}^{N-1} \, i^{m-n} \; T_{mn}(p,k) \, S_n(k) \; , \nn
\end{eqnarray}
with
\begin{eqnarray}
T_{mn}(p,k) \, &=& \frac{2}{\pi} \int_{-1}^1 \,
\sqrt{1-z^2} dz \\
&& \hskip -1.5cm
\frac{1}{x_s + m_s^2} \; \int_{-1}^1 dx
\, \frac{2}{N} \sum_{k=1}^{N} \, \biggl(P(x_1) P(x_2)\biggr)_{y
= y_k} \nn\\
&& \hskip .8cm T(p,y_k,k,z,u(x,y_k,z)) \; T_m(y_k) T_n(z) \; , \nn
\end{eqnarray}
where $x_s = k^2 + 2i (1\!-\!\eta) M_n k z - (1\!-\!\eta)^2 M_n^2 $ is the
invariant momentum of the free scalar propagator $D$ of mass $m_s$.
\section{Introduction}
To the precision accessible to current measurements, the proton is the
only known hadron that is stable under the effect of all interactions.
Protons are thus ideal for use in beams or targets for scattering
experiments designed to explore the fundamental dynamics of the strong
interaction. From this, it follows that more is known about the proton and
its nearly stable isospin partner, the neutron, than any other hadron.
There is an abundance of observable properties of the nucleon, from
elastic scattering form factors to electromagnetic form factors and parton
distributions, which are being measured with increasing precision at
various accelerator facilities around the world.
In particular, the ratio of the electric and the magnetic form factor of the
proton is subject to current measurements at TJNAF, and current
experiments at MAMI~\cite{MAMI} and NIKHEF~\cite{NIKHEF} show great
promise that soon we will have a precise knowledge of the electromagnetic
properties of neutron as well. Hence, the development of an accurate and
tractable covariant framework for the nucleon in terms of the underlying
quarks and gluons is clearly desirable.
Models of the nucleon and other baryons are numerous and have had varying
degrees of success as they are usually designed to describe particular
properties of baryons. Some of the frameworks that have been
employed are non-rela\-tivis\-tic~\cite{Gel64,Kar68,Fai68} and
relativistic~\cite{Fey71} quark potential models, bag
models~\cite{Cho74,Has78}, skyrmion models~\cite{Sky61,Adk83,Schw89,Hol93} or
the chiral soliton of the Nambu-Jona-Lasinio (NJL)
model~\cite{Alk96a,Chr96}.
The relativistic bound state problem of 3-quark Faddeev type was studied
extensively within the NJL model~\cite{Rei90,Buc92,Ish93,Hua94,Buc95,Ish95}
and its non-local generalization, the global color model~\cite{Cah89,Bur89}.
In addition, some complementary aspects of these models have been
combined, {\it e.g.}, the chiral bag model~\cite{Cho75,Rho94} and
a hybrid model that implements the NJL-soliton picture of baryons within
the quark-diquark Bethe-\-Salpe\-ter (BS) framework \cite{Zue97}.
The present study is concerned with the further development of a description of
baryons (and the nucleon, in particular) as bound states of quarks and
gluons in a fully-covariant quantum field theoretic framework based on the
Dyson-Schwinger equations of QCD. Such a framework has already reached a high
level of sophistication for mesons.
In these studies, the quark-antiquark scattering kernel is modeled as a
confined, non-perturbative gluon exchange. Such a gluon exchange can be
provided by solutions to the Dyson-Schwinger equations for the propagators
of QCD in the covariant gauge~\cite{Sme97,Sme98}. In phenomenological studies
the gross features of such solutions are mimicked by the use of a
phenomenological quark-antiquark kernel. With this kernel, the
dressed-quark propagator is
obtained as the solution of its (quenched) Dyson-Schwinger equation (DSE),
and the same kernel is then employed in the quark-antiquark Bethe-Salpeter
equation (BSE)\footnote{
We warn the reader not to confuse the meaning of this acronym with another one
adopted recently. See, {\it e.g.}, Ref.~\cite{Lancet}.}
from which one obtains the masses of the meson bound states
and their BS amplitudes. Once the dressed-quark propagators and meson BS
amplitudes have been obtained, observables can be calculated in a
straight-forward manner. The success of this approach has been demonstrated
in many phenomenological applications, such as
electromagnetic form factors~\cite{Bur96}, decay widths,
$\pi$-$\pi$ scattering~\cite{Rob96}, vector-meson
electroproduction~\cite{Pic97}, to name a few.
Summaries of this Dyson-\-Schwinger/\-Bethe-\-Salpeter description of
mesons may be found in Refs.~\cite{Miranski,Rob94,Tan97}.
The significantly more complicated framework required for
an analogous description of baryons based on the quantum field theoretic
description of bound states in 3-quark correlations has meant that baryons
have received far less attention than mesons.
Consequently, the utility of such a description for the baryons
is considerably less understood, even on the phenomenological level.
(For example, the ramifications of various truncation schemes,
required in any quantum field theoretic treatment of bound states, which
have been explored extensively in the meson sector are completely unknown
in the baryon sector.) The developments in this direction employ a
picture based on separable 2-quark correlations, {\it i.e.}, diquarks,
interacting with the 3rd quark which allows a treatment of the
relativistic bound state in a manner analogous to the 2-body BS
problem~\cite{Kus97,Hel97b,Oet98}. Although these studies employ
simplified model assumptions for the quark propagators and diquark
correlators, they de\-mon\-strate the utility of the approach in general, and
these simplified model assumptions can be replaced by more realistic ones in
the future as more is understood about the underlying dynamics of quarks and
gluons.
In the present article we generalize this framework for the 3-quark bound
state problem in quantum field theory to accommodate the quark-substructure
of the separable diquark correlations. This necessary extension has
important implications on the electromagnetic properties of the nucleons. We
explicitly construct a conserved current for their electromagnetic couplings,
and we verify analytically that it yields the correct charges of both
nucleons.
To achieve this we employ Ward and Ward-Takahashi identities for the
quark correlations which arise from electromagnetic gauge invariance.
In order to avoid unnecessary complications,
our present study considers only simple constituent-quark and diquark
propagators. However, the generalization to include dressed
propagators, most importantly to account for confined nature of quarks and
diquarks, requires only minor modifications.
The organization of the article is as follows.
In Sec.~\ref{dq_corrs}, some general properties of the two-quark correlations
(or diquarks) are discussed. The importance of constructing
a diquark correlator that is antisymmetric under quark exchange is
emphasized. The parametrisations of the Lorentz-scalar isoscalar diquark
correlations which are employed in the numerical calculations of the subsequent
sections are introduced to reflect these properties in deliberately simple
ways. The framework is sufficiently general to accommodate the results for
the diquark correlators as they become available from studies of the underlying
dynamics of quark and gluon correlations in QCD.
Of course, diquark correlations other than the scalar diquark will be
important for a more complete description of the nucleon. Certainly necessary
is the inclusion of other channels, such as the axialvector diquark, when the
description is extended to the octet and the decuplet baryons.
Nonetheless, for the purposes of the present study, only scalar
diquarks are considered; the generalization to include other diquarks is
straight-forward~\cite{Oet98}. In Sec.~\ref{QDBSE}, the nucleon BS equation
and the quark-exchange kernel are introduced. One particular feature of this
kernel is that it necessarily depends on the total momentum of the nucleon
bound state $P_n$. This conclusion is based on general arguments, such as
the exchange-antisymmetry of the diquark correlations. It is important to
obtain the correct normalizations and charges of the nucleon bound states.
In particular, to ensure current conservation requires a considerable
extension beyond the impulse approximation in the calculations of
electromagnetic form factors for baryons.\footnote{The
$P_n$-dependence of the exchange-kernel violates one of the
necessary conditions for current conservation in the {\em impulse}
approximation. See Sec.~\ref{TECO}.} The numerical solutions for the
nucleon BS amplitudes obtained herein preserve the invariance
of observables under a re-routing of the relative momentum, a requirement
that follows from the translational invariance of the relativistic bound
state problem.
In Sec.~\ref{SecNucNorm}, the normalization condition for
the nucleon BS amplitude is derived and the nucleon electromagnetic current
is obtained in Sec.~\ref{TECO}. In Sec.~\ref{WIaS}, the ``seagull''
contributions to the electromagnetic current, necessary for current
conservation, are derived from the Ward and Ward-Takahashi
identities. In Sec.~\ref{ElmFFs}, expressions for the electromagnetic
form factors of the nucleon are derived, the numerical calculations are
described in Sec.~\ref{NC}, and the results for the form factors are
presented and discussed in Sec.~\ref{Res}. The conclusions of this study are
provided in Sec.~\ref{Conclusions}.
Several appendices have also been included with further
details which may provide the interested reader with addition insight into
the framework.
\section{Diquark Correlations}
\label{dq_corrs}
To obtain a solution of the 3-particle Faddeev equations in quantum field
theory requires a truncation of the interaction kernel.
A widely-employed truncation scheme is to neglect contributions to
the Faddeev kernel which arise from irreducible 3-quark interactions.
This allows one to rewrite the Dyson series for the 3-quark Green function
as a coupled system of equations. The first being the BS equation for the
2-quark scattering amplitude and the other being the Faddeev equation,
which describes the coupling of the 3rd quark to these 2-quark correlations.
As a result, the nature of the gluonic interactions of quarks enters only in
the BSE of the 2-quark subsystem, {\it i.e.}, in the quark-quark
interaction kernel. The solution of the full inhomogeneous BS problem for
this 2-quark system is simplified by assuming separable contributions
(explained below), hereafter referred to as diquarks, to account for the
relevant 2-quark correlations within the hadronic bound state.
Herein, a ``diquark correlation'' thus refers to the use of a separable
4-quark function in the $\bar{3}$ representation of the $SU(3)$ color group.
The utility of such diquark correlations for a description of baryon bound
states is a central element of the present approach.
While in the NJL model the 2-quark scattering amplitude has the property
of being separable, in general this is an additional assumption
useful to simplify the 3-quark bound state problem in quantum field
theory.
An example for separable contributions would be (a finite sum of)
isolated poles at timelike total momenta $P^2$ of the diquark.
Such poles allow the use of homogeneous BSEs to obtain the respective
amplitudes from the gluonic interaction kernel of the quarks, and thus to
calculate these separable contributions to the 2-quark scattering
amplitude.
However, the validity of using a diquark correlator to parametrise the
2-quark correlations phenomenologically, does not rely on the existence of
asymptotic diquark states. Rather, the diquark correlator may be devoid of
singularities for timelike momenta, which may be interpreted as one possible
realization of diquark confinement. In principle, one may appeal to models
employing a general, separable diquark correlator which need not have any
simple analytic structure, in which case no particle interpretation for the
diquark would be possible. The implementation of this model of
confined diquark is straight-forward and does not introduce
significant changes to the framework. The use of diquark correlations in this
capacity is quite general and does not necessarily imply the existence
diquarks, which have not been observed experimentally.
The absence of asymptotic-diquark states may be explained in a number
of ways. Although, it has been observed that solutions of the BSE in ladder
approximation yield asymptotic color-$\bar{3}$ diquark states
\cite{Pra89}, when terms {\em beyond ladder approximation} are maintained,
the diquarks cease to be bound~\cite{Ben96}. That is, the addition of terms
beyond the ladder approximation to the BS kernel, in a way which preserves
Goldstone's theorem at every order, has a minimal impact on
solutions for the color-singlet meson channels. In contrast, such terms have a
significant impact on the color-$\bar{3}$ diquark channels.
In Ref.~\cite{Ben96}, it was demonstrated that these
contributions to BS kernel beyond ladder approximation are predominantly
repulsive in the color-$\bar 3$ diquark channel. It was furthermore
demonstrated in a simple confining quark model that the strength of these
repulsive contributions suffices to move the diquark poles from
the timelike $P^2$-axis and far into the complex $P^2$-plane.
While the particular, confining quark model is in conflict with locality,
the same effect was later verified within the NJL model~\cite{Hel97a}.
This suggests that this mechanism for diquark confinement, which
eliminates the possibility of producing asymptotic diquark states,
might hold independent of the particular realization of quark confinement.
In a local quantum field theory, on the other hand, colored
asymptotic states do exist, for the elementary fields as well as possible
colored composites such as the diquarks, but not in the physical subspace of
the indefinite metric space of covariant gauge theories.
The analytic structure of correlation functions, the holomorphic envelope of
extended permuted tubes in coordinate space, is much the same in this
description as in quantum field theory with a positive definite inner product
(Hilbert) space. In particular, 2-point correlations in momentum space
are generally analytic functions in the cut complex $P^2$-plane with the cut
along the timelike real axis. Confinement is interpreted as the
observation that both absorptive as well as anomalous thresholds in hadronic
amplitudes are due only to other hadronic states~\cite{Oeh95}. However, the
implementation of this algebraic notion of confinement
seems much harder to realize in phenomenological applications.
For the present purposes of developing a general framework for the
description of baryon bound states, the question as to whether one should
or should not model the diquark correlations in terms of functions with or
without singularities for total timelike momentum $P^2$ is irrelevant.
However, we reiterate that the present framework is able to accommodate
both of these descriptions of confinement with only straight-forward
modifications.
The goal of the present study is to assess the utility of
describing baryons as bound states of quark and diquark correlations, in a
framework in which the diquark correlations are assumed to be
{\em separable}. (The term separable refers to the property that a 4-point
Green function $G(p,q;P)$ be independent of the scalar $qp$, where $q$
and $p$ are the relative momenta of the two incoming and outgoing particles,
respectively, and $P$ is the total momentum.)
To provide a simple demonstration of the general formalism, and its
application to the calculation of nucleon form factors, we assume that the
diquark correlator corresponds to a single scalar-diquark pole at $P^2 =
m_s^2$ which is both separable and constituent-like. In this description, the
(color-singlet) baryon thus is a bound state of a color-$3$ quark and a
color-$\bar{3}$ (scalar) diquark correlation.
\input pictex/Fig1
Assuming identical quarks, consider the 4-point quark Green function in
coordinate space given by
\begin{eqnarray}
G_{\alpha\beta\gamma\delta}(x_1,x_2,x_3,x_4) &=& \label{Gen4q-G} \\
&& \hskip -1cm \langle
T(q_\gamma(x_3) q_\alpha(x_1) \bar q_\beta(x_2) \bar q_\delta(x_4)) \rangle
\; , \nn
\end{eqnarray}
where $\alpha, \beta, \gamma$, and $\delta$ denote the Dirac indices of
the quarks and $T$ denotes time-ordering of the quark fields
$q_{\alpha}(x)$.
The assumption of a separable diquark correlator,
corresponding to the diagram shown in Fig.~\ref{dqpole}, can be written in
momentum space as
$G^{\hbox{\tiny sep}}_{\alpha \beta \gamma \delta}(p,q,P)$ where
\begin{eqnarray}
G_{\alpha\gamma , \beta\delta}^{\hbox{\tiny sep}}(p,q,P) \, &:=&
e^{-iPY} \, \int d^4\!X\, d^4\!y\, d^4\!z \,\, e^{iqz}
e^{-ipy} \nn \\
&& \hskip 1.5cm e^{iPX} G_{\alpha\beta\gamma\delta}^{\hbox{\tiny sep}} (x_1,x_2,x_3,x_4) \nn
\\
&=& D(P)
\; \chi_{\gamma\alpha}(p,P) \bar \chi_{\beta\delta}(q,P) \;
\; , \label{dq_pole_ms}
\end{eqnarray}
where $D(P)$ is the diquark propagator,
$X = \sigma x_1 + (1\!-\!\sigma) x_3 $, $ Y =
(1\!-\!\sigma') x_2 + \sigma' x_4 $, the total momentum partitioning of
the outgoing and incoming quark pairs are given respectively
by $ \sigma $ and $ \sigma' $ both in $ [0,1]$, and $y = x_1 - x_3$, $z = x_2
- x_4$.
As described above, the separable form of the diquark correlation of
Eq.~(\ref{dq_pole_ms}) does not necessarily entail the existence of an
asymptotic diquark state.
The framework developed herein makes no restrictions on the particular
choice of the diquark propagator $D(P)$. Technically, model
correlations which mimic confinement through the absence of timelike poles
are easy to implement as shown in Ref.~\cite{Hel97b}.
Nonetheless, for the purpose of demonstration, here we employ the simplest
form for this propagator corresponding to a simple (scalar) diquark bound
state in the 2-quark Green function $G$ of Eq.~(\ref{Gen4q-G}).
The appearance of such an asymptotic diquark state
requires the diquark propagator to have a simple pole at $P^2 = m_s^2$,
where $m_s$ is the mass of the diquark bound state, {\it i.e.},
\begin{eqnarray}
D(P) = \frac{i}{P^2 - m_s^2 + i \epsilon} \; . \label{pole_dqcorr}
\end{eqnarray}
Then $\chi(p,P)$ and its adjoint $\bar\chi(p,P) = \gamma_0
\chi^\dagger(p,P) \gamma_0 $ are the BS wave functions of the (scalar) diquark
bound state which are defined as the matrix elements of two quark fields or two
antiquark fields between the bound state and the vacuum,
respectively. Further details concerning these definitions are given in
Appendix~\ref{AppDqDetails}.
It is convenient to define the {\em truncated} diquark BS amplitudes
(sometimes referred to as BS vertex functions) $\widetilde\chi$ and
$\widetilde{\bar\chi}$ from the BS wave functions in
Eq.~(\ref{dq_pole_ms}) by amputating the external quark propagators,
\begin{eqnarray}
\widetilde\chi (p,P) &:=& S^{-1}((1\!-\!\sigma)P+p)
\; \chi(p,P) \; {S^{-1}}^T\!(\sigma P-p) \, , \\
\widetilde{\bar\chi} (p,P) &:=& {S^{-1}}^T\!(\sigma P-p)
\; \bar\chi (p,P) \; S^{-1}((1\!-\!\sigma) P+p) \, .
\end{eqnarray}
The convention employed here is to use the same symbols for both the BS
wave functions and the truncated BS amplitudes, with the latter denoted by
a tilde.
An important observation made in Appendix~\ref{AppDqDetails}, which is of
use in the following discussions, is that the for identical quarks,
antisymmetry under quark exchange constrains the diquark BS
amplitudes $\widetilde\chi$ to satisfy:
\begin{eqnarray}
\widetilde\chi(p, P) \, = \, \left. - \widetilde\chi^T(-p,P)\right|_{\sigma
\leftrightarrow (1-\sigma)} \! .
\label{dq_asym}
\end{eqnarray}
Note that $\sigma $ and $(1\!-\!\sigma)$ have been interchanged here.
In principle, at this point one could go ahead and specify the form of the
kernel for the quark-quark BSE in ladder approximation, and obtain diquark BS
amplitudes in much the same manner in which solutions for the mesons are
obtained.
However, as discussed above, the appearance of stable bound state solutions
might be an artifact of the ladder approximation rather than
the true nature of the quark-quark scattering amplitude.
Therefore, for our present purposes various simple model
parametrisations for diquark BS amplitudes are explored, rather than
using a particular solution of the diquark homogeneous BSE.
The motivation is to explore the general aspects and implications of using
a separable diquark correlation for the description of the nucleon bound
state.
For separable contributions of the pole type~(\ref{pole_dqcorr}), but
possibly complex mass (with Re$(m_s^2) > 0$), one readily obtains standard
BS normalization conditions to fix the overall strength of the quark-quark
coupling to the diquark for a given parametrisation of the diquark
structure. These are obtained from the inhomogeneous quark-quark BSE for the
Green function of Eq.~(\ref{Gen4q-G}) employing pole dominance for $P^2$
sufficiently close to $m_s^2$.\footnote{For other separable contributions,
{\it e.g.}, of the form of non-trivial entire functions for which one
necessarily has a singularity at $|P^2| \to \infty $, the
use of the inhomogeneous BSE to derive normalization conditions
relies on the existence of full solutions to (\protect\ref{dqIBSE}) of the
separable type.} To sketch their derivation consider the
inhomogeneous BSE which is of the general form,
\begin{eqnarray}
G(p,q,P) \, = \, \left( {G^{(0)}}^{-1} (p,q,P) - K(p,q,P) \right)^{-1} \; .
\label{dqIBSE}
\end{eqnarray}
Here $G^{(0)}$ denotes the antisymmetric Green function for the
disconnected propagation of two identical quarks. The definition of its
inverse ${G^{(0)}}^{-1}$ and a brief discussion of how to derive it, may
be found in Appendix~\ref{AppDqDetails}.
With the simplifying assumption that the quark-quark interaction kernel
$K$ be independent on the total diquark momentum $P$; that is,
$K(p,q,P) \equiv K(p,q)$ (which is satisfied in the ladder approximation for
example), the derivative of $G$ with respect to the total momentum $P$, gives
the relation
\begin{eqnarray}
&& - P^\mu\frac{\partial}{\partial P^\mu} \, G(p,q,P) \, = \,
\int \frac{d^4k}{(2\pi)^4}\frac{d^4k'}{(2\pi)^4} \\
&& \hskip .6cm
G(p,k,P) \left(
P^\mu\frac{\partial}{\partial P^\mu} \, {G^{(0)}}^{-1}\!(k,k',P) \right)
G(k',q,P) \nn \; .
\end{eqnarray}
Upon substitution of $G(p,q,P)$ as given by
Eqs.~(\ref{dq_pole_ms}) and (\ref{pole_dqcorr}), and equating the residues of
the most singular terms, one obtains the non-linear constraint for the
normalization of the diquark BS amplitudes $\chi$ and $\bar\chi$,
\begin{eqnarray}
1 & \, \stackrel{!}{=} \, & \frac{-i}{4 m_s^2} \int
\frac{d^4p}{(2\pi)^4} \label{dq_norm} \\
&& \left\{
\hbox{tr} \left( S^T(p_\beta) \widetilde{\bar\chi}(p,P) \left(
P\frac{\partial}{\partial P} S(p_\alpha)\right) \widetilde\chi(p,P) \right)
\right. \nn\\
&& \hskip .5cm + \, \left.
\hbox{tr} \left(\widetilde{\bar\chi}(p,P) S(p_\alpha) \widetilde\chi(p,P)
\left( P\frac{\partial}{\partial P} S^T(p_\beta)\right) \right)
\right\} \; , \nn
\end{eqnarray}
where $p_\alpha = p + (1\!-\!\sigma)P$ and $ p_\beta = -p + \sigma P$, {\it
i.e.}, $P= p_\alpha + p_\beta$ and $p = \sigma p_\alpha - (1\!-\!\sigma)
p_\beta $.
The scalar diquark contribution, relevant to the present study of
the nucleon bound state, is color-$\bar{3}$ and isosinglet.
Lorentz covariance requires its Dirac structure to be the sum of four
independent contributions, each proportional to a function of two
independent momenta.
For simplicity only a single term is maintained here, which has the
following structure:
\begin{eqnarray}
\widetilde\chi(p,P) = \gamma_5C \, \frac{1}{N_s}
\tilde P(p^2,pP) \; ,
\label{sdq_amp_def}
\end{eqnarray}
where $C$ is the charge conjugation matrix ($C = i\gamma_2 \gamma_0$ in the
standard representation).
The normalization constant $N_s$ is fixed from the condition given in
Eq.~(\ref{dq_norm}) for a given form of $\tilde P(p^2,pP)$.
It may seem reasonable to neglect the dependence of this invariant
function on the scalar $pP$;
such a simplification would yield the leading moment of an
expansion of the angular dependence in terms of orthogonal polynomials,
which has been shown to provide the dominant contribution to the meson
bound state amplitudes in many circumstances.
However, in the present case the antisymmetry of the diquark
wave function, {\it c.f.}, Eqs. (\ref{dq_asym}), entails that
\begin{eqnarray}
\tilde P(p^2, pP) \, =\, \left.
\tilde P(p^2, - pP) \right|_{\sigma
\leftrightarrow (1-\sigma)} \; . \label{sym_P}
\end{eqnarray}
For $ \sigma \not= 1/2 $ and thus for $\bar p := p
\big|_{\sigma \leftrightarrow (1-\sigma)} \not= p$, it is not
possible to neglect the $pP$ dependence in the amplitude
without violating the quark-exchange antisymmetry.
To maintain the correct quark-exchange antisymmetry, we assume
instead that the amplitude depends on both scalars, $p^2$ and $pP$
in a specific way. In particular, we assume the diquark BS amplitude is
given by a function that depends on the scalar
\begin{eqnarray}
x &:=& p_\alpha p_\beta - \sigma (1-\sigma) \, m_s^2
= - (1 - 2\sigma) pP - p^2 \nn \\
&=& (1 - 2\sigma) \bar pP - \bar p^2 \label{x_def}
\end{eqnarray}
with $\bar p = (1-\sigma) p_\alpha - \sigma p_\beta $ and
$p_{\{\alpha , \beta\}}$ as given above.
For equal momentum partitioning between the quarks in the
diquark correlation $\sigma = 1/2$, the scalar $x$ reduces to the negative
square of the relative momentum, $x = -p^2$.
The two scalars that may be constructed from the available momenta $p$ and
$P$ (noting that $P^2 = m^2_{s}$ is fixed) which have definite symmetries
under quark exchange are given by the two independent combinations
$p_\alpha p_\beta$ (which is essentially the same as above $x$) and
$p_\alpha^2 - p_\beta^2 $.
The latter may only appear in odd powers which are associated with
higher moments of the BS amplitude. Hence, these are neglected by setting
\begin{eqnarray}
\tilde P(p^2, pP) \equiv P(x)
\end{eqnarray}
which can be shown to satisfy the antisymmetry constraint given by
Eq.~(\ref{sym_P}) $\forall \sigma \in [0,1]$.
Finally, the diquark BS normalization $N_s $, as obtained from
Eq.~(\ref{dq_norm}), is given by
\begin{eqnarray}
N_s^2 \, &=& \, \frac{-i}{4 m_s^2} \int
\frac{d^4p}{(2\pi)^4} \; P^2(x)
\label{Ns} \\
&& \hskip 1cm P\frac{\partial}{\partial P} \;
\hbox{tr} \big[ S(p+(1\!-\!\sigma)P) S(-p+\sigma P) \big] \; . \nn
\end{eqnarray}
The numerical results presented in the following sections explore the
ramifications of several Ans{\"a}tze for $P(x)$,
\begin{eqnarray}
P_{n\mbox{\tiny-P}}(x) = \left(\frac{\gamma_n}{x + \gamma_n} \right)^n
\hskip -.1cm \mbox{or} \;
P_{\hbox{\tiny EXP}}(x) = \exp \, \{ - x/\gamma_{\hbox{\tiny EXP}} \}
, \label{P_amps}
\end{eqnarray}
where the integer $n= 1,2 ...$ and corresponds to monopole,
dipole,... diquark BS amplitudes.
Their widths $\gamma_n$, $\gamma_{\hbox{\tiny EXP}}$ are determined
from the nucleon BSE by varying them until the diquark normalization given
by Eq.~(\ref{Ns}) and coupling strength $g_s^2$ necessary to produce the
correct nucleon bound-state mass are equal.
This is carried out numerically and described in detail in Sec.~\ref{QDBSE}.
For completeness, various Gaussian forms that peak for values of $x = x_0
\ge 0 $,
\begin{eqnarray}
P_{\hbox{\tiny GAU}}(x) \, = \, \exp \, \{ -
(x-x_0)^2 /\gamma_{\hbox{\tiny GAU}}^2 \} \;
, \label{P_gau}
\end{eqnarray}
are also explored.
Such forms with finite $x_0$ were suggested for diquark amplitudes as
a result of a variational calculation of an approximate diquark BSE in
Ref.~\cite{Pra88} and have been used in the nucleon calculations of
Ref.~\cite{Bur89}.
Therein, a fit to the Gaussian form given by Ref.~\cite{Pra88}
(and Eq.~(\ref{P_gau}) above) was employed with a width of
$\gamma_{\hbox{\tiny GAU}} \simeq 0.11$GeV$^2$ and
$x_0/\gamma_{\hbox{\tiny GAU}}\simeq 1.7$.
From a calculation within the present framework, which is described in
Sec.~\ref{QDBSE}, we observe that the necessary value for
$\gamma_{\hbox{\tiny GAU}}$ to obtain a reasonable nucleon mass is about
an order of magnitude smaller than the value given in Ref.~\cite{Bur89}.
Furthermore in Sec.~\ref{ElmFFs}, we observe that the effect of a finite
$x_0$ on the electric form factor of the neutron rules out the use of a
Gaussian form with $x_0 \not = 0$.
Some of the parametrisations used for diquark correlations in previous
quark-diquark model studies of the nucleon~\cite{Kus97,Hel97b,Oet98},
correspond to neglecting the substructure of diquarks entirely;
this may be reproduced in the present framework by setting
$\tilde P(p^2,pP)=1$.
By neglecting the diquark substructure, the diquark BS normalizations such as
$N_s$ here are not well-defined. The strengths of the
quark-diquark couplings are undetermined and chosen as free
adjustable parameters of the model (one for each diquark channel
maintained). In the present, more general approach,
these couplings are determined by the normalizations of the respective
diquark amplitudes. At present this is the scalar diquark normalization
$N_s$ alone which in turn determines the coupling strength $g_s=1/N_s$
between the quark and diquark which binds the nucleon.
The aim of our present study is to generalize the notion of diquark
correlations by going beyond the use of a point-like diquark and include a
diquark substructure in a form similar to that of mesons.
In the most general calculation of the three-body bound state,
the strength of the quark-diquark coupling is not at one's disposal; it
arises from the elementary interactions between the quarks.
By using the diquark BS normalization condition of Eq.~(\ref{Ns}), we
can assess whether the quark-diquark picture of the nucleon is still
able to provide a reasonable description of the nucleon bound state if the
coupling strength is obtained from the diquark BSE rather than forcing the
quark and diquark to bind by adjusting their couplings freely.
Whether the quark-diquark coupling is sufficiently strong to produce bound
baryons will eventually be determined by the strength of the
quark-quark interaction kernel that leads to the diquark BS amplitude.
Finally, the use of a diquark BS amplitude with a {\em finite} width improves
on the previous calculations in yet another respect.
Without the substructure of the diquark, an additional ultraviolet
regularization had to be introduced in the exchange kernel of the BSE for the
nucleon in Refs.~\cite{Kus97,Hel97b,Oet98}.
In the present study, the finite-sized substructure of the diquark
leads to a nucleon BSE which is completely regular in the ultraviolet
in a natural way.
In this section, we have discussed some general features of the
diquark amplitude employed in our present study. The important observations
are the implications of its antisymmetry under quark exchange which
constrains the functional dependence on the quark momenta, and the
derivation of the BS normalization condition, Eq.~(\ref{Ns}), to fix the
quark-diquark coupling strength.
By taking these into account, we will find that the precise form of
the Ansatz for the diquark BS amplitude $P(x)$ has little qualitative
influence on the resulting nucleon amplitudes as long as $P(x)$ falls off
by at least one power of $x$ for large $x$ corresponding to a large spacelike
relative momentum.
\section{The Quark-Diquark Bethe-Salpeter Equation of the Nucleon}
\label{QDBSE}
By neglecting irreducible 3-quark interactions in the
kernel of the Faddeev equation giving the 6-point Green function that
describes the fully-interacting propagation of 3 quarks, the Dyson series
for this 6-point Green function reduces to a coupled set of two-body
Bethe-Salpeter equations, see for example, Ref.~\cite{Hua94}.
As discussed in the previous section, for the purpose of demonstrating the
framework developed herein, we choose to employ constituent quark and
diquark propagators. However, the framework itself is much more general.
The assumptions under which it is developed require only that
the irreducible 3-quark interactions are neglected in the nucleon Faddeev
kernel and that the diquark correlations be well-parametrised by a sum of
separable terms (which may or may not have poles associated with asymptotic
diquark states).
Maintaining only the (flavor-singlet, color-$\bar 3$) scalar diquark
channel in these correlations, corresponding to Eqs. (\ref{dq_pole_ms})
and (\ref{pole_dqcorr}), the appearance of a stable nucleon bound state
coincides with the development of a pole in the Green function
$G_{\alpha\beta}$ describing the fully-interacting (spin-${1}/{2}$) quark
and (scalar) diquark correlations which is of the form,
\begin{eqnarray}
G^{\hbox{\tiny pole}}_{\alpha\beta} (p,k,P_n) &=& \label{nuc_pole_cont} \\
&& \left(\psi
(p,P_n) \frac{ i ( \fslash P_n + M_n ) }{P_n^2 - M_n^2 + i \epsilon}
\bar\psi(-k,P_n) \right)_{\alpha\beta} . \nn
\end{eqnarray}
Here $k$ and $p$ are the relative momenta between the quark and diquark,
incoming and outgoing, respectively, $P_n$ is the four-momentum of
the nucleon with mass $M_n$, $\alpha$ and $\beta$ denote the Dirac indices
of the quark, and the nucleon BS wave function $\psi(p,P_n)$ is related to
it's adjoint according to
\begin{eqnarray}
\bar \psi(p,P_n) \, = \, \gamma_0 \psi^\dagger(-p,P_n) \gamma_0
\, = \, C \psi^T(p,-P_n) C^{-1} \, .
\end{eqnarray}
The truncated nucleon BS amplitude $\widetilde\psi$ is defined as
\begin{eqnarray}
\psi(p,P_n) = D((1\!-\!\eta)P_n - p) S(\eta P_n + p) \widetilde\psi(p,P_n)
\; ,
\end{eqnarray}
where $\eta \in [0,1]$ is the fraction of the nucleon momentum $P_n$
carried by the quark.
The resulting homogeneous Bethe-Salpeter equation for the nucleon bound
state reads,
\begin{eqnarray}
\widetilde\psi_{\alpha\beta}(p,P_n) = \int \frac{d^4k}{(2\pi)^4} \;
K_{\alpha\gamma}(p,k;P_n) \, \psi_{\gamma\beta}(k,P_n)
\; , \label{hom_nuc_BSE}
\end{eqnarray}
where $K_{\alpha\beta}(p,k;P_n)$ is the kernel of the nucleon BSE
which represents the exchange of one of the quarks within the diquark
with the spectator quark. Maintaining the quark-exchange antisymmetry
of the diquark correlations, this kernel provides the full antisymmetry
within the nucleon ({\it i.e.}, Pauli's principle) in the quark-diquark
model \cite{Rei90}.
\input{pictex/Fig3}
The exchange kernel is shown in Fig.~\ref{kernel} and given by
\begin{eqnarray}
K_{\alpha\beta}(p,k;P_n)\, &=& \label{xker} \\
&& \hskip -1cm \,- \frac{1}{2} \,
\widetilde \chi_{\alpha\gamma}(p_1,q+p_\alpha)
\, S^T_{\gamma\delta}(q) \,
\widetilde{\bar\chi}^T_{\delta\beta}(p_2,q+p_\beta) \nn \\
&=& \, \frac{1}{2 N_s^2}\, P(x_1) P(x_2) \;
S_{\alpha\beta}(q) \; , \label{x_kern_par}
\end{eqnarray}
where the factor -1/2 arises from the flavor coupling between the quark and
diquark and $p_1$ and $p_2$ are the relative momenta of the quarks within
the incoming and outgoing diquark, respectively, such that
\begin{eqnarray}
p_1 = \sigma p_\alpha - (1\!-\!\sigma) q \; ,
\quad \hbox{and} \quad p_2 = (1\!-\!\sigma') q - \sigma' p_\beta \; .
\end{eqnarray}
The respective momentum partitionings are $\sigma$ and $\sigma'$
and need not be equal.
The total momenta of the incoming and the outgoing diquark are $q+p_\alpha$
and $q+p_\beta$.
The momentum $q$ of the exchanged quark
is expressed in terms of the total nucleon momentum $P_n$ and relative
momenta $k$ and $p$ as
\begin{eqnarray}
q \, =\, ( 1\!-\!2\eta) P_n - p -k \; . \label{ex_qk_mom}
\end{eqnarray}
Using the definitions of $q$ and the quark momenta $p_\alpha = \eta P_n
+p$ and $ p_\beta = \eta P_n + k$, the relative momenta in the diquark
BS amplitudes can be expressed as
\begin{eqnarray}
p_1 \, & =&\, (\sigma \eta - (1\!-\!\sigma)(1\!-\!2\eta)) P_n +
(1\!-\!\sigma) k + p
\; , \label{p_1} \\
p_2 \, & =&\, ((1\!-\!\sigma') (1\!-\!2\eta) - \sigma'\eta) P_n - k -
(1\!-\!\sigma')p
\; .
\end{eqnarray}
The corresponding arguments of the quark-exchange-anti\-sym\-metric diquark
BS amplitudes follow readily from their definition in Eq.~(\ref{x_def}),
\begin{eqnarray}
x_1 \, &=& \, - p_1^2 - (1\!-\!2\sigma) ((1\!-\!\eta) p_1 P_n
- p_1 k)
\; , \label{x_1}\\
x_2 \, &=& \, - p_2^2 + (1\!-\!2\sigma') ((1\!-\!\eta) p_2 P_n
- p_2 p)
\; . \label{x_2}
\end{eqnarray}
Before proceeding, it is worth summarizing some of the important aspects
of this framework:
\begin{enumerate}
\item The dependence of the exchange kernel on the total
nucleon bound-state momentum $P_n$ is {\em crucial} in order to obtain the
correct relationship between the electromagnetic charges of the proton and
neutron bound states and the normalizations of their BS amplitudes.
\item The momentum $q$ of the exchanged quark is {\em independent} of the
nucleon momentum $P_n$ {\em only for} the particular value of momentum
partitioning $\eta = 1/2$.
\item The relative momenta $p_1$ and $p_2$ of the quarks within the diquarks
are only independent of the total momentum $P_n$ of the nucleon for the
particular choice:
\begin{eqnarray}
\sigma = \sigma' = \frac{1-2\eta}{1-\eta} \; . \label{dq_mom_part}
\end{eqnarray}
The {\em symmetric} arguments of $x_{1}$ and $x_2$ in the diquark BS
amplitudes are independent of the total momentum $P_n$ only if, in
addition to the above criteria, $\sigma = \sigma' = 1/2$ as well, and
hence $\eta = 1/3$.
In fact, this conclusion can be further generalized:
Any exchange-symmetric argument of the diquark amplitude can differ
from the definition of Eq.~(\ref{x_def}) only by a term that is
proportional to the square of the total diquark momentum.
From this, it is possible to show that the diquark BS amplitudes can be
independent of $P_n$ only if $\eta = 1/3$. This is shown in
Appendix~\ref{SuppNBSE}.
\end{enumerate}
Unlike the ladder-approximate kernels commonly employed in
phenomenological studies of the meson BSE, the quark-exchange kernels
of the BSEs for baryon bound states, such as the one depicted in
Fig.~\ref{kernel}, {\em necessarily} depend on the total momentum of the
baryon $P_n$ for {\em all} values of $\eta \in [0,1]$.
This has important implications on the normalization of the bound-state BS
amplitudes as well as on the calculation of the electromagnetic charges of
the bound states. In particular, considerable extensions of the framework
beyond the impulse approximation are required in the calculation of
electromagnetic form factors in order to ensure that the electromagnetic
current is conserved for baryons. This issue is explored in detail in
Sec.~\ref{TECO}.
The form of the pole contribution arising from the nucleon bound state to
the quark-diquark 4-point correlation function in
Eq.~(\ref{nuc_pole_cont}) constrains the nucleon BS amplitude to obey the
identities:
\begin{eqnarray}
\widetilde\psi(p ,P_n) \Lambda^+(P_n) &=& \widetilde\psi(p ,P_n)\; , \\
\Lambda^+(P_n) \widetilde{\bar\psi}(p,P_n) &=&
\widetilde{\bar\psi}(p,P_n) \; ,
\end{eqnarray}
where $\Lambda^+(P_n) \, =\, (\fslash P_n + M_n)/2M_n $.
It follows from this that the most general Lorentz-covariant
form of the nucleon BS amplitude can be parametrised by
\begin{eqnarray}
\widetilde\psi(p ,P_n)\, &=& \label{psiDec} \\
&& \hskip -1.5cm S_1(p,P_n)\, \Lambda^+(P_n) \, +\,
S_2(p,P_n) \, \Xi(p,P_n)\, \Lambda^+(P_n) \; , \nn \\
\widetilde{\bar\psi}(p,P_n) \, &=& \label{psibarDec} \\
&&\hskip -1.5cm S_1(-p,P_n)\, \Lambda^+(P_n) \, +\,
S_2(-p,P_n)\, \Lambda^+(P_n)\, \Xi(-p,P_n) \; , \nn
\end{eqnarray}
where $\Xi(p,P_n) = (\fslash p - p P_n/M_n)/M_n $, and $S_1(p,P_n)$ and
$S_2(p,P_n)$ are Lorentz-invariant functions of $p$ and $P_n$.
This provides a separation of the positive and negative energy
components of the nucleon BS amplitude.
In Appendix~\ref{SuppNBSE}, some of the consequences of this decomposition
are explored. In particular, it allows one to rewrite the
homogeneous BSE of Eq.~(\ref{hom_nuc_BSE}) in a compact manner, in terms
of a 2-vector $S^T(p,P_n) := (S_1(p,P_n),\, S_2(p,P_n))$ as
\begin{eqnarray}
S(p,P_n) &=& \label{EucSBSE} \\
&& \hskip -1.2cm \frac{1}{2N_s^2} \int \frac{d^4k}{(2\pi)^4} \; P(x_1) P(x_2)
\, D(k_{s}) \, T(p,k,P_n) \, S(k,P_n) \; , \nn
\end{eqnarray}
where $k_s = (1\!-\!\eta)P_n-k\, $ and $T(p,k,P_n)$
is a $2\times 2$ matrix in which each of the four elements is
given by a trace over the Dirac indices of the quark
propagator $S(k_q)$ (with $k_q = \eta P_n + k$),
the propagator $S(q)$ of the exchanged quark and
a particular combination of Dirac structures derived from the
decomposition of the nucleon amplitude in Eq.~(\ref{psiDec}), see
Appendix~\ref{SuppNBSE}.
\begin{figure}[t]
\leftline{\hskip -.1cm
\epsfig{file=plot/S1.eps,width=0.49\linewidth}
\hskip .2cm \epsfig{file=plot/S2.eps,width=0.49\linewidth} }
\caption{The first 5 moments of the nucleon BS amplitudes $S_1$ (left)
and $S_2$ (right).}
\label{S1_S2}
\end{figure}
Upon carrying out these traces, the resulting BSE is
transformed into the Euclidean metric by introducing 4-dimensional polar
coordinates corresponding to the rest frame of the nucleon
according to the following prescriptions (where ``$\to $'' denotes the formal
transition from the Minkowski to the Euclidean metric):
\begin{eqnarray}
&&p^2 \, \to \, -p^2\; , \quad P_n^2 \to M_n^2 \; , \quad
p P_n \, \to \, i M_n p\, y \; ,\label{WickRot} \\
&& \mbox{and} \quad S(p,P_n) \, \to \, S(p,y) \; . \nn
\end{eqnarray}
Written in terms of these variables, the nucleon BS amplitude is a
function of the square of the relative momentum $p^2$ and the cosine
$y \in [-1,+1]$ of the azimuthal angle between the four-vectors $p$ and
$P_n$. The dependence of the nucleon BS amplitudes $S(p,y)$ on the
angular variable $y$ is approximated by an expansion to the order $N$ in
terms of the complete set of Chebyshev polynomials $T_n(y)$,
\begin{eqnarray}
S(p,y) \, &\simeq& \, \sum_{n=0}^{N-1} (-i)^n \, S_n(p) T_n(y)
\label{ChebyS} \\
S_n(p) &=& i^n \, \frac{2}{N} \sum_{k=1}^N S(p,y_k) T_n(y_k) \; ,
\label{ChebyM} \\
\hbox{where} \; y_k &=& \cos\left( \frac{\pi (k- 1/2)}{N} \right)
\nn \end{eqnarray}
are the zeros of the Chebyshev polynomial $T_N(y)$ of degree $N$ in $y$.
Here, we employ Chebyshev polynomials of the first kind with a
convenient (albeit non-standard) normalization for the zeroth Chebyshev
moment from setting $T_0 = 1/\sqrt{2}$.
The explicit factor of $(-i)^n$ is introduced into Eq.~(\ref{ChebyS}) to
ensure that the moments $S_n(p)$ are real functions of positive $p \equiv
\sqrt{p^2}$ for all $n$.
The BSE for these moments of the nucleon BS amplitude is then
\begin{eqnarray}
S_m(p) &=& \label{Eq:38}\\
&& \, - \frac{1}{2N_s^2} \, \int \frac{k^3 dk}{(4\pi)^2} \,
\sum_{n=0}^{N-1} \, i^{m-n} \; T_{mn}(p,k) \, S_n(k) \; , \nn
\end{eqnarray}
where the ($2N\times 2N$) matrix $T_{mn}(p,k)$ is obtained from $T(p,k,P_n)$
by expanding in terms of Chebyshev polynomials both amplitudes $S$ in the
nucleon BSE of Eq.~(\ref{EucSBSE}); that is, summing over the
$y_k$ on both sides of Eq.~(\ref{EucSBSE}) according to
Eq.~(\ref{ChebyM}) and using Eq.~(\ref{ChebyS}) for the BS amplitude on the
right-hand side.
The definition of the matrix $T_{mn}$ appearing in Eq.~(\ref{Eq:38})
furthermore includes the explicit
diquark propagator and BS amplitudes $P(x_1)$ and $P(x_2)$
of the integrand in Eq.~(\ref{EucSBSE}) as well as the integrations over all
angular variables. Its exact form is provided in Appendix~\ref{SuppNBSE}.
\begin{figure}[t]
\centerline{\epsfig{file=plot/eta.eps,width=0.6\linewidth}}
\caption{Dependence of $\lambda/N_s^2$ on
$\eta $ for fixed $\sigma = \sigma' = 1/2$ (solid) and with $\sigma $,
$\sigma'$ from Eq.~(\protect\ref{dq_mom_part}) (dashed). The width
$\gamma_2$ is fixed to yield $\lambda/N_s^2 = 1 $ at $\eta = 1/3$.}
\label{eta_fig}
\end{figure}
In the calculations presented herein, we restrict ourselves to the use of
free propagators for constituent quark and diquark, with masses
$m_q$ and $m_s$ respectively, as the simplest model to parametrise
the quark and diquark correlations within the nucleon. Measuring
all momenta in units of the nucleon bound-state mass $M_n$
leaves $m_q/M_n$ and $m_s/M_n$ as the only free parameters.
Using for the scalar diquark amplitude $P(x)$ the forms of
Eqs.~(\ref{P_amps}) or (\ref{P_gau}) with fixed widths $\gamma$, the
homogeneous BSE for the nucleon in Eq.~(\ref{hom_nuc_BSE}) is converted
into an eigenvalue equation of the form,
\begin{eqnarray}
\lambda \,\widetilde\psi(p,P_n) \, =\, \int
\frac{d^4k}{(2\pi)^4} \; N_s^2 K(p,k,P_n) \, \psi(p,P_n) \; ,
\end{eqnarray}
with the additional constraint that $\lambda = N_s^2$ (note that $N_s^2 K$ is
independent of $N_s$). This equation is solved iteratively and the largest
eigenvalue $\lambda$ is found which corresponds to the nucleon ground-state.
To implement the constraint, we calculate the diquark normalization $N_s^2$
from Eq.~(\ref{Ns}) and compare it to the eigenvalue $\lambda$. This
procedure is repeated with a new value for the width $\gamma$ of the diquark
BS amplitude until the eigenvalue obtained from the BSE agrees with the
normalization integral in Eq.~(\ref{Ns}); that is, until the constraint
$\lambda = N_s^2$ is satisfied.
A typical solution of the nucleon BSE is shown in Fig.~\ref{S1_S2}.
Five orders were retained in the Chebyshev expansion.
The figure depicts the relative importance of the first four Chebyshev
moments of the nucleon amplitudes $S_1(p,P_n)$ and $ S_2(p,P_n)$.
The respective results for their fifth moments are too small ($\le 10^{-4}$)
to be distinguished from zero on the scale of Fig.~\ref{S1_S2}.
This provides an indication for the high accuracy obtained from the Chebyshev
expansion to this order. This particular solution, which will be shown to
provide a good description of the electric form factors in Sec.~\ref{ElmFFs},
was obtained for the dipole form of the diquark amplitude, {\it i.e.}, from
Eq.~(\ref{P_amps}) with $n=2$, using $m_q = 0.62 M_n$ and $m_s = 0.7 M_n$
(with $\sigma = \sigma' = 1/2 $, $\eta = 1/3$).
The value for the corresponding diquark width $\gamma_2$, necessary to yield
$\lambda = N_s^2 $ (to an accuracy of $10^{-3}$), resulted thereby to be
$\gamma_2 = (0.294\, M_n)^2$.
The dependence of the BS eigenvalue on the momentum partitioning parameter
$\eta$ is shown in Fig.~\ref{eta_fig}. The complex domains of the constituent
quark and diquark momentum variables, $k_q^2$ and
$k_s^2$ respectively, that are sampled by the integration over the relative
momentum $k$ in the nucleon BSE are devoid of poles for
$ 1 - m_s/M_n < \eta < m_q /M_n$. The pole in the momentum $q^2$ of the
exchanged quark of the kernel in Eq.~(\ref{x_kern_par}) is avoided by
imposing the further bound $\eta > (1- m_q/M_n)/2$ on the momentum partitioning
parameter. Therefore, with the present choice of constituent masses it is for
values of $\eta$ in the range $ 0.3 < \eta < 0.6$, for which our numerical
methods can yield stable results.
In principle, the integrations necessary to solve the nucleon BSE should
always be real and never lead to an imaginary result (since $m_q + m_s
>M_n$). More refined numerical methods would be necessary, however,
when integrations were to be performed in presence of
the poles in the constituent propagators that occur within the integration
domain for values of $\eta$ outside the above limits.
\begin{figure}[t]
\leftline{\hskip -.1cm
\epsfig{file=plot/S11b.eps,width=0.49\linewidth}
\hskip .2cm \epsfig{file=plot/S21b.eps,width=0.49\linewidth} }
\caption{The leading moments of the nucleon BS amplitudes $S_1$ (left)
and $S_2$ (right) for the diquark amplitudes
(\protect\ref{P_amps},\protect\ref{P_gau}) with $m_q$ adjusted to yield
$S_{1,0}(p)\vert_{p = 0.2M_n} = 1/2$.}
\label{S1b_S2b}
\end{figure}
The momentum partitioning within the nucleon $\eta$ is not an observable.
Hence, for any value of $\eta$ for which our numerical results are stable,
we expect the eigenvalue of the nucleon BSE $\lambda$ (which implicitly
determines the nucleon mass) to be independent of $\eta$.
In Fig.~\ref{eta_fig}, the dependence on $\eta$ of the nucleon BSE
eigenvalue $\lambda$ is explored using a fixed value for $\sigma = \sigma'
= 1/2$ (solid curve) and using the values of $\sigma$ and $\sigma'$ from
Eq.~(\protect\ref{dq_mom_part}) (dashed curve).
In the first case, the arguments of the diquark amplitudes simplify to
$x_i = - p_i^2$ with $p_1 = -(1\!-\!3\eta)P_n/2 + p + k/2$
and $p_2 = (1\!-\!3\eta)P_n/2 - k - p/2 $.
This implies that the real parts of $x_i$ are guaranteed to be positive only
for $\eta = 1/3$. For values of $\eta\not=1/3$, a negative contribution
arises from the timelike nucleon bound-state momentum $P_n$.
If the $n$-pole forms for the diquark amplitudes are employed, this results
in the appearance of artificial poles whenever $(1\!-\!3\eta)^2 M_n^2/4 \ge
\gamma_n $. The value of $\gamma_2 = (0.294\, M_n)^2$ as it results
here, would entail the appearance of a pole for $\eta \ge 0.53$. This
explains why our numerical procedure becomes unstable as $\eta$ approaches the
value $1/2$ in this case.
No such timelike contribution to the $x_i$ arises in the second
case shown in Fig.~\ref{eta_fig}.
Here, for $\sigma = \sigma' = (1\!-\!2\eta)/(1\!-\!\eta)$
the relative momenta $p_i$ are independent of $P_n$, and there are no terms
$\propto M_n^2 $ in $x_i$; see, for example, Eqs.~(\ref{x_1},\ref{x_2}).
However, as $\eta \to 1/2$, we find that $\sigma , \sigma' \to 0$ and the
diquark normalization integrals are then affected by singularities.
\begin{table}[b]
\begin{center}
\begin{tabular}{lcccc}
\noalign{\leftline{Fixed width $S_1(p)\vert_{p = 0.2M_n} = 1/2$,
see Fig.~\protect\ref{S1b_S2b}:}}
$P(x)$ & $m_s\; [M_n]$ & $m_q\; [M_n]$ & $\sqrt{\gamma} \; [M_n]$ &
$ g_s = 1/N_s $ \\ \hline
$n= 1$ & 0.7 & 0.685 & 0.162 & 117.1\\[-2pt]
$n= 2$ & 0.7 & 0.620 & 0.294 & 91.80\\[-2pt]
$n= 4$ & 0.7 & 0.605 & 0.458 & 85.47\\[-2pt]
$n= 6$ & 0.7 & 0.600 & 0.574 & 84.37\\[-2pt]
$n= 8$ & 0.7 & 0.598 & 0.671 & 83.76\\[-2pt]
EXP & 0.7 & 0.593 & 0.246 & 82.16\\[-2pt]
GAU & 0.7 & 0.572 & 0.238 & 71.47\\
\noalign{\leftline{Fixed masses, see Fig.~\ref{S1a_S2a}:}}
$n= 1$ & 0.7 & 0.62 & 0.113 & 155.8\\[-2pt]
$n= 2$ & 0.7 & 0.62 & 0.294 & 91.80\\[-2pt]
$n= 4$ & 0.7 & 0.62 & 0.495 & 81.08\\[-2pt]
$n= 6$ & 0.7 & 0.62 & 0.637 & 78.61\\[-2pt]
EXP & 0.7 & 0.62 & 0.283 & 74.71
\end{tabular}
\end{center}
\caption{Summary of parameters used for the various diquark amplitudes
$P(x)$, {\it c.f.},
Eqs.~(\protect\ref{P_amps},\protect\ref{P_gau}). \label{FWHM_table}}
\end{table}
In the restricted range allowed to $\eta$, the results for the nucleon BSE
obtained herein are found to be independent of $\eta$ to very good
accuracy when the equal momentum partitioning between the quark in the
diquarks is used, $\sigma = \sigma' = 1/2 $.
In this case, the diquark normalization in Eq.~(\ref{Ns}) yields a fixed
value for $N^2_s$ and any $\eta$ dependence of the ratio
$\lambda/N_s^2$ has to arise entirely from the nucleon BSE. The $\eta$
independence of $\lambda$ thus demonstrates that the solutions to the nucleon
BSE are under good control.
The more considerable $\eta$-dependence observed for $\sigma = \sigma' =
(1\!-\!2\eta)/(1\!-\!\eta)$ arises from the dependence of $N_s^2$ on
$\sigma$, $\sigma'$. Here, the limitations of the model assumptions
for the {\em diquark} BS amplitudes are manifest. The dependence of the
diquark BS amplitudes on the diquark bound-state mass, higher Chebyshev
moments (which have been neglected) and the Lorentz structures
(which we have neglected), are all responsible for this observed $\sigma$
dependence. Such a dependence would not occur, had the diquark
amplitudes employed herein been calculated from the diquark BSE.
For the numerical calculations of electromagnetic form factors presented
in Sec.~\ref{TECO}, we therefore choose to restrict the model to
$\sigma = \sigma' = 1/2 $ and vary $\eta$.
From the observed independence of the nucleon BSE solutions to $\eta$
(when $\sigma = \sigma' = 1/2$), one expects to find that calculated
nucleon observables, such as the electromagnetic form factors, will display
a similar independence of $\eta$ as well.
\begin{figure}[t]
\leftline{\hskip -.1cm
\epsfig{file=plot/S11a.eps,width=0.49\linewidth}
\hskip .2cm \epsfig{file=plot/S21a.eps,width=0.49\linewidth} }
\caption{The moments $S_{1,0}$ (left) and $S_{2,0}$ (right) of the nucleon
amplitudes for the diquark amplitudes of
Eqs.~(\protect\ref{P_amps},\protect\ref{P_gau})
with fixed masses, $m_s = 0.7 M_n$, $m_q = 0.62 M_n$.}
\label{S1a_S2a}
\end{figure}
In Figure~\ref{S1b_S2b} the zeroth Chebyshev moments $S_{1,0}(p)$
and $S_{2,0}(p)$ obtained from a numerical solution of the nucleon BSE
with $\sigma = \sigma' =1/2 $ and $\eta = 1/3$ are shown, for various
diquark amplitudes of the forms given in Eqs.~(\ref{P_amps}) and
Eq.~(\ref{P_gau}) with $x_0 =0$.
To provide a comparison between these different forms of diquark BS
amplitude, we have chosen to keep the diquark mass fixed and vary the
quark mass until a solution of the nucleon BSE was found with the property
that
\begin{eqnarray} S_{1,0}(p)\vert_{p = 0.2M_n}
\stackrel{!}{=} 1/2 \; .\label{FWHMC}
\end{eqnarray}
The resulting values for $m_q$ along with the corresponding diquark widths
and couplings $g_s \equiv 1/N_s$ are summarized in Table~\ref{FWHM_table}.
We observe that the mass of the quark required to satisfy the condition in
Eq.~(\ref{FWHMC}) tends to smaller values for
higher $n$-pole diquark amplitudes.
On the other hand, for fixed constituent quark (and diquark) mass
the higher $n$-pole diquark amplitudes lead to wider nucleon amplitudes.
This is shown in Fig.~\ref{S1a_S2a} and the corresponding diquark widths
and couplings are given in the lower part of Table~\ref{FWHM_table}.
The mass values hereby correspond to the results shown in the previous
Figure~\ref{S1b_S2b} for $n= 2$.
The qualitative effect of shifting the maximum in the Gaussian forms in
Eq.~(\ref{P_gau}) for the diquark amplitudes by a finite amount $x_0$, is
shown in Fig.~\ref{S1a_S1gau}.
The curve for $x_0 = 0$ resembles the Gaussian result given in
Fig.~\ref{S1b_S2b} with masses $m_s = 0.7 M_n$ and $m_q = 0.572 M_n$,
which are kept fixed in the results for finite shifts $x_0$.
The corresponding widths and normalizations of these forms for the diquark
amplitudes are given in Table~\ref{Sgau_table}.
While the widths $\gamma$ and the couplings $g_s$ are not free parameters
in our approach, the additional parameter
$x_0/\gamma_{\hbox{\tiny GAU}}$ introduced into the form of the diquark BS
amplitude is free to vary.
In contrast to the Gaussian form of the model diquark BS amplitude
employed in Ref.~\cite{Bur89} with $x_0/\gamma_{\hbox{\tiny GAU}} =
0.19/0.11 \simeq 1.73$, by implementing the diquark normalization condition,
we find that the width must be about 30 times smaller to provide a sufficient
interaction strength $g_s = 1/N_s^2$ necessary to bind the nucleon.
To provide for a closer comparison with the results of Ref.~\cite{Bur89},
we have also solved the nucleon BSE using the values of parameters
employed in that study. That is, we have solved the nucleon BSE using
$P_{\hbox{\tiny GAU}}(x)$ with the parameter $x_0/\gamma_{\hbox{\tiny
GAU}} = 0.19/0.11$ and using the constituent quark and diquark masses
$m_q = 0.555 M_n$ and $m_s = 0.7 M_n$, respectively.
(Note that we use the nucleon mass as the intrinsic momentum scale of our
framework, whereas the momentum scale employed in Ref.~\cite{Bur89} was
1~GeV.)
The value employed in Ref.~\cite{Bur89} for the diquark mass
$m_s = $ 0.568~GeV implies that $M_n =$ 811~MeV in order to compare to
our calculations (with $m_s = 0.7 M_n$), and our value for the quark mass
($m_q = 0.555 M_n$) thus corresponds to $m_q = $ 0.45~GeV.
With these para\-meters, we obtain $\gamma_{\hbox{\tiny GAU}} = (0.216 \,
M_n)^2 =$ $ 0.308 \, 10^{-2} \mbox{GeV}^2$,
which may be compared to $\gamma_{\hbox{\tiny GAU}}
= (0.409\, M_n)^2 = 0.11\, \mbox{GeV}^2$ used in Ref.~\cite{Bur89}.
\begin{figure}[t]
\leftline{\hskip -.1cm
\epsfig{file=plot/S1_gau.eps,width=0.49\linewidth}
\hskip .2cm \epsfig{file=plot/vgl.eps,width=0.49\linewidth} }
\caption{The moment $S_{1,0}$ for the Gaussian diquark amplitudes of
Eq.~(\protect\ref{P_gau}) with various shifts $x_0/\gamma_{\mbox{\tiny GAU}}$
(left), see Table~\protect\ref{Sgau_table}, and a comparison (right) of our
result for $x_0/\gamma_{\mbox{\tiny GAU}} = 19/11$ (with $m_s = 0.7 M_n$,
$m_q = 0.555 M_n$) with a result we obtain from using
$\gamma_{\mbox{\tiny GAU}} = 0.11 $GeV$^2$ as in Ref.~\protect\cite{Bur89}.}
\label{S1a_S1gau}
\end{figure}
\begin{wraptable}[15]{l}{6cm}
\begin{tabular}{ccc}
\noalign{\leftline{$m_s = 0.7 M_n$, $m_q = 0.572 M_n$:}}
$x_0/\gamma_{\hbox{\tiny GAU}} $ & $\sqrt{\gamma_{\hbox{\tiny GAU}}}\; [M_n]$
& $ g_s = 1/N_s $ \\ \hline
0 & 0.238 & 71.47\\[-2pt]
1/4 & 0.209 & 69.09\\[-2pt]
1/2 & 0.181 & 72.64\\[-2pt]
3/4 & 0.154 & 81.96\\[-2pt]
1 & 0.130 & 97.75\\[-2pt]
5/4 & 0.109 & 121.3\\[-2pt]
3/2 & 0.091 & 154.3\\[-2pt]
7/4 & 0.077 & 198.5
\end{tabular}
\caption{Width and normalizations of Gaussian amplitudes,
Eq.~(\protect\ref{P_gau}) for various shifts $x_0$, see
Fig.~\protect\ref{S1a_S1gau} (right). \label{Sgau_table}}
\end{wraptable}
\noindent
The same results for the nucleon ampli\-tu\-des as shown above are used for the
calculations of the electromagnetic form factors of the proton and the
neutron in Sec.~\ref{ElmFFs}.
We will show that when $n$-pole diquark BS amplitudes with $n=2$ and $n=4$
are employed, one obtains results for the electromagnetic form factors
that are in good agreement with the phenomenological dipole fit for the
electric proton form factor, while higher powers or exponential forms tend
to produce form factors which decrease too fast with increasing momentum
transfer $Q^2$.
We also show that use of a Gaussian form for the diquark BS amplitude
with $x_0 \not = 0$ leads to neutron electric form factor
that has a {\em qualitatively} different behavior than that obtained by
using the other model forms of diquark BS amplitudes.
Furthermore, this form leads to nodes in the neutron electric form factor
for small $Q^2$, a feature for which there is {\em no} experimental
evidence.
\section{Normalization of Nucleon Bethe-Salpeter Amplitude}
\label{SecNucNorm}
In order to reproduce the correct electromagnetic charges of the physical
asymptotic states, it is required that their Bethe-Salpeter (or Faddeev)
amplitudes be normalized according to the normalization conditions obtained
from the fully-interacting Green functions of the elementary fields.
These normalization conditions are
derived from {\em inhomogeneous} BS (or Faddeev) equations.
In the present framework, the quark-diquark 4-point Green function
is given by (suppressing Dirac indices)
\begin{eqnarray}
G(p,q,P) \, = \, \left( {G^{(0)}}^{-1} (p,q,P) - K(p,q,P) \right)^{-1} \; .
\end{eqnarray}
Here, $K(p,q,P)$ is the quark-exchange kernel of Eq.~(\ref{x_kern_par})
depicted in Fig.~\ref{kernel}, and $G^{(0)}(p,q,P)$ is the disconnected
contribution to this Green function, given by
\begin{eqnarray}
G^{(0)}_{\alpha\beta}(p,k,P) \, := \, (2\pi)^4 \delta^4(p-k) \,
S_{\alpha\beta}(k_q) \, D(k_s) \delta_{\eta \eta'} \; ,
\end{eqnarray}
with $k_q =\eta P + k$ and $k_s = (1\!-\!\eta)P - k$.
Taking the derivative of $G(p,k,P)$ with respect to the total momentum $P$
and then examining the leading contributions in the limit that
$P^2 \rightarrow M_n^2$ which arise from the nucleon pole contribution
from Eq.~(\ref{nuc_pole_cont}), one finds
\begin{eqnarray}
M_n \, \Lambda^+(P_n) &\stackrel{!}{=}&
\; i \int \frac{d^4p}{(2\pi)^4}
\frac{d^4k}{(2\pi)^4} \label{gen_nuc_norm} \\
&& \hskip -1.2cm \bar\psi(-p,P_n) \left(
P_\mu\frac{\partial}{\partial P_\mu} G^{-1}(p,k,P) \right)_{P = P_n}
\! \psi(k,P_n) \; . \nn
\end{eqnarray}
The most important difference in normalizing the amplitudes in the
Bethe-Salpeter framework developed here and a genuine {two-body}
BSE, is the dependence of the BS kernel on the total momentum of the
bound-state $P$. In phenomenological studies of 2-body bound states
BS kernels, such as ladder-approximate ones for example, are most commonly
employed in a form which does not depend on the bound-state momentum.
However, in the present approach, the original 3-body nature of the
nucleon bound state {\em requires} the exchange kernel of the reduced
BS equation for quark and diquark to depend on the total momentum $P_n$ of
the nucleon.
In particular, when $\eta \not= 1/2$, the propagator for the
exchanged quark in the kernel depends on the total momentum $P_n$ and for
$\eta \not= 1/3$, the diquark BS amplitudes depend on $P_n$.
This added complication, which is easily
avoided by ladder-approximate studies of meson bound states is unavoidable
for studies of baryons.
Thus, the normalization condition for the nucleon will always contain
contributions from the derivative of the kernel.
Following the above procedure, one obtains a normalization condition of
the form
\begin{eqnarray}
1 \, &\stackrel{!}{=}& \label{nuc_norm} \\
&& \hskip -.1cm \eta N_q + (1-\eta) N_D + (1-2\eta) N_X + (1-3\eta)
N_P \; , \nn
\end{eqnarray}
where
\begin{eqnarray}
N_q \, &=& \, - \frac{P^\mu}{2M_n} i \int \frac{d^4k}{(2\pi)^4}
\; D(k_s)
\label{NqDef} \\
&&
\hskip 2cm
\hbox{tr} \bigg[ \widetilde{\bar\psi}(-k,P) \left(\frac{\partial}{\partial
k_q^\mu} S(k_q) \right) \widetilde \psi(k,P) \bigg] \; ,\nn \\
N_D \, &=& \, - \frac{P^\mu}{2M_n} i \int \frac{d^4k}{(2\pi)^4}
\; \left( \frac{\partial}{\partial k_s^\mu} D(k_s) \right)
\label{NDDef} \\
&&\hskip 3cm
\hbox{tr} \bigg[ \widetilde{\bar\psi}(-k,P) S(k_q) \widetilde
\psi(k,P)\bigg] \; ,\nn \\
N_X \, & = & \, - \frac{P^\mu}{2M_n} i \int
\frac{d^4p}{(2\pi)^4}\frac{d^4k}{(2\pi)^4}
\; \frac{1}{2N_s^2} \, P(-p^2_1)
P(-p^2_2)
\label{NXDef} \\
&& \hskip 2cm
\hbox{tr}\bigg[ \bar\psi(-p,P) \left(\frac{\partial}{\partial
q^\mu} S(q) \right) \psi(k,P) \bigg] \; , \nn \\
N_P \, & = & \, - \frac{P^\mu}{2M_n} i \int
\frac{d^4p}{(2\pi)^4}\frac{d^4k}{(2\pi)^4}\;
\frac{1}{2N_s^2}
\label{NPDef} \\
&&
\hskip .5cm \bigg( \, p_{1\mu} \, P'(-p^2_1) P(-p^2_2)
\, - \, p_{2\mu} \, P(-p^2_1) P'(-p^2_2) \, \bigg) \nn\\
&& \hskip 3cm
\hbox{tr}\bigg[ \bar\psi(-p,P) S(q) \psi(k,P) \bigg] \; , \nn
\end{eqnarray}
with $ p_1 = -(1\!-\!3\eta) P/2 + p + k/2 \, , \;
p_2 = (1\!-\!3\eta) P/2 - p/2 - k \, , \; \mbox{for} \; \sigma\! =\!\sigma'
\!=\!1/2 $.
It is clear from Eq.~(\ref{nuc_norm}) that either the exchanged quark or the
presence of a diquark substructure, or in general both, provide
non-vanishing contributions to the nucleon BS normalization.
Maintaining these terms is critical for the correct
determination of the electromagnetic charges of baryons since the nucleon
normalization and electromagnetic form factors are intimately related by
the differential Ward identities.
In the following section, we show that in calculations of the electromagnetic
form factors of the nucleon, use of the usual impulse
approximation~\cite{Man55} (which includes only contributions arising from
the coupling of the photon to the quark and diquark, that are related to
the terms $N_q$ and $N_D$ in the normalization condition), by itself is
insufficient to
guarantee electromagnetic current and charge conservation of the nucleon.
One of the additional contributions required for the proper calculation of
electromagnetic form factors that goes beyond the usual
impulse approximation is the coupling of the photon to the exchanged-quark
in the kernel from Eq.~(\ref{xker}).
We will show that this term provides a crucial
contribution to the nucleon electromagnetic form factors and helps maintain
electromagnetic current conservation of the nucleon.
It is interesting to note that the contribution of this term is important
even in the special case that $\eta = 1/2$ in which the term $N_X$ does not
contribute to the normalization condition of Eq.~(\ref{nuc_norm}).
We will show in the next section that in the presence of a diquark with
a non-trivial substructure, additional, direct couplings of the photon to
this substructure are required to maintain current conservation.
Similar contributions have previously been found important
in several other contexts \cite{Oth89,Wan96}. These couplings are commonly
referred to as ``sea\-gulls''.
We conclude this section by summarizing that the respective
contributions to the nucleon normalization condition of Eq.~(\ref{nuc_norm})
given in Eqs. (\ref{NqDef}) to (\ref{NDDef}) correspond to the usual impulse
approximate contributions arising from the constituent quark and diquark,
plus the contribution from the exchange-quark in the BS kernel, and the
seagull contributions which arise from the quark-substructure in the diquark
correlations.
\section{The Electromagnetic Current in Impulse Approximation and Beyond}
\label{TECO}
The electromagnetic current operator $J^{\hbox{\tiny em}}_\mu(x)$ in impulse
approximation is determined by the disconnected contributions from
the electromagnetic couplings of the spectator quark ($J^\mu_{q}$) and the
scalar diquark ($J^\mu_{D}$) which in the Mandelstam formalism are calculated
from the following momentum space kernels~\cite{Man55},
\begin{eqnarray}
J^\mu_{q}(p,P';k,P) \, &=& \, (2\pi)^4 \delta^4(p-k-\hat\eta Q) \\
&&\hskip -1cm q_{q} \, \Gamma^\mu_{q} (\eta P' + p, \eta P +
k) \; D^{-1}(\hat\eta P - k) \; , \nn \\
J^\mu_{D}(p,P';k,P) \, &=& \, (2\pi)^4 \delta^4(p-k+\eta Q) \\
&&\hskip -1cm q_{D} \, \Gamma^\mu_{D} (\hat\eta P' - p, \hat\eta
P - k) \; S^{-1}(\eta P + k) \; , \nn
\end{eqnarray}
with $Q = P' - P$ and $\eta + \hat\eta = 1$. The charge of
the spectator quark in the nucleon is denoted $q_{q}$, the charge of the
scalar diquark is $q_{D}$, and $q_{q} + q_{D} = $ 1 and 0 for the
proton and neutron, respectively.
The Ward-Takahashi identities for the quark and diquark electromagnetic
vertices are
\begin{eqnarray}
Q_\mu \Gamma^\mu_{q}(p+Q,p) \, &=& \, iS^{-1}(p+Q) \, - \, iS^{-1}(p)
\label{qk_WTI_nuc} \; , \\
Q_\mu \Gamma^\mu_{D}(p+Q,p) \, &=& \, iD^{-1}(p+Q) \, - \, iD^{-1}(p) \; .
\label{dq_WTI_nuc}
\end{eqnarray}
From these one can immediately write down the nucleon matrix elements for
the divergences of the corresponding Mandelstam currents between initial
and final nucleon states with momentum and spin $P, \, s$ and $P', \, s'$
respectively, as
\begin{eqnarray}
\langle P',s' | \partial_\mu J^\mu_{q}(0) | P,s \rangle &=&
\, q_{q} \int \frac{d^4k}{(2\pi)^4} \, \Big\{ \label{dsqc} \\
&& \hskip -2.5cm
\bar u(P',s')\, \widetilde{\bar\psi}(-(k+\hat\eta Q), P') \psi(k,P) \,
u(P,s) \nn\\
&& \hskip -2cm
-\, \bar u(P',s') \, \bar\psi(-(k+\hat\eta
Q), P') \widetilde{\psi}(k,P) \, u(P,s) \Big\} , \nn \\
\langle P',s' | \partial_\mu J^\mu_{D}(0) | P,s \rangle &=&
\, q_{D} \int \frac{d^4k}{(2\pi)^4} \, \Big\{ \label{dsdc} \\
&& \hskip -2.5cm
\bar u(P',s') \, \widetilde{\bar\psi}(-(k -\eta Q), P') \psi(k,P) \,
u(P,s) \nn\\
&& \hskip -2cm
- \, \bar u(P',s') \, \bar\psi(-(k-\eta
Q), P') \widetilde{\psi}(k,P) \, u(P,s) \Big\} \; . \nn
\end{eqnarray}
Here, we insert the BSEs for the
amplitudes $\widetilde\psi$ and
$\widetilde{\bar\psi}$ which can be written in the
compact form,
\begin{eqnarray}
\widetilde\psi(p,P) &=& \int \frac{d^4k}{(2\pi)^4} \, K(p,k,P)
\, \psi(p,P) \; , \label{BSEpsi} \\
\widetilde{\bar\psi}(-p,P) &=& \int \frac{d^4k}{(2\pi)^4} \,
\bar\psi(-k,P)\, K(p,k,P) \; . \label{BSEpsibar}
\end{eqnarray}
After shifting the integration momentum by $p+\hat\eta Q \to p$
in the second terms of Eqs.~(\ref{dsqc}) and (\ref{dsdc}), one obtains,
\begin{eqnarray}
\langle P',s' | \partial_\mu J^\mu_q(0) | P,s \rangle \, &=& \, q_q
\int \frac{d^4p}{(2\pi)^4} \frac{d^4k}{(2\pi)^4} \, \Big\{ \label{mac1} \\
&& \hskip -3.5cm
\bar u(P',s') \, \bar\psi(-p, P') K(p, k+\hat\eta Q, P')
\psi(k,P) \, u(P,s) \nn \\
&& \hskip -3.2cm
- \, \bar u(P',s')\, \bar\psi(-p, P') K(p-\hat\eta Q, k,
P) \psi(k,P) \, u(P,s) \Big\} \; , \nn \\
\langle P',s' | \partial_\mu J^\mu_D(0) | P,s \rangle \, &=& \, q_D
\int \frac{d^4p}{(2\pi)^4} \frac{d^4k}{(2\pi)^4} \, \Big\{ \label{mac2} \\
&& \hskip -3.5cm
\bar u(P',s') \, \bar\psi(-p,
P') K(p, k-\eta Q, P') \psi(k,P) \, u(P,s) \nn\\
&& \hskip -3.2cm
-\, \bar u(P',s') \, \bar\psi(-p, P') K(p+\eta Q,k,P) \psi(k,P) \,
u(P,s) \Big\} \; . \nn
\end{eqnarray}
Examination of the terms in Eqs.~(\ref{mac1}) and (\ref{mac2}) reveals
that their sum gives rise to a conserved current only if $K(p,k,P) $ $\equiv
K(p-k) $. That is, if the nucleon BS
kernel is independent of the total nucleon bound-state momentum $P$ and, in
addition, it only depends on the {\em difference} of the relative momenta
$p-k$.
These criteria are satisfied in studies of meson bound states within the
ladder approximation, see for example Ref.~\cite{Tan97}.
However, we observe that even in absence of an explicit
dependence on the total nucleon momentum $P$, the exchange kernel of the
BSE, as obtained from the nucleon Faddeev equation, {\em necessarily}
depends on the sum of the relative momenta $p+k$ and not their difference.
It follows that even with approximating the exchange kernel by a
$P$-independent one, which corresponds to neglecting diquark substructure
together with using $\eta = 1/2$ as in Refs.~\cite{Hel97b,Oet98}, the
electromagnetic current of the nucleon is not conserved in the impulse
approximation and it is already necessary to include an additional coupling
of the photon to the quark-exchange kernel.
It is interesting to note that similar photon-kernel couplings are
required in other systems as well. For example, while the nucleon-nucleon
scattering kernels due to meson exchanges depend only on the difference of
the relative momenta ({\it i.e.}, $K \equiv K(p-k)$), the isospin dependence of
a charged meson that is exchanged between the two nucleons requires
one to introduce additional photon couplings to the exchanged meson to
maintain current conservation, in this case the charged pion.
Such contributions play an important role in determining the
electromagnetic form factors of the deuteron \cite{Gro87}. For the importance
of meson-exchange currents in few-nucleon systems, see also the recent
review, Ref.~\cite{Car98}.
The coupling of the exchange quark (with electromagnetic charge $q_X$)
in the kernel of Eq.~(\ref{xker}) gives rise to the additional
contribution $J_X$ to the nucleon current:
\begin{eqnarray}
&& \hskip -.9cm J^\mu_X(p,P';k,P)\, = \,\hbox{\hfill} \,
- q_X \, \frac{1}{2} \label{J_exchange} \\
&&\hskip -.5cm \widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \, {\Gamma^\mu_q}^T(q',q) \, S^T(q') \,
\widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \; , \nn \\
\hbox{with} &&
q \, = \, \hat\eta P - \eta P' - p - k \; ,
\; \; q' \, = \, q + Q \; ,\nn \\
&& \hskip -.9cm
p_1 \,=\, \sigma (\eta P' + p) - \hat\sigma q \; , \;
\; p_2' \, = \, - \sigma' (\eta P+k) + \hat\sigma' q' \; , \nn \\
\hbox{and~} && \sigma + \hat\sigma = \sigma' + \hat\sigma' = 1 \; . \nn
\end{eqnarray}
From the Ward-Takahashi identity for the quark-photon vertex in
Eq.~(\ref{qk_WTI_nuc}), one finds that the divergence of the
exchanged-quark contribution to the nucleon electromagnetic current is
\begin{eqnarray}
Q_\mu J^\mu_X(p,P';k,P)\, &=& \label{QJ_X} \\
&&\hskip -2.5cm \,- i q_X \, \frac{1}{2} \, \biggl(
\widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \, \widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \nn \\
&&\hskip -1cm - \,
\widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q') \,\widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \, \biggr) \; . \nn
\end{eqnarray}
To provide a comparison with the quark and diquark electromagnetic
currents given above for the quark-exchange kernel of Eq.~(\ref{xker})
written using the same momentum conventions as used
in Eq.~(\ref{J_exchange}), we rewrite the divergences of the Mandelstam
currents given in Eqs.~(\ref{mac1}) and (\ref{mac2}) as
\begin{eqnarray}
Q_\mu J^\mu_q(p,P';k,P)\, &=& \label{e5.13} \\
&&\hskip -3cm \,- i q_q \, \frac{1}{2} \, \biggl(
\widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \, \widetilde{\bar\chi}^T(p_2'-Q,\hat\eta P' -p) \nn \\
&&\hskip -2cm - \,
\widetilde \chi(p_1-Q,\hat\eta P-k)
\, S^T(q') \,\widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \, \biggr) \; , \nn \\
Q_\mu J^\mu_D(p,P';k,P)\, &=& \label{e5.14} \\
&&\hskip -3cm \,- i q_D \, \frac{1}{2} \, \biggl(
\widetilde \chi(p_1-\hat\sigma Q,\hat\eta P-k +Q)
\, S^T(q') \, \widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \nn \\
&&\hskip -2.5cm
- \, \widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \,\widetilde{\bar\chi}^T(p_2' - \hat\sigma' Q ,\hat\eta P' -p -Q)
\, \biggr) \; . \nn
\end{eqnarray}
Since $ q_q \, - \, q_D \, + \, q_X \, = \, 0 $,
one thus finds that, in the case of a point-like diquark BS amplitude,
({\it i.e.}, neglecting any momentum dependence in the diquark BS amplitudes
$\widetilde\chi$ and $\widetilde{\bar\chi}$), the sum of the three
currents given above now yields a conserved electromagnetic current for
the nucleon; that is,
\begin{eqnarray}
Q_\mu \, \biggl( J^\mu_q \, + \, J^\mu_D \, + \, J^\mu_X \biggr) \,
= \, 0 \; .
\end{eqnarray}
For this cancellation all three contributions are crucial.
In particular, the contributions from the photon coupling to the exchanged
quark $J^{\mu}_{X}$, as well as the impulse approximate contributions
$J^\mu_q \, + \, J^\mu_D$ must all be included.
For the quark-diquark model of baryons, these three contributions to the
current correspond to those given in Ref. \cite{Bla99a} for the general
structure of the {\em one-particle contributions} to the current of a
3-particle Faddeev bound state (when the interactions are due to
a separable 2-particle scattering kernel).\footnote{Here, {\em one-particle}
refers to a contribution that arises from a one-particle irreducible {\em
3-point vertex} for the photon coupling.} In the NJL model the one-particle
contributions yield the complete current of the 3-particle bound
state~\cite{Ish95}.
However, if the substructure of the diquark BS amplitudes is taken
into account and the diquark BS amplitude is dependent on any momentum,
the cancellation of the longitudinal pieces in the one-particle
contributions, Eqs.~(\ref{QJ_X}) to~(\ref{e5.14}), is destroyed. Additional
photon couplings which are not of the one-particle type become necessary.
The violations to current conservation from the one-particle contributions
can be displayed in the present framework in a way which will become useful in
following sections. Rearranging the six terms from the
Eqs.~(\ref{QJ_X}),~(\ref{e5.13}) and (\ref{e5.14}) in such a way as to
factor out two terms,
\begin{eqnarray}
S_1(p,P';k,P) \, &:=& \, - i q_q \, \widetilde \chi(p_1-Q,\hat\eta P-k)
\, + \\
&& \hskip -2cm i q_D \, \widetilde \chi(p_1-\hat\sigma Q,\hat\eta
P-k +Q) \, -
\, i q_X \, \widetilde\chi(p_1,\hat\eta P-k) \; ,\nn \\[4pt]
S_2(p,P';k,P) \, &:=& \, - i q_q \, \widetilde{\bar\chi}(p_2'-Q,\hat\eta P'
-p) \, + \\
&& \hskip -2cm i q_D \, \widetilde{\bar\chi}(p_2'-\hat\sigma'
Q,\hat\eta P'-p -Q)
\, - \,i q_X \,\widetilde{\bar\chi}(p_2',\hat\eta P' -p) \; , \nn
\end{eqnarray}
one obtains
\begin{eqnarray}
Q_\mu \, \biggl( J^\mu_q \, + \, J^\mu_D \, + \, J^\mu_X \biggr) \,
&=& \\
&& \hskip -2.5cm
- \frac{1}{2} \biggl( S_1(p,P';k,P) S^T(q')
\,\widetilde{\bar\chi}^T(p_2',\hat\eta P' -p) \nn\\
&& \hskip -2cm - \, \widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \, S_2^T(p,P';k,P) \, \biggr) \; . \nn
\end{eqnarray}
It will be demonstrated in the subsection below that these contributions are
exactly canceled by the so-called ``seagull'' contributions which arise
from one-particle irreducible {\em 4-point couplings} of the two quarks, the
diquark and photon. Such terms must be included whenever the substructure
of a diquark bound state (in form of a momentum-dependent diquark BS
amplitude) is included in the description of the nucleon.
Analogous seagull contributions were previously found necessary
in $\gamma$-meson-baryon-baryon couplings to satisfy the corresponding
Ward-Takahashi identities \cite{Oth89,Wan96}.
Note that terms analogous to the explicit one-particle contributions to the
bound state currents presented in this section can also be obtained by
employing a ``generalized'' impulse approximation for the 3-particle Faddeev
amplitudes. This procedure was recently adopted in an exploratory study of the
electromagnetic nucleon form factors \cite{Blo99}. In this study five
distinct (one-particle) contributions to the form factors arose which were
calculated using parameterizations of a simplified nucleon Faddeev
amplitude. From the separable-kernel Faddeev equation one readily verifies,
however, that only three of these five contributions are independent. These
three have the exact same topology as the one-particle contributions
presented above. Starting from the generalized impulse approximation, the
relative weights of these contributions differ, however, from those needed
for current conservation. The latter can be systematically derived from a
gauging technique~\cite{Ish98}. The discrepancy in the weights is due to an
overcounting of the (generalized) impulse approximation which can therefore
not lead to a conserved current~\cite{Bla99b}. This problem is independent of
the necessity for the additional seagull contributions which persists when
non-pointlike diquarks are used. We will now address these contributions.
\subsection{Ward Identities and Seagulls} \label{WIaS}
The Ward-Takahashi identity for the quark-photon vertex,
Eq.~(\ref{qk_WTI_nuc}), follows from the equal-time commutation relation for
the electromagnetic quark-current operator $j_\mu(x)$ with the quark field
(with charge $q_q$),
\begin{eqnarray}
&&\ [j^0(x), q(y) ] \, \delta(x_0 - y_0) \, = \, - q_{q} \, q(x) \,
\delta^4 (x-y) \; , \nonumber\\
&&\ [j^0(x), \bar q(y) ] \, \delta(x_0 - y_0) \, = \, q_{q} \, \bar q(x)
\, \delta^4 (x-y) \; . \label{charge_con}
\end{eqnarray}
Formal problems with equal-time commutation relations for interacting fields
can be avoided by replacing the canonical formalism with a Lagrangian
formulation based on relativistic causality rather than to single out a sharp
timelike surface \cite{Pei52}.
However, as an operational device for the derivation of Ward identities,
the equal-time commutation relations of Eqs.~(\ref{charge_con}) will
nevertheless give the correct result.
Consider the 5-point Green function that describes the photon coupling to
four quarks $G^\mu_{\alpha\gamma ,\beta\delta}$.
Using the notation that $q_{q\alpha}$, $q_{q\beta}$, $q_{q\gamma}$, and
$q_{q\delta}$ denote the charges of the quark fields with Dirac indices
denoted by $\alpha$, $\beta$, $\gamma$ and $\delta$, respectively, the
Ward identity is given by
\begin{eqnarray}
&& \hskip -.2cm
\partial_\mu^{z} \, \langle T\big( q_\gamma(x_3) q_\alpha(x_1) \bar
q_\beta(x_2) \bar q_\delta(x_4) \, j^\mu(z) \big) \rangle \, =
\label{wti5pt} \\
&& \hskip .1cm
- \, \big( \, q_{q\alpha} \, \delta^4(x_1 - z) \, +\, q_{q\gamma} \, \delta^4(x_3 - z) - \, q_{q\beta}
\, \delta^4(x_2 - z) \, \nn \\
&& \hskip 1cm -\, q_{q\delta} \, \delta^4(x_4-z) \, \big) \;
\langle T\big( q_\gamma(x_3) q_\alpha(x_1) \bar q_\beta(x_2) \bar
q_\delta(x_4) \big) \rangle \; . \nn
\end{eqnarray}
The 4-point function on the right-hand side has the diquark pole
contribution given in Eq.~(\ref{dq_pole_ms}) and depicted in
Fig.~\ref{dqpole}.
The Fourier transformation of the left-hand side of Eq.~(\ref{wti5pt})
allows one to define,
\begin{eqnarray}
G^\mu_{\alpha\gamma , \beta\delta}(p,P';k,P) \, &:=& \\
&& \hskip -2cm
\int d^4\!x_1\, d^4\!x_2\, d^4\!x_3 \, d^4\!x_4 \,\; e^{ip_\alpha x_1}\,
e^{ip_\beta x_2} \, e^{ip_\gamma x_3} \, e^{ip_\delta x_4} \nn \\
&& \hskip -.6cm \langle T\big( q_\gamma(x_3) q_\alpha(x_1) \bar q_\beta(x_2)
\bar q_\delta(x_4) \, j^\mu(0) \big) \rangle \; . \nn
\end{eqnarray}
Here, $p = \sigma p_\gamma - \hat\sigma p_\alpha$, $k = \sigma' p_\beta -
\hat\sigma' p_\delta$ and $P' = P + Q $ as before.
It is straight forward to verify the following Ward identity for this
5-point Green function from the pole contribution to the 4-quark Green
function, given in Eq.~(\ref{dq_pole_ms}), which determines the dominant
contribution when the diquark momenta are close to the diquark pole at
$P^2 = {P'}^2 = m_s^2$.
\begin{eqnarray}
i Q_\mu G^\mu_{\alpha\gamma , \beta\delta}(p,P';k,P) \, &:=& \\
&& \hskip -2.5cm
q_{q\alpha} \, \frac{i}{P^2 -m_s^2 + i\epsilon} \, \chi_{\gamma\alpha}(p +
\hat\sigma Q, P) \, \bar\chi_{\beta\delta}(k,P) \nn\\
&& \hskip -3cm
+ \, q_{q\gamma} \, \frac{i}{P^2 -m_s^2 + i\epsilon} \,
\chi_{\gamma\alpha}(p -\sigma Q, P) \, \bar\chi_{\beta\delta}(k,P) \nn \\
&& \hskip -3cm
- \, q_{q\beta} \, \frac{i}{{P'}^2 -m_s^2 + i\epsilon} \,
\chi_{\gamma\alpha}(p,P') \, \bar\chi_{\beta\delta}(k-\sigma' Q,P') \nn\\
&& \hskip -3cm
- \, q_{q\delta} \, \frac{i}{{P'}^2 -m_s^2 + i\epsilon} \,
\chi_{\gamma\alpha}(p,P') \, \bar\chi_{\beta\delta}(k+\hat\sigma'Q,P') \; .\nn
\end{eqnarray}
To explicitly demonstrate that this does indeed give the additional
contributions necessary to current conservation of the BSE solution for
the nucleon, one needs to consider the irreducible 4-point coupling of the
photon to the quarks and the diquark derived from the following definition:
\begin{eqnarray}
\big( S(p_\gamma) \, M^\mu(p_\gamma,p_\alpha,P_d) \, S^T(p_\alpha)
\big)_{\gamma\alpha} \, D(P_d) \, & := & \\
&&
\hskip -5.5cm Z^{-1} \, \int \frac{d^4k}{(2\pi)^4} \, G^\mu_{\alpha\gamma ,
\beta\delta}(p,P_d+Q;k,P_d) \, \widetilde\chi_{\delta\beta}(k,P_d) \; , \nn
\end{eqnarray}
with $p_\alpha = - p + \sigma (P_d +Q) $, $p_\gamma = p + \hat\sigma (P_d
+Q) $, and
\begin{eqnarray}
Z \, := \, \, \int \frac{d^4k}{(2\pi)^4} \, \hbox{tr}\big[
\bar\chi(k,P_d) \, \widetilde\chi(k,P_d)\, \big] \; .
\end{eqnarray}
The Ward identity for the 5-point Green function then entails,
\begin{eqnarray}
iQ_\mu M^\mu(p_\gamma,p_\alpha,P_d) \, & = & \\
&& \hskip -2cm q_{q\alpha} \,
\widetilde\chi(p + \hat\sigma Q, P_d ) \, S^T(p_\alpha - Q) \, S^{-1\,
T}(p_\alpha) \nn \\
&& \hskip -1.6cm + \, q_{q\beta} \, S^{-1}(p_\gamma) \, S(p_\gamma -Q) \,
\widetilde\chi(p - \sigma Q , P_d )
\nn\\
&& \hskip -1.6cm - \, \Delta_\Phi(Q^2) \; \widetilde\chi(p, P_d + Q ) \,
\frac{P_d^2 - m_{s}^2}{(P_d+Q)^2 - m_s^2 + i \epsilon} \; , \nn
\end{eqnarray}
with
\begin{eqnarray}
Q \, = \, p_\gamma + p_\alpha - P_d \; , \;\;
p \, = \, \sigma p_\gamma - \hat\sigma p_\alpha \; , \nn
\end{eqnarray}
($\sigma + \hat\sigma= 1$) and $\Delta_\Phi(Q^2)$ is defined by
\begin{eqnarray}
\Delta_\Phi(Q^2) &:=& \, Z^{-1} \, \int \frac{d^4k}{(2\pi)^4} \, \Big\{ \\
&& \hskip -1cm
q_{q\beta}\, \hbox{tr}\big[ S^T(-k +\hat\sigma' P_d +Q) \times \nn \\
&& \hskip -.3cm
\widetilde{\bar\chi}(k-\sigma' Q,P_d+Q) \, S(k + \sigma'P_d) \,
\widetilde\chi(k,P_d) \, \big] \nn \\
&& \hskip -1cm
+ \,q_{q\delta} \,
\hbox{tr}\big[ S^T(-k +\hat\sigma' P_d ) \times \nn\\
&& \hskip -.3cm
\widetilde{\bar\chi}(k+\hat\sigma'
Q,P_d+Q) \, S(k+\sigma'P_d+Q) \,\widetilde\chi(k,P_d) \big] \Big\} \; . \nn
\end{eqnarray}
In the limit $Q\to 0$, this is normalized in such a way as to yield the
charge of the scalar diquark; that is,
$\Delta_\Phi(0) = q_{q\beta} + q_{q\delta} \equiv q_{\Phi} $.
In a more detailed and complete calculation, the coupling of the diquark
to the photon would itself have to be done within a Mandelstam formalism.
To achieve this, the Ward identity of Eq.~(\ref{qk_WTI_nuc}) would be used
in the Mandelstam formalism to construct the Ward identity for the quark
substructure of the diquark, thereby replacing the naive Ward identity of
Eq.~(\ref{dq_WTI_nuc}) with a more accurate identity which accurately
depicts the quark substructure of the diquark.
This added complication can be worked out in a straight-forward manner by
introducing a few additional technical details.
However, the basic principle of such couplings, as derived from Ward
identities, can be seen from the following simplifying assumption, which
will be used in the following sections.
Assume that $\Delta_\Phi(Q^2)$ is independent of the photon momentum,
such that $\Delta_\Phi(Q^2) = \Delta_{\Phi}(0) = q_{\Phi}$.
This assumption is sufficient in
order to obtain the correct charges for the nucleon bound state.
From this starting point, the electromagnetic diquark form factor can be
easily included in a minor extension of the framework and follows simply
from the inclusion of a dependence on the photon momentum $Q^2$ of
$\Delta_\Phi(Q^2)$.
By including the effect of only the charge of the diquark, that is setting
$\Delta_\Phi(Q^2) \equiv q_\Phi$ for all photon momenta $Q$, the divergence
of the amplitude $M^\mu$ is written as
\begin{eqnarray}
Q_\mu M^\mu(p_\alpha,p_\beta,P_d) &=& \\
&& \hskip -1.5cm Q^\mu M^{legs}_\mu(p_\alpha,p_\beta,P_d)
\, + \, Q^\mu M^{sg}_\mu(p_\alpha,p_\beta,P_d) \; , \nn
\end{eqnarray}
where $M^{legs}$ contains the couplings of the photon to the amputated quark
and diquark legs according to their respective Ward identities,
\begin{eqnarray}
iQ^\mu M^{legs}_\mu(p_\alpha,p_\beta,P_d) \, & = & \\
&& \hskip -2cm \phantom{+} \, q_{q\alpha} \left(
S^{-1}(p_\alpha) - S^{-1}(p_\alpha - Q) \right) \times \nn\\
&& S(p_\alpha - Q)
\widetilde\chi(p - \hat\sigma Q , P_d ) \nn\\
&& \hskip -2cm + \, q_{q\beta} \widetilde\chi(p + \sigma Q , P_d )
S^T(p_\beta - Q) \times \nn\\
&& \left( S^{-1\, T}(p_\beta) - S^{-1\, T} (p_\beta -
Q) \right) \nn\\
&& \hskip -2cm - \, q_\Phi \widetilde\chi(p,
P_d + Q ) D(P_d+Q) \nn\\
&& \left(D^{-1}(P_d) - D^{-1}(P_d+Q) \right) \; , \nn
\end{eqnarray}
where the term $M^{sg}$ describes the one-particle irreducible seagull
couplings and its divergence is given by
\begin{eqnarray}
iQ^\mu M^{sg}_\mu(p_\alpha,p_\beta,P_d) \, & = & \label{sgWI}\\
&& \hskip -2cm q_{q\alpha} \, \widetilde\chi(p - \hat\sigma
Q , P_d ) \, + \, q_{q\beta} \, \widetilde\chi(p + \sigma Q , P_d ) \nn\\
&& \hskip 1cm
- \, q_\Phi \widetilde\chi(p, P_d + Q ) \; . \nn
\end{eqnarray}
These seagull couplings are exactly what is needed to arrive at a conserved
electromagnetic current for the nucleon.
Upon substitution of the charges of the spectator and exchanged quark and
the scalar diquark, $q_{q\alpha} = q_{q}$, $q_{q\beta} = q_X$ and $q_\Phi
= q_D$, respectively, one finds
\begin{eqnarray}
S_1(p,P';k,P) \, = \, Q^\mu M^{sg}_\mu(\eta P'+p,q+Q,\eta P - k) \, ,
\end{eqnarray}
with $ Q = P' - P $.
A solution to this Ward identity, with transverse terms added so as to keep
the limit $Q\to 0$ regular, which follows from a standard
construction, {\it c.f.}, Refs.~\cite{Oth89,Wan96}, is provided by
\begin{eqnarray}
i M^{sg}_\mu(p_\alpha,p_\beta,P_d) \, & = & \label{sgM} \\
&& \hskip -3cm \phantom{+}
q_{q} \frac{(2p_\alpha -
Q)_\mu }{p_\alpha^2 - (p_\alpha - Q)^2}
\bigg( \widetilde{\chi}(p_\alpha -
Q,p_\beta,P_d) - \widetilde{\chi}(p_\alpha ,p_\beta ,P_d) \bigg) \nn \\
&& \hskip -3cm
+ q_X \frac{(2p_\beta - Q)_\mu }{p_\beta^2 - (p_\beta -
Q)^2} \bigg( \widetilde{\chi}(p_\alpha ,p_\beta -Q,P_d) -
\widetilde{\chi}(p_\alpha ,p_\beta ,P_d) \bigg) \nn \\
&& \hskip -3cm - q_D \frac{(2P_d + Q)_\mu }{(P_d+Q)^2 - P_d^2} \bigg(
\widetilde{\chi}(p_\alpha ,p_\beta ,P_d+Q) -
\widetilde{\chi}(p_\alpha ,p_\beta ,P_d) \bigg) \nn
\end{eqnarray}
with $Q = p_\alpha + p_\beta -P_d$.
Analogously, for the other seagull, one finds
\begin{eqnarray}
i \bar M^{sg}_\mu(p_\alpha,p_\beta,P_d) \, & = & \label{sgMbar}\\
&& \hskip -3cm \phantom{+}
q_{q} \frac{(2p_\alpha - Q)_\mu }{p_\alpha^2 - (p_\alpha - Q)^2}
\bigg( \widetilde{\bar\chi}(p_\alpha -
Q,p_\beta,P_d) - \widetilde{\bar\chi}(p_\alpha ,p_\beta ,P_d) \bigg) \nn \\
&& \hskip -3cm + q_X \frac{(2p_\beta - Q)_\mu }{p_\beta^2 - (p_\beta -
Q)^2} \bigg( \widetilde{\bar\chi}(p_\alpha ,p_\beta -Q,P_d) -
\widetilde{\bar\chi}(p_\alpha ,p_\beta ,P_d) \bigg) \nn \\
&&\hskip -3cm
- q_D \frac{(2P_d - Q)_\mu }{P_d^2 - (P_d-Q)^2} \bigg(
\widetilde{\bar\chi}(p_\alpha ,p_\beta ,P_d-Q) -
\widetilde{\bar\chi}(p_\alpha ,p_\beta ,P_d) \bigg)
. \nn
\end{eqnarray}
The amplitudes
$\widetilde{\chi}(p_\alpha ,p_\beta ,P_d)$ herein
need to be constructed from the BS amplitude of the scalar diquark by
removing the overall momentum conserving constraint $p_\alpha + p_\beta = P_d
$. With these seagull couplings, a conserved electromagnetic current operator
is obtained by including the seagull contribution
\begin{eqnarray}
J_\mu^{sg} (p,P';k,P) \,
&=& \\
&& \hskip -2.6cm
\frac{1}{2} \biggl( M^{sg}_\mu(\eta P'+p,q',\hat\eta P - k) \,
S^T(q') \,\widetilde{\bar\chi}^T(p_2,\hat\eta P' -p) \nn\\
&& \hskip -2.6cm - \, \widetilde \chi(p_1,\hat\eta P-k)
\, S^T(q) \, \bar{M}_\mu^{sg\, T}\!(-(\eta P +k),-q,\hat\eta P' -p) \,
\biggr) \; . \nn
\end{eqnarray}
The total conserved electromagnetic current of the nucleon is therefore
given by $ J_{\hbox{\tiny em}}^\mu := J^\mu_{q} \, + \,
J^\mu_D \, + \, J^\mu_X \, + \, J^\mu_{sg} $.
Note that this explicit construction of the conserved current
complies with the general gauging formalism presented in Ref.~\cite{Bla99a}.
In the reduction of Faddeev equations with separable 2-particle interactions,
the seagull couplings arise from 2-particle contributions to the bound state
current. These contributions describe the irreducible coupling of the photon
to the 2-particle scattering kernel.
Technically, the additional contributions to the electromagnetic nucleon
current arising from exchanged quark in the nucleon BSE kernel $J^\mu_X$,
as well as the seagull term $J^\mu_{sg} $, involve two
4-dimensional loop integrations to calculate the electromagnetic form
factors from the nucleon BS amplitudes.
As demonstrated above, this considerable extension to the Mandelstam
formalism (involving only single loop integrations) is absolutely necessary
to correctly include the non-trivial substructure of the diquark
correlations and maintain current conservation of the nucleon.
While seagull contributions are not necessary in an approximation that employs
point-like diquarks, as is the case in Ref.~\cite{Hel97b},
beyond-impulse contributions, such as the coupling of the exchanged-quark
to the photon are necessary!
In the study of Ref.~\cite{Hel97b}, it was observed that neglecting this
contribution produced negligible violations to the charges of proton and
neutron and so it was dismissed as unimportant.
However, in the case of the present study this contribution is
significant. The reason for this is the larger value of the coupling
strength $g_s$ (obtained from the diquark normalization condition $g_s =
1/N_s^2$) used herein. For a point-like diquark, the coupling need not be
as large.
Furthermore, in contrast to the impulse-approximate Mandelstam currents,
which are independent of $g_s$, the contribution to the electromagnetic
current due to the exchanged quark is proportional to $g_s^2$
(see Eq. (\ref{J_exchange})).
Hence, use of a smaller coupling strength $g_s^2$ in the BSE, reduces the
importance of going beyond the impulse approximation.
As an example of the importance of the contributions beyond the impulse
terms of $J_q$ and $J_D$, we consider the results for the proton
and neutron electromagnetic charges using the amplitudes plotted in
Fig.~\ref{S1_S2} from the Mandelstam current $ J^\mu_{q} \, + \, J^\mu_D $
alone. This leads to charges $Q_P = 0.85$ for the proton and $Q_N =
0.15$ for the neutron.
A theorem constrains the charges of the proton and neutrons to obey
$Q_P+Q_N=1$.
The theorem relies on using $\eta = 1/3$ and is derived from the
nucleon normalization condition in Eq.~(\ref{nuc_norm}).
However, the way it is realized here is not very satisfying.
In Sec.~\ref{ElmFFs}, we discuss in detail the relevance of the various
contributions to the electromagnetic form factors of the proton and
neutron due to exchanged-quark-photon coupling and seagull couplings.
Having proven that the present framework conserves the electromagnetic
current, one might think that the results for the proton and neutron
charges, $Q_P$ = 1 and $Q_N = 0$, must follow trivially.
However, the verification that this framework provides the correct
charges for the proton and neutron is not entirely trivial as is demonstrated
in the next section.
For finite momentum transfer $Q = P'\! -\! P>0$, the complete
Mandelstam couplings for the diquark will still require modifications.
If the photon is coupled to the elementary carriers of charge only,
that is, to the quarks within the diquark (with quark charges
$q_{q\alpha}$, $q_{q\beta}$), the photon-diquark vertex will itself be of
the form,
\begin{eqnarray}
F_\Phi^\mu(Q) &=& \\
&&\hskip -1cm
Z^{-1} \,q_{q\alpha} \, \int \frac{d^4k}{(2\pi)^4} \,
\hbox{tr}\bigg[ S^T(- k+\hat\sigma P_d +Q)\times \nn\\
&&
\widetilde{\bar\chi}(k-\sigma Q,P_d+Q)
S(k+\sigma P_d) \, \widetilde\chi(k,P_d) \times
\nn\\
&& \hskip -.5cm
S^T(-k+\hat\sigma P_d )
\Gamma^{\mu\, T}(-k+\hat\sigma P_d +Q, -k+\hat\sigma P_d ) \bigg] \nn \\
&& \hskip -1cm
+ \, Z^{-1} \,q_{q\beta} \, \int \frac{d^4k}{(2\pi)^4} \,
\hbox{tr}\bigg[ S^T(-k+\hat\sigma P_d ) \times \nn\\
&&
\widetilde{\bar\chi}(k+\hat\sigma' Q,P_d+Q)
S(k+\sigma P_d +Q) \times \nn\\
&& \hskip -.5cm
\Gamma^{\mu}(k+\sigma P_d +Q,k+\sigma P_d) \,
S(k+\sigma P_d) \,\widetilde\chi(k,P_d) \bigg] \; . \nn
\end{eqnarray}
This gives the correct diquark charge in the limit the photon momentum $Q
\to 0$.
In this limit, as far as the electric form factors of the nucleons are
concerned this detail is irrelevant since they are constrained to be
proportional to the charge of the nucleon. On the other hand, the anomalous
magnetic moments of the nucleons may receive additional contributions from
the quark substructure of the diquark.
\section{Electromagnetic Form Factors of the Nucleon}
\label{ElmFFs}
The matrix elements of the nucleon current can be para\-me\-trised as
\begin{eqnarray}
\langle P',s'| J_{\hbox{\tiny em}}^\mu(0) | P,s \rangle &=& \\
&& \hskip -2cm \bar u(P',s')
\left[\gamma^\mu {\mathcal F}_1
+\frac{i \kappa {\mathcal F}_2}{2M} \sigma^{\mu\nu} Q_\nu \right]
u(P,\sigma)\; . \nn
\end{eqnarray}
Here, ${\mathcal F}_1$ and ${\mathcal F}_2$ are the Dirac charge and
the Pauli anomalous magnetic form factors, respectively \cite{Aitchison}.
$Q^\mu = P'-P$ is the spacelike momentum of the virtual photon probing
the nucleon ($-Q^2\ge 0$). Using the Gordon decomposition
\begin{eqnarray}
\bar u(P',s') \, \frac{i\sigma^{\mu\nu} Q_\nu}{2M} \, u(P,s) &=& \\
&& \hskip -2cm
\bar{u}(P',s') \, \left[\, \gamma^\mu\, - \, \frac{P_{\hbox{\tiny
BF}}^\mu} {M} \, \right]
u(P,s) \; , \nn
\end{eqnarray}
with the definition of the Breit momentum
$P_{\hbox{\tiny BF}} :=(P'+P)/2$, the current can be rewritten as
\begin{eqnarray}
\langle P',s'|J_{\hbox{\tiny em}}^\mu(0) | P,s \rangle &=& \\
&& \hskip -2cm \bar{u}(P',s')
\left[ \, \gamma^\mu \, ({\mathcal F}_1+
\kappa {\mathcal F}_2)\, - \, \frac{P_{\hbox{\tiny BF}}^\mu}{M} \,
\kappa {\mathcal F}_2 \, \right] \, u(P,s) \; .\nn
\end{eqnarray}
It is convenient in the following to introduce (matrix valued)
matrix elements by initial and final spin-summations,
\begin{eqnarray}
\langle P'| \widehat J^\mu |P \rangle \, := \,
\langle P',s' | J^\mu | P,s \rangle \, \sum_{s,s'}
u(P',s') \bar{u}(P,s) \; , \label{Jhat}
\end{eqnarray}
to remove the nucleon spinors. The frequently used Sachs electric and
magnetic form factors $G_E$ and $G_M$, are introduced via
\begin{eqnarray}
G_E &=& {\mathcal F}_1 + \frac{Q^2}{4M^2}\kappa {\mathcal F}_2 \;
,\label{ge_0} \\
G_M &=& {\mathcal F}_1 + \kappa {\mathcal F}_2 \, . \label{gm_0}
\end{eqnarray}
These can be extracted from Eq.~(\ref{Jhat}),
\begin{eqnarray}
\langle P' | \widehat J_{\hbox{\tiny em}}^\mu(0) |P \rangle &=& \\
&& \hskip -1.5cm \Lambda^+(P') \left[ \, \gamma^\mu \, G_M \, +\,
M \, \frac{P_{\hbox{\tiny BF}}^\mu}{P_{\hbox{\tiny BF}}^2} \; (G_E-G_M) \,
\right] \Lambda^+(P)\, , \nn
\end{eqnarray}
by taking traces of $\langle \widehat J_{\hbox{\tiny em}}^\mu \rangle \equiv
\langle P'| \widehat J_{\hbox{\tiny em}}^\mu(0) |P \rangle $ as follows:
\begin{eqnarray}
G_E &=& \frac{M}{2P_{\hbox{\tiny BF}}^2} \, \mbox{tr} \,
\langle \widehat J^{\hbox{\tiny em}}_\mu \rangle P_{\hbox{\tiny BF}}^\mu \; ,
\label{ge} \\
G_M &=& \frac{M^2}{Q^2} \, \left( \, \mbox{tr} \, \langle \widehat
J^{\hbox{\tiny em}}_\mu
\rangle \gamma^\mu \, - \, \frac{M}{P_{\hbox{\tiny BF}}^2} \, \mbox{tr} \,
\langle \widehat J^{\hbox{\tiny em}}_\mu \rangle P_{\hbox{\tiny
BF}}^\mu \right) \; .
\label{gm}
\end{eqnarray}
We calculate the current matrix elements using Mandelstam's approach with
the current operators defined in the previous sections, such that
\begin{eqnarray}
\langle P' |\widehat J_{\hbox{\tiny em}}^\mu(0) |P \rangle &=&
\label{Eq:6.10}\\
&& \hskip -1cm \int \frac{d^4p}{(2\pi)^4}
\frac{d^4k}{(2\pi)^4} \; \bar{\psi}(-p,P') \, J_{\hbox{\tiny
em}}^\mu(p,P';k,P) \, \psi(k,P) \nn
\end{eqnarray}
The current operator $J_{\hbox{\tiny em}}^\mu$ consists of the four parts
which describe the coupling of the photon to quark or diquark, to the
exchanged-quark and the seagull contributions which arise from the
coupling of the photon to the diquark BS amplitudes.
These are determined by the following kernels,
\begin{eqnarray}
J^\mu_{q} \, &=& q_{q} \, \Gamma^\mu_{q} (p_q, k_q) \; D^{-1}(k_s)\,
(2\pi)^4 \delta^4(p-k-\hat\eta Q) \; , \label{jq}\\
J^\mu_{D} \, &=& q_{D} \, \Gamma^\mu_{D} (p_s, k_s) \;
S^{-1}(k_q)\, (2\pi)^4 \delta^4(p-k+\eta Q) \; , \label{jd}\\
J^\mu_X \, &=& - q_X \, \frac{1}{2} \\
&& \widetilde \chi(p_1,k_s)
\, S^T(q) \, {\Gamma^\mu_q}^T(q',q) \, S^T(q') \,
\widetilde{\bar\chi}^T(p_2',p_s) \; , \nn \\
J^\mu_{sg} \, &=& \, \frac{1}{2}\, \big( \, M_{sg}^\mu(p_q,q',k_s) \,
S^T(q') \,\widetilde{\bar\chi}^T(p_2',p_s) \\
&& \hskip 1.5cm - \, \widetilde \chi(p_1,k_s)
\, S^T(q) \, \bar{M}^{\mu\, T}_{sg}\!(-k_q,-q,p_s) \,
\big) \; . \nn
\end{eqnarray}
The abbreviations for the various momenta are summarized in the following
table:
\vskip -.6cm
\begin{eqnarray}
\label{mom_table} \\[-8pt]
\renewcommand{\arraystretch}{.9}
\begin{array}{>{$ \small }l<{$\hskip .25cm} |
>{\hskip .25cm$\small }l<{$\hskip .25cm} >{\hskip .25cm $ \small }l<{$} }
& incoming & outgoing \\
\hline \\[-8pt]
quark: & $\scriptstyle k_q = \eta P + k $
& $\scriptstyle p_q = \eta P' + p $ \\
diquark: & $\scriptstyle k_s = \hat\eta P - k$
& $\scriptstyle p_s = \hat\eta P' - p$ \\
exchange quark: & $\scriptstyle q = \hat\eta P -\eta P' -p-k$
& $\scriptstyle q' = \hat\eta P' -\eta P -p-k$ \\
\hline \\[-8pt]
relative momenta &($\scriptstyle \sigma = \sigma' = 1/2$) & \\
within diquark: & $\scriptstyle p_1 = \frac{1}{2} \, (p_q - q) $
& $\scriptstyle p_2' = \frac{1}{2} \, ( -k_q + q' )$ \\
seagull quark-pair: & $\scriptstyle p_1' = \frac{1}{2} \, (p_q - q') $
& $\scriptstyle p_2 = \frac{1}{2} \, (-k_q + q) $
\end{array} \nn
\end{eqnarray}
The contributions to the form factors in the impulse approximation are
depicted in Fig.~\ref{IAD}, while the exchange-quark and seagull
contributions are shown in Fig.~\ref{momrout}.
\begin{figure}[t]
\begin{minipage}{0.49\linewidth}
\epsfig{file=plot/ffquark.eps, width=\linewidth}
\centerline{$\scriptstyle p_s = k_s \; , \; \; p_q = k_q + Q $.}
\end{minipage}
\hfill
\begin{minipage}{0.49\linewidth}
\epsfig{file=plot/ffdiquark.eps, width=\linewidth}
\centerline{$\scriptstyle p_s = k_s + Q \; , \; \; p_q = k_q $.}
\end{minipage}
\caption{Impulse approximation diagrams.}
\label{IAD}
\vspace{5mm}
\end{figure}
We use $\sigma = \sigma' = 1/2$ in the diquark amplitudes.
As discussed in Sec.~\ref{dq_corrs}, this implies that the relative momenta
$p_i$ within the diquarks are exchange symmetric and that our
parameterizations of the diquark BS amplitudes are independent of the mass of
the diquark. We thus set,
\begin{eqnarray}
\widetilde\chi(p_1,k_s) &\to& \widetilde\chi(p_1^2) = \frac{\gamma_5C}{N_s}
P(-p_1^2) \; , \label{sg_momdep} \\
\widetilde{\bar\chi}(p_2',p_s) &\to&
\widetilde{\bar\chi}(p'_2 \!^2) = \frac{\gamma_5 C^{-1}\!\!}{N_s}
P(-p'_2 \!^2) \; . \nn
\end{eqnarray}
This also simplifies the seagull terms, as the seagull couplings to the
diquark legs do not contribute to the seagulls in this case.
The amplitudes in the brackets of the last lines of Eqs.~(\ref{sgM}) and
(\ref{sgMbar}) cancel.
The construction of such vertex functions from the Ward-Takahashi
identities is not unique. In particular,
the forms for the irreducible seagull couplings $M^\mu_{sq}$ and $\bar
M^\mu_{sq}$ given in Eqs.~(\ref{sgM}) and (\ref{sgMbar}), respectively,
are designed for amplitudes $\widetilde\chi$ and $\widetilde{\bar\chi}$
which are functions of the scalars $p_\alpha^2$, $p_\beta^2$
and, in general, $P_d^2$.
For $Q\to 0$, the possibility that the denominators in each of the three
terms may vanish entails that the prefactors that arise
from expanding the amplitudes in brackets must also vanish.
\begin{figure}[t]
\begin{minipage}{0.49\linewidth}
\epsfig{file=plot/ffexch.eps,width=\linewidth}
\vspace{-.2cm}
\parbox{5.6cm}{
$\scriptstyle
q = (\hat\eta\!-\!\eta) P_{\hbox{\tiny BF}} - p - k - Q/2$,\\
$\scriptstyle
q'\!= (\hat\eta\!-\!\eta) P_{\hbox{\tiny BF}} - p - k + Q/2$,\\
$\scriptstyle
p_1 = (\eta\!+\!1)Q/4-(1\!-\!3\eta)P_{\hbox{\tiny BF}}/2 + p + k/2$,\\
$\scriptstyle
p_1' = (\eta\!-\!1)Q/4-(1\!-\!3\eta)P_{\hbox{\tiny BF}}/2 + p + k/2$,\\
$\scriptstyle
p_2 = (\eta\!-\!1)Q/4+(1\!-\!3\eta)P_{\hbox{\tiny BF}}/2 - p/2 - k$,\\
$\scriptstyle
p_2' = (\eta\!+\!1)Q/4+(1\!-\!3\eta)P_{\hbox{\tiny BF}}/2 - p/2 - k$.
}
\vfill
\end{minipage}
\hfill
\begin{minipage}{0.49\linewidth}
\epsfig{file=plot/ffseagull1.eps,width=\linewidth} \\
\epsfig{file=plot/ffseagull2.eps,width=\linewidth}
\end{minipage}
\caption{Exchange quark and seagull diagrams.}
\label{momrout}
\end{figure}
\begin{eqnarray}
\widetilde\chi( (p_\alpha-Q)^2, p_\beta^2 ) - \widetilde\chi(
p_\alpha^2, p_\beta^2 ) &\to& \\
&& \hskip -1cm - 2 (p_\alpha Q) \,
\frac{\partial}{\partial p_\alpha^2 } \widetilde\chi( p_\alpha^2, p_\beta^2)
\; . \nn
\end{eqnarray}
Our present assumption on the dominant momentum dependence of these
amplitudes is slightly different though. For amplitudes
$\widetilde\chi \equiv \widetilde\chi((p_\alpha - p_\beta)^2/4)$,
see~(\ref{sg_momdep}), the prefactors of the seagulls in the form given
in Eqs.~(\ref{sgM}) and (\ref{sgMbar}), corresponding to factors $\propto
1/(p_\alpha Q)$ and $\propto 1/(p_\beta Q)$ respectively, are not canceled in
an analogous way. To cure this, we replace the quark momenta $p_\alpha $ and
$p_\beta $ in these prefactors by (plus/minus) the relative momentum,
$\pm (p_\alpha - p_\beta)/2 $. This yields,
\begin{eqnarray}
iM^\mu_{sg}&=& \, {q_q} \, \frac{(4p_1'-Q)^\mu}{4p_1'Q-Q^2}
\big(\widetilde\chi((p_1'-Q/2)^2)-\widetilde\chi(p_1' \!^2)\big)
\label{good_sg} \\
& &\hskip .5cm + {q_X}\, \frac{(4p_1'+Q)^\mu}{4p_1'Q+Q^2}
\big(\widetilde\chi((p_1'+Q/2)^2)-\widetilde\chi(p_1' \!^2)\big) \nonumber \\
i\bar M^\mu_{sg}&=& \, {q_q}\, \frac{(4p_2-Q)^\mu}{4p_2Q-Q^2}
\big(\widetilde{\bar\chi}((p_2-Q/2)^2)-\widetilde{\bar\chi}(p_2^2)\big)
\label{good_sgbar} \\
& & \hskip .5cm + {q_X} \, \frac{(4p_2+Q)^\mu}{4p_2Q+Q^2}
\big(\widetilde{\bar\chi}((p_2+Q/2)^2)-\widetilde{\bar\chi}(p_2^2)\big)
\nonumber
\end{eqnarray}
Since Ward-Takahashi identities do not completely constrain the form
of the vertices, such modification of this type is within the freedom
allowed by this ambiguity.
The forms~(\ref{good_sg},\ref{good_sgbar}) solve the
corresponding Ward identities at finite $Q$, see Eq.~(\ref{sgWI}) (with
$ q_q + q_X = q_D$). In addition, the smoothness of the limit $ Q\to 0 $ for
the given model assumptions is ensured. In this limit,
\begin{eqnarray}
iM^\mu_{sg}\, &\to& \; - \, ( q_q - q_X ) \; p_1^\mu \;
\widetilde\chi'(p_1^2) \; ,\nn \\
i\bar M^\mu_{sg}\, &\to& \; - \, (q_q - q_X ) \; p_2^\mu \;
\widetilde{\bar\chi}'(p_2^2) \;
. \label{diff_sg_WI}
\end{eqnarray}
We emphasize that this limit is unambiguous.
It must coincide with the form required by the differential form of the
Ward identity for the seagull couplings.
As such it restricts contributions that are both longitudinal and
transverse to the photon four-momentum $Q_{\mu}$.
The form above follows necessarily for the model diquark amplitudes
employed in the present study, and this form provides the crucial
condition on the seagull couplings that ensure charge conservation for
the nucleon bound state.
\noindent The nucleon charges are obtained by calculating
\begin{eqnarray}
G_E(0) &=& \frac{1}{2M} \, \int \frac{d^4p}{(2\pi)^4}
\frac{d^4k}{(2\pi)^4} \\
&& \hskip 1.2cm \hbox{tr} \big[ \bar{\psi}(-p,P) \, P_\mu J_{\hbox{\tiny
em}}^\mu(p,P;k,P) \, \psi(k,P) \big] \; . \nn
\end{eqnarray}
The various contributions to the electromagnetic current for $Q \to 0$ ({\it
i.e.}, $P' = P$) are given by
\begin{eqnarray}
J^\mu_{q} \, &\to & iq_{q} \left(\frac{\partial}{\partial k^\mu_q}
S^{-1}\!(k_q)\right) D^{-1}(k_s)\,
(2\pi)^4 \delta^4(p-k) , \\
J^\mu_{D} \, &\to& iq_{D} \left(\frac{\partial}{\partial k^\mu_s}
D^{-1}\!(k_s)\right)
S^{-1}(k_q)\, (2\pi)^4 \delta^4(p-k) , \\
J^\mu_X \, &\to& - \, i q_X \, \frac{1}{2N_s^2} \,
P(-p_1^2) P(-p_2^2)
\, \left( \frac{\partial}{\partial q^\mu} S(q) \right) , \\
J^\mu_{sg} \, &\to& \, i (q_q - q_X) \; \frac{1}{2N_s^2}\; S(q) \\
&& \left(
p_{1\mu} \, P'(-p_1^2) P(-p_2^2) \, - \, p_{2\mu} \, P(-p_1^2)
P'(-p_2^2) \right) \, . \nn
\end{eqnarray}
Comparing this to the normalization integrals given in
Sec.~\ref{SecNucNorm}, one finds
\begin{eqnarray}
G_E(0) &=& \\
&& q_q \, N_q \, + \, q_D \, N_D \, + \, q_X \, N_X \, - \, (q_q
- q_X) \, N_P \; . \nn
\end{eqnarray}
Using $q_q = 2/3$, $q_D = 1/3$, $q_X = -1/3 $ for the proton, and $q_q =
-1/3$, $q_D = 1/3$, $q_X = 2/3 $ for the neutron, together with
Eq.~(\ref{nuc_norm}), one therefore has,
\begin{eqnarray}
1 \; &=& \, \eta N_q + (1-\eta) N_{D} + (1-2\eta) N_{X} + (1-3\eta) N_{P} \;
,\nn \\
Q_P &=& \, \frac{2}{3} N_q + \frac{1}{3} N_{D} -\frac{1}{3} N_{X} - N_{P}
\; , \label{chargeconds} \\
Q_N &=& \, - \frac{1}{3} N_q + \frac{1}{3} N_{D} + \frac{2}{3} N_{X} +
N_{P}\label{Qn} \; . \nn
\end{eqnarray}
However, these three equations are not independent.
Rewriting the normalization condition for the nucleon BS amplitudes, we
find that
\begin{eqnarray}
1 \; &=& \, \frac{2}{3} N_q + \frac{1}{3} N_{D} -\frac{1}{3} N_{X} - N_{P} \\
&& \hskip 1cm + \, (\eta - \frac{2}{3}) \, \big( N_q - N_D -
2N_X -3 N_P \big) \; , \nn
\end{eqnarray}
which entails that
\begin{eqnarray}
1 \, = \, Q_P \, + \, (2 - 3\eta )\, Q_N \; . \label{chargesum}
\end{eqnarray}
To verify that we do in fact obtain the correct charges of the proton and
neutron, it suffices to show that $N_q - N_D = 2 N_X + 3 N_P$;
that is, it suffices to show that the neutron is neutral, $Q_N = 0$.
The proof of this is straightforward and is given in
Appendix~\ref{supplCC}.
\subsection{Numerical Computation}
\label{NC}
The numerical computation of the form factors is done in the Breit frame,
where
\begin{eqnarray}
&& Q^\mu \, = \, (0, \vec Q ) \; , \nn\\
&& P^\mu \, = \, (\omega_Q,- \vec Q/2) \; , \label{BF_def}\\
&& P'\ \!\!^\mu \, = \, (\omega_Q,\vec Q/2) \; , \nn \\
&& P_{\hbox{\tiny BF}}^\mu \, = \,
(\omega_Q, 0) \; , \nn
\end{eqnarray}
with $\omega_Q = \sqrt{ M^2 + \vec Q^2/4 }$.
The transformation of these variables to 4-dimensional Euclidean polar
coordinates follows the same prescriptions as those employed in
Sec.~\ref{QDBSE} (see Eqs.~(\ref{WickRot})), namely
\begin{eqnarray}
&& \hskip -.2cm
\{p^2,\, k^2, \, Q^2 \} \, \to \, \{-p^2, \, -k^2, \, -Q^2\} \; ,
\quad P_{\hbox{\tiny BF}}^2 \to \omega_Q^2 \; , \label{WickRot2} \\
&& \hskip -.2cm
pQ \, \to \, - \, p \, |\vec Q| \, y_Q \; , \quad kQ \, \to \, - \, k \,
|\vec Q| \, z_Q \; , \nn\\
&& \hskip -.2cm
pP_{\hbox{\tiny BF}} \, \to \, i \omega_Q \, p\, y_{\hbox{\tiny BF}}
\; , \quad
kP_{\hbox{\tiny BF}} \, \to \, i \omega_Q \, k \, z_{\hbox{\tiny BF}} \;
, \nn \\
&&\hskip -.2cm
pP' = pP_{\hbox{\tiny BF}} + pQ/2 \,
\to \nn\\
&& \hskip 2cm
i \omega_Q \, p\, y_{\hbox{\tiny BF}} \, - \, p \, |\vec Q| \, y_Q/2
\, =: \, i \, M\, p \, y \; , \nn \\
&& \hskip -.2cm
kP = kP_{\hbox{\tiny BF}} - kQ/2 \,
\to \nn\\
&& \hskip 2cm
i \omega_Q \, k\, z_{\hbox{\tiny BF}} \, + \, k \, |\vec Q| \, z_Q/2
\, =: \, i \, M \, k \, z \; . \nn
\end{eqnarray}
In the presence of two independent external momenta, $Q$ and
$P_{\hbox{\tiny BF}}$, we are left with 5 independent angular variables.
Together with the absolute values of the integration momenta $p$ and $k$
the exchange-quark and seagull contributions to the form factors at finite
momentum transfer $Q$ require performing 7-dimensional integrations.
These are computed numerically using Monte Carlo integrations.
For the impulse approximation diagrams, the number of necessary
integrations collapses to three due to the momentum-conserving delta
functions in Eqs.~(\ref{jq}) and (\ref{jd}).
One of the integrations is over the absolute value of the loop momentum
$k$ and two are the angular integrations over $z_{\tiny BF}$ and $z_Q$,
the cosines of the angles between $k$ and $P_{\tiny BF}$ and $k$ and $Q$,
respectively.
The BS amplitudes for the nucleon bound states are given in terms of the
two scalar functions $S_1(p,P)$ and $S_2(p,P) $, {\it c.f.},
Eqs.~(\ref{psiDec})
and~(\ref{psibarDec}) in Sec.~\ref{QDBSE}, which, we recall Eqs.~(\ref{ChebyS})
and~(\ref{ChebyM}), are expanded in terms of Chebyshev polynomials $T_n$ to
account for their dependence on the azimuthal Euclidean variable,
\begin{eqnarray}
S(p,y) \, \simeq \, \sum_{n=0}^{N-1} (-i)^n \, S_n(p) T_n(y) \;
. \label{BSACE}
\end{eqnarray}
While the argument $y$ of the Chebyshev polynomials, the cosine between
relative and total momentum, is in $[-1,1]$ in the rest-frame of the nucleon,
this cannot be simultaneously true for the corresponding arguments in the
initial and final nucleon bound-state amplitudes at finite (spacelike)
momentum transfer $Q$. In the Breit frame, these arguments are,
\begin{eqnarray}
z \, &=&\, \frac{\omega_Q}{M} \, z_{\hbox{\tiny BF}} \, - \, i \,
\frac{1}{2} \, \frac{|\vec Q|}{M} \, z_Q \quad \hbox{and} \nn \\
y &=&\, \frac{\omega_Q}{M} \, y_{\hbox{\tiny BF}} \, + \, i \,
\frac{1}{2} \, \frac{|\vec Q|}{M} \, y_Q \; , \label{compl_zy}
\end{eqnarray}
for the initial and final nucleon BS amplitudes respectively (with the angular
variables $z_Q$, $y_Q$ and $z_{\hbox{\tiny BF}}$, $y_{\hbox{\tiny BF}}$ all in
$[-1,1]$). In order to use the nucleon amplitudes computed from the BSE in
the rest frame, analytical continuation into a complex domain is necessary.
This can be justified for the bound-state BS wave functions $\psi$ (with
legs attached). These can be expressed as vacuum expectation values of local
and almost local operators and we can resort to the domain of holomorphy of
such expectation values to continue the relative momenta of the bound-state
BS wave function $\psi (p,P) $ into the 4-dimensional complex Euclidean
space necessary for the computation of Breit-frame matrix elements from
rest-frame nucleon wave functions. The necessary analyticity properties are
manifest in the expansion in terms of Chebyshev polynomials with complex
arguments.
There are, in general however, singularities associated with the constituent
propagators attached to the legs of the bound state amplitudes, here given by
the free particle poles on the constituent mass shells. For sufficiently small
$Q^2$ these are outside the complex integration domain. For larger $Q^2$,
these singularities enter the integration domain. As the general
analyticity arguments apply to wave functions $\psi$ rather than the truncated
BS amplitudes $\widetilde\psi$ with 2-component structure $R(p,y)$, it is
advantageous to expand these untruncated BS wave functions directly in terms
of Chebyshev polynomials (introducing moments $R_n(p)$),
\begin{eqnarray}
R(p,y) \, \simeq \, \sum_{n=0}^{N-1} (-i)^n \, R_n(p) T_n(y) \; ,
\label{BSWFCE}
\end{eqnarray}
and employ the analyticity of Chebyshev polynomials for the BS wave
function $R$. This can be written in terms of the two Lorentz-invariant
functions $R_1$ and $R_2$ by
\begin{eqnarray}
\psi(p,P) & = & D(p_s) S(p_q) \widetilde\psi(p,P)\, = \\
&& \hskip -1.5cm D(p_s) S(p_q) \Big( S_1(p,P)\, \Lambda^+(P) \, +\,
S_2(p,P) \, \Xi(p,P)\, \Lambda^+(P) \Big) \, \nn \\
&& \hskip -.5cm =: R_1(p,P)\, \Lambda^+(P) \, +\, R_2(p,P)
\, \Xi(p,P) \, \Lambda^+(P) \; . \nn
\end{eqnarray}
The price one must pay, however, is a considerably slower suppression of
the higher Chebyshev moments in the expansions in Eq.~(\ref{BSWFCE}) for
the BS wave functions compared to the much faster suppression observed for
the truncated amplitudes $\widetilde\psi$.
For example, the fourth moments of the truncated nucleon amplitudes
$\widetilde\psi $ are shown in Fig.~\ref{S1_S2} of Sec.~\ref{QDBSE}.
Their magnitudes are less than 2 orders of magnitude smaller than the
leading Chebyshev moments. One must include up to 8 Chebyshev
moments in the expansion for the untruncated BS wave functions in order to
achieve a comparable reduction.
If the truncated BS amplitudes $\widetilde\psi$ are used in the expansion,
one must account for the singularities of the quark and diquark
legs explicitly when these enter the integration domain for some finite
values of $Q^2$.
A naive transformation to the Euclidean metric, such as the one given by
Eqs.~(\ref{WickRot2}), is insufficient.
Rather the proper treatment of these singularities is required when they
come into the integration domain.
For the impulse-approximate contributions to the form factors
we are able to take the corresponding residues into account explicitly in
the integration.
Although, this is somewhat involved, it is described in
Appendix~\ref{residue}.
For these contributions, one can compare both procedures and verify
numerically that they yield the same, unique results.
This is demonstrated in App.~\ref{residue}.
We have to resort to the BS wave function expansion for calculating the
exchange quark and seagull diagrams, however. The residue structure
entailed by the structure singularities in the constituent quark and diquark
propagators is too complicated in these cases (involving the 7-dimensional
integrations). The weaker suppression of the higher Chebyshev moments and
the numerical demands of the multidimensional integrals thus lead to
limitations on the accuracy of these contributions at large $Q^2$ by the
available computer resources.
The Dirac algebra necessary to compute $G_E$ and $G_M$, according
to Eqs.~(\ref{ge}) and (\ref{gm}), can be implemented directly into our
numerical routines.
We use the moments $S_n(p)$ or $R_n(p)$ obtained from the nucleon
BSE as described in Sec.~\ref{QDBSE}, which are real scalar functions with
positive arguments. These functions are computed on a one-dimensional
grid of varying momenta with typically $n_p$ = 80 points.
Then spline interpolations are used to obtain the values of these
functions at intermediate values.
The scalar functions $S(p,P)$ and $R(p,P)$ are then easily
reconstructed from Eqs.~(\ref{BSACE}) and (\ref{BSWFCE}), respectively.
Then complex arguments, as given in Eqs.~(\ref{compl_zy}), appear in the
Chebyshev polynomials when the electromagnetic form factors are calculated.
In the results shown below, the 3-dimensional integrations of the impulse
approximation diagrams are performed using Gauss-Legendre or
Gauss-Chebyshev quadratures, while the 7-dimensional integrations
necessary for the calculation of the exchange-quark and seagull
contributions are carried out by means of stochastic Monte-Carlo
integrations with 1.5$\times$10$^7$ grid points.
We find that beyond $Q^2=3$ GeV$^2$, numerical errors for the stochastic
integrations becomes larger than 1\% of the numerical result.
This we attribute to the continuation of the Chebyshev polynomials
$T_n(z)$ to complex values of $z$ as described above.
In addition to the aforementioned complications, there is another bound on
the value of $Q^2$ above which the exchange and seagull diagrams can not be
evaluated. It is due to the singularities in the diquark amplitudes
$\widetilde\chi (p_i)$ and in the exchange quark propagator. The rational
$n$-pole forms of the diquark amplitude,
$P_{n\mbox{\tiny-P}}(p)=(\gamma_n/(\gamma_n+p^2))^n$ for example yield the
following upper bound,
\begin{eqnarray}
Q^2 < 4 \left( \frac{4\gamma_n}{(1-3\eta)^2} -M^2 \right) \; .
\end{eqnarray}
A free constituent propagator for the exchange quark gives the additional
constraint,
\begin{eqnarray}
Q^2 < 4 \left( \frac{m_q^2}{(1-2\eta)^2-M^2} \right ) \; .
\end{eqnarray}
It turns out, however, that these bounds on $Q^2$ are insignificant for the
model parameters employed in the calculations described herein.
\subsection{Results}
\label{Res}
\begin{figure*}[t]
\begin{minipage}{\linewidth}
\vspace{1cm}
\hskip -.2cm \epsfig{file=plot/gep_s.eps,width=2\figwidth}
\hfill
\hskip -.2cm \epsfig{file=plot/gen_s.eps,width=2\figwidth}
\end{minipage}
\bigskip
\begin{minipage}{\linewidth}
\hskip -.2cm \epsfig{file=plot/gmp_s.eps,width=2\figwidth}
\hfill
\hskip -.2cm \epsfig{file=plot/gmn_s.eps,width=2\figwidth}
\end{minipage}
\refstepcounter{figure}
\centerline{\parbox{0.8\linewidth}{\small\textbf{Fig.~\thefigure.}
Nucleon electric and magnetic form-factors for fixed widths of
$S_{1,0}$. The label {\sf Gauss, shifted} refers to
a calculation with diquark vertex function $P_{\mbox{\tiny GAU}}
=\exp(-(p^2/\gamma_{\mbox{\tiny GAU}} -1)^2)$. The parameters are
listed in Tabs.~\protect\ref{FWHM_table} and~\protect\ref{Sgau_table} in
Sec.~\protect\ref{QDBSE}. \label{ffs}
}}
\end{figure*}
In Fig.~\ref{ffs}, we show the electric and magnetic Sachs
form factors of the proton and neutron using the parameter sets given in
Table~\ref{FWHM_table} in Sec.~\ref{QDBSE} which correspond to a fixed
value for the diquark mass $m_s = $ 0.66~GeV.
The nucleon amplitudes used in these calculations correspond to those
shown in Fig.~\ref{S1b_S2b} of Sec.~\ref{QDBSE}.
The charge radii obtained by using the model forms of the diquark BS
amplitude given in Table~\ref{FWHM_table} are given in Table~\ref{charger}.
\begin{figure*}[t]
\begin{minipage}{\linewidth}
\vspace{1cm}
\hskip -.2cm \epsfig{file=plot/gepdipole+data.eps,width=2\figwidth}
\hfill
\hskip -.2cm \epsfig{file=plot/gendipole+data.eps,width=2\figwidth}
\end{minipage}
\refstepcounter{figure}
\centerline{\parbox{0.8\linewidth}{\small\textbf{Fig.~\thefigure.} Comparison
of the electric form factors of proton and neutron with the experimental data
of Refs.~\cite{Hoe76,Bos92} and Ref.~\cite{Pla90}, respectively. Here, the
dipole form of the diquark amplitude was used corresponding to the $n=2$
result of Fig.~\ref{ffs}.\label{ffdata}}}
\end{figure*}
\begin{table}[b]
\begin{tabular}{ll|ll}
& form of diquark & $r_p$ (fm) & $r_n^2$ (fm$^2$) \\
& amplitude $P$ & ($\pm 0.02$) & ($\pm 0.02$) \\
\hline
&&& \\[-4pt]
fixed $S_{1,0}$-width :
& $n$=1 & 0.78 & -0.17\\
& $n$=2 & 0.82 & -0.14\\
& $n$=4 & 0.84 & -0.12\\
& EXP & 0.83 & -0.04\\
& GAU & 0.92 & 0.01\\
& GAU shifted & 1.03 & 0.37\\[2pt]
\hline
&&& \\[-4pt]
fixed masses:
&$n$=1& 0.97 & -0.24\\
&$n$=2& 0.82 & -0.14\\
&$n$=4& 0.75 & -0.03\\
&EXP & 0.73 & -0.01\\
\end{tabular}
\caption{The electric charge radii for proton and neutron for parameter sets
having either the $S_{1,0}$-width or the quark mass fixed. Error estimates
come from the uncertainty in the 7-dimensional integration.
The corresponding experimental values are about $r_p \simeq 0.85$fm for the
proton and $r_n^2 \simeq -0.12$fm$^2$ for the neutron.
\label{charger}}
\end{table}
Examination of the charge radii given in Table~\ref{charger} reveals that
the width obtained for the nucleon BS amplitude is closely
correlated with the obtained value for the charge radius of the proton.
The dipole, quadrupole and exponential forms for the diquark BS amplitude,
all give reasonable values for the charge radius of the proton.
The accepted value of the proton charge radius is $r_p \simeq$ 0.85~fm.
However, the width of $S_{1,0}(p)$ does not determine the behavior of the
form factors away from $Q^2=0$. This is especially clear when the
exponential and Gaussian forms are employed for the diquark BS amplitude.
In this case, the proton electric form factor ceases to even vaguely
resemble the phenomenological dipole fit to experimental data over most of
the range of $Q^2$ shown in Fig.~\ref{ffs}.
The neutron electric form factor is even more sensitive to the functional
form of the diquark amplitudes. The square of charge radius of the neutron
depends strongly on the chosen form of the diquark BS amplitude.
For the exponential and Gaussian forms, the obtained value of $r^2_{n}$ is
close to zero, and it is positive when the Gaussian form of the diquark
amplitude is used with its peak away from zero ({\it i.e.}, $x_0 \not = 0$).
In Fig.~\ref{ffs}, we show the form factors that result from using the
shifted Gaussian form for the diquark BS amplitude with
$x_0/\gamma_{\hbox{\tiny GAU}} = 1$.
In fact, the shifted-Gaussian form also produces a node in the electric form
factor of the neutron, for which there is no experimental evidence.
We conclude that to obtain a realistic description of nucleons,
\pagebreak[-4] one must rule out the use of forms for the diquark BS
amplitude which peak away from the origin.
In Figure~\ref{ffdata} we compare the $n=2$ results, obtained from employing
the diquark BS amplitude of dipole form, to the experimental data of
Refs.~\cite{Hoe76,Bos92} and Ref.~\cite{Pla90} for the electric form factors
of proton and neutron, respectively. In Ref.~\cite{Pla90}, the neutron $G_E$
is extracted from data taken on an unpolarised deuteron target and with
employing various $N$--$N$ potentials. As pointed out in
Ref.~\cite{NIKHEF}, due to possible systematic errors of this procedure,
these results should not be over-interpreted. They can serve to give us a
feeling for the qualitative behavior and rough size of the electric form
factor of the neutron, however. With resembling the phenomenological
dipole fit for the proton fairly well, as seen in Fig.~\ref{ffs}, it might
not be too surprising to discover good agreement also with the experimental
results for the proton. However, with the special emphasis of our present
study being put on charge conservation, such compelling agreement also for
finite photon momentum transfer and over the considerable range of $Q^2$
(from 0 up to 3.5 GeV$^2$), seems quite encouraging. Also the neutron
electric form factor compares reasonably good with the data, especially
considering that we did deliberately not put much effort in adjusting the
free parameters in our present model.
The obtained magnetic moments, which range from 0.95 \dots 1.26 nuclear
magnetons for the proton and from -0.80\dots -1.13 nuclear magnetons
for the neutron, are too small.
The accepted values for the proton and neutron magnetic moments are
2.79 and -1.91 nuclear magnetons, respectively.
The essential reasons for this discrepancy are summarized as follows:
\begin{figure*}[t]
\begin{minipage}{\linewidth}
\vspace{1cm}
\hskip -.2cm \epsfig{file=plot/gep+dp_m.eps,width=2\figwidth}
\hfill
\hskip -.2cm \epsfig{file=plot/gen_m.eps,width=2\figwidth}
\end{minipage}
\refstepcounter{figure}
\centerline{\parbox{0.8\linewidth}{\small\textbf{Fig.~\thefigure.}
Nucleon electric form factors for fixed quark and
diquark masses.
In the left plot, the bottom set of curves represent the results
obtained for the proton electromagnetic form factor $G_{E}(Q^2)$ and the
top set of curves depict the ratios of these form factors over the dipole fit
$ \big(1+Q^2/(0.84\mbox{GeV})^2\big)^{-2}$.
\label{ffm}}}
\end{figure*}
\begin{enumerate}
\item
The next important diquark correlations which should be included in the
present framework are those of axialvector diquarks. These are necessary for
an extension of the quark-diquark model to the decuplet baryons \cite{Oet98}.
The contribution of their magnetic moments to the anomalous magnetic moments
of the nucleons was assessed in Ref.~\cite{Wei93}. There, the NJL model was
employed to calculate the electromagnetic form factors of on-shell diquarks,
and their influence on the nucleon magnetic moments was estimated from an
additive diquark-quark picture. The conclusion from this study was that
including the magnetic moment of the axialvector diquark alone did not
improve the nucleon magnetic moments. The additionally possible transitions
between scalar to axialvector diquarks, however, were found to raise them
substantially. Whether this finding persists in the fully relativistic
treatment, is subject to current investigations \cite{Oet00}.
Furthermore, since the axialvector diquark enhances the binding ({\it i.e.},
lowers the coupling $g_s$), it tends to lower the quark mass required to
produce the same nucleon bound state mass and as will be discussed below, a
smaller quark mass would also serve to improve the obtained values for the
magnetic moments.
\item
In our present study, the diquark-photon vertex is that of a free scalar
particle. The contribution of the corresponding impulse approximation diagram
to the magnetic moments (in the right panel of Fig. \ref{IAD}) is small,
below 0.01 nuclear magnetons, but non-vanishing.
Resolving the diquark substructure by coupling the photon
directly to the quarks within the diquark, and taking into account its
sub-leading Dirac structure, will dress this vertex as discussed (at the end
of Sec.~\ref{WIaS}) and it might increase that contribution. We do not expect
the gain to be substantial though.
\item
The consistency requirement is that the strength of the
coupling $g_s = 1/N_s$ is given by the normalization of the diquark $N_s$.
At present, this leads to a rather narrow nucleon BS amplitude in
combination with somewhat large values for constituent quark mass.
Both of these effects tend to suppress the quark contribution to the form
factors arising from the impulse-approximate terms.
If we had assumed a quark mass $m_q$ of 450~MeV, and artificially
increased the width of the diquark BS amplitude, the quark diagram alone
would easily contribute 1.3 \dots 1.5 nuclear magnetons to the magnetic
moment of the proton. That is, a small change to the dynamics of the
diquark BS amplitude or mass of the quark can have a significant impact on
the magnetic moment of the proton. A similar sensitivity of magnetic
moments of vector mesons to the scales in the quark propagator and bound
state vector-meson BS amplitude was also observed in Ref.~\cite{Haw99}.
\end{enumerate}
The electric form factors obtained using the fixed values for quark and
diquark masses of $m_q = 0.58$ GeV and $m_s = 0.66$ GeV,
respectively, ({\it i.e.}, for fixed values of the binding energy) are shown in
Fig.~\ref{ffm}.
The corresponding charge radii are given in the right half of
Table~\ref{charger}. The differences between the various diquark amplitude
parameterizations considered herein do not lead to such dramatic differing
behaviors of the nucleon form factors in this case.
Nevertheless, we observed that use of the exponential form of the diquark
BS amplitude still produces a proton electric form factor that falls off
too fast when compared to the phenomenological dipole fit.
It also results in a tiny value for the square of the charge radius of the
neutron.
\begin{figure*}[t]
\begin{minipage}{\linewidth}
\vspace{1cm}
\hskip -.2cm \epsfig{file=plot/gepdipole.eps,width=2\figwidth}
\hfill
\hskip -.2cm \epsfig{file=plot/gendipole.eps,width=2\figwidth}
\end{minipage}
\refstepcounter{figure}
\centerline{\parbox{0.8\linewidth}{\small\textbf{Fig.~\thefigure.}
Contribution of the single diagrams to neutron and proton electric form
factors. The diquark amplitude is the dipole. The sum of these contributions
{\sf (SUM)} corresponds to the result of Fig.~\ref{ffdata}.\label{gedipole}}}
\end{figure*}
Finally, we compare the relative importance of the various contributions
to the form factors that arise from the impulse-approximate,
exchange-quark, and seagull diagrams.
We separately plot each of these contributions to the total proton and
neutron electric form factors in Fig.~\ref{gedipole} for comparison.
For the purposes of comparison, we used the parameter set for the
dipole form of the diquark BS amplitude. (This form of the diquark BS
amplitude is used because of the excellent description it provides for the
the electric form factor of the proton.)
Here, it is interesting to note that the seagull couplings contribute
up to about 2/5 of the proton electric form factor $G_E(Q^2)$!
It is clear that in the nucleon, these beyond-the-impulse contributions are
certainly not negligible.
As in the previous discussion of the magnetic moments, where we observed
a suppression of the impulse-approximate diagrams arising from the narrow
momentum distribution of the nucleon BS amplitude which was a result of
employing narrow diquark BS amplitude.
Again, we find that employing a diquark BS amplitude that is wider in
momentum space leads to a wider nucleon BS amplitude and therefore implies
that the impulse approximate contributions are more dominant than the
other contributions. This is the case when point-like diquark BS
amplitudes are employed, such as in Ref.~\cite{Hel97b}.
For the neutron, we observe that its electric form factor arises
from a sum of large terms which strongly cancel each other to produce a
small effect. This cancellation is the explanation for the appearance of
``wiggles'' in the results for the neutron $G_E(Q^2)$ at moderate $Q^2$
shown in Figs.~\ref{ffs}, and \ref{ffm}.
The wiggles are artifacts of the numerical procedure employed.
\section{Conclusions}
\label{Conclusions}
We have introduced an extension of the covariant
quark-diquark model of baryon bound states. The framework developed
herein allows for the inclusion of finite-sized diquark correlations
in the description of the nucleon bound state in a manner which preserves
electromagnetic current conservation for the first time.
For such a framework to maintain current conservation of the nucleon, it
is necessary to include contributions to the electromagnetic current which
arise from the couplings of the photon to the quark-exchange kernel of the
nucleon BSE.
These contributions are derived from the Ward-Takahashi identities of
QED and include the coupling of the photon to the exchanged quark in the
kernel and the photon coupling directly to the BS amplitude of the
diquark (the so-called seagull contributions).
It was shown analytically that the resulting nucleon current is conserved
and these additions are sufficient to ensure the framework provides the
correct proton and neutron charges independent of the details of the model
parameters.
To explore the utility of this framework under the most simple model
assumptions, simple constituent-quark and constituent-diquark propagators
and one-parameter model diquark BS amplitudes were employed for the
numerical application of the framework.
The BS amplitude for the scalar diquark was parametrized by the leading
Dirac structure and various forms for the momentum dependence of the
amplitude were investigated.
It was shown that the antisymmetry of the diquark
amplitude under quark exchange places tight constraints on the form of the
diquark BS amplitude and that the incorporation of these constraints have
the effect of removing much of the model dependence of the diquark BS
amplitude parameters from the calculation of the nucleon electromagnetic
form factors.
Calculations of the electromagnetic form factors away from $Q^2 = 0$
require that the nucleon BS amplitudes and wave functions be boosted.
In this Euclidean-space formulation, this amounts to a continuation
of amplitudes and wave functions into complex plane. Two procedures to
account for proper handling of poles arising from the constituent
particles in the nucleon are described and compared. The feasibility of
each procedure is discussed and explored in detail.
It is shown analytically and explicitly in a numerical calculation that
the two approaches produce the same results for the electromagnetic form
factors. Thus, demonstrating that the framework properly accounts for the
non-trivial analytic structures of the constituent propagators.
For the particular choice of simple dynamical models explored herein, the
masses of the quark and diquark propagators along with the width of the
model diquark BS amplitude are the only free parameters in the framework.
The latter width is thereby implicitly determined from fixing the nucleon mass.
It is shown, for the case of the dipole (and possibly quadrupole) form of
the diquark BS amplitude, that these few parameters are sufficient
to provide an excellent description of the electric form factors for both
the neutron and proton.
Other forms of the diquark BS amplitude that were explored, such as the
exponential and Gaussian forms, may be ruled out on
phenomenological grounds as they lead to nucleon electromagnetic form
factors which are inconsistent with the experimental data.
It was found that the framework is at present unable to reproduce the nucleon
magnetic moments. The calculated magnetic moments are smaller than those
obtained in experiment by about 50\%.
Possible explanations for this are the effect of an inclusion of axialvector
diquark correlations, the addition of more complex
structures in the diquark amplitudes, and a resolution of the
quark substructure in the diquark-photon coupling.
These improvements are the subject of work currently in progress.
In conclusion, we find that the covariant quark-diquark model of the
nucleon provides a framework that is sufficiently rich to describe the
electromagnetic properties of the nucleon. However, to ensure that
the framework satisfies electromagnetic current conservation one must
go beyond the usual impulse approximation diagrams and include
contributions that arise from the photon couplings to the nucleon BSE kernel.
In a numerical application of this framework, it was found that these
contributions provide a significant part of the electromagnetic form
factors of the nucleon and can not be neglected.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,060
|
Q: cannor resolve symbol "baidu" android studio import com.baidu.android.pushservice.PushConstants;
import com.baidu.android.pushservice.PushManager;
I am unable to import the above libraries into my android application.
What dependencies should I add in the build.gradle in order to import them?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,373
|
\section{Introduction}
Overdiagnosis consists of identifying abnormalities that meet disease definitions but will never cause symptoms or death during a patient`s ordinarily expected lifetime~\cite{ref1}. It causes significant harm and increases costs for healthcare systems~\cite{ref2}. The drawbacks of overdiagnosis are unnecessary labelling, adverse effects of additional health examinations and treatments, long term-medication intake, anxiety, and waste of resources which are needed to treat or prevent genuine illnesses~\cite{ref1, ref2, ref3}. According to a previous study, from 24.4\% to 48.3\% of women aged 35 to 84 years were overdiagnosed with breast cancer (including ductal carcinoma \textit{in situ}) in 2010~\cite{ref9}. Another work shows that 30.2\% of individuals, participating in a study and diagnosed with asthma, did not present the condition and, among them, 65.5\% did not need any medical treatment~\cite{ref10}.
There are two main scenarios where overdiagnosis can take place: (i) overdetection of mild "abnormalities" that likely will never compromise a patient's life~\cite{ref1} and (ii) overdefinition – inclusion of patients with minor or uncertain symptoms by broadening definitions and lowering thresholds for risk factors~\cite{ref2, ref3}. The increased availability of digital patient data and machine learning models provide a perfect substrate for overdiagnosis due to several reasons. Firstly, the definition of a disease is translated by a developer to an algorithm. Secondly, access to fast and cheap diagnostic tools might induce its broader application by clinicians, increasing chances for overdiagnosis. Thirdly, combining diagnostic algorithms and self-trackers and wearables leads to application of these technologies to broader and healthier segments of the population, aggravating the risk of overdiagnosis.
Overdiagnosis is usually observed \textit{post hoc}, during randomised controlled trials, with the majority of efforts taking from years or decades, when the harm to patients has already been done.
The purpose of this study is to propose novel methods to preemptively estimate the proportion of overdiagnosed patients generated by a new digital diagnostic algorithm, as means to mitigate these during development stages.
\begin{figure*}
\centering
\includegraphics[width=0.715\linewidth]{Workflow_2.png}
\caption{\textbf{A.} Workflow of the study: building sepsis prediction model, creating, and clustering clinical trajectories, characterizing potential cases of overdiagnosis among true positive patients.
\textbf{B.} Medical trajectory for cluster 6 obtained from ProM 6.1.
\textbf{C.} Comparison of SOFA scores between sepsis positive and negative groups in cluster 6. Wilcoxon test shows that difference is statistically insignificant (P > 0.05).
\textbf{D.} Death rate in cluster 6. Test of Equal or Given Proportions indicated p-value > 0.05.}
\label{fig:teaser}
\end{figure*}
\section{Methods}
\subsection{Data collection and inclusion criteria}
We chose sepsis as a test case for this study since the likelihood of overdiagnosis is high~\cite{ref5, ref6, ref7}, sepsis is managed in critical care settings where large amounts of data are collected, and cases are fully resolved during hospitalization, thus allowing to quickly determine disease trajectories and prognosis.
For the purpose of this study, we used the MIMIC-IV ~\cite{ref11} database. MIMIC-IV contains de-identified clinical data from patients admitted to the Beth Israel Deaconess Medical Center (BIDMC, Boston Massachusetts, USA) between 2008 – 2019. This dataset includes clinical and administrative information such as patient demographics, transfers, vital signs, laboratory tests, procedures, medications, and diagnoses.
We included adult patients without an early suspicion of infection, defined as the absence of ordered cultures or antibiotic prescriptions during the first 24 hours of being admitted to the ICU. To avoid data duplication, we used information collected during the first ICU stay if there were more than one ICU stays during one hospital admission. The final cohort contained 39,216 ICU stays with 1,988 positive and 37,228 negative cases.
\subsection{Method description}
The overall workflow of the study consists of the following steps and is illustrated in (Figure~\ref{fig:teaser}A):
\begin{itemize}
\item {Sepsis prediction model}: we built a sepsis prediction model employing supervised machine learning using \textit{sepsis-3} as the ground truth as described in~\cite{ref18}. For further analysis we kept only patients with true positive and true negative labels.
\item {Clinical Trajectories}: we created clinical trajectories for each patient using relevant activities from the surviving sepsis campaign guideline~\cite{ref8}.
\item {Trajectory Clusters}: we then clustered clinical trajectories using active trace clustering (ActiTraC) algorithm~\cite{ref14}.
\item {Case Analysis}: to identify potential overdiagnosis, we assessed true positive patients with disease trajectories that clustered with true negative ones.
\end{itemize}
\subsection{Sepsis prediction model}
Data collected during the first 24 hours of a patient's stay in ICU were used as an evidence for training of the sepsis prediction model. Four groups of features were used for the modelling: vital signs, laboratory test results, SOFA-related features and scores, and medication administration. Based on greedy feature selection, 13 best features were included into the final model.
Since the final cohort has significant class imbalance (there are only 4.6\% of sepsis positive cases), an undersampling strategy was used for building a balanced dataset. Experiments showed that the best performance was achieved by employing Light Gradient Boosting Machine (LightGBM) algorithm.
\subsection{Event Log and Clinical Trajectories}
We created an event log for patients classified as true positives or true negatives by the predictive model and the ground truth and whose trajectories contained at least three different events. We chose three events as a minimum length to provide enough information for analysis. The dataset for clustering consisted of 1,421 different traces (680 sepsis positive and 741 sepsis negative) with a mean length of 4 events per trace. Total number of events was 13: 'lactate' test, 'fluids crystalloids', 5 events associated with vasopressor therapy: 'norepinephrine', 'epinephrine', 'vasopressin', 'dopamine', 'dobutamine' and 6 events related to antibiotic administration: 'vancomycin', 'other antibiotics', 'cefepime', 'piperacillin-tazobactam', 'ceftriaxone', 'cefazolin'.
\subsection{Clustering}
To cluster trajectories, we used active trace clustering (ActiTraC) plugin in ProM 6.1~\cite{ref15}. ActiTraC finds an optimal distribution of execution traces over a given number of clusters, maximizing combined accuracy of the associated process models, also employing principles of active learning. This algorithm has two selective sampling strategies~\cite{ref14}. For this work we chose 'Distance based selective sampling' strategy with MRA-based (maximum repeat alphabet) euclidean distance function as defined in~\cite{ref16}. We employed an iterative approach using overall accuracy of the process model as evaluation metric to define parameters for the ActiTraC algorithm (target ICS-fitness was 95\%, maximum number of clusters was 24, minimum cluster size was 4 traces and window size for selective sampling was 0.5).
\section{Results}
Our balanced sepsis prediction model achieved an AUROC of 0.86 and MCC of 0.57, under cross validation, which was consistent with performance on a blind test (AUROC=0.86 and MCC=0.57), demonstrating generalisation capabilities. For an imbalanced blind dataset, the model showed an AUROC of 0.86 and MCC of 0.28.
After constructing the event log for patients correctly labelled by the sepsis prediction model and applying the clustering algorithm, we got the following results. For the event log of 1,421 cases, 24 clusters were obtained with the average cluster size of 59 traces. The largest and the smallest clusters consisted of 374 (26.31\%) and 6 (0,42\%) traces, respectively. All remaining 288 (20.26\%) unassigned traces were put into a separated cluster. Most clusters had size of 16-72 traces (1-5\% of the event log).
We then evaluated clustering composition. Positive cases are distributed as follows: 10 clusters (41\%) among 24 have less than 50\% of sepsis positive instances. There are two clusters with more than 75\% of positive incidents and one cluster that consists of sepsis negative instances only. Clusters with relatively small shares of sepsis cases presented vasopressor medication, blood test on lactate level and antibiotics administration in primary sequencies of their trajectories. For clusters with high proportion of sepsis cases, antibiotics administration is the first or second action in a trajectory. We expect that potential cases of overdiagnosis to be in clusters skewed towards sepsis negative subgroup and that contain at least 10 sepsis positive cases.
To evaluate similarity of patients in a cluster we analysed combination of several factors: mortality rate, SOFA scores during the first 24 hours in ICU and discharge location.
Using Wilcoxon test for a SOFA score and test of Equal or Given Proportions for mortality rate we assessed similarity of patients in terms of outcome in each cluster. Among all clusters with lower than 50\% of sepsis positive cases, two clusters (6 and 12) showed no statistically significant difference in SOFA and death rate with 95\% of confidence (Figure~\ref{fig:teaser} B,C,D). In cluster 6, the proportions of patients in positive and negative groups who were discharged to rehabilitation centres or skilled nursing facilities are roughly the same (18\% and 20\% in each of the locations). Since medical trajectories inside these clusters demonstrated no statistical difference in treatment outcomes, we can expect that cases of overdiagnosis are represented in these two clusters. Using the described method, we found 29 cases of potential overdiagnosis, which are 4.3\% of all cases of sepsis included in the study.
\section{Conclusion}
In this paper we proposed for the first time, a computational approach for preemptive identification of potential cases of overdiagnosis, which showed promising results. The developed method is a strategy to quantify the problem and to improve tool development which, with the dissemination of machine learning prediction models, will become more prominent. This method has the advantage over other approaches as it can be used to inform model engineering and potentially actively mitigate machine learning-induced overdiagnosis. A limitation of our approach is the available metrics to measure treatment outcomes and can be complemented as more data becomes available (\textit{e.g.}, discharge summary). In future works we intend to supplement medical trajectories with larger number of events and take into consideration alternative parameters describing a patient's recovery and treatment outcomes.
\bibliographystyle{ACM-Reference-Format}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Calculate size of record column If i have a table like this then insert it 16 digit for each column except the PK
CREATE TABLE x
(
id BIGINT AUTO_INCREMENT PRIMARY KEY,
col1 BIGINT,
col2 CHAR(16)
) Engine=InnoDB;
INSERT INTO x
VALUES (1234567890123456, '1234567890123456');
Then the size of col1 it will stored 8 byte, and the size of col2 it will stored 16 byte.
Is my understanding correct?
A: MySQL database has not a function for calculating the size of columns, but you can do this manually by writing your own function. Now I will tell you how to do it. Firstly you can get data types of columns by this query:
select
c.COLUMN_NAME,
c.DATA_TYPE,
c.CHARACTER_MAXIMUM_LENGTH
from information_schema.`COLUMNS` c
where c.TABLE_SCHEMA = 'test' and c.TABLE_NAME = 'test_table';
You will get data
id bigint null
col1 bigint null
col2 char 31
col3 varchar 58
So, you can write your own function that is function get_size_column(column_name varchar) input parameter. Inside the function, you must control data types of the column, for example if datatype = bigint then size = 8, such datatypes as int, smallint, bigint are static sizes. If datatype = varchar or char you must use CHARACTER_MAXIMUM_LENGTH field. I wrote this for examples, but you have to write if conditions for all types. I must also say that there are many tutorials about MySQL data types and store sizes. After all this process you can use your function in the select statements of your queries.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The process of path aggregation
is a ubiquitous phenomenon in nature. Some examples of such
phenomena are river basin formation\cite{scheidegger},
aggregation of trails of liquid droplets moving down a window pane,
formation of insect pheromone trails\cite{ant1, ant2, pheromone},
and of pedestrian trail systems\cite{human,helbing}.
Path aggregation has been mathematically studied mainly in two classes
of models. One of them is known as the active-walker
models\cite{helbing}
in which each walker in course of its passage through the system
changes the surrounding environment locally, which in turn influences the later
walkers. An example of such a process is the ant trail
formation\cite{ant1,ant2}.
While walking, an ant leaves a chemical trail of pheromones which other
ants can sense and follow.
The mechanism of human and animal trail formations is mediated by the
deformation of vegetation that generates an interaction between
earlier and later walkers\cite{helbing}. A mathematical formalism to
study the formation of
such trails has been developed in Ref.\cite{human,helbing}.
The other class of models showing path aggregation deals with non-interacting
random walkers moving through a fluctuating environment.
In Ref.\cite{wilk}, condensation of trails of particles moving in an
environment with Gaussian spatial and temporal correlation is
demonstrated analytically.
Another example of this model class is
the Scheidegger river model\cite{scheidegger}
(and related models\cite{river-book})
which describes the formation of a stream network by aggregation of streams
flowing downhill on a slope with local random elevations.
In this Letter we analyze the dynamics of path aggregation using
a simple model that belongs to the class of active walker systems
discussed above.
The model is similar to the one used to study path localization
in Ref.\cite{schulz}. In our model, however, we take into account the
aging of the paths, an important aspect of the active walker models.
For instance,
in ant trail systems, the pheromone trails age due to evaporation.
In the mammalian trail formation the deformations of the
vegetation due to the movement of a mammal
decays continuously with time\cite{helbing}.
In our model, the individual paths do not age gradually, but rather
maintain their full identity until they are abruptly removed from
the system. This particular rule for path aging is chosen to allow
application of our model to the process of axon fasciculation, which
we discuss next.
During the development of an organism,
neurons located at peripheral tissues
(e.g. the retina or the olfactory epithelium)
establish connections to the brain via growing axons.
The {\it growth cone} structure at the tip of the axon interacts with
other axons or external chemical signals and can
bias the direction of growth when spatially distributed chemical signals are
present \cite{Gilbert}. In the absence of directional
signals, the growth cone maintains an approximately constant average
growth direction, while exploring stochastically the environment in the
transverse direction \cite{katz1985}.
The interaction of growth cones with the shafts of other axons commonly
leads to fasciculation of axon shafts\cite{Mombaerts}.
During development a significant portion of fully grown neurons die and
get replaced by newborn neurons with newly growing axons.
For certain types of neurons (such as the sensory neurons of the mammalian
olfactory system) the turnover persists throughout the lifespan
of an animal.
In mice, the average lifetime of an olfactory
sensory neuron is 1--2 months \cite{Crews}, which is less
than one tenth of the mouse lifespan.
The mature connectivity pattern is fully established only after several
turnover periods \cite{Zou04}.
The model we propose in this Letter captures the basic ingredients
of the process of axon fasciculation, i.e. attractive interaction of
growth cones with axon shafts, as well as neuronal turnover.
The main contribution of this Letter is a detailed discussion of the slow
time scales that emerge from the dynamics of our model.
Using Monte-Carlo simulations we characterize the time scale for the approach
to steady state and the correlation time within the steady state, and show
that they can exceed the average axonal life time by orders of magnitude.
To understand these results we formulate
an analytically tractable effective single fascicle dynamics.
This allows us to relate the observed slow time scales to the dynamics of
the basins of the fascicles.
From the effective fascicle dynamics we derive three time scales which
we compare to the time scales extracted from the Monte-Carlo simulation
of the full system.
For clarity, we stress that the dynamics of our model differs
substantially from one-dimensional coalescence
($A + A \to A$)\cite{benAvraham} or
aggregation ($mA + nA \to (m+n)A$)\cite{redner}.
In our model, there is no direct
inter-walker interaction; rather, each random walker interacts locally
with the {\em trails} of other walkers.
While the stationary
properties of the system (such as the fascicle size distribution in the
steady state) may be approximately understood using an analogy to
one-dimensional diffusion with aggregation, the dynamical properties
(such as the time scale of approach to the steady state,
and the correlation time in the steady state)
are undefined in the one dimensional analogy, and require understanding
based on the full two dimensional model.
This is in contrast to the situation for path aggregation
models without turnover: e.g., the Scheidegger river
network model can be mapped onto the Takayasu model of
diffusion-aggregation in presence of injection of mass
in one dimension\cite{takayasu}.
\section{Model and numerical implementation}
Each growing axon is represented as a directed random walk in two spatial
dimensions (Fig.~\ref{conf}$a$). The random walkers (representing the growth
cones) are initiated at the periphery ($y=0$, random $x$) with a birth rate
$\alpha$, and move towards the target area (large $y$)
with constant velocity $v_y=1$. In
the numerical implementation on a tilted square lattice, at each time step
the growth cone at ($x,y$) can move to ($x-1,y+1$) (left) or ($x+1,y+1$)
(right). (Note that the sites are labeled alternatively by even $x$ and
odd $x$ at successive $y$ levels.)
The probability $p_{\{L,R\}}$ to move left/right is evaluated based
on the axon occupancy at the ($x-1,y+1$) and ($x+1,y+1$) sites and their
nearest neighbours (see Fig.~\ref{conf}$a$). In the simplest version of the
model, the interaction is governed
by the ``always attach, never detach'' rule: $p_L = 1$ when among the sites
$(x \pm 1,y+1)$, $(x \pm 3,y+1)$ only $(x -3,y+1)$ is already occupied;
$p_R =1$ when only $(x+3,y+1)$ is occupied; $p_L = p_R = 1/2$ in all other
cases.
Periodic boundary conditions are used in the $x$-direction.
\begin{figure}[t]
\psfrag{x}{$x$}
\psfrag{y}{$y$}
\psfrag{(a)}{$(a)$}
\includegraphics[width=4cm]{randwalkf.eps}
\psfrag{Lb}{$D$}
\psfrag{Eb}{$E$}
\psfrag{y}{$y$}
\psfrag{(b)}{($b$)}
\includegraphics[width=4cm]{conf.eps}
\caption{(Color online)
($a$)~Interacting directed random walks on a tilted square lattice.
A random walker
($+$) represents a growth cone. For one walker, the possible future sites
($\Box$) and their nearest neighbours ($\circ$) are marked. The trail of
a walker (line) models an axon shaft.
($b$)~Typical late-time configuration ($t=25T$) in a system with
$L=800$ and $N_0=100$.
For the fascicle identified at $y=6000$ (arrow), $D$ indicates its
basin, i.e. the interval at the level $y=0$ between the right-most and
left-most axons belonging to the fascicle. The gap $E$ is the inter-basin free
space at $y=0$.
Note that the $y$-coordinate cannot be understood as equivalent to time
(see the main text).
}
\label{conf}
\end{figure}
To capture the effect of neuronal turnover, each random walker is assigned
a lifetime from an exponential distribution with mean $T$. When the lifetime
expires, the random walker and its entire trail is removed from the system.
The mean number of axons in the system therefore reaches the steady state
value $N=N_0\exp(-\beta y)$ where
$N_0=\alpha/\beta$, and $\beta = 1/T$ is the death rate per axon.
In the simulations, we use $T = 10^{5}$ time steps, and restrict
our attention to $y \le T/10$. The birth rate $\alpha$ is
chosen so as to obtain the desired number of axons $N_0$, or equivalently, the
desired axon density $\rho = N_0/L$ ($\rho=1/2$ implies an average
occupancy of one axon per site), where $L$ is the system size in
$x$-direction. The presence of turnover distinguishes our model from
previous theoretical works on axon fasciculation \cite{AvanOoyen1,AvanOoyen2}.
The $y$-coordinate in
Fig.\ref{conf}$a$ cannot be
viewed as equivalent to time, and the dynamics at fixed $y$ (which
is the main focus of this Letter) has no analogy in one dimensional models of
aggregation or coalescence.
{\em Mean fascicle size:}
A typical late-time configuration for a system with $L=800$ and $N_0=100$
is shown in Fig.~\ref{conf}$b$. With increasing fasciculation distance $y$,
the axons aggregate into a decreasing number $m(y)$ of {\em fascicles}.
(At a given $y$, two axons are considered to be part of the same fascicle
if they are not separated by any unoccupied sites.) The number of axons in
the fascicle is referred to as the fascicle size $n$. The mean
fascicle size $\bar n$ at level $y$ may be estimated using the following
mean-field argument. Each of the $m$ fascicles collects axons that
were initiated on an interval of length $D \simeq L/m = L\bar n/N$ at the
level $y=0$ (see Fig.~\ref{conf}$b$). The axons initiated at opposite edges of
the interval are expected to meet within $y\simeq (D/2)^2$ steps of the
random walk in $x$-direction. Consequently,
$\bar n\simeq DN/L \simeq 2 \rho y^{1/2} \exp(-\beta y)$
for $y$ up to $y \simeq (L/2)^2$,
where complete fasciculation ($\bar n=N$) is expected.
Thus for $y\ll (L/2)^2$ and $\beta y\ll 1$ (which are satisfied in our
simulations) one obtains the power law growth $\bar n \simeq 2\rho y^{1/2}$.
\begin{figure}[t]
\begin{center}
\psfrag{t/T}{$t/T$}
\psfrag{ninf-n}{$n_\infty - \langle \bar n \rangle $}
\psfrag{ninf-c}{$n_\infty-c$}
\psfrag{1y0.48}{\tiny{$~~~~~~y^{0.48}$}}
\psfrag{0.25y0.48}{\tiny{$0.25\, y^{0.48}$}}
\psfrag{y}{$y$}
\psfrag{ y=102x}{\tiny{$y=10^2$}}
\psfrag{ y=103x}{\tiny{$y=10^3$}}
\psfrag{ y=104x}{\tiny{$y=10^4$}}
\psfrag{ 5000*x}{\tiny{$\rho=1/8$}}
\includegraphics[width=6cm]{nuf33.eps}
\end{center}
\caption{(Color online)
Approach to the steady state in the $L=400$, $N_0=200$ system
at representative $y$-levels indicated in the legend.
The mean fascicle size $\langle \bar n (t;y)\rangle $
[averaged over $10^3$ initial conditions]
approaches $n_\infty(y)$ as $t\to\infty$.
The data set labeled as $\rho=1/8$ is from the $L=400$, $N_0=50$
system, collected at $y=5000$.
Inset: Power-law growth of mean fascicle size
$n_\infty - c = 2\rho y^{b}$ in the steady state.
The values of the growth exponent and offset are:
$b=0.48$, $c=1.3$
for the $L=800$, $N_0=100$ system($\times$), and
$b=0.48$, $c=6.9$ for the $L=400$, $N_0=200$ system($+$).}
\label{ap}
\end{figure}
\section{Slow time scales}
The measured mean fascicle size, obtained by averaging over all the
existing fascicles at a given $y$ (Fig.~\ref{ap}), grows
with time as $\bar n = n_\infty - p \exp (-\beta t) - q \exp (-t/\tau_{ap}) $,
where $\tau_{ap}(y)$ defines the time scale of
approach to the steady state value $n_\infty (y)$.
We find that $\tau_{ap}$
is an increasing function of $y$ (up to $y \simeq (L/2)^2$ where a
single fascicle remains and $\tau_{ap}$ drops to $T$), and can exceed the
axon lifetime $T$ by orders of magnitude (Figs.~\ref{ap} and \ref{tscale}$b$).
Asymptotically in $y$, we find $n_\infty = c+2\rho y^b$, with
$b=0.480\pm0.018$ (Fig.~\ref{ap} inset) -- in reasonable agreement with the
mean-field prediction.
The dynamics in the steady state is characterized by the
auto-correlation function for the mean fascicle size
$\bar n(t)$ at a fixed $y$-level:
$c(t)=\langle \bar n(t) \bar n(0)\rangle$ which
fits to the form $p+q\exp(-\beta t)+r\exp(-t/\tau_c)$.
In Fig.~\ref{tscale}$a$ we plot the subtracted correlation function
$g(t)=[c(t)-p]/(q+r)$. The correlation time $\tau_c$
increases with $y$ and significantly exceeds the axon lifetime $T$
(Fig.~\ref{tscale}$b$).
\begin{figure}[t]
\psfrag{c(t)}{$g(t)$}
\psfrag{t/T}{$t/T$}
\psfrag{(a)}{$(a)$}
\psfrag{y=10xxxx}{\tiny{$y=10~~$}}
\psfrag{y=100xxx}{\tiny{$y=10^2~~$}}
\psfrag{y=1000xx}{\tiny{$y=10^3~~$}}
\includegraphics[width=4cm]{f5a.eps}
\psfrag{t/T}{$\tau_c/T$}
\psfrag{y}{$y$}
\psfrag{(b)}{$(b)$}
\includegraphics[width=4cm]{f5b.eps}
\caption{(Color online)
($a$) Subtracted auto-correlation $g(t)$
at $y$-levels indicated in the legend for the $L=100$, $N_0=50$ system.
The time series $\bar n(t)$ is collected over $t=200T$ to $2\times 10^4 T$;
$g(t)$ is further averaged over $30$ initial conditions.
($b$)~The correlation time $\tau_c$ ($\Box$) and approach-to-steady-state
time scale $\tau_{ap}$ ($\circ$) as a function of $y$.
The theoretical time-scales $\tau$ (filled $\triangle$),
$\tau_f$ (filled $\nabla$)
(see text) are evaluated from the values of
$a_+,\,b_+,\,c_+$ in Table I.
The solid line is $y^b$ with $b=1/2$.
$\tau_c$ ($\Diamond$) for the $L=400$, $N_0=50$ system is also shown.
}
\label{tscale}
\end{figure}
\section{Effective fascicle dynamics at fixed $y$}
We next examine the dynamics of
individual fascicles. These are typically long lived,
but at a given $y$-level the fascicles very rarely merge
or split (data not shown). Consequently, the number $n(t)$
of axons in each fascicle may be viewed as given by a stochastic
process specified by the rates $u_\pm(n,y)$ (for transitions
$n \rightarrow n\pm1$).
At fixed $y$, a fascicle can loose
an axon only when the axon dies, thus $u_-(n)=\beta n$,
independent of $y$.
The gain rate $u_+(n,y)$ is governed by
the properties of the fascicle {\em basin} (see Fig.~\ref{conf}$b$).
Under the ``always attach, never detach'' rule, new axons
initiated anywhere within the basin of size $D$ cannot escape the fascicle.
In addition, some of the axons
born in the two gaps (of size $E$) flanking the basin contribute.
Therefore,
\begin{eqnarray}
u_+=\alpha D/L+(\alpha E/L)[1-\Pi(E,y)]
\label{up}
\end{eqnarray}
where $\Pi(E,y)$ is the probability that
an axon born within the gap of size $E$ survives as a single axon
at level $y$.
The two stochastic variables that fully characterize the dynamics of a
fascicle are the number of axons $n(t)$ and the basin size $D(t)$.
In Fig.~\ref{track}$a$, we plot $n(t)$ and
$D(t)$ for a specific fascicle followed over $200 T$.
It is seen that $n$ and $D$ tend to co-vary (cross correlation
coefficient $c(D,n)=0.74$).
In the following we treat the dynamics of $D$ as slave to the
dynamics of $n$, i.e. we assume that the average separation
$S = D/(n-1)$ between two neighbouring axons within the basin is
time-independent. This implies
$u_+(n) = a + b n$
with $b=\beta \rho S$.
The measured average gain rate $u_+(n)$ in the steady state \cite{rates}
deviates from linearity at high $n$, but is well fit by
$u_+(n)=a_+ +b_+ n -c_+n^2$ (Fig.~\ref{track}$b$). The quadratic correction
may be understood as a saturation effect which reflects that basins of size
$D$ can not exceed $2 y$ or $L$ and $D > 2 y^{1/2}$ occur with low
probability.
The quadratic correction to the linear growth of $u_+$ with $n$
becomes significant at $n \gtrsim 2y^{1/2}\rho\approx \bar n$.
Note that the coefficients
$a_+,~b_+$ and $c_+$ are functions of $y$.
\begin{figure}[t]
\psfrag{n}{$n$}
\psfrag{t/T}{$t/T$}
\psfrag{n}{{$n$}}
\psfrag{Lb}{{$D$}}
\psfrag{a}{{$S$}}
\psfrag{Lb, n}{$D,~n$}
\psfrag{(a)}{$(a)$}
\includegraphics[width=4cm]{Lbyn.eps}
\psfrag{n}{$n$}
\psfrag{upbybeta}{$u_+/\beta$}
\psfrag{u1}{$u_+^{(1)}$}
\psfrag{u2}{$u_+^{(2)}$}
\psfrag{un}{$u_-$}
\psfrag{up, um}{$u_+, u_-$}
\psfrag{(b)}{$(b)$}
\includegraphics[width=4cm]{updens.eps}
\caption{(Color online)
($a$)~Time series of number of axons $n$ and basin size $D$
for an individual fascicle in a system of
$L=100$ and $N_0=50$ at $y=400$.
The equal time cross correlation coefficient averaged over this
time-span,
$c(D,n)=[\langle D n\rangle -\langle D\rangle \langle n\rangle]/
\sqrt{[\langle D^2\rangle - \langle D\rangle^2][\langle n^2\rangle - \langle n\rangle^2]}=0.74$.
($b$)~Mean gain and loss rates (in units of $\beta$)
[averaged over $10^3$ initial conditions and the interval
$100T\leq t\leq 150T$]
as a function of fascicle size $n$. The rates are measured at $y=400$ in
a system with $L=100$.
The fits (lines) are $u_- = n$,
$u_+^{(1)}=1.92+0.974n-0.002n^2$ (for $N_0=50$),
and $u_+^{(2)}=1.95+0.927n-0.008n^2$ (for $N_0=25$).}
\label{track}
\end{figure}
{\em Master equation at fixed $y$:}
Let $P(n,t)$ be the distribution of fascicles of size $n$
at time $t$ at a fasciculation distance $y$.
The master equation of the effective birth-death process may be written as
\begin{eqnarray}
\dot P(n,t) &=& u_+(n-1)P(n-1,t)+ u_-(n+1) P(n+1,t)\nonumber\\
&-& [u_+(n)+ u_-(n)] P(n,t),
\label{master}
\end{eqnarray}
for $n>1$.
For the boundary state ($n=1$)
$$\dot P(1,t) = J_+(y) + u_-(2) P(2,t) - [u_+(1)+u_-(1)]P(1,t)$$
where $J_+(y)$ represents the rate with which new single
axons appear between existing fascicles at $y$.
In the steady state, $J_+(y)$
is balanced by the rate with which existing fascicles disappear from the
system, i.e. $J_+(y)=u_-(1)P_s(1)$. The steady state condition
$\dot P(n,t)=0$ then implies pairwise balance,
$u_+(n-1) P_s(n-1) = u_-(n) P_s(n)$, for all $n>1$.
Thus the steady state distribution is given by
$P_s(n)=J_+(y) \frac{1}{u_-(n)} \prod_{k=1}^{n-1} \frac{u_+(k)}{u_-(k)}$,
with the normalization condition $\sum_{n=1}^{\infty} n P_s(n,y)=N$.
In order to obtain a closed-form expression for $P_s(n,y)$, we
expand the pairwise balance condition up to linear order in $1/N$
and solve to
find,
\begin{eqnarray}
\beta P_s(n,y) \simeq J_+(y)~ n^\gamma \exp[-\ell (n-1) - \kappa (n-1)^2],
\label{lin2}
\end{eqnarray}
where $\gamma=a_+/\beta-1$,
$\ell=1-b_+/\beta$ and
$\kappa=c_+/2 \beta$.
{\em Time scales:}
Three distinct time scales may be extracted from the effective
birth-death process.
The correlation time $\tau$ for the fascicle size $n$,
near the macroscopic stationary point $n_s$ [$u_+(n_s)=u_-(n_s)$]
can be expressed\cite{vanKampen} as
$\tau=1/(u'_-(n_s)-u'_+(n_s))
=1/(\beta - b_+ + 2c_+ n_s)$.
With a linear approximation
of $u_+(n)=a_+ +b_+ n$ the approach-to-steady-state
time scale for $\langle n\rangle$ is $\tau_{ap}=1/(\beta - b_+)$ \cite{vanKampen}.
We note that the long time scales do not simply arise as a consequence of
fascicles containing many axons, but are due to the dynamics of the
fascicle basins. To see that, imagine a fascicle with frozen boundary
axons, for which $u_+ \approx (\alpha/L)D$ with $D$
constant. In this case $u'_+(n_s)=0$ and
the correlation time $\tau\simeq T$.
To obtain $\tau >T$, $D$ must co-vary with $n$.
At high $y$, $u'_+\approx 1/T$ resulting in $\tau \gg T$.
We note that in the full system, the dynamics of the basin
size can be viewed as arising from the competetion between
neighbouring fascicles for basin space.
A third time scale $\tau_f$ can be defined by $J_+(y)=m/\tau_f$,
and reflects the rate of turnover of {\em fascicles} in the
full system. Evaluating,
$\tau_f=[\int_1^\infty P_s(n,y)dn]/J_+(y)$ using $P_s(n,y)$
from Eq.~\ref{lin2} (with $\gamma=1$), we obtain
\begin{eqnarray}
\tau_f=(T/2\kappa)\left[1-\left(\sqrt\pi e^{\frac{\ell^2}{4\kappa}}(\ell-2\kappa) \text{erfc}(\ell/2\sqrt\kappa)\right)/2\sqrt\kappa\right].
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\psfrag{Ps(n;y)}{$P_s(n,y)$}
\psfrag{n}{$n$}
\psfrag{APsny}{$AP_s(n,y)$}
\psfrag{Bn}{$Bn$}
\includegraphics[width=6cm]{clps2.eps}
\end{center}
\caption{(Color online)
Steady state distribution of fascicle sizes $P_s(n,y)$
[averaged over $10^4$ initial conditions and the time interval
$10T\leq t\leq 25T$] for the $N_0=100$, $L=800$ system at $y$-levels
indicated in the legend.
Inset: A scaling with $B=1/\langle n\rangle$ and $A=\langle n\rangle^{2.1}$ collapses all data
obtained for $y=1585,\, 1995,\, 3162,\, 5012,\, 6310,\, 7943,\, 10^4$
onto a single curve
$\phi(u)={\cal N} u \exp(-\nu u-\lambda u^2)$
with $u=n/\langle n\rangle$ and
${\cal N}=274$, $\nu=0.78$, $\lambda=0.45$.
}
\label{pkg-clps}
\end{figure}
\section{Fascicle size distribution in the steady state}
Following the discussions on effective single fascicle dynamics,
we now return to the simulation results for the dynamics of the
whole system.
The steady state is characterized by the stationary distribution
of fascicle sizes $P_s(n,y)$, defined as the number of fascicles of size $n$
at level $y$. For a system with $L=800$ and $N_0=100$, $P_s(n,y)$ is shown
at a series of $y$-levels in Fig.~\ref{pkg-clps}.
Within the range $y=10^3-10^4$ all data collapse onto a single curve after
appropriate rescaling (Fig.~\ref{pkg-clps}).
This data collapse implies the scaling law
\begin{eqnarray}
P_s(n,y)= \langle n(y)\rangle ^{-2.1}\phi(n/\langle n (y)\rangle)
\label{scale}
\end{eqnarray}
where the scaling function $\phi(u) = {\cal N} u \exp(-\nu u - \lambda u^2)$.
\section{Analogy to particle aggregation in one dimension}
As we have stressed in the introduction,
the full dynamics of our system can not be mapped
onto the particle dynamics of a one-dimensional reaction-diffusion system.
At fixed time $t$ (within the steady state), however,
the aggregation of fascicles with increasing $y$ may be formally viewed as
the evolution of an irreversible aggregation process
$mA+nA\to (m+n)A$ in one spatial dimension, where the $y$-coordinate
(Fig.\ref{conf}($b$)) takes the meaning of time.
This exhibits analogous scaling properties, but
with a different scaling function $u\exp(-\lambda u^2)$ \cite{redner}, which
lacks the exponential part $\exp(-\nu u)$.
It is interesting to note at this point that
to remove the exponential
part in the expression of the steady state distribution (Eq.\ref{lin2})
one would require $b_+=\beta$ which in turn implies
$\tau_{ap}=\infty$. In other words, without the exponential part
$\exp(-\nu u)$
the emerging long time scale $\tau_{ap}$ is lost.
This is consistent with the fact that the time scales arising
from turnover are undefined in the one dimensional analogy.
\section{Scaling of $y$-dependent parameters}
The expansion coefficients for $u_+(n,y)$, measured at selected $y$-levels in a system with $L=100$ and $N_0=50$, are listed
in Table I. The scaling property of the distribution of fascicle sizes
(Eq.~\ref{scale}) implies
$P_s(n,y)
= {\cal N} y^{-3.1\, b}\, n \exp(-\nu n y^{-b} - \lambda n^2 y^{-2b})$.
Consistency with Eq.~\ref{lin2} requires $J_+(y)\sim y^{-3.1\, b}$,
$\kappa\sim y^{-2b}$ [i.e. $c_+\sim y^{-2b}$],
and $\ell\sim y^{-b}$ [i.e. $(\beta -b_+)\sim y^{-b}$].
This is in reasonable agreement with the data in Table I.
The coefficient $a_+ \simeq 2 \beta$ for all $y$
reflects $\gamma\simeq 1$.
Notice that $J_+(y)\propto \Pi(E,y)$ the probability that an
axon born in the gap remains single at the level $y$.
$\Pi(E,y)$ can be approximated as the survival probability of a
random walker moving in between two absorbing boundaries (basin borders)
each of which is undergoing random walk;
thus $\Pi\sim y^{-3/2}$\cite{redner}. This is consistent with the
measured exponent in $J_+(y)\sim y^{-3.1 b}$.
The gradual approach of $b_+\approx\beta \rho S$ to $\beta$ with increasing $y$
may be understood as follows. At low $y$, fascicles are formed preferentially
by axons that started with small separation $S$ at the $y=0$ level, and the
typical gap size $E$ exceeds $S$. With increasing $y$, the smallest gaps
are removed to become part of fascicles; consequently both
$E$ and $S$ grow with $y$, and $S$ approaches $1/\rho$ at high $y$. The time
averaged $S$ for selected fascicles is shown in the last two columns of
Table I.
\begin{table}[t]
\begin{tabular}{|c|c|c|c|}
\hline
$y$ & $a_+/\beta$ & $b_+/\beta$ & $c_+/\beta$ \\ \hline
20 & 1.96 & 0.862 & 0.0116 \\
40 & 1.94 & 0.909 & 0.0082 \\
100 & 1.96 & 0.947 & 0.0056 \\
400 & 1.92 & 0.974 & 0.0022 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline
$y$ & $\langle S\rangle$\\ \hline
200 & 1.14\\
400 & 1.60\\
1000 & 2.02\\
10000 & 2.21 \\
\hline
\end{tabular}
\caption{$y$-dependence of the fascicle parameters defined in the main text.
System parameters are $L=100$ and $N_0=50$.}
\end{table}
{\em $y$-dependence of time scales:}
The $y$-dependence of parameters discussed above implies the
following scaling for the theoretically derived time scales:
the approach-to-steady-state time $\tau_{ap}=1/(\beta -b_+)\sim y^b$,
the correlation time $\tau\sim\tau_{ap}\sim y^b$ and
the life time of fascicles $\tau_f\sim 1/\kappa\sim y^{2b}$.
As seen in Fig.~\ref{tscale}$b$,
the measured correlation time $\tau_c$ falls in between the
computed time scales $\tau$ and $\tau_f$, and the three time scales
show distinct $y$-dependence.
In a system with lower axon density, the time scales are
reduced (Fig.~\ref{tscale}$b$) as the increased inhomogeneity of
axon separations at $y=0$ leads to a stronger deviation of $\rho S$
from $1$.
\section{Conclusion}
To summarize, we have proposed a simple model for axon fasciculation
that shows rich dynamical properties. We identified multiple time scales
that grow with the fasciculation distance $y$ and become $\gg T$,
the average lifetime of individual axons.
The slow time scales do not simply arise as a consequence of
fascicles containing many axons, but are due to the dynamics of the
fascicle basins.
Our theoretical results have wider relevance for existing related
models (e.g. of insect pheromone trails \cite{ant1, ant2, pheromone}
and pedestrian trail formation \cite{human,helbing}),
in which the slow maturation and turnover
of the connectivity pattern have not been analyzed in detail.
To conclude, we comment on the applicability of our model to
the biological system which we described in the introduction,
i.e. the developing mammalian
olfactory system. The model presented in this Letter is readily
generalized to
include multiple axon types, type-specific interactions,
and detachment of axons from fascicles \cite{long_paper}.
This is important as in the olfactory system the nasal epithelium
contains neurons belonging to multiple types, distinguished by the
expressed olfactory-receptor\cite{Mombaerts}, with type specific interaction
strengths\cite{homotypic1}.
(We note, however, that preparing transgenic mice expressing only a single
olfactory-receptor has recently become a reality\cite{sakano}).
A further modification of the basic model that might be
required is the introduction of history-dependent axonal
lifetimes\cite{Paul}.
A comparatively
simpler task is to relate our model to experimental studies
of axon growth in neuronal cell culture \cite{honig1998, hamlin}.
\acknowledgments
We gratefully acknowledge extensive discussions with Paul Feinstein
on olfactory development and on the formulation of our model.
|
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{"url":"https:\/\/math.stackexchange.com\/questions\/1408747\/does-the-alternating-harmonic-series-where-only-prime-terms-are-negative-conve?noredirect=1","text":"# Does the \u201calternating\u201d harmonic series where only prime terms are negative converge?\n\nWe know that the harmonic series $\\sum \\frac{1}{n}$ diverges, yet the alternating harmonic series $\\sum \\frac{(-1)^n}{n}$ converges.\n\nEuler famously gave a proof of the infinitude (and of the \"density\") of primes by showing that the series $\\sum_p \\frac{1}{p}$ diverges; this reasoning can be applied more generally to show Dirichlet's theorem on primes in arithmetic progression.\n\nCan this reasoning lead to the stronger statement that $\\sum \\frac{\\epsilon(n)}{n}$ converges, where $\\epsilon(n)$ is $1$ if $n$ is prime and $-1$ otherwise? If this isn't true (which I suspect but can't easily prove), is there any relationship between whether an \"alternating\" harmonic series like this, where we give a $+$ sign to numbers in some set $S$ and a $-$ to others, converged and whether the sum of the harmonic series restricted to $S$ converges? Perhaps the case where $S$ is an arithmetic progression is easy?\n\nWhat if we generalize further to partitions of the positive integers into $n$ parts and consider the harmonic sum where there's a different $n^{th}$ root of unity for each part? This feels like something related to Dirichlet zeta functions and characters of finite groups, but I don't know how to make this precise.\n\n\u2022 If you had $\\epsilon(n) = +1$ when $n$ is a multiple of three and $-1$ otherwise, this would diverge because the negative terms outgrow the positive ones. The same thing is true for your question, only the primes are so rare that not even 0.0001% of all terms will be positive (in the long run). This is kind of like hoping $\\lim_{x\\to\\infty} x - e^x$ will exist just because $x$ and $e^x$ both diverge to infinity. \u2013\u00a0Erick Wong Aug 25 '15 at 5:20\n\u2022 For a similar problem that actually has somewhat balanced behavior, try Liouville's function $-1$ to the power of the number of prime factors of $n$. This paper also discusses a root-of-unity analogue which you speculate near the end of your question. \u2013\u00a0Erick Wong Aug 25 '15 at 5:31\n\u2022 If you want to quantify exactly how far away you got carried, you could read up on Mertens' second theorem... \u2013\u00a0Micah Aug 25 '15 at 5:34\n\u2022 Wow, so I guess you could say the asymptotic behavior of $\\sum \\frac{1}{p}$ is as different from that of $\\sum \\frac{1}{n}$ as $\\sum \\frac{1}{n}$ is to $\\sum 1$! (since the former is between $\\log \\log n$ and $\\log n$ and the latter between $\\log n$ and $n$) \u2013\u00a0Dorebell Aug 25 '15 at 5:48\n\u2022 @Dorebell In fact it is sometimes said that $\\sum 1\/p$ diverges so slowly that $\\sum 1\/p$ over all known primes is less than $4$. There was a question on MSE that discussed the validity of this claim, but I can't seem to find it right now. \u2013\u00a0Erick Wong Aug 25 '15 at 6:21\n\nWe have two well-known asymptotics,\n\n\\begin{align} \\sum_{n \\leq N} \\frac{1}{n} &\\sim \\log N \\\\ \\sum_{p \\leq N} \\frac{1}{p} &\\sim \\log \\log N. \\end{align}\n\nThe difference in magnitude between these is so large that $$\\sum_{n \\leq X} \\frac{\\epsilon(n)}{n} \\gg \\log N$$ still, and so your sum still diverges.\n\nMore generally, you can hope for convergence if you choose $\\epsilon(n)$ in such a way that the partial sums $$\\sum_{n \\leq N} \\epsilon(n)$$ are uniformly bounded by some number $M$. This is the case in the regular harmonic series, as the partial sums are bounded by $1$. In the case of primes, the resulting partial sums diverge to infinity as \"most\" integers are not prime.\n\nHowever, if the partial sums are uniformly bounded, then one can use Dirichlet's convergence test to show that the resulting pseudo-alternating harmonic series converges. This applies even if one chooses the weights to be roots of unity instead of $\\pm 1$.","date":"2019-06-20 09:33:14","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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9615688920021057, \"perplexity\": 146.2678035794394}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627999200.89\/warc\/CC-MAIN-20190620085246-20190620111246-00498.warc.gz\"}"}
| null | null |
What procedure do I use to get the Plasma Tube to work for Frozen Shoulder?
Look through these, read notes for them, determine if any of these meet your needs... If so, click them to load into the Loaded Programs block, lower left of the screen - currently empty as I don't know what you'll choose.
Once you have chosen the programs desired, move to the Control tab, click "Allow Generator Overwrite" box, then the generator box for the generator connected to the Central unit, and if the Central is On, click Start to run the program.
2. If you question is about pathogens, you can kill Legionella, Mycobacterium leprae, Gonorrhoea and Siphilis.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,448
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Sales >
Lists Serve: Use A List Service to Help Find Leads
New homeowners offer a great source of remodeling leads
By Stacey Freed
Edwin Fotheringham
During the boom, Sherman Webb was busy remodeling hotels. "Once everything went south, we reinvented ourselves," says Webb, whose Dawg Construction, in North Hollywood, Calif., began to focus on residential projects. To find good prospects, Webb turned to Homeowners Marketing Services, which specializes in new-homeowner lists.
"A new homeowner, especially of an older home, has immediate needs, is in remodeling and spending mode, and is loaded with credit. They're a perfect audience," says Homeowners Marketing Services vice president and sales manager Laura Friedman.
Set Parameters
HMS updates its lists daily and puts out a fresh list each week at the close of escrow. As a compiler, not a reseller, HMS does all the research. Clients determine their list parameters, e.g., selecting a whole county or several ZIP codes, or focusing on specific home types or the highest home value.
HMS covers 1,200 U.S. counties and 105 SIC codes. "We provide about 50,000 new homeowners [nationwide] a week," Friedman says. "If someone subscribes to El Paso, for example, every week they get a list of new homeowners in that area. They don't get the same names duplicated."
First In
List buyers pay a monthly fee. An area serving up about 20 to 40 new homeowners a week costs $49 per month; 100 new homeowners a week costs $98; and for an area with 300 new names a week it's $196 per month. HMS makes an effort not to allow more than two companies that offer the same services to buy the same lists.
Webb gets about 120 names each week. "I put the names on my mailer and send it out," he says. "There's no better audience than someone moving into a new area." He gets three to six calls per week based on the leads and did three projects from them in 2011.
"The strategy here is reaching the right person at the right time; that's the only time marketing really works," says HMS president Barry Weiner. "It's all about timing. You've got to get there first."
—Stacey Freed, senior editor, REMODELING.
Stacey Freed
Formerly a senior editor for REMODELING, Stacey Freed is now a contributing editor based in Rochester, N.Y.
Houston-Sugar Land-Baytown, TX
Chicago-Naperville-Joliet, IL-IN-WI
Atlanta-Sandy Springs-Marietta, GA
Sherman Webb
Laura Friedman
Barry Weiner
Bryan Sebring
Dawg Construction
Sebring Services
Best Pick
EBSCO Research
Year in the Rearview
Remodeling What Every Remodeling Prospect Wants
There's a Right (and a Wrong) Way to Be Competitive
Remodeling Managing the Big Three
Beware of Tie-Down Questions
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Coventry City recorded their first 0-0 draw of the season against Blackpool at the Ricoh Arena in a game of few real clear cut chances.
Adam Armstrong had arguably the best chance of the game in the second half but he could not make contact with the ball when it was squared to him six yards out.
City top scorer Armstrong made it straight back into the starting line up after his international duty. He came into the side along with Ryan Kent, the two coming in to replace Jim O'Brien and Ruben Lamerais in the Sky Blues' only changes from last week's Fleetwood game.
The game was three minutes old when the Sky Blues had their first sight of goal. Marc-Antoine Fortune received the ball in the area with his back to goal. He turned his man and fired a shot across the face of goal, but the ball flashed wide of the far post.
City were seeing much of the ball in the opening ten minutes but the visitors started making some chances of their own soon after, with Mark Cullen twice having shots blocked in the area.
The Seasiders were looking to hit City on the break, but were largely untesting the City goal, save for a couple of weak efforts held well by Reice-Charles Cook.
Midway through the half Sam Ricketts managed to get his head on a Kent corner but could only direct his header well over the bar.
Armstrong hit a 30-yard curled effort well over a couple of minutes later as the Sky Blues looked to gain a stranglehold on what had been an even game.
Despite not being known for his goalscoring, Chris Stokes nearly dispatched a sublime 25-yard effort ten minutes from the break, with goalkeeper Doyle at full stretch to turn the ball behind for a corner.
From the flag kick, the ball eventually found its way to John Fleck, who hit a cross-cum-shot that unfortunately evaded everybody.
The game was in danger of dying down as the half came to a close but it sprang back into life in the 43rd minute when an incisive passing attack from Stokes, Fleck and Murphy saw Fortune fling himself at a whipped cross in the area, but the ball didn't sit right and Doyle managed to claim.
The half time whistle came and after a tense first 45, both teams went in level.
The second half began at an end to end pace, with the best chance of the first five minutes falling to Armstrong, who somehow managed to miss the ball six yards out as it was played across to him.
By far the best chance of the game fell to Cullen with just over 20 minutes to go. He managed to evade Johnson's lunge and was one-on-one, but he dragged his shot wide of the post with the goal gaping.
With twenty minutes to go it was still anyones game and Murphy did his best to change that when he fired a fierce effort from 20 yards that Doyle spilled, but Armstrong couldn't quite capitalise.
Sixty seconds later Fleck got his name in the referee's book and will subsequently miss Tuesday's trip to Rochdale.
There was saddening scenes as Johnson was taken off injured with seven minutes remaining with what looked like a leg injury, he will be assessed during the week.
Five minutes was added on by the fourth official, but there was to be no more chances as the visitors kept the ball in the corner to cling on to an away point.
Focus now turns to Rochdale, where the Sky Blues will go next on Tuesday night.
Attendance: 12,094 (450 Away)
Coventry City: Charles-Cook, Ricketts ©, Martin, Johnson (Phillips 83'), Stokes, Murphy, Vincelot (Lamerais 68'), Fleck, Kent (Morris 78'), Fortune, Armstrong.
Subs: Burge, O'Brien, G.Thomas, Tudgay.
Blackpool: Doyle, Ferguson, McAlister, Robertson, Potts, Cullen, Redshaw (Thomas 63'), Cameron (Osayi-Samuel 79'), Aldred, Boyce, Cubero.
Subs: Rivers, Oliver, Herron, Dunne, Letheren.
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\section*{supplementary material}
See online Supplementary Material for further details on PEEM measurements, for the sample fabrication procedure and for the determination of the SiO$_2$, TiO$_2$ and hBN VB offsets.
\section*{acknowledgement}
S. U. acknowledges financial support from VILLUM FONDEN (Grant. No. 15375). R. J. K. is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). D. S. acknowledges financial support from the Netherlands Organisation for Scientific Research under the Rubicon Program (Grant 680-50-1305). The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was supported by IBS-R009-D1. The work at NRL was supported by core programs and the Nanoscience Institute.
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Q: AtomicKotlin course adds additonal spaces and doesn't accept answers I'm doing Atomic Kotlin course and in one of the first exercises I get something like this:
And this is my code:
It should accept my answer but it doesn't. What seems to be the problem?
A: I solved it by going to build.gradle file and modifying printOutput from this:
def printOutput(def output) {
return tasks.create("printOutput") {
println "#educational_plugin_checker_version 1"
def lines = output.toString().split("(?<=\n)|(?=\n)")
for (line in lines) {
println "#educational_plugin" + line
}
}
}
to this:
def printOutput(def output) {
return tasks.create("printOutput") {
def lines = output.toString().split("\n")
for (line in lines) {
println "#educational_plugin" + line
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,912
|
\section{Introduction}
Percolation is one of the most important models in statistical mechanics~\cite{Stauffer94}.
Recently, there has been increasing interest in correlated percolation models, where the activity (occupation) of a node (site) depends on the states of its neighbors (see Refs.~\cite{Araujo14,Saberi15} and references therein).
In this paper, we study the diffusion percolation (DP) and bootstrap percolation (BP) models, which belong to the correlated site percolation model, in complex networks.
In the DP~\cite{Adler88}, each node is activated with probability $p$, and then inactive nodes that have at least $k$ active neighbors are activated recursively until all the inactive nodes have less than $k$ active neighbors;
In the BP~\cite{Chalupa79}, each node is activated with probability $p$ and then nodes that do not have at least $m$ active neighbors are deactivated recursively until all the active nodes have at least $m$ active neighbors.
The two models are closely related and sometimes both are called BP as well.
The DP of $k$ larger than the maximum degree and the BP of $m=0$ are the same as the classical percolation (CP).
The DP and BP have been studied intensively on lattices~
\cite{Kogut81,Branco84,Khan85,Branco86,Adler88,Adler90,Adler91,Chaves95,Medeiros97,Branco99,Gravner12,Choi19,Choi20}.
In the $\Delta$-regular Archimedean lattices and three-dimensional lattices (simple cubic, body-centered cubic, and face-centered cubic lattices), it was shown that the DP and BP have first-order percolation transitions with the percolation thresholds $p_c=1$ and $p_c=0$, respectively, if $m>m_c$ and $k<(\Delta+1-m_c)$, where $m_c=\lfloor (\Delta+1)/2 \rfloor$ (the only exception is the bounce lattice, which has $m_c=\lfloor (\Delta+1)/2 \rfloor+1$)~\cite{Kogut81,Enter87,Adler88,Branco99,Choi19,Choi20};
otherwise, the DP and BP have second-order percolation transitions at finite $p_c$ ($0<p_c<1$) with the same critical exponents as the CP~\cite{Choi19,Choi20} (exceptionally, the BP transition with $m = m_c = 6$ in the face-centered cubic lattice is first-order~\cite{Choi20}).
The BP was originally proposed to understand disordered dilute magnetic systems~\cite{Chalupa79}, and later it turned out that many interesting phenomena of complex systems are also closely related to the DP and BP.
Neuronal activity~\cite{Soriano08}, jamming transition~\cite{Gregorio04,Gregorio05}, and opinion formation~\cite{Ramos15} can be modeled by the DP.
The diffusion of innovations, which is the diffusion process of a new idea, technology, or product through a social network~\cite{Rogers03}, can be modeled by the DP or BP depending on the properties of the innovative option.
The DP can describe the situation where an agent adopts the innovative option when at least $k$ neighbors have adopted it~\cite{Centola10,Jin21}.
On the other hand, the BP is a more appropriate model for a situation that an agent abandons the innovative option whenever there are less than $m$ neighbors that keep the innovative option~\cite{Helbing12}.
Recently, there have been studies about the network robustness~\cite{Buldyrev10,Parshani10,Hu11,Gao12,Havlin15}, which adopt the culling process of the BP.
They considered two networks that interact with each other; a failure of a node in one network leads to the failure of some other nodes in the other network. The iterative cascade of failures may make the global connectivity of the two networks broken.
Interestingly, they found a critical point that separates first- and second-order percolation transitions~\cite{Parshani10,Hu11,Gao12}.
Studies of the DP and BP in complex networks, however, are still insufficient and mostly restricted to the DP.
Balogh and Pittel studied the minimum value of $p$ that can make all the nodes active by the DP in regular random networks (RRNs)~\cite{Balogh07}.
Baxter et al. discovered a double transition (continuous percolation transition followed by a discontinuous hybrid transition) for the DP of $k=3$ in the Erd\H{o}s-R\'{e}nyi network (ERN) with an average degree $\langle\Delta\rangle=5$~\cite{Baxter10}.
The double transition of the DP was also observed in small-world networks and an RRN~\cite{Gao15}.
Multiple hybrid phase transition of the DP was reported in ERNs with community structure~\cite{Wu14}.
In this paper, we present comprehensive results on percolation transitions of the DP and BP in RRNs and ERNs using modified Newman-Ziff algorithms. The percolation transitions are classified by the behavior of the order parameter and its derivatives near the percolation threshold;
first-order, second-order, and double transitions are observed.
Notably, we also discovered third-order percolation transitions for the BP with $m=2$ in RRNs.
\section{Methods}
The strength of the largest cluster ($P_{\infty}$), which is the probability that a node belongs to the largest cluster, is the order parameter of the percolation transition~\cite{Stauffer94}.
We calculated $P_{\infty}$ for the DP and BP using the modified Newman-Ziff algorithms.
In the Newman-Ziff algorithm~\cite{Newman00,Newman01}, the average strength of the largest cluster $P_{\infty}(n)$ is calculated as a function of the number of initially active nodes ($n$), and $P_{\infty}(p)$ as a function of the initial probability of active nodes $p$ is obtained using the following convolution:
\begin{eqnarray}
P_{\infty}(p) = \sum_{n=0}^{N} B_p(n,N) \; P_{\infty}(n) , \label{canonical_transf}
\end{eqnarray}
where $B_p(n,N)={_NC_n} \, p^n (1-p)^{N-n}$ is the binomial distribution with ${_NC_n}={N!}/[n! (N-n)!]$. Here $N$ is the total number of nodes.
Since $P_{\infty}(p)$ for any value of $p$ can be calculated easily if $P_{\infty}(n)$ is obtained once, the Newman-Ziff algorithm is much more efficient than traditional brute-force methods.
Another advantage of the Newman-Ziff algorithm is that derivatives of a physical quantity can be obtained without numerical differentiation, which has inevitably a large error.
The first, second, and third derivatives of $P_{\infty}(p)$ can be obtained by
\begin{eqnarray}
\frac{d P_{\infty}(p)}{dp} &=& \frac{d}{dp} \left[\sum_{n=0}^{N} B_p(n,N)
\, P_{\infty}(n) \right] = \sum_{n=0}^{N} B_p(n,N) \frac{n-Np}{p(1-p)} \,
P_{\infty}(n) \\
\frac{d^2 P_{\infty}(p)}{dp^2} &=& \frac{d}{dp} \left[\sum_{n=0}^{N} B_p(n,N) \, \frac{n-Np}{p(1-p)} \,
P_{\infty}(n) \right] = \sum_{n=0}^{N} B_p(n,N) \, \frac{p(N-1)(Np-2n) + n(n-1)}{p^2(1-p)^2} \,
P_{\infty}(n) \\
\frac{d^3 P_{\infty}(p)}{dp^3} &=& \frac{d}{dp}
\left[ \sum_{n=0}^{N} B_p(n,N) \, \frac{p(N-1)(Np-2n) + n(n-1)}{p^2(1-p)^2} \,
P_{\infty}(n) \right] \nonumber \\
&=& \sum_{n=0}^{N} B_p(n,N) \, \frac{p^2(N-1)(N-2)(3n-Np)+n(n-1)[3p(2-N)+n-2]}{p^3(1-p)^3} \,
P_{\infty}(n) . \label{canonical_diff}
\end{eqnarray}
The derivatives of $P_{\infty}(p)$ are used to determine the percolation threshold and the order of percolation transitions.
Equations~(\ref{canonical_transf})-(\ref{canonical_diff}) are generally applied to other physical quantities.
The Newman-Ziff algorithm was modified for the DP and BP~\cite{Choi19}.
In the case of the DP, at each step of the Newman-Ziff algorithm, all nodes with at least $k$ active neighbors are activated recursively.
Pre-active (pre-occupied) state is introduced for the BP: the chosen node at each Newman-Ziff step is not activated but is set to be a pre-active state, which becomes active only when it has at least $m$ active neighbors.
Using these algorithms, the DP and BP can be studied as efficiently as the CP except for the BP with $m\geq3$.
As for the BP with $m=3$, it takes about ten times more CPU time than the CP~\cite{Choi19}, and it becomes worse for larger $m$.
To solve this problem, we proposed another algorithm for the BP in regular networks using the close relation of the DP and BP~\cite{Choi20}.
The algorithm was used for three-dimensional lattices successfully but it is inapplicable to nonregular networks.
In this paper, we extend the algorithm of Ref.~\cite{Choi20} to simulate BP with a large $m$ in nonregular networks.
Within the Newman-Ziff algorithm, the BP process begins from the initial state ${S_n}$ with $n$ active nodes at random. Then any node with less than $m$ active neighbors is deactivated (culled out) recursively.
This culling process of active nodes can be regarded as a diffusion process of inactive nodes with the new rule that active node with less than $m$ active neighbors is deactivated.
Therefore, the BP process from $S_n$ is equivalent to the diffusion process from $S_{N-n}$ with the new diffusion rule.
Note that nodes with less than $m$ neighbors should be excluded in the BP activation. The new algorithm can be implemented as follows.
\renewcommand{\labelenumi}{(\arabic{enumi})}
\begin{enumerate}
\item Initially, all nodes are set active. \label{item1}
\item Make an array of all the nodes in random order. \label{item2}
\item Set the step number to be $n'=1$.
\item Deactivate the $n'$-th node of the array made in step~(\ref{item2}). Deactivate any node that has less than $m$ active neighbors recursively.
If the newly deactivated node has at least $m$ neighbors, push the node in the stack with the step number $n'$.
Increase $n'$ by one. \label{item4}
\item Repeat step~(\ref{item4}) until all the nodes are deactivated. \label{item5}
\item Pop all the nodes with $n'$ from the stack and activate them.
Calculate $P_{\infty}(n)$ with $n=N-n'$. \label{item6}
\item Repeat step~(\ref{item6}) until the stack is empty.
\end{enumerate}
From step~(\ref{item1}) to step~(\ref{item5}), the diffusion process from $S_{N-n}$ with the new diffusion rule is saved in the stack; the stack is a linear data structure in the last-in-first-out (LIFO) order.
The information of the stack is used to simulate the BP process from $S_n$.
The whole steps are repeated to make an average,
and the transformations of Eqs.~(\ref{canonical_transf})-(\ref{canonical_diff}) give $P_{\infty}(p)$ and its derivatives.
This algorithm gives mathematically the same results as that in Ref.~\cite{Choi19}.
For $\Delta$-regular networks, the algorithm of Ref.~\cite{Choi20} is the most efficient since the DP and BP can be simulated simultaneously by one calculation for $m+k=\Delta+1$.
For nonregular networks, on the other hand, the algorithm of Ref.~\cite{Choi19} and the new algorithm should be used separately for the DP and BP, respectively.
The CPU time required for this algorithm is of the same order as the Newman-Ziff algorithm for the original percolation; it is proportional to the network size $N$ and average degree $\langle\Delta\rangle$ as $T_{\mathrm{cpu}} \sim \langle\Delta\rangle^{0.5} N^{1.2}$~\cite{Choi19}. It is notable that this algorithm can be efficiently parallelized within distributed memory techniques such as message passing interface (MPI).
In this work, we consider two kinds of complex networks: RRNs and ERNs.
RRNs with $N$ nodes of degree $\Delta$ are generated by the Steger-Wormald algorithm~\cite{Steger99,Choi20PhA}, which produces uniform RRNs asymptotically in the limit of large $N$~\cite{Kim03}.
ERNs are formed by the standard method~\cite{Erdos59}: arbitrary two nodes are connected by a link with probability $p_{\mathrm{ER}}$ and so the average degree is $\langle\Delta\rangle=(N-1) p_{\mathrm{ER}}$.
We studied RRNs and ERNs with $3\leq\Delta\leq10$ and $3\leq\langle\Delta\rangle\leq10$, respectively.
We made twenty realizations of networks for each parameter set; we confirmed that every realization gives equivalent results within statistical error bars.
\begin{figure}
\includegraphics[width=15.0cm]{figure1.pdf}
\caption{Strength of the largest cluster ($P_{\infty}$) and its derivatives as a function of initial filling probability $p$ for the DP, CP, and BP in regular random networks with $\Delta=4$. $N$ represents the number of nodes in the network.}
\label{Fig1}
\end{figure}
\begin{figure}
\includegraphics[width=15.0cm]{figure2.pdf}
\caption{Strength of the largest cluster ($P_{\infty}$) and its derivatives as a function of initial filling probability $p$ for the DP, CP, and BP in Erd\H{o}s-R\'{e}nyi networks with $\langle\Delta\rangle=4$.}
\label{Fig2}
\end{figure}
\section{Results and discussions}
Figures~\ref{Fig1} and \ref{Fig2} show strength of the largest cluster ($P_{\infty}$) and its first and second derivatives as a function of initial filling probability $p$ for the DP, CP, and BP in RRN of $\Delta=4$ and ERNs of $\langle\Delta\rangle=4$, respectively.
For the DP, CP, and BP of $m\leq2$, the percolation transition is continuous, and it is discontinuous (first-order) for the BP of $m=3$.
Note the double transition in the DP with $k=2$:
the second transition, which shows a discontinuous jump of $P_{\infty}$, appears at $p_{c2}$ with $p_{c2}>p_c$.
The transition at $p_{c2}$ is the hybrid transition, which combines a discontinuity and a singularity~\cite{Baxter10,Lee16}.
The three kinds of transition type (double transition, continuous, and first-order transition) are also observed in RRNs and ERNs of the other average degree.
The phase transition can be classified, in general, by the discontinuity of derivatives of the free energy~\cite{Ehrenfest33,Sauer17}.
The discontinuity can appear in two types.
In the magnetic phase transition of the two-dimensional Ising model~\cite{Ising25}, for example, the magnetic susceptibility and specific heat (second-order derivatives of the free energy) diverge at the transition temperature~\cite{Onsager44}.
In the case of the lambda transition~\cite{Keesom32}, to the contrary, there is a jump discontinuity in the specific heat.
If a discontinuity appears in the $n$th-order derivative of the free energy, it is called the $n$th-order phase transition.
If the first derivative of the free energy (e.g., order parameter) has a discontinuity at the transition point,
it is the first-order phase transition; otherwise, the phase transition is classified as the continuous phase transition.
With rare cases such as the Berezinskii-Kosterlitz-Thouless transition~\cite{BKT1,BKT2}, which is an infinite-order phase transition, most of the continuous phase transitions are of second-order.
Recently, third-order phase transitions were proposed concerning large-$N$ four-dimensional lattice gauge theory~\cite{Gross80}, spin glass~\cite{Crisanti03}, supercritical fluids~\cite{Koga09}, the domino tilings of an Aztec diamond~\cite{Colomo13}, and constrained Coulomb gas~\cite{Cunden17}.
In some superconducting transitions, third-order~\cite{Junod99,Werner99} and fourth-order~\cite{Kumar99,Hall00} phase transitions were suggested.
However, there are still controversies about the existence of higher-order phase transitions~\cite{Woodfield99,Wang11}.
In the case of the percolation transition, the order of the phase transition can be defined by the discontinuity of the order parameter $P_{\infty}$ and its derivatives:
if $P_{\infty}$ has a discontinuity at the transition, the percolation transition is classified as first-order~\cite{Gao12,Lee16}.
It is well-known that the classical percolation transition is of second-order~\cite{Stauffer94}.
However, recently, explosive percolation models~\cite{Achlioptas09} show very abrupt continuous or discontinuous phase transition depending on the spatial dimension and detailed percolation rules~\cite{Ziff09,daCosta10,Riordan11,Cho13}. (For a review, see Refs.~\cite{Boccaletti16,Lee16}.)
As for the DP and BP in lattices, the percolation transition is also second-order when the percolation threshold is between 0 and 1~\cite{Choi19,Choi20}. (Only one exception is known: the BP transition with $m=6$ in the face-centered cubic lattice at $p_c=0.75243(6)$ is first-order~\cite{Choi20}.)
\begin{figure}
\includegraphics[width=15.0cm]{figure3.pdf}
\caption{Histogram of the strength of the largest cluster ($P_{\infty}$) for the bootstrap percolation with $m=1$ in (a) and $m=3$ in (b) when initial filling probability $p$ is $p_c\pm 0.005$, $p_c\pm 0.002$, and $p_c$, where $p_c$ is the percolation threshold. The network is regular random with $\Delta=9$ and $N=160\,000$.}
\label{Fig3}
\end{figure}
\begin{figure}
\includegraphics[width=15.0cm]{figure4.pdf}
\caption{The maximum value of the second derivative of the strength of the largest cluster ($d^2P_{\infty}/dp^2$) as a function of the network size ($N$) of the BP with $m=2$ in regular random networks (RRNs) and Erd\H{o}s-R\'{e}nyi networks (ERNs) in log-log scale. Dashed lines serve as an eye guide.}
\label{Fig4}
\end{figure}
\begin{figure}
\includegraphics[width=15.0cm]{figure5.pdf}
\setlength{\fboxsep}{1pt}
\caption{Types of the percolation phase transition of the DP, CP, and BP in regular random networks (RRNs) and Erd\H{o}s-R\'{e}nyi networks (ERNs).
\fbox{D}, \fbox{1}, \fbox{2}, and \fbox{3} represent double, first-order, second-order, and third-order transitions, respectively.
\fbox{?} means that it is not clear whether the transition is of second-order or of third-order.}
\label{Fig5}
\end{figure}
As shown in Figs.~\ref{Fig1} and \ref{Fig2}, the DP and BP transitions in RRNs and ERNs are continuous except for the BP of $m\geq 3$, which is first-order.
Figure~\ref{Fig3} shows the histogram of the order parameter near the percolation threshold, which distinguishes continuous and first-order transitions clearly~\cite{Selke10,Azhari20}.
In the case of a continuous transition (the BP of $m=1$), there is only one peak in the histogram and the peak moves to higher $P_{\infty}$ as $p$ increases.
Near the first-order transition (the BP of $m=3$), to the contrary, there are two peaks, and the weights of the two peaks change with $p$.
Note that the BP of $m=2$ in RRNs shows the discontinuity not in $dP_{\infty}/dp$ but in $d^2 P_{\infty}/dp^2$, which means that the percolation transition is of third-order.
The discontinuity can be clarified by the maximum value of $d^2 P_{\infty}/dp^2$ as a function of the network size.
For second-order transitions, the maximum value of $d^2 P_{\infty}/dp^2$ diverges as $d^2 P_{\infty}/dp^2 \propto N^{\lambda}$, where $\lambda>0$ is a fitting parameter; this shows the discontinuity of $dP_{\infty}/dp$ at $p=p_c$.
For the BP of $m=2$ in RRNs, however, the maximum value of $d^2 P_{\infty}/dp^2$ is saturated as shown in Fig.~\ref{Fig4}(a), which implies the absence of discontinuity in $dP_{\infty}/dp$;
we confirmed that the maximum value of $d^3 P_{\infty}/dp^3$ diverges as $N$ increases.
In the case of the BP of $m=2$ in ERNs, as shown in Fig.~\ref{Fig4}(b), $d^2 P_{\infty}/dp^2$ behaves like in RRNs for small networks ($N<N_c$), but for $N>N_c$, it shows a diverging behavior as $N$ increases. The critical size $N_c$ increases as average degree $\langle\Delta\rangle$ increases roughly as $N_c \approx 6\exp(2\langle\Delta\rangle)$, and the divergence was not observed up to $N=2\,000\,000$ for $\langle\Delta\rangle \geq 7$.
Therefore, it would be reasonable to assume that the transition is of second-order in infinite networks irrespective of $\langle\Delta\rangle$.
Figure~\ref{Fig5} summarizes the types of the percolation transitions.
As for the RRNs, double transition behavior appears for the DP with $k \leq \Delta-2$.
Second-order transition is for the DP with $k \geq \Delta-1$, for the CP, and for the BP with $m=1$.
The BP of $m=2$ has a third-order transition and the BP with $m\geq3$ shows the first-order transition.
In the ERNs, the same sequence of phase transition type is also observed but the existence of a third-order percolation transition is questionable.
\begin{figure}
\includegraphics[width=15.0cm]{figure6.pdf}
\caption{Percolation threshold for the DP, CP, and BP in regular random networks (RRNs) and Erd\H{o}s-R\'{e}nyi networks (ERNs). ``1st'', ``2nd'', and ``3rd'' represent first-, second-, and third-order percolation transitions, respectively.
In the left panels, ``Double'' means second-order percolation followed by a discontinuous jump of $P_{\infty}$.}
\label{Fig6}
\end{figure}
\begin{table}
\caption{\label{tableRR} Percolation threshold ($p_{c}$) as a function of average degree ($\langle\Delta\rangle$) of the diffusion percolation (DP) and bootstrap percolation (BP) in regular random networks (RRNs) and Erd\H{o}s-R\'{e}nyi networks (ERNs).
The second transition probability ($p_{c2}$) is also provided for the double transition.}
\centering
\begin{tabular}{|c|c|cc|cc|c|c|c|}
\hline
& & \multicolumn{2}{|c|}{RRN-DP} & \multicolumn{2}{|c|}{ERN-DP} & & RRN-BP & ERN-BP\\
$\langle\Delta\rangle$ & $k$ & $p_{c}$ & $p_{c2}$ & $p_{c}$ & $p_{c2}$ & $m$ & $p_{c}$ & $p_{c}$\\ \hline
3 & $2$ & 0.1250(1) & - & 0.0374(4) & 0.0940(2) & $1$ & 0.5000(1) & 0.3335(4) \\
& $3$ & 0.3444(1) & - & 0.1399(4) & - & $2$ & 0.5000(1) & 0.3336(4) \\ \hline
4 & $2$ & 0.0472(1) & 0.1111(1) & 0.0201(3) & 0.0408(2) & $1$ & 0.3332(1) & 0.2500(2) \\
& $3$ & 0.1723(1) & - & 0.0886(4) & - & $2$ & 0.3332(1) & 0.2503(4) \\
& $4$ & 0.2840(1) & - & 0.2840(1) & - & $3$ & 0.8889(2) & 0.7644(4) \\ \hline
5 & $2$ & 0.0245(1) & 0.0508(2) & 0.0129(1) & 0.0242(1) & $1$ & 0.2499(1) & 0.2000(1) \\
& $3$ & 0.1060(1) & 0.2751(1) & 0.0630(1) & 0.1366(3) & $2$ & 0.2499(1) & 0.2000(3) \\
& $4$ & 0.1892(1) & - & 0.1190(1) & - & $3$ & 0.7249(2) & 0.6475(2) \\
& $5$ & 0.2399(1) & - & 0.1630(4) & - & $4$ & 0.9492(2) & 0.9326(2) \\ \hline
6 & $2$ & 0.0150(1) & 0.0291(1) & 0.0089(1) & 0.0161(2) & $1$ & 0.2000(1) & 0.1666(2) \\
& $3$ & 0.0732(1) & 0.1651(2) & 0.0476(2) & 0.0938(3) & $2$ & 0.1999(1) & 0.1668(5) \\
& $4$ & 0.1380(1) & 0.3972(2) & 0.0935(2) & 0.2298(2) & $3$ & 0.6028(2) & 0.5514(3) \\
& $5$ & 0.1837(1) & - & 0.1314(2) & - & $4$ & 0.8349(2) & 0.8267(4) \\
& $6$ & 0.1987(1) & - & 0.1545(1) & - & $5$ & 0.9709(1) & 0.9956(4) \\ \hline
7 & $2$ & 0.0101(1) & 0.0188(1) & 0.0065(1) & 0.0115(1) & $1$ & 0.1666(1) & 0.1428(1) \\
& $3$ & 0.0542(1) & 0.1129(1) & 0.0376(1) & 0.0703(2) & $2$ & 0.1665(1) & 0.1429(2) \\
& $4$ & 0.1067(1) & 0.2690(2) & 0.0763(1) & 0.1668(3) & $3$ & 0.5137(2) & 0.4764(1) \\
& $5$ & 0.1471(1) & 0.4863(2) & 0.1096(2) & - & $4$ & 0.7310(2) & 0.7255(1) \\
& $6$ & 0.1641(1) & - & 0.1309(2) & - & $5$ & 0.8871(2) & 0.9267(2) \\
& $7$ & 0.1665(1) & - & 0.1398(2) & - & $6$ & 0.9812(1) & 1.0000(1) \\ \hline
8 & $2$ & 0.00726(1)& 0.0132(1) & 0.00500(4)& 0.0086(1) & $1$ & 0.1429(1) & 0.1250(2) \\
& $3$ & 0.0422(1) & 0.0832(2) & 0.0308(4) & 0.0555(1) & $2$ & 0.1428(1) & 0.1250(3) \\
& $4$ & 0.0859(1) & 0.1992(2) & 0.0641(2) & 0.1317(2) & $3$ & 0.4468(2) & 0.4181(4) \\
& $5$ & 0.1217(1) & 0.3540(2) & 0.0938(2) & 0.2335(2) & $4$ & 0.6459(2) & 0.6402(2) \\
& $6$ & 0.1391(1) & 0.5532(2) & 0.1135(4) & - & $5$ & 0.8008(2) & 0.8350(2) \\
& $7$ & 0.1426(1) & - & 0.1222(2) & - & $6$ & 0.9168(2) & 0.9816(2) \\
& $8$ & 0.1428(1) & - & 0.1246(2) & - & $7$ & 0.9868(1) & 1.0000(1) \\ \hline
9 & $2$ & 0.00548(1)& 0.0098(1) & 0.0039(2) & 0.0067(2) & $1$ & 0.1250(1) & 0.1111(1) \\
& $3$ & 0.0340(1) & 0.0645(2) & 0.0257(1) & 0.0452(2) & $2$ & 0.1249(1) & 0.1110(2) \\
& $4$ & 0.0713(1) & 0.1557(2) & 0.0548(2) & 0.1081(3) & $3$ & 0.3949(2) & 0.3723(3) \\
& $5$ & 0.1032(1) & 0.2755(2) & 0.0815(3) & 0.1896(4) & $4$ & 0.5771(2) & 0.5713(5) \\
& $6$ & 0.1203(1) & 0.4229(2) & 0.0999(2) & 0.2925(2) & $5$ & 0.7245(2) & 0.7510(6) \\
& $7$ & 0.1245(1) & 0.6051(2) & 0.1083(3) & - & $6$ & 0.8443(2) & 0.9091(8) \\
& $8$ & 0.1250(1) & - & 0.1107(1) & - & $7$ & 0.9355(2) & 1.0000(1) \\
& $9$ & 0.1250(1) & - & 0.1111(2) & - & $8$ & 0.9902(1) & 1.0000(1) \\ \hline
10& $2$ & 0.00428(5)& 0.0075(1) & 0.0032(1) & 0.0054(1) & $1$ & 0.1111(1) & 0.1000(2) \\
& $3$ & 0.0282(1) & 0.0519(1) & 0.0219(1) & 0.0378(2) & $2$ & 0.1110(1) & 0.0998(2) \\
& $4$ & 0.0604(1) & 0.1263(2) & 0.0478(2) & 0.0912(2) & $3$ & 0.3537(2) & 0.3351(3) \\
& $5$ & 0.0891(1) & 0.2234(2) & 0.0720(2) & 0.1597(1) & $4$ & 0.5207(1) & 0.5145(4) \\
& $6$ & 0.1058(1) & 0.3407(2) & 0.0891(2) & 0.2422(1) & $5$ & 0.6593(2) & 0.6783(4) \\
& $7$ & 0.1105(1) & 0.4793(1) & 0.0972(3) & 0.3448(2) & $6$ & 0.7766(2) & 0.8296(3) \\
& $8$ & 0.1111(1) & 0.6463(2) & 0.0995(2) & - & $7$ & 0.8737(2) & 0.9585(2) \\
& $9$ & 0.1111(1) & - & 0.0999(2) & - & $8$ & 0.9481(1) & 1.0000(1) \\
& $10$& 0.1111(1) & - & 0.1000(2) & - & $9$ & 0.9925(1) & 1.0000(1) \\ \hline
\end{tabular}
\end{table}
We used two kinds of methods to calculate the percolation threshold $p_c$.
For $n$th-order percolation transitions, the finite-size percolation threshold $p_c(N)$ for a finite network of size $N$ is defined by the initial probability that maximizes $d^n P_{\infty}/dp^n$.
The percolation threshold of infinite network ($p_c$) can be obtained by the scaling relation $[p_c(N) - p_c] \propto N^{-a}$ with a positive fitting parameter $a$.
The other method uses the fact that values $d^{n-1} P_{\infty}/dp^{n-1}$ of different sizes coincide at $p_c$ for $n$th-order percolation transitions, which is confirmed in Figs.~\ref{Fig1} and \ref{Fig2}.
The two methods give equivalent results within error bars.
The percolation threshold of the DP, CP, and BP obtained by these methods are in Fig.~\ref{Fig6} and Table~\ref{tableRR}.
The percolation threshold of the CP in sparse, undirected, uncorrelated complex networks of infinite size is known analytically to be $p_c = \langle\Delta\rangle/(\langle\Delta^2\rangle - \langle\Delta\rangle)$~\cite{Cohen00}.
In the cases of RRNs, $\langle\Delta\rangle=\Delta$ and $\langle\Delta^2\rangle=\Delta^2$ and so $p_c = 1/(\Delta-1)$ for the CP.
The percolation threshold of the CP in ERNs is $p_c = 1/\langle\Delta\rangle$ since $\langle\Delta^2\rangle=\langle\Delta\rangle^2+\langle\Delta\rangle$ in ERNs~\cite{Newman00}.
We confirmed that all $p_c$ values obtained numerically in this work are consistent with these analytic results within 0.05\%.
Besides, Table~\ref{tableRR} verifies that the BP with $m=1$ and $m=2$ has the same percolation threshold as the CP~\cite{Adler90}.
\section{Summary}
We presented a new efficient algorithm that applies to the BP model in nonregular networks within the Newman-Ziff algorithm.
With the new algorithm and the algorithms of Refs.~\cite{Choi19,Choi20}, we calculated the strength of the giant cluster $P_{\infty}(p)$ and its derivatives for the CP, DP, and BP in RRNs and ERNs.
Based on the results, we classified the percolation transitions.
The DP with a small $k$ has the double transition, which is a second-order percolation transition followed by the discontinuous jump of $P_{\infty}$ at higher $p$.
The DP with a large $k$, CP, and BP with a small $m$ shows the second-order percolation transition.
The BP with $m\geq3$ has the first-order percolation transition.
The third-order percolation transition occurs in the BP of $m=2$ in RRNs.
The DP and BP are basic models for various real phenomena in complex systems such as neuronal activity, jamming transition, opinion formation, and diffusion of innovations.
Therefore, the results of this work, especially, the detailed analysis of the percolation transitions and the discovery of third-order phase transition would help to further understand dynamics and critical transformations in complex systems.
\section*{Acknowledgments}
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1F1A1052117).
\bibliographystyle{elsarticle-num}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Q: Orika - list to list conversion This is probably an easy one, but I cant find it in the docs.
I have a person class
class BasicPerson {
private String name;
private int age;
private Date birthDate;
// getters/setters omitted
}
and a list of it
ArrayList<Person>
I want to change them to change them to
ArrayList<PersonDTO>
but with out an explicit loop.
Is there a way to use MapperFacade.map for a list to list one line conversion ?
A: If you use the MapperFacade interface, Orika can perform the mapping multiple times on the collection:
final MapperFacade mapperFacade = mapperFactory.getMapperFacade();
final List<Person> people = // Get the person instances
final List<PersonDto> personDtos = mapperFacade.mapAsList(people, PersonDto.class);
On the other hand, if you use the BoundMapperFacade interface, it doesn't contain such a convenience method.
And lastly, if you choose to use the ConfigurableMapper approach, it also includes a mapAsList method which in fact delegates to the MapperFacade.mapAsList method.
A: It has this functionality built-in. Did you tried using the method
List<D> ma.glasnost.orika.impl.ConfigurableMapper.mapAsList(Iterable<S> source, Class<D> destinationClass)?
I tried to find a updated version of the Javadoc, but here is one of the 1.3.5. The current version is 1.4.5.
MapperFacade Class
A: package com.miya.takeaway.common.util.orika;
import com.miya.takeaway.common.util.orika.converter.ConverterHelper;
import ma.glasnost.orika.MapperFactory;
import ma.glasnost.orika.converter.ConverterFactory;
import ma.glasnost.orika.impl.DefaultMapperFactory;
import ma.glasnost.orika.metadata.ClassMapBuilder;
import org.apache.commons.collections4.CollectionUtils;
import java.util.Collections;
import java.util.List;
public class OrikaUtils {
private static MapperFactory mapperFactory = new DefaultMapperFactory.Builder().build();
public static <T> List<T> map(List<?> objects, Class<T> target) {
if (CollectionUtils.isEmpty(objects)) {
return Collections.EMPTY_LIST;
}
return mapperFactory.getMapperFacade().mapAsList(objects.toArray(), target);
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,834
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Horn is the Hornets Hero in D1 Quarterfinal
Olivia Horn (8) scored the game-winning goal with 19 minutes remaining to send Mansfield into the Div. 1 South semifinal. (Josh Perry/HockomockSports.com)
By Josh Perry, Managing Editor
MANSFIELD, Mass. – With 19:37 left in the a scoreless game, Wellesley coach Erin Stickle called a timeout. Up until that point, the Raiders had dominated play, albeit without finding the back of the net. Twenty-six seconds after the timeout, Mansfield scored.
Olivia Horn, playing her final game before going on a mission trip to Africa, swept home a centering pass by Caitlin Whitman on one of the few clear chances the Hornets had up to that point.
Mansfield controlled play for the final 19 minutes and, with help from a last-minute save by goalie Brenna Prior, pulled out a 1-0 win over the Raiders at Alumni Field and advanced to the Div. 1 South semifinal to face rival Franklin.
The Raiders had started the second half strongly and Prior was forced into several clutch saves to stop Ling Groccia and Danielle Hickman from opening the scoring. Groccia also whiffed on a good look from close-range.
The Hornets had escaped, but head coach Theresa Nyhan was hesitant to call a timeout. She smiled when asked about Wellesley taking a timeout and allowing her to settle down her team.
"I'm glad she called the timeout," said Nyhan. "The clock doesn't stop so you want to keep those in your pocket."
Out of the timeout, Mansfield started to push forward and Whitman got room on the left side of the attack. She played a ball across the middle of the crease where Horn was stationed and the junior attacker redirected the pass inside the post for the goal.
"We have a player in her last game because she's going on a mission trip," said Nyhan, indicating that it was Horn that would miss the remaining tournament games. "She played out of her mind today."
For the first 41 minutes of the game, it was the Raiders that had the territorial edge. Wellesley was bottling up the Hornets attack, particularly limiting the touches of senior Caroline Maher, who was moved from the middle out to the wing to try and find space.
"When you score four goals in a game (against Duxbury in the first round) you're going to get attention," said Nyhan of the Wellesley defense keying on Maher. "It's great to see other players step up and make plays."
The Raiders thought that they had taken the lead in the ninth minute when a long shot from nearly midfield took a deflection and snuck past Prior. The referee waved it off because the deflection was by a Mansfield defender not a Wellesley player and thus did not count.
Horn had the two best chances for the Hornets in the opening half, but both were saved at the near post. Following the Mansfield goal, Julia Todesco had a shot from the middle of the crease kicked aside and Horn had a shot from a Todesco pass that was saved.
Amalia Todesco had a chance off a corner that was saved and then Brianna Puccia had the follow up blocked off the line by a Wellesley defender.
With five minutes remaining, Mairead Dunn was given a card for blocking a restart, which left the Hornets down a player. Amalia Todesco and Jennifer McCabe were immense down the stretch intercepting Wellesley passes towards goal.
The Raiders continued to press until the final whistle and created one last chance with only 59 seconds remaining, but Prior once again kicked out a leg to stop the shot and the rebound was cleared over half.
"Everyone needs to make plays and she stepped up with her leg and her arm and came through with a big stop at the end," said Nyhan.
The final whistle blew and the Hornets mobbed the goalie and celebrated another big win to advance to the sectional semifinal. Standing in their way will be Franklin, their rival and the team that they edged for the league title this season.
The semifinal will also provide the chance to determine the winner of the season series with the two teams splitting their regular season games.
"They're a strong team, well-coached and…I love the Hockomock," said Nyhan, "and I'm glad they weren't knocked out."
The semifinal will be played on Wednesday at 2 p.m. at Alumni Field.
Josh Perry can be contacted at JoshPerry@hockomocksports.com and followed on Twitter at @Josh_Perry10.
Ryan Lanigan
Ryan Lanigan is the founder and Editor-in-Chief of HockomockSports.com. You can contact him at RyanLanigan@hockomocksports.com and follow him on Twitter at @R_Lanigan.
Latest posts by Ryan Lanigan (see all)
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Canton Extends Unbeaten Streak With Win Over KP - 01/11/2020
Amalia Todesco Brenna Prior Caitlin Whitman Caroline Maher Jennifer McCabe Julia Todesco Mansfield Olivia Horn Theresa Nyhan Wellesley
← Sunday's Schedule & Scoreboard – 11/08/15
Field Hockey Photos: Mansfield vs. Wellesley →
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,741
|
Beyond ETFs - investing by you 4+
Buy what stocks, when to sell
Digital Hermosa
Beyond ETFs gives the two most important pieces of information an investor needs to know - what to buy, and when to sell. We do this in a way that leads the subscriber to beat the S&P100 Index. We offer an instant scorecard to show how it all shakes out. This is a truly innovative, all-mobile and inexpensive way to manage investing by yourself.
Why the S&P100? The S&P100 stocks are the finest of all US companies - they're name-brands. You've heard of (virtually) every one of them. They're not going away. Some have dividends, some don't. Each of them put billions (sometimes trillions) of dollars of investor's capital to work, every day.
The trouble with ETFs is they hold ALL the stocks in their category. So, a typical S&P100 index, such as OEF, holds the winner stocks in the S&P100, and at the same time, they hold the loser stocks in the S&P100.
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In Beyond ETFs, we use PRICE MOMENTUM to rank order each of the stock in the Index. We automatically apply the Brockmann Method to place each stock in the BUY* ZONE, the DON'T BUY MORE ZONE and below the SELL* THRESHOLD, into the AVOID ZONE.
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Beyond ETFs Pro offers a scorecard where users can compare their portfolio performance with the S&P 100 ETF (OEF), instantly and over time. Graphs. Stats. News. Insights. Coaching Tips.
Disclaimer: Brockmann & Company are not a registered broker-dealer or investment adviser; consult your own investment adviser for any financial advice. Peter and Wilfred Brockmann may own some or all of the stocks discussed. There are always risks associated with owning stock including loss of capital and high, low or no returns.
Investing was never this clear - what to buy, when to sell.
Background tasks analyze Zone Changes report and provide insights on the lock screen.
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Beyond ETFs
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,991
|
The section in which smartphone manufacturers are mostly focusing efforts is to improve their cameras. Especially in the high end models, because that is the feature that marks the difference between high end and low end phones.
It is perhaps because of this, Samsung faced with the S5, the challenge to overcome the quality and performance of the camera from its previous version, the Galaxy S4, one of the most balanced we've seen on a mobile device. We must also overcome, or at least match, the photographic abilities of the top rivals: iPhone 5s, HTC One (M8), the high end Nokia Lumias, the Sony Xperia Z2 and some Chinese devices.
The problem for the South Korean and other manufacturers is that with current imaging technologies, mobile cameras are hitting the roof. Therefore, increasing innovation cause higher costs when it comes to upgrading that component if the traditional design of the device is not altered, which only Nokia and Samsung have dared to achieve. The first was Nokia, giving their Lumia 1020 an astounding sensor protruding from the back body of the phone, and the second was Samsung, when they equipped the Galaxy S4 Zoom with a 10x lens. Although it is no secret that these two excellent devices have not exactly been bestsellers.
The main innovation on the Galaxy S5 camera, is the self-made sensor. It uses an exclusive technology called Isocell, which consists in placing a kind of barrier between the photodiodes that capture the light to turn it into pixels. This way avoiding the light rays that impacting each photodiode from contaminating the surrounding.
With this method Samsung says it is possible to achieve more accurate and detail-rich images.
On paper it sounds good, but in reality we don't find that there has been a qualitative leap. The fault is most likely to be sought in the high resolution sensor: 16 megapixels, something that many potential buyers will see as an evolution from the 13 megapixel Galaxy S4, when in fact it is rather a drag.
It is true that several direct competitors, exceed the resolution of the Galaxy S5: the Sony Xperia Z1 and Z2 and Lumia 1020, 930 and 1520. But what all those devices have in common, is that their megapixels are motivated by pixel oversampling technology used, which gives the main advantage of using digital zoom, without a great loss in image quality. This requires reducing the resolution to 8 megapixels on Sony devices, and 5 on Nokia devices.
The high resolution on the Galaxy S5 seems to be more to marketing fact than anything else. So we are left with the desire to see what could have been achieved by the Isocell technology if Samsung had opted to give it a lower megapixel sensor.
In any case, it is true that the sensor is larger than other phones, including the Galaxy S4 with dimensions of 1/2.6", very similar to most mobile cameras. Due to this fact, despite 16 megapixels, the pixel size is 1.12 microns, the same value as the 13 megapixel camera on the Galaxy S4.
Another fault on the Galaxy S5 can be seen in the 31mm. lens, since the maximum aperture is limited to f/2.4. Not too bad, but there are now many mobile phones that offer greater apertures, which is very useful when you are shooting in low light situations and want to preserve image quality.
It also lacks the addition of a stabilized lens like the one included in many Nokias, which would avoid having to raise the ISO when ambient light is not at its best. In fact, on tests made with the Galaxy S5, we have noticed that working with high ISO levels of 400 in interiors, causes images to blur.
It was also convincing that Samsung limited the ISO to 800, and, although other manufacturers offer higher values, the results result extremely low. The best example is an image posted on Flickr a few months back to appreciate the real size detail of a photo shot with the Sony Xperia Z1 at 3200 ISO. Speechless.
The only mobile device that we have tested that has beat any other shooting photos at ISO levels higher than 800, is the Nokia Lumia 1020. In any case, if we were to use the Galaxy S5 to shoot at low light levels, recurring to ISO levels of 800 should be our last option, seeing that the results are not too convincing.
Where we do see an excellent behavior is on the Galaxy S5 focus, being that the focus speed is fast enough to be considered the fastest he have ever tested on a mobile phone. To manage this, it depends on a hybrid technology that combines focus and contrast, typical of all mobile phone cameras, with phase detection, which is the system used by reflex cameras and some cameras without interchangeable lens mirrors.
But its rapid shooting is not only seen on the focus. The Galaxy S5 has a Snapdragon 801 2.5GHz Quad Core that allows it to shoot many frames simultaneously to obtain High Dynamic Range photos (HDR). It is so fast at this feature, that during our tests with it, despite the fact that there were many moving objects in the scene, including moving vehicles, we did not see ghost blur in any of the images that we shot while testing this feature. Results can be seen in the video with image tests.
We also have the ability to initially view on the screen what our HDR will look like after shot. The best thing that we noticed about this feature, in our opinion, is the outstanding color treatment on the HDR images. These do not suffer from the over-saturated colors that are the result on many other phones.
The Galaxy S5 doesn't quit in providing various functions to the camera application, as we have seen in previous Samsung phones. To be exact, these functions were already abundant, Samsung now offers the possibility of downloading features into the camera application directly from their app store. This is why the default functions do not include creating animated images, which is one of the preferred functions of GIF animators.
Last, but not least, another great feature of the Galaxy S5 camera, is the ability to record in 4k resolution, a feature that is already included in other devices such as the Note 3.This resolution results to be the less used, due to the fact that there are not many screens that can play this High Resolution, but it is still possible to practice curious experiments with these clips, as we did in a self-recorded video we made, and printing a single 4K frame recorded with a Note 3.
What Samsung has achieved by adding this function is the fact that you can record using the HDR, something that no other device can do.
It is definite, that in Image Processing, this Samsung will not fail you and is very much ahead of the Galaxy S4, but it will also have us in the worry that this manufacturer still has some bugs and details that it must fix, like being a bit more detailed with the resolution that the device reaches to give us better resulting images, adding a lens with higher aperture and stability and maybe even add a camera shutter release.
Previous The Fuji X -Pro2 will have a APS-C format sensor.
Next Oppo N1 Mini, more compact, maintaining a Rotating Camera.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,326
|
Some curious numbers from my HP-67
Palmer O. Hanson, Jr.
A recent thread compared the capability of the HP-67 and the SR-52. While looking through the HP-67 Owner's Handbook and Programming Guide I found the following statement on page 109
"..Unlike storage register arithmetic, the Sum+ function allows overflows (i.e., numbers whose magnitudes are greater than 9.999999999x10^99) in storage registers RS4 through RS9 without registering Error in the display. ..."
I tested the statistics routine of my HP-67 with the following sequence:
1. Start with clear .registers.
2. Press 1 ENTER Sum+ to enter the first data pair. See 1.00 in the display.
3. Press 2 ENTER Sum+ to enter the second data pair. See 2.00 in the display.
4. Press 3 ENTER Sum+ to enter the third data pair. See 3.00 in the display.
5. Press 6 EE 99 ENTER 8 EE 99 Sum+ to enter a fourth data pair. See 4.00 in the display. There will not be an error indication.
6. Press f xbar to calculate the means. See 2.000000000 99 in the display. Press h x<>y and see 1.500000000 99 in the display. These number makes sense as the mean values and can be duplicated by pressing f P<>S and calculating RCL4 4 / and RCL6 4 / .
7. If you interchanged the primary and secondary registers to cross check the calculation of means in step 6 then interchange them again.
8. Press g s to calculate the standard deviations. See 2.309401077 99 in the display. Press h x<>y and see 1.732050808 99 in the display. Those numbers are the square root of 16/3 times 1 E 99 and the square root of 3 times 1E99.
If I interchange the primary and secondary registers and attempt to verify the numbers by calculating the standard deviations using the formulas on pages 113-114 of the manual I get 5.000000000 49 for both standard deviations.
I have tried calculating the standard deviations without interchanging the primary and secondary registers by using indirect arithmetic and I still get 5.000000000 49 as the values.
So my question is "Where do the numbers in step 6 come from?,
Karl Schneider
Hi, Palmer --
The mean values in Step 6 are fine, as you stated. Maybe you should be asking about the standard-deviation calculations in Step 8, which, if based on summation methods (as expected), would utilize overflowed values.
The HP-15C does not allow overflow summation values in the summation registers. Flag 9 is set (display blinks) and "9.999999999 99" is stored in registers R4, R6, and R7 (x2, y2, and xy). Means can be calcluated, but standard deviations cannot, due to capped summation values.
I don't understand the rationale behind the "special treatment" of summation. Is there a different data format for the summation registers than for stack and other storage registers? If so, recall of "special" stored numbers could lead to erroneous and unpredictable results in calculations.
-- KS
Karl:
You are correct. I meant to write Step 8, but somehow went to print with step 6.
You also wrote:
The only thing that I can think of at the moment is that the statistics routines use something similar to the so-called "tricky properties" available in the HP-41 and in subsequent machines; however, when I tried to observe the "tricky properties" in the HP-67 a few years ago I wasn't able to see them. Maybe I will have to try again. But, clearly, the standard deviation calculation with g s does something different from what happens when I do the supposed equivalent from the keyboard.
The methodology for handling overflow has varied a lot over the years. My HP-33C and HP-38C will accept a summation input which causes an overflow in the summations of squares without warning the user. But, when the user tries to do the standard deviation calculations the machines return an error indication.
The TI-59 flashes an error after a summation input has caused an overflow but has done the summations.
I like what the TI-66 does. If the summation input will cause an overflow the summations are not performed and "Error" is displayed to tell the user that there is a problem.
The TI-68 does something entirely different. If an overflow occurs the statistics registers are cleared!
...so-called "tricky properties" available in the HP-41 and in subsequent machines...
"tricky properties", of course, is a term from the HP-15C Advanced Functions Handbook for extended-precision internal arithmetic in statistical-summation calculations, exploited using "Sigma-" to calculate a near-zero quadratic-function discriminant of the form B2-AC with increased accuracy.
http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv018.cgi?read=130481#130481
Not quite the same thing as storing the summation results to numbered registers in a different format that allows larger exponents.
J-F Garnier
Hi Palmer,
Interesting observations on the HP-67!
Try this even simpler test:
1. Start with clear registers and stack.
2. Press 1 EE 80 Sum+ . See 1.00 in the display.
3. Press 3 EE 80 Sum+ to enter the second data. See 2.00 in the display.
4. Press f xbar to calculate the mean. See 2.000000000 80 in the display (OK). Press g s to calculate the standard deviation. See 2.8284.... 80 in the display. The correct answer should be 1.4142.... 80.
I believe that even if there is no overflow error and that internal math may be able to compute with exponents larger than 99, then the results are not reliable due to the fact than the square sum registers are 'clamped' to 9.999... 99 during data input.
J.F.
I tend to agree; however, if you do my sequence with x or y values for the fourth data point with integer mantissas ranging from 1 through 10 and the exponent as 99 you will find that the mantissa of the standard deviation result will be the mantissa of the input divided by the square root of 12, where of course 12 is n(n-1). This suggests that although the summation of the squares may read out to the display as 9.9999999999 E 99 the number actually in the summation register may be correct. Strange indeed!
Here is my analysis, with your test case:
The formula used to calculate Sx is:
Sx=sqrt((R5-R4^2/R9)/(R9-1))
if the HP67 registers would be able to hold exponents above 1e99, the registers would be:
R4=8E99
R5=6.4E199
R9=4
and the result would be:
Sx=4E99
My hypothesis is that R5 is actually clamped to 9.99..E99 and is negligible compared to R4^2/R9, and that the HP-67 seems to ignore the resulting negative sign so Sx is evaluated (more or less) as Sx=R4/sqrt(R9*(R9-1)) i.e. 2.30..E99.
Edited: 26 May 2008, 7:29 a.m.
You are correct that the sum of the squares of the inputs overflows and becomes 9.999999999E99.
As I understand what you have written (8E99)2/4 must be much larger than 9.999999999E99 . But it won't be because the square of the sum of the input values (a little bit larger than 8E99) also overflows. Thus, the square of 8E99 becomes 9.999999999E99. That value divided by 4 becomes 2.500000000E99. Subtracting that from the 9.999999999E99 from R5 yields 7.499999999E99. Dividing by 3 yields 2.500000000E99 and taking the square root yields 5.000000000E49.
You will get that answer if you used 6E99 instead of 8E99 at the beginning, or for that matter, if you use any value which overflows the summation of squares, since it will also overflow the square of the sum of input values.
You will get the same answer whether you use P<>S so you can do the recalls directly,or if you don't do P<>S and use indirect recalls.
My only idea so far is at least implausible. Suppose that the statistics calculations in the HP-67, while not having the "tricky properties" of the HP-41 and subsequent machines per se, have their own set of "tricky properties" which can somehow keep track of large numbers like th square of 8E99, and that the statistics calculations can somehow use those values even though I can't see them in keyboard calculations. Then it would be possible to get the strange results. That sounds far-fetched but it is not too far removed from the situation with the "tricky properties" of the HP-41 et al
Thatt's the best I can come up with so far. Bizarre, to say the least.
I tested overflow in the statistics routines with other HP machines in my collection using the four data pairs (1,1), (2,2), (3,3) and (6E99, 8E99).
My HP-27 yields results which are similar to those from my HP-67, but with an important difference. When the fourth pair is entered the calculator displays OF. Page 48 of the HP-27 Owner's Handbook states "If the value in a storage register exceeds 9.9999999 x 1099 the following indicator appears on the display: OF". The summations are made. If I clear the display and proceed I find that the means are correct. The standard deviations are displayed as 2.3094011 99 and 1.7320508 99 . If I divide the displayed values by 1E99 the displays become 2.309401077 and 1.732050808 which are the same ten digits as in the mantissas of the standard deviations returned from the HP-67. The linear regression function (L.R.) yields an intercept of 0.3125 and a slope of 0.75 . If I calculate the standard deviations from the keyboard using sums recalled from the statistics registers I get the 5.0000000 49 display.
Other ten digit mantissa machines in my collection such as the HP-33C, HP-38C, HP-11C, HP-12C and HP-41 do not give a warning of overflow when the fourth pair is entered and do make the summations. Each machine calculates the means correctly. Each machine returns an error message when asked for standard deviations.
Twelve digit mantissa machines in my collection such as the HP32S, HP-33S and HP-35S flash OVERFLOW when the fourth pair is entered. NOTE: to get overflow the fourth pair must be (6E499, 8E499) not (6E99, 8E99) as with the ten digit mantissa machines. The means are calculated correctly. The response to a request for standard deviations is STAT ERROR.
Finally, my HP-10B flashes OFLO when the fourth pair is entered. The summations are made. The means are correct. When I ask for the standard deviation the machine responds with Error - StAt.
So, at least for the machines in my collection, the HP-67 is unique in not providing any indication of overflow to the user. A bit bizarre! One wonders why the designers did that.
In my understanding, the HP-67 is unique not in providing no overflow indication (other machines do the same), but in providing no error indication in case of invalid standard deviation.
For instance if, after accumulating some data, you clear the square sum register (P<>S, 0 STO 5, P<>S), you will get a result for the standard deviation instead of an error indication. We could call it a bug.
Yes, the '67 can work with exponenents up to 499 internally. That's what is happening here with the '67 statistics bug.
I hate to burst anybody's bubble, but these "non-normalized numbers" (NNN's) caused by the '67 statistics functions were thoroughly explored (and written about) over 30 years ago, in the "65 Notes" newsletter (which soon afterwards changed its name to the "PPC Journal").
The whole story behind HP-67 NNN's can be found here:
http://holyjoe.net/hp/nnn.htm
-jkh-
Hello Joe,
Thanks for this valuable information! It was really a pleasure to read your post, and your page about the NNN's on the HP67. My first personal HP calculator was a HP-41CV in the beginning of the 80's so I'm not a HP-67 expert, but I always considered this machine as one of the most important (and beautiful) model from HP and I was very happy when I got one some years ago.
Creating NNN's that are beyond the 9.99...E99 limit with the x-bar function is indeed very easy, for instance (just to describe the process to all):
1E60 STO 6 1/x STO 9 P<>S x-bar
creates the 1E120 NNN in Y, that is displayed as "-1.000000000 -20" by the STK command.
Jean-Francois Garnier
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 9,198
|
Contemporary, United Kingdom
On the market: Architect-designed five-bedroomed hillside house in Totley, Sheffield, Yorkshire
Architect-designed five-bedroomed hillside house in Totley, Sheffield
Now here's something, this architect-designed five-bedroomed hillside house in Totley, Sheffield, which perhaps has a smaller selling price that you would initially think.
I'd have nailed this as a period property, probably from the 1960s, But it's actually billed as a 'contemporary' design, although the agent claims it has been 'undergone considerable refurbishment to the highest specification in the last 12 months' – which leads us to believe it does actually have some years on the clock. Either way, it's an impressive build.
It's also got all plenty of modern technology too from that refit – a remote controlled multi-zone sound and media system which includes Bose speakers, Porsche designed intelligent lighting system, wired and wireless networking for high speed internet connections, as well as an upgraded heating system incorporating air filtration and cooling.
In terms of accommodation, you enter via double opening double glazed reception doors with matching arched double glazed windows above, using a secure video and voice entry system. Once in there's a an atrium-style reception hall with 'water feature', along with a high-end fitted 'living kitchen', a large with floor-to-ceiling windows, five bedrooms (one currently doubling up as an office), utility area, dayroom, an additional reception room, a family bathroom (with Philippe Starck fittings), an en-suite for the master bedroom, a separate shower room and a separate WC to name just some of the things on offer here.
Outside, there's a wraparound external decking area, illuminated paths, mature garden and lawn areas, two brick-built garages, a timber shed and a timber summerhouse. There's also some amazing views of the nearby Derbyshire countryside too.
To be honest, that's only touching the surface, check out the agent's site for the full details and more images. £545,000 is the asking price, which certainly isn't cheap. But perhaps cheaper than we would think for a property of this style.
Tags: architect, contemporary, design home, house, property, sale, Sheffield, style, Totley, Yorkshire
On the market: Donal Hutchinson-designed contemporary modernist property in Hove, East Sussex
Six-bedroom Grand Designs property in Brighton, Sussex
On the market: Hans Klaentschi and Paula Klaentschi-designed Long Barn property in Berwick St. James, Wiltshire
On the market: Two-bedroom art deco-style apartment in Leigh-On-Sea, Essex
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,786
|
{"url":"http:\/\/dml.cz\/dmlcz\/143612","text":"# Article\n\nFull entry | PDF \u00a0 (0.1 MB)\nKeywords:\nreflexive algebra; reflexive lattice; subspace lattice; bilattice\nSummary:\nWe study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice $\\mathcal {L}$ we may associate a bilattice $\\Sigma _{\\mathcal {L}}$. Similarly, having a bilattice $\\Sigma$ we may construct a subspace lattice $\\mathcal {L}_{\\Sigma }$. Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.\nReferences:\n[1] Davidson, K. R., Harrison, K. J.: Distance formulae for subspace lattices. J. Lond. Math. Soc., II. Ser. 39 (1989), 309-323. DOI\u00a010.1112\/jlms\/s2-39.2.309 | MR\u00a00991664 | Zbl\u00a00723.47003\n[2] Hadwin, D.: General view of reflexivity. Trans. Am. Math. Soc. 344 (1994), 325-360. DOI\u00a010.1090\/S0002-9947-1994-1239639-4 | MR\u00a01239639 | Zbl\u00a00802.46010\n[3] Halmos, P. R.: Two subspaces. Trans. Am. Math. Soc. 144 (1969), 381-389. DOI\u00a010.1090\/S0002-9947-1969-0251519-5 | MR\u00a00251519 | Zbl\u00a00187.05503\n[4] Loginov, A. I., Shulman, V. S.: Hereditary and intermediate reflexivity of $W^*$-algebras. Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 1260-1273 Russian. MR\u00a00405124\n[5] Sarason, D.: Invariant subspaces and unstarred operator algebras. Pac. J. Math. 17 (1966), 511-517. DOI\u00a010.2140\/pjm.1966.17.511 | MR\u00a00192365 | Zbl\u00a00171.33703\n[6] Shulman, V. S.: Nest algebras by K. R. Davidson: a review. Algebra Anal. 2 (1990), 236-255.\n[7] Shulman, V. S., Turowska, L.: Operator synthesis. I. Synthetic sets, bilattices and tensor algebras. J. Funct. Anal. 209 (2004), 293-331. DOI\u00a010.1016\/S0022-1236(03)00270-2 | MR\u00a02044225 | Zbl\u00a01071.47066\n\nPartner of","date":"2016-10-27 14:52:33","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\": 0.9011839032173157, \"perplexity\": 3789.5864206059036}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988721347.98\/warc\/CC-MAIN-20161020183841-00341-ip-10-171-6-4.ec2.internal.warc.gz\"}"}
| null | null |
angular.module("wordApp").factory("synsetService", ["$q", "dataService", function ($q, dataService) {
function parseSynsetsIds(input) {
var wordIds = [];
_.each(input, function (word) {
_.each(word.synsets, function (synset) {
wordIds.push(synset.id);
});
});
return {
synsetIds: wordIds
}
};
function parseSynsetsIdsAndNames(input) {
var wordIds = [];
_.each(input, function (word) {
_.each(word.synsets, function (synset) {
wordIds.push(synset);
});
});
return wordIds
};
function fetchAndParseSynsets(input) {
return $q(function (resolve, reject) {
var synsetsList = [];
var wordIds = parseSynsetsIds(input);
dataService.getSynsets(wordIds).then(function (synsets) {
var allSysnsets = parseSynsetsIdsAndNames(input);
_.each(synsets.data, function (synset) {
if(synset.results){
var word = _.find(allSysnsets, function(x){
return x.id == synset.results.id;
});
word.name = synset.results.str;
}
});
resolve(input);
});
});
};
return {
parseSynsetsIds: parseSynsetsIds,
fetchAndParseSynsets: fetchAndParseSynsets
}
}]);
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 733
|
Q: Win32Com Error Running with Task Manager I have a python script that calls open an Excel file, calls a macro in the file, then closes. If I run the file from the CLI, it works. If i put it in Task Scheduler, I get an error from win32com.
Method opening the Excel file:
import win32com.client as WinCom
if os.path.exists(reportGeneratorFileName):
try:
xl = WinCom.Dispatch("Excel.Application")
xl.Workbooks.Open(Filename=os.path.abspath(reportGeneratorFileName))
xl.Application.Visible = False
xl.Application.Run("'{}'!Runner.Runner".format(reportGeneratorFileName))
l.info('Start Sleeping')
# Async mode of pythonw causes this to finish before the file is made
time.sleep(300)
l.info('Done Sleeping')
xl.Application.Quit()
except Exception as e:
l.error('Error updating file')
l.error(e, exc_info=True)
This is the Error i get:
06/04/2018 06:56:19 AM ERROR: (-2146959355, 'Server execution failed', None, None)
Traceback (most recent call last):
File "LAW Report.py", line 846, in createReport
xl = WinCom.Dispatch("Excel.Application")
File "c:\python27\lib\site-packages\win32com\client\__init__.py", line 95, in Dispatch
dispatch, userName = dynamic._GetGoodDispatchAndUserName(dispatch,userName,clsctx)
File "c:\python27\lib\site-packages\win32com\client\dynamic.py", line 114, in _GetGoodDispatchAndUserName
return (_GetGoodDispatch(IDispatch, clsctx), userName)
File "c:\python27\lib\site-packages\win32com\client\dynamic.py", line 91, in _GetGoodDispatch
IDispatch = pythoncom.CoCreateInstance(IDispatch, None, clsctx, pythoncom.IID_IDispatch)
com_error: (-2146959355, 'Server execution failed', None, None)
Now i am running 64 bit python 2.7 and 64 bit win32com while office is 32 bit, but as I said above, if I just run the script from the CLI it runs fine, just not from task manager. I am running this on a Windows Server 2012R2. I have tried configuring the task for 2008, 2008r2, and 2012r2. I also tried with highest privileges. I do need this to be able to run whether user is logged on or not. Everytime I have tested, the user has been logged on.
A: You need to
1) setup permissions in DCOMCnfg utility for user under which you're running -or-
2) change user to your desktop user account.
But first you need to get proper error code, not a generic message.
Pay attention that this should be considered as temporary solution since DCOM is badly supported by MS for automation as service. Consider switching to any library as soon as possible. There're numerous problems with Excel interop: you will have to reboot server from time to time, performance degradation with subsequent calls, you can't run Excels in parallel because of a good chance of errors, etc...
A: I encountered the same error of which my program can run from command prompt but not from task scheduler.
I solved this issue with these settings of task properties in task scheduler:
1. General->Security Options
a. make sure same user account you used to run python/microsoft program.
b. checked "Run only when user is logged on.
c. unchecked "Hidden"
For 1a, I'm using DOMAIN\userid that I used to login my laptop/PC.
This is the only way I found in order for the task scheduler to run successfully. Hope this helps.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 9,665
|
Q: Flutter - unboundedHeight error within ListView Flutter newbie here!
Currently, my Scaffold has 2 listview builders and the bottom one is giving me the unbounded height (!constraints.hasBoundedHeight error) issue.
I have tried using shrinkWrap on all 3 list views, as suggested in similar questions but I get the same error.
The only thing that works is wrapping the FutureBuilder in a SizedBox. But that seems unintuitive to me, as I would want it to ideally expand as needed and be scrollable.
My rough solution is to maybe dynamically calculate the height based on the number of items the FutureBuilder needs, but again, I feel there should be a better solution.
My code snippet is attached below:
return Scaffold(
appBar: AppBar(),
body: ListView(
children: [
ListView.builder(
scrollDirection: Axis.horizontal,
itemCount: 2,
itemBuilder: (context, index) {
return const SuggestCard(
indexKey: 'takt',
);
}),
FutureBuilder<AnimeDetails>(
future: _animeDetail,
builder: (context, snapshot) {
if (snapshot.hasData) {
return ListView.builder(
itemCount: 2, //Number of anime from list
itemBuilder: (context, index) {
var anime = snapshot.data; //Get the data from the index
return AnimeCard(
anime: anime,
);
});
} else {
return const Center(child: CircularProgressIndicator());
}
},
),
],
),
);
A: As per your comment I think below link will helpful to you.
Lists
A: I just added the below lines to your code. You can try the below code.
shrinkWrap: true,
physics: const NeverScrollableScrollPhysics(),
return Scaffold(
appBar: AppBar(),
body: ListView(
children: [
ListView.builder(
scrollDirection: Axis.horizontal,
itemCount: 2,
shrinkWrap: true,
physics: const NeverScrollableScrollPhysics(),
itemBuilder: (context, index) {
return const SuggestCard(
indexKey: 'takt',
);
}),
FutureBuilder<AnimeDetails>(
future: _animeDetail,
builder: (context, snapshot) {
if (snapshot.hasData) {
return ListView.builder(
itemCount: 2, //Number of anime from list
shrinkWrap: true,
physics: const NeverScrollableScrollPhysics(),
itemBuilder: (context, index) {
var anime = snapshot.data; //Get the data from the index
return AnimeCard(
anime: anime,
);
});
} else {
return const Center(child: CircularProgressIndicator());
}
},
),
],
),
);
A: The parent ListView handling scroll event for body and while second ListView will have fixed height, you can do it like this
return Scaffold(
body: LayoutBuilder(
builder: (context, constraints) => ListView(
children: [
SizedBox(
height: constraints.maxHeight * .3,
child: ListView.builder(
itemCount: 122,
shrinkWrap: true,
scrollDirection: Axis.horizontal,
itemBuilder: (context, index) => Text("H $index"),
),
),
ListView.builder(
shrinkWrap: true,
physics:
const NeverScrollableScrollPhysics(), // parent controll the scroll event
itemCount: 44,
itemBuilder: (context, index) => Text("$index"),
),
],
),
));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,140
|
Iselsberg-Stronach ist eine Gemeinde mit Einwohnern (Stand ) im Bezirk Lienz in Tirol, Österreich.
Geographie
Iselsberg liegt etwas südlich des Iselsbergpasses an der Straßenverbindung vom Lienzer Talboden in das Mölltal, der Ortsteil Stronach etwas abseits vom Durchzugsverkehr. Die nordöstliche Gemeindegrenze ist zugleich die Grenze zu Kärnten.
Iselsberg-Stronach liegt im Norden (Sonnenseite) des Lienzer Talbodens nordöstlich der Bezirkshauptstadt Lienz auf Das Gemeindegebiet erstreckt sich über den Süd- und Ostabhang des Geiersbichls bis auf den Südabhang des Stronachkogels und umfasst 17,96 km², womit Iselsberg-Stronach die siebtkleinste Gemeinde des Bezirks Lienz ist.
Berge
Iselsberg-Stronach liegt an der Grenze zweier Gebirgsgruppen, wobei der Mühlbach die Schobergruppe im Westen von der Kreuzeckgruppe im Osten trennt. Im Bereich der Schobergruppe umfasst das Gemeindegebiet von Iselsberg-Stronach Teile der Sonnenseite des Debanttals, wobei der Anteil der Gemeinde über rund 1.600 Meter oder höher beginnt. Der höchste Berg von Iselsberg-Stronach befindet sich mit dem Mulleter Seichenkopf () an der Grenze zum Bundesland Kärnten. In nächster Nähe befinden sich der Mitteregg (), der Törlkopf () und der Spitze Seichenkopf (). Danach folgen von Norden nach Süden der Roßbichl (), der Winkelkopf (), der Großbodenkopf () und der Straßkopf (). Im Bereich der Kreuzeckgruppe finden sich der Stronachkogel () und der Ederplan ().
Gewässer
Im Bereich der Schobergruppe hat die Gemeinde Iselsberg-Stronach auf Grund der Höhe ihres Anteils am Debanttal lediglich Anteil an den Quellgebieten des Geißlitzenbaches, des Weißenbaches, des Dietenbaches und des Eggbaches. Unterhalb des Schwarzkofels liegt mit dem Schwarzkofelsee zudem der einzige See der Gemeinde. Wichtigstes Gewässer ist der Mühlbach, der Iselsberg im Westen von Stronach im Osten trennt und unterhalb des Iselsberger Sattels entspringt. In der Kreuzeckgruppe westlich von Stronach befinden sich des Weiteren der Gödnachbach und der Frühaufbach.
Gemeindegliederung
Die Gemeinde Iselsberg-Stronach besteht aus den Katastralgemeinden Stronach im südöstlichen Gemeindegebiet und Iselsberg im nordwestlichen Gemeindegebiet, wobei Iselsberg mit einer Größe von 1.147,65 Hektar wesentlich größer als Stronach mit 648,28 Hektar ist. In Iselsberg lebten 2001 409 Menschen, in Stronach 161 Menschen. In Iselsberg bestehen neben dem Dorf Iselsberg () und den Einzelhöfen Plautz und Reiter die Streusiedlungen Oberberg () und Pass Iselsberg (). Stronach besteht aus den Rotten Stronach-Säge () und Stronach-Kirche () sowie den Einzelhöfen Deutsch, Moar und Wiesflecker. Zudem gehören zu Iselsberg die Iselsberger Alm, die Luggeralm, die Moseralm, die Plautzalm, die Reiteralm und die Straganzalm, zu Stronach des Weiteren die Ederalm, Groje, Oberegger und die Stronachalm. Der tiefstgelegene Bauernhof ist der Moarhof in Stronach auf 835 m Seehöhe, der höchstgelegene der Plautzhof in Iselsberg auf 1330 m.
Die Gemeinde liegt im Gerichtsbezirk Lienz.
Nachbargemeinden
Geschichte
Im 13. Jahrhundert errichteten die Grafen von Görz südwestlich der heutigen Stronach die Burg Walchenstein zur Sicherung der Straße über den Iselsberg. Urkundlich erwähnt wurde die Festung erstmals 1293. Nach dem Ende des Geschlechts der Görzer verlor die Burg an Bedeutung und verfiel rasch. Am Ende des 20. Jahrhunderts wurden die Mauerreste gesichert.
Die Ortsnamen tauchen im Mittelalter das erste Mal schriftlich auf. In den Regesten von 1346 findet sich Stranach und Yselsperg. Die beiden Namen waren immer eng verbunden: ein Guet auf Iselsperg, gehaissen auf Stronach (1518). Stronach war eine Flur auf dem Iselsberg. Der Name leitet sich von 'bei den Bewohnern der Leite' ab. Iselsberg geht auf den Gewässernamen Isel zurück.
Die Gemeinde in der heutigen Form entstand 1850 durch Zusammenlegung der Orte Iselsberg und Stronach.
Demographie
Bevölkerungsstruktur
2012 lebten in der Gemeinde Iselsberg-Stronach 617 Menschen. Gemessen an der Einwohnerzahl war Iselsberg-Stronach zu diesem Zeitpunkt die sechstkleinste Gemeinde im Bezirk Lienz. Nach der Volkszählung 2001 waren 96,8 % der Bevölkerung österreichische Staatsbürger (Tirol: 90,6 %), bis zum Jahresbeginn 2012 sank der Wert nur unbedeutend auf 95,1 %. Von der Bevölkerung waren 7,1 % im Ausland geboren, wobei der Großteil der nicht in Österreich geborenen Einwohner aus den EU-Mitgliedstaaten stammte. Zur römisch-katholischen Kirche bekannten sich 2001 94,0 % der Einwohner (Tirol: 83,4 %), 3,3 % hatten kein religiöses Bekenntnis, 1,4 % waren evangelisch.
Der Altersdurchschnitt der Gemeindebevölkerung lag 2001 in etwa im Landesdurchschnitt. 17,7 % der Einwohner von Iselsberg-Stronach waren jünger als 15 Jahre (Tirol: 18,4 %), 63,5 % zwischen 15 und 59 Jahre alt (Tirol: 63,0 %). Der Anteil der Einwohner über 59 Jahre lag mit 18,8 % leicht über dem Landesdurchschnitt von 18,6 %. Der Altersdurchschnitt der Bevölkerung von Iselsberg-Stronach stieg in der Folge deutlich. Der Anteil der unter 15-Jährigen sank per 1. Jänner 2012 auf 14,1 %, während der Anteil der Menschen zwischen 15 und 59 Jahren auf 69,9 % nach oben ging. Der Anteil der über 59-Jährigen sank hingegen auf 16,0 %. Nach dem Familienstand waren 2001 50,5 % der Einwohner von Iselsberg-Stronach ledig, 40,0 % verheiratet, 6,1 % verwitwet und 3,3 % geschieden.
Bevölkerungsentwicklung
Die Einwohnerzahl von Iselsberg-Stronach entwickelte sich anfangs wesentlich schwächer als das Bevölkerungswachstum von Tirol bzw. des Bezirks Lienz. So stieg die Einwohnerzahl im späten 19. Jahrhundert zwar zunächst leicht an, ging jedoch nach 1880 wieder zurück. Die geringste Einwohnerzahl wurde in Iselsberg-Stronach 1923 mit 283 Einwohnern ausgewiesen, womit die Gemeinde seit 1880 rund 20 % ihrer Einwohner verloren hatte. In der Folge verlief die Bevölkerungsentwicklung weitgehend parallel zu jener im Bezirk, wobei sich die Einwohnerzahl bis 1971 sukzessive erhöhte. Das stärkste Wachstum verzeichnete die Gemeinde dabei in den 1960er Jahren. Danach kam es nach einem kurzen Einbruch in den 1970er Jahren zu einem stärkeren Bevölkerungswachstum als im Bezirksgebiet, wobei es insbesondere in den 1990er Jahren zu einem starken Zuwachs kam. Ihren bisherigen Höchststand erreichte die Einwohnerzahl von Iselsberg-Stronach Anfang 2009, als 623 Einwohner gezählt wurden. Während die Geburtenbilanz der Gemeinde seit 2002 nur leicht positiv war, profitierte Iselsberg-Stronach von einer höheren, positiven Wanderungsbilanz.
Kultur und Sehenswürdigkeiten
Katholische Filialkirche Iselsberg hl. Schutzengel
Katholische Kapelle Stronach Mariä Heimsuchung
Wirtschaft und Infrastruktur
Arbeitsstätten und Beschäftigte
Die im Rahmen der Volkszählung durchgeführte Arbeitsstättenzählung ergab 2001 in Iselsberg-Stronach 23 Arbeitsstätten mit 86 Beschäftigten (ohne Landwirtschaft), wobei 80 % unselbstständig Beschäftigte waren. Die Anzahl der Arbeitsstätten war dabei gegenüber dem Jahr 1991 um acht Betriebe (plus 53 %) gestiegen, die Anzahl der Beschäftigten um 34 Personen (65 %) gewachsen. Wichtigster Wirtschaftszweig war 2001 das Beherbergungs- und Gaststättenwesen mit 11 Betrieben und 37 Beschäftigten (43 % der Beschäftigten in der Gemeinde) gefolgt vom Gesundheits-, Veterinär- und Sozialwesen mit zwei Arbeitsstätten und 22 Beschäftigten (26 % der Beschäftigten).
Von den 287 erwerbstätigen Einwohnern aus Iselsberg-Stronach gingen 2001 lediglich 79 Personen in der Gemeinde ihrer Beschäftigung nach. 208 mussten zur Arbeit auspendeln. Von den Auspendlern hatten 35 % ihre Arbeitsstätte im benachbarten Ballungszentrum Lienz. Weitere 23 % hatten im übrigen Bezirksgebiet eine Arbeitsstelle gefunden, weitere 15 bzw. 26 % pendelten nach Nordtirol oder in ein anderes Bundesland aus, einer der Einwohner musste ins Ausland auspendeln. Unter den Arbeitsorten der Einwohner von Iselsberg-Stronach spielte 2001 auch Spittal an der Drau eine Rolle, wo 11 % aller Beschäftigten arbeiteten.
Tourismus
Wie die Beschäftigungszahlen belegen, hat der Tourismus in Iselsberg-Stronach für die Gemeinde eine wichtige wirtschaftliche Bedeutung, wenngleich viele Osttiroler Gemeinden wesentlich höhere Nächtigungszahlen aufweisen. Die Gemeinde konnte im Tourismusjahr 2011/12 rund 30.000 Übernachtung zählen. Sowohl für die Wintersaison als auch für die Sommersaison verzeichnete Iselsberg-Stronach seit der Jahrtausendwende leichte Nächtigungssteigerungen. So stiegen die Übernachtungen in der Sommersaison zwischen 2000 und 2012 von 16.110 auf 19.555 Übernachtungen, in der Wintersaison zwischen 1999/2000 und 2011/12 von 7.669 auf 9.335 Übernachtungen. Von den 19.555 Übernachtungen im Sommer 2012 entfielen lediglich 12 % auf Österreicher, 67 % auf Deutsche und 4 % auf Italiener. Den Gästen stehen zwei Hotels, zwei Gasthöfe sowie verschiedene Pensionen und Privatzimmervermieter zur Verfügung. Die Gemeinde gehört wie alle übrigen Osttiroler Gemeinden zum Tourismusverband Osttirol, wobei Iselsberg-Stronach in der "Ferienregion Lienzer Dolomiten" organisiert ist.
Verkehr und Infrastruktur
Iselsberg-Stronach wird von der Großglocknerstraße (B 107) durchquert, die von der Gemeinde Dölsach durch Iselsberg bis zum Pass Iselsberg und weiter auf Kärntner Seite nach Heiligenblut führt. Der Ort Stronach ist durch eine Straße von der Großglocknerstraße sowie von Dölsach erreichbar. An das öffentliche Verkehrsnetz ist Iselsberg-Stronach mittels Linienbussen der ÖBB-Postbus GmbH angeschlossen. Die Linie 5002 bindet Iselsberg-Stronach an Werktagen rund zwölf Mal an die Bezirkshauptstadt Lienz an, wobei die Fahrzeit rund 20 Minuten beträgt und die Streckenführung von Lienz über Debant, Stribach und Dölsach nach Iselsberg erfolgt und weiter über Winklern, Mörtschach, Großkirchheim und Heiligenblut geführt wird. Der nächstgelegene Anschluss an das Bahnnetz der Drautalbahn befindet sich in der Nachbargemeinde Dölsach.
Mit den benachbarten Gemeinden des Lienzer Beckens hat sich Iselsberg-Stronach zum "Abwasserverband Lienzer Talboden" zusammengeschlossen, wobei die Kanalisierung der Gemeinde bis auf wenige Randobjekte Anfang des 21. Jahrhunderts abgeschlossen war und 96 % der abwasserproduzierende Objekte an Kanal und Kläranlage angeschlossen waren. Der Abfall, der in der Gemeinde anfällt, wird über den Abfallwirtschaftsverband Osttirol (AWVO) entsorgt.
Bildung
Seit dem Jahre 1810 besteht in Iselsberg ein regelmäßiger Schulunterricht, wobei der Unterricht bis 1827 in verschiedenen Stuben der örtlichen Bauernhöfe erfolgte. Erst in diesem Jahr errichtete die Gemeinde einen kleinen Holzbau, der in der Folge als Schule diente. Nachdem die Kinder von Stronach zunächst noch die Schule in Dölsach besucht hatte, erfolgte 1900 die Umsprengelung von Stronach nach Iselsberg. Daher wurde der Bau eines neuen Schulgebäudes notwendig, das jedoch im Jahr 1957 im Zuge des Baus der Iselsbergstraße abgerissen werden musste. Die heutige Volksschule wurde 1957 bezogen und 2000 umgebaut sowie in das neue Dorfzentrum integriert. Seit dem Schuljahr 1953/54 ist die Schule zweiklassig. Zum Hauptschulbesuch pendeln die Kinder der Gemeinde nach Nußdorf-Debant, das nächstgelegene Gymnasium befindet sich in Lienz. Ein Kindergarten besteht in Iselsberg-Stronach seit 1988. Er befindet sich ebenfalls im Schul- und Gemeindezentrum.
Sicherheit und Gesundheitswesen
Die Freiwillige Feuerwehr Iselsberg-Stronach wurde im Jahr 1928 gegründet und war anfangs im Schulhaus untergebracht. Zwischen 1949 und 1953 erfolgte der Bau des ersten eigenen Feuerwehrhauses unterhalb der Schutzengelkirche. Nachdem am Ort des Feuerwehrhauses 1980 der Friedhof mit Totenkapelle entstand, wurde 1983 das Feuerwehrhaus in der "Roanerreide" eröffnet. 1983 erfolgte auch die Weihe des ersten Kleinlöschfahrzeuges, 1990 folgte das erste Tanklöschfahrzeug, 1999 wurde ein neues Tanklöschfahrzeug angeschafft. Nachdem 2003 ein Um- und Zubau des Feuerwehrhauses erfolgte, schaffte die FF Iselsberg-Stronach 2006 eine Tragkraftspritze und 2010 ein gebrauchtes Mannschaftstransportfahrzeug (MTF) an.
Bezüglich des Gesundheitswesens ist Iselsberg-Stronach gemeinsam mit den Gemeinden Dölsach, Lavant, Nikolsdorf und Nußdorf-Debant im "Sozialsprengel Nußdorf-Debant und Umgebung" organisiert. Im Gesundheitssprengel werden beispielsweise Gesundheitsleistungen wie Alten- und Pflegehilfe, Heim- und Haushaltshilfe, Hospiz und Essen auf Rädern organisiert. Der nächstgelegene praktische Arzt befindet sich in der Nachbargemeinde Dölsach, die nächstgelegene Apotheke in Nußdorf-Debant. In polizeilicher Hinsicht ist die Polizeiinspektion Dölsach für die Gemeinde zuständig.
Politik
Gemeinderat
Die Gemeinderat hat insgesamt 11 Mitglieder.
Mit den Gemeinderats- und Bürgermeisterwahlen in Tirol 1998 hatte der Gemeinderat folgende Verteilung: 5 Unabhängige Gemeinschaftsliste Iselsberg-Stronach, 4 Für unser Iselsberg-Stronach und 2 Bauern – Landarbeiterliste.
Mit den Gemeinderats- und Bürgermeisterwahlen in Tirol 2004 hatte der Gemeinderat folgende Verteilung: 7 Für Iselsberg-Stronach und 4 Unabhängige Gemeinschaftsliste Iselsberg-Stronach.
Mit den Gemeinderats- und Bürgermeisterwahlen in Tirol 2010 hatte der Gemeinderat folgende Verteilung: ? Für Iselsberg und Stronach und ? Neue Kraft für Iselsberg-Stronach.
Mit den Gemeinderats- und Bürgermeisterwahlen in Tirol 2016 hatte der Gemeinderat folgende Verteilung: 7 Für Iselsberg und Stronach und 4 Neuer Schwung für unsern Berg.
Mit den Gemeinderats- und Bürgermeisterwahlen in Tirol 2022 hat der Gemeinderat folgende Verteilung: 6 Für Iselsberg und Stronach und 5 Team ZUKUNFT Iselsberg-Stronach.
Bürgermeister
bis 2004 Jürgen Kropp
2004–2022 Thomas Tschapeller
seit 2022 Gerhard Wallensteiner
Wappen
Das Wappen wurde der Gemeinde im Jahr 1987 verliehen. Es stellt die Zinnen der Burg Walchenstein sowie den alten Übergang über den Iselsberger Sattel dar. Die Zweiteilung weist auf die beiden Orte Iselsberg und Stronach hin.
Beschreibung: In von Rot und Silber gespaltenem Schild ein oben schräggezinnter, unten eingebogener Flachsparren in verwechselten Farben.
Persönlichkeiten
Söhne und Töchter der Gemeinde
Franz Defregger (1835–1921), Maler
Weblinks
Website der Gemeinde
Einzelnachweise
Schobergruppe
Kreuzeckgruppe
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 3,914
|
System.register(['domLogger'], function(exports_1) {
'use strict';
var __extends = (this && this.__extends) || function (d, b) {
for (var p in b) if (b.hasOwnProperty(p)) d[p] = b[p];
function __() { this.constructor = d; }
d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());
};
var domLogger_1;
var logger, MyDictionary, Animal, Snake, Horse, sam, tom, Greeter, greeter1, greeterMaker, greeter2, BeeKeeper, ZooKeeper, Animal2, Bee, Lion;
function greeter(person) {
return "Hello, " + person.firstName + " " + person.lastName;
}
function logging(func) {
func('Hello from logger');
}
function findKeeper(a) {
return a.prototype.keeper;
}
return {
setters:[
function (domLogger_1_1) {
domLogger_1 = domLogger_1_1;
}],
execute: function() {
logger = new domLogger_1.DomLogger('content');
logger.log(greeter({ firstName: 'Pesho', lastName: 'Peshov' }));
logger.log('Hello from typescript');
MyDictionary = (function () {
function MyDictionary() {
this.data = { 'a': 'aaa', 'b': 'bbb' };
this.lenght = this.getLength(this.data);
}
MyDictionary.prototype.getLength = function (dict) {
var count = 0;
for (var key in dict) {
if (dict.hasOwnProperty(key)) {
count++;
}
}
return count;
};
return MyDictionary;
})();
Animal = (function () {
function Animal(theName) {
this.name = theName;
}
Animal.prototype.move = function (meters) {
if (meters === void 0) { meters = 0; }
logger.log(this.name + ' moved ' + meters + 'm.');
};
return Animal;
})();
Snake = (function (_super) {
__extends(Snake, _super);
function Snake(name) {
_super.call(this, name);
}
Snake.prototype.move = function (meters) {
if (meters === void 0) { meters = 5; }
logger.log('Slithering...');
_super.prototype.move.call(this, meters);
};
return Snake;
})(Animal);
Horse = (function (_super) {
__extends(Horse, _super);
function Horse(name) {
_super.call(this, name);
}
Horse.prototype.move = function (meters) {
if (meters === void 0) { meters = 45; }
logger.log('Galloping...');
_super.prototype.move.call(this, meters);
};
return Horse;
})(Animal);
sam = new Snake('Sammy the Python');
tom = new Horse('Tommy the Palomino');
sam.move();
tom.move(34);
Greeter = (function () {
function Greeter() {
}
Greeter.prototype.greet = function () {
if (this.greeting) {
return "Hello, " + this.greeting;
}
else {
return Greeter.standardGreeting;
}
};
Greeter.standardGreeting = 'Hello, there';
return Greeter;
})();
greeter1 = new Greeter();
logger.log(greeter1.greet());
greeterMaker = Greeter;
greeterMaker.standardGreeting = 'Hey there!';
greeter2 = new greeterMaker();
logger.log(greeter2.greet());
BeeKeeper = (function () {
function BeeKeeper() {
}
return BeeKeeper;
})();
ZooKeeper = (function () {
function ZooKeeper() {
}
return ZooKeeper;
})();
Animal2 = (function () {
function Animal2() {
}
return Animal2;
})();
Bee = (function (_super) {
__extends(Bee, _super);
function Bee() {
_super.apply(this, arguments);
}
return Bee;
})(Animal2);
Lion = (function (_super) {
__extends(Lion, _super);
function Lion() {
_super.apply(this, arguments);
}
return Lion;
})(Animal2);
}
}
});
//findKeeper(Animal2).nametag; // typechecks!
//# sourceMappingURL=app.js.map
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,418
|
<?php
namespace Omea\GestionTelco\EvenementBundle\Clients\Eligibilite;
class GestionClientPassReturn
{
/**
* @var int $codeRetour
*/
protected $codeRetour = null;
/**
* @var string $labelRetour
*/
protected $labelRetour = null;
/**
* @var string $dateFin
*/
protected $dateFin = null;
/**
* @var boolean $isEligibleOption
*/
protected $isEligibleOption = null;
/**
* @var int $idSms
*/
protected $idSms = null;
/**
* @var string $labelPass
*/
protected $labelPass = null;
public function __construct()
{
}
/**
* @return int
*/
public function getCodeRetour()
{
return $this->codeRetour;
}
/**
* @param int $codeRetour
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setCodeRetour($codeRetour)
{
$this->codeRetour = $codeRetour;
return $this;
}
/**
* @return string
*/
public function getLabelRetour()
{
return $this->labelRetour;
}
/**
* @param string $labelRetour
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setLabelRetour($labelRetour)
{
$this->labelRetour = $labelRetour;
return $this;
}
/**
* @return string
*/
public function getDateFin()
{
return $this->dateFin;
}
/**
* @param string $dateFin
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setDateFin($dateFin)
{
$this->dateFin = $dateFin;
return $this;
}
/**
* @return boolean
*/
public function getIsEligibleOption()
{
return $this->isEligibleOption;
}
/**
* @param boolean $isEligibleOption
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setIsEligibleOption($isEligibleOption)
{
$this->isEligibleOption = $isEligibleOption;
return $this;
}
/**
* @return int
*/
public function getIdSms()
{
return $this->idSms;
}
/**
* @param int $idSms
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setIdSms($idSms)
{
$this->idSms = $idSms;
return $this;
}
/**
* @return string
*/
public function getLabelPass()
{
return $this->labelPass;
}
/**
* @param string $labelPass
* @return \Omea\GestionTelco\EvenementBundle\Clients\Eligibilite\GestionClientPassReturn
*/
public function setLabelPass($labelPass)
{
$this->labelPass = $labelPass;
return $this;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,129
|
The BBC, Feminism and the Demonisation of the Male
17th March 2016 Colin Crosss Arts, Culture, Media, Politics, Radio, Television 0
24×7 Disinformation and Social Engineering
I have long considered the BBC to be an undemocratic and agenda driven monopoly that has become extremely skilled at hiding some of its real intent behind slick production, and clever writing/editing.
At its heart, as we all know, it is politically biased and unashamedly plays its favourites, allocating far too much time to the Jones, Mandlesons, Toynbees etc of the world to the exclusion of many social and political thinkers that would offer a far more balanced and erudite view of things.
Political reporting and comment is one thing however, the use of drama as a political and social engineering tool is quite another.
The recent series of Happy Valley demonstrates this cleverness quite well.
Taken at face value it is a gritty police drama set in a "deprived" part of semi rural West Yorkshire, some of the acting is extremely good and the story lines, mixing the comically criminal with the truly evil probably depict, albeit in microcosm, the types of things that the police have to deal with on a daily basis.
But, look a little deeper and you will see that this series, written by a woman and with strong female characters actually promotes a very different and far from subtle agenda.
All the women are both strong and forthright with a human vulnerable side, or victims who have overcome trauma and tribulation or both.
On the other hand the men in the series are depicted as either feckless, sexually profligate, downright evil or a combination of the three.
Sarah Lancashire, the star of the series, plays a police sergeant, a single mother bringing up a grandchild who is the son of a psychopath. She is portrayed as a flawed but essentially honest and reliable woman, betrayed by her husband and bemused by her eldest son who, needless to say is living at home after having an affair.
This character is surrounded by females who have triumphed over adversity or who are "victims" of men be they philanderers, drug dealers, pimps, rapists or simply stupid and unthinking.
The policemen in the show are depicted as misogynistic, stupid or both.
A woman who kills her own son, who just happens to be the product of an incestuous relationship, is depicted clearly as nothing more than another victim of male abuse and so it goes on.
Not one man who is in the show for more than a bit part is portrayed as having any redeeming features.
The other thing that is striking, for an organisation which has a supposedly limitless pool of talent is the fact that the female actors playing the bigger roles are obviously sourced from a small group, no doubt a group that has the same views as the writer.
coloniescross ©
Coloniescross
George Osborne's Sweet Spot
Adam Deacon & Bashy Ft. Paloma Faith: 'Keep Moving'
Question Time with Going Postal, 19th October 2017
19th October 2017 Going Postal Uncategorized 0
Graphic EJ David Dimbleby chairs topical debate from Dunstable in Bedfordshire. On the panel are Conservative transport secretary Chris Grayling, Labour MP Lisa Nandy, president of the Liberal Democrats Sal Brinton, the chief executive of [more…]
Question Time with Going Postal, 26th May 2016
26th May 2016 Going Postal Uncategorized 0
Graphic EJ Question Time from ? On the panel: Labour's former leader Ed Miliband MP, Conservative former shadow home secretary David Davis MP, the Green Party's former leader Caroline Lucas MP, crime writer Dreda Say [more…]
The Rise of the Uber-Twat, A Phenomenon of our Times
29th July 2016 Colin Crosss Uncategorized 0
Colin Cross had something of a fascination for words, he was by nature inquisitive, a bit of a loner and as a child he was quickly recognised as someone that could "go places". He had [more…]
We Want Our Country Back: Leeds with Going Postal
1st June 2016 0
We Want Our Country Back public meeting with UKIP Leader Nigel Farage MEP, Jill Seymour MEP and Brendan Chilton from Labour Leave. 7:30pm start.
Medicine – Part One
We are often regaled with tales of Our NHS and the wonders it can produce. Indeed it reached a high point recently when it managed to cure the Skripals, who had been poisoned with a [more...]
Radioactivity And The Geiger Counter – Part 1
7th February 2019 0
I would not say that I am afraid of science and technology but I do make an exception in certain areas: nuclear physics being one example. Firstly, it's very dangerous to play with (duh!). Secondly, [more...]
The Communists
24th October 2018 0
The twenties were dominated by the Bolsheviks in Moscow trying to spread their revolution world wide. Marx had predicted Great Britain would be the first to succumb to his fantasies so those in Moscow made [more...]
Donald Trump Rally in Phoenix, Arizona (29th October, 2016)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,056
|
(1) A party is not liable for a failure to perform any of its obligations if he proves that the failure was due to an impediment beyond his control and that he could not reasonably be expected to have taken the impediment into account at the time of the conclusion of the contract or to have avoided or overcome it or its consequences.
1. Article 79 specifies the circumstances in which a party "is not liable" for failing to perform its obligations, as well as the remedial consequences if the exemption from liability applies. Paragraph (1) relieves a party of liability for "a failure to perform any of his obligations" if the following requirements are fulfilled: the party's non-performance was "due to an impediment"; the impediment was "beyond his control"; the impediment is one that the party "could not reasonably be expected to have taken into account at the time of the conclusion of the contract"; the party could not reasonably have "avoided" the impediment; and the party could not reasonably have "overcome" the impediment "or its consequences".
2. Article 79(2) applies where a party engages a third person "to perform the whole or a part of the contract" and the third person fails to perform.
3. Article 79(3), which has not been the subject of significant attention in case law, limits the duration of an exemption to the time during which an impediment continues to exist. Article 79(4) requires a party that wishes to claim an exemption for non-performance "to give notice to the other party of the impediment and its effect on his ability to perform". The second sentence of article 79(4) specifies that failure to give such notice "within a reasonable time after the party who fails to perform knew or ought to have known of the impediment" will make the party who failed to give proper notice "liable for damages resulting from such non-receipt". Article 79(4) also appears not to have attracted significant attention in case law, although one decision did note that the party claiming exemption in that case had satisfied the notice requirement.
4. Paragraph (5) makes it clear that article 79 has only a limited effect on the remedies available to a party aggrieved by a failure of performance for which the non-performing party enjoys an exemption. Specifically, article 79(5) declares that an exemption precludes only the aggrieved party's right to claim damages, and not any other rights of either party under the Convention.
5. Several decisions have suggested that exemption under article 79 requires satisfaction of something in the nature of an "impossibility" standard. One decision has compared the standard for exemption under article 79 to those for excuse under national legal doctrines of force majeure, economic impossibility, and excessive onerousness -- although another decision asserted that article 79 was of a different nature than the domestic Italian hardship doctrine of eccessiva onerosità sopravvenuta. It has also been stated that, where the CISG governs a transaction, article 79 pre-empts and displaces similar national doctrines such as Wegfall der Geschäftsgrundlage in German law and eccesiva onerosità sopravvenuta. Another decision has emphasized that article 79 should be interpreted in a fashion that does not undermine the Conventions basic approach of imposing liability for a seller's delivery of non-conforming goods without regard to whether the failure to perform resulted from the seller's fault. And a court has linked a party's right to claim exemption under article 79 to the absence of bad faith conduct by that party.
6. Many decisions have suggested that the application of article 79 focuses on an assessment of the risks that a party claiming exemption assumed when it concluded the contract. The decisions suggest, in other words, that the essential issue is to determine whether the party claiming an exemption assumed the risk of the event that caused the party to fail to perform. In one case, a seller had failed to make a delivery because the seller's supplier could not supply the goods without an immediate infusion of substantial cash, and the seller did not have the funds because the buyer had justifiably (but unexpectedly) refused to pay for earlier deliveries. The seller's claim of exemption under article 79 was denied because the buyer, as per the contract, had pre-paid for the missing delivery and the tribunal found that this arrangement clearly allocated to the seller risks relating to the procurement of goods. The risk analysis approach to exemption under article 79 is also evident in cases raising issues concerning the relationship between article 79 and risk of loss rules. Thus where the seller delivered caviar and risk of loss had passed to the buyer, but international sanctions against the seller's State prevented the buyer from taking immediate possession and control of the caviar so that it had to be destroyed, an arbitral tribunal held that the buyer was not entitled to an exemption when it failed to pay the price: the tribunal emphasized that the loss had to be sustained by the party who bore the risk at the moment the force majeure occurred. And where a seller complied with its obligations under CISG article 31 by timely delivering goods to the carrier (so that, presumably, risk of loss had passed to the buyer), a court found that the seller was exempt under article 79 from liability for damages caused when the carrier delayed delivering the goods.
7. Article 79 has been invoked with some frequency in litigation, but with limited success. In two cases, a seller successfully claimed exemption for a failure to perform, but in at least nine other cases a seller's claim of exemption was denied. Buyers have also twice been granted an exemption under article 79 but have been rebuffed in at least six other cases.
8. It has been questioned whether a seller that has delivered non-conforming goods is eligible to claim an exemption under article 79. On appeal of a decision expressly asserting that such a seller could claim an exemption (although it denied the exemption on the particular facts of the case), a court recognized that the situation raised an issue concerning the scope of article 79. The court, however, reserved decision on the issue because the particular appeal could be disposed of on other grounds. More recently, that court again noted that it had not yet resolved this issue, although its discussion suggests that article 79 might well apply when a seller delivers non-conforming goods. Nevertheless, at least one case has in fact granted an article 79 exemption to a seller that delivered non-conforming goods.
9. Decisions have granted exemptions for the following breaches: a seller's late delivery of goods; a seller's delivery of non-conforming goods, a buyer's late payment of the price; and a buyer's failure to take delivery after paying the price. Parties have also claimed exemption for the following breaches, although the claim was denied on the particular facts of the case: a buyer's failure to pay the price; a buyer's failure to open a letter of credit; a seller's failure to deliver goods; and a seller's delivery of non-conforming goods.
10. As a prerequisite to exemption, article 79(1) requires that a party's failure to perform be due to an impediment that meets certain additional requirements (e.g., that it was beyond the control of the party, that the party could not reasonably be expected to have taken it into account at the time of the conclusion of the contract, etc.). One decision has used language suggesting that an impediment must be an unmanageable risk or a totally exceptional event, such as force majeure, economic impossibility or excessive onerousness. Another decision asserted that conditions leading to the delivery of defective goods can constitute an impediment under article 79; on appeal to a higher court, however, the exemption was denied on other grounds and the lower courts discussion of the impediment requirement was declared moot. More recently, a court appeared to suggest that the non-existence of means to prevent or detect a lack of conformity in the goods may well constitute a sufficient impediment for exemption of the seller under article 79. yet another decision indicated that a prohibition on exports by the seller's country constituted an "impediment" within the meaning of article 79 for a seller who failed to deliver the full quantity of goods, although the tribunal denied the exemption because the impediment was foreseeable when the contract was concluded.
11. Other available decisions apparently have not focused on the question of what constitutes an "impediment" within the meaning of article 79(1). Where a party was deemed exempt under article 79, however, the tribunal presumably was satisfied that the impediment requirement had been met. The impediments to performance in those cases were: refusal by state officials to permit importation of the goods into the buyer's country (found to exempt the buyer, who had paid for the goods, from liability for damages for failure to take delivery); the manufacture of defective goods by the seller's supplier (found to exempt the seller from damages for delivery of non-conforming goods where there was no evidence the seller acted in bad faith); the failure of a carrier to meet a guarantee that the goods would be delivered on time (found, as an alternative ground for denying the buyer's claim to damages, to exempt the seller from damages for late delivery where the seller had completed its performance by duly arranging for carriage and turning the goods over to the carrier); seller's delivery of non-conforming goods (found to exempt the buyer from liability for interest for a delay in paying the price).
12. In certain other cases, tribunals that refused to find an exemption use language suggesting that there was not an impediment within the meaning of article 79(1), although it is often not clear whether the result was actually based on failure of the impediment requirement or on one of the additional elements going to the character of the required impediment (e.g., that it be beyond the control of the party claiming an exemption). Decisions dealing with the following situations fall into this category: a buyer who claimed exemption for failing to pay the price because of inadequate reserves of any currency that was freely convertible into the currency of payment, where this situation did not appear in the exhaustive list of excusing circumstances catalogued in the written contract's force majeure clause; a seller who claimed exemption for failing to deliver based on an emergency halt to production at the plant of the supplier who manufactured the goods; a buyer who claimed exemption for refusing to pay for delivered goods because of negative market developments, problems with storing the goods, revaluation of the currency of payment, and decreased trade in the buyer's industry; a seller who claimed exemption for failing to deliver because its supplier had run into extreme financial difficulty, causing it to discontinue producing the goods unless the seller provided it a considerable amount of financing.
13. Most decisions that have denied a claimed exemption do so on the basis of requirements other than the impediment requirement, and without making clear whether the tribunal judged that the impediment requirement had been satisfied. The claimed impediments in such cases include the following: theft of the buyer's payment from a foreign bank to which it had been transferred; import regulations on radioactivity in food that the seller could not satisfy; increased market prices for tomatoes caused by adverse weather in the seller's country; significantly decreased market prices for the goods occurring after conclusion of the contract but before the buyer opened a letter of credit; an international embargo against the seller's country that prevented the buyer from clearing the goods (caviar) through customs or making any other use of the goods until after their expiration date had passed and they had to be destroyed; a remarkable and unforeseen rise in international market prices for the goods that upset the equilibrium of the contract but did not render the seller's performance impossible; failure of the seller's supplier to deliver the goods to seller and a tripling of the market price for the goods after the conclusion of the contract; failure of the seller's supplier to deliver the goods because the shipping bags supplied by the buyer (made to specifications provided by the seller) did not comply with regulatory requirements of the suppliers government; failure of a third party to whom buyer had paid the price (but who was not an authorized collection agent of the seller) to transmit the payment to the seller; an order by the buyer's government suspending payment of foreign debts; chemical contamination of the goods (paprika) from an unknown source; a substantial lowering of the price that the buyer's customer was willing to pay for products in which the goods were incorporated as a component.
14. Certain claimed impediments appear with some frequency in the available decisions. One such impediment is failure to perform by a third-party supplier on whom the seller relied to provide the goods. In a number of cases seller's have invoked their suppliers default as an impediment that, they argued, should exempt the seller from liability for its own resulting failure to deliver the goods or for its delivery of non-conforming goods. Several decisions have suggested that the seller normally bears the risk that its supplier will breach, and that the seller will not generally receive an exemption when its failure to perform was caused by its suppliers default. In a detailed discussion of the issue, a court explicitly stated that under the CISG the seller bears the acquisition risk -- the risk that its supplier will not timely deliver the goods or will deliver non-conforming goods -- unless the parties agreed to a different allocation of risk in their contract, and that a seller therefore cannot normally invoke its suppliers default as a basis for an exemption under article 79. The court, which linked its analysis to the Conventions no-fault approach to liability for damages for breach of contract, therefore held that the seller in the case before it could not claim an exemption for delivering non-conforming goods furnished by a third-party supplier. It disapproved of a lower courts reasoning which had suggested that the only reason the seller did not qualify for an exemption was because a proper inspection of the goods would have revealed the defect. Nevertheless, another court has granted a seller an exemption from damages for delivery of non-conforming goods on the basis that the defective merchandise was manufactured by a third party, which the court found was an exempting impediment as long as the seller had acted in good faith.
15. Claims that a change in the financial aspects of a contract should exempt a breaching party from liability for damages have also appeared repeatedly in the available decisions. Thus seller's have argued that an increase in the cost of performing the contract should excuse them from damages for failing to deliver the goods, and buyer's have asserted that a decrease in the value of the goods being sold should exempt them from damages for refusing to take delivery of and pay for the goods. These arguments have not been successful, and several courts have expressly commented that a party is deemed to assume the risk of market fluctuations and other cost factors affecting the financial consequences of the contract. Thus in denying a buyer's claim to an exemption after the market price for the goods dropped significantly, one court asserted the such price fluctuations are foreseeable aspects of international trade, and the losses they produce are part of the "normal risk of commercial activities". Another court denied a seller an exemption after the market price for the goods tripled, commenting that "it was incumbent upon the seller to bear the risk of increasing market prices ...". Another decision indicated that article 79 did not provide for an exemption for hardship as defined in the domestic Italian doctrine of eccesiva onerosità sopravvenuta, and thus under the CISG a seller could not have claimed exemption from liability for non-delivery where the market price of the goods rose "remarkably and unforeseeably" after the contract was concluded. Other reasons advanced for denying exemptions because of a change in financial circumstances are that the consequences of the change could have been overcome, and that the possibility of the change should have been taken into account when the contract was concluded.
16. In order for a non-performing party to qualify for an exemption, article 79(1) requires that the non-performance be due to an impediment that was beyond his control. It has been held that this requirement was not satisfied, and thus it was proper to deny an exemption, where a buyer paid the price of the goods to a foreign bank from which the funds were stolen, and as a consequence were never transmitted to the seller. On the other hand, some decisions have found an impediment beyond the control of a party where governmental regulations or the actions of governmental officials prevented a party's performance. Thus a buyer that had paid for the goods was held exempt from liability for damages for failing to take delivery where the goods could not be imported into the buyer's country because officials would not certify their safety. Similarly, an arbitral tribunal found that a prohibition on the export of coal implemented by the seller's State constituted an impediment beyond the control of the seller, although it denied the seller an exemption on other grounds. Several decisions have focused on the question whether a failure of performance by a third party who was to supply the goods to the seller constituted an impediment beyond the seller's control. One court found that this requirement was satisfied where defective goods had been manufactured by the seller's third-party supplier, provided the seller had not acted in bad faith. Where the seller's supplier could not continue production of the goods unless the seller advanced it a considerable amount of cash, however, an arbitral tribunal found that the impediment to the seller's performance was not beyond its control, stating that a seller must guarantee its financial ability to perform even in the face of subsequent, unforeseeable events, and that this principle also applied to the seller's relationship with its suppliers. And where the seller's supplier shipped directly to the buyer, on the seller's behalf, a newly-developed type of vine wax that proved to be defective, the situation was found not to involve an impediment beyond the seller's control: a lower court held that the requirements for exemption were not satisfied because the seller would have discovered the problem had it fulfilled it obligation to test the wax before it was shipped to its buyer; on appeal, a higher court affirmed the result but rejected the lower courts reasoning, stating that the seller would not qualify for an exemption regardless of whether it breached an obligation to examine the goods.
17. To satisfy the requirements for exemption under article 79, a party's failure to perform must be due to an impediment that the party "could not reasonably be expected to have taken ... into account at the time of the conclusion of the contract." Failure to satisfy this requirement was one reason cited by an arbitral tribunal for denying an exemption to a seller that had failed to deliver the goods because of an emergency production stoppage at the plant of a supplier that was manufacturing the goods for the seller. Several decisions have denied an exemption when the impediment was in existence and should have been known to the party at the time the contract was concluded. Thus where a seller claimed an exemption because it was unable to procure milk powder that complied with import regulations of the buyer's state, the court held that the seller was aware of such regulations when it entered into the contract and thus took the risk of locating suitable goods. Similarly, a seller's claim of exemption based on regulations prohibiting the export of coal and a buyer's claim of exemption based on regulations suspending payment of foreign debts were both denied because, in each case, the regulations were in existence (and thus should have been taken into account) at the time of the conclusion of the contract. Parties have been charged with responsibility for taking into account the possibility of changes in the market value of goods because such developments were foreseeable when the contract was formed, and claims that such changes constitute impediments that should exempt the adversely-affected party have been denied.
18. In order for a non-performing party to satisfy the prerequisites for exemption under article 79(1), the failure to perform must be due to an impediment that the party could not reasonably be expected to have avoided. In addition, it must not reasonably have been expected that the party would overcome the impediment or its consequences. Failure to satisfy these requirements were cited by several tribunals in denying exemptions to sellers whose non-performance was allegedly caused by the default of their suppliers. Thus it has been held that a seller whose supplier shipped defective vine wax (on the seller's behalf) directly to the buyer, as well as a seller whose supplier failed to produce the goods due to an emergency shut-down of its plant, should reasonably have been expected to have avoided or surmounted these impediments, and thus to have ulfilled their contractual obligations. Similarly, it has been held that a seller of tomatoes was not exempt for its failure to deliver when heavy rainfalls damaged the tomato crop in the seller's country, causing an increase in market prices: because the entire tomato crop had not been destroyed, the court ruled, the seller's performance was still possible, and the reduction of tomato supplies as well as their increased cost were impediments that seller could overcome. Where a seller claimed exemption because the used equipment the contract called for had not been manufactured with the components that the contract specified, the court denied exemption because the seller regularly overhauled and refurbished used equipment and thus was capable of supplying goods equipped with components not offered by the original manufacturer.
19. In order for a non-performing party to qualify for an exemption under article 79(1), the failure to perform must be "due to" an impediment meeting the requirements discussed in the preceding paragraphs. This causation requirement has been invoked as a reason to deny a party's claim to exemption, as where a buyer failed to prove that its default (failure to open a documentary credit) was caused by its governments suspension of payment of foreign debt. The operation of the causation requirement may also be illustrated by an appeal in litigation involving a seller's claim of exemption under article 79 from liability for damages for delivering defective vine wax. The seller argued it was exempt because the wax was produced by a third party supplier that had shipped the goods directly to the buyer. A lower court denied the seller's claim because it found that the seller should have tested the wax, which was a new product, in which event it would have discovered the problem; hence, the court reasoned, the suppliers faulty production was not an impediment beyond its control. On appeal to a higher court, the seller argued that all vine wax produced by its supplier was defective that year, so that even if it had sold a traditional type (which it presumably would not have had to examine) the buyer would have suffered the same loss. The court dismissed the argument because it rejected the lower courts reasoning: according to the higher court, the seller's responsibility for defective goods supplied by a third party did not depend on its failure to fulfil an obligation to examine the goods; rather, the seller's liability arose from the fact that, unless agreed otherwise, sellers bear the "risk of acquisition", and the seller would have been liable for the non-conforming goods even if it was not obliged to examine them before delivery. Thus even if the seller had sold defective vine wax that it was not obliged to examine, the default would still not have been caused by an impediment that met the requirements of article 79.
Convention. The approach or language of several other decisions strongly imply that the burden of proving the elements of an exemption falls to the party claiming the exemption.
21. Article 79(2) imposes special requirements if a party claims exemption because its own failure to perform was "due to the failure by a third person whom he has engaged to perform the whole or a part of the contract." Where it applies, article 79(2) demands that the requirements for exemption under article 79(1) be satisfied with respect to both the party claiming exemption and the third party before an exemption should be granted. This is so even though the third party may not be involved in the dispute between the seller and the buyer (and hence the third party is not claiming an exemption), and even though the third party's obligations may not be governed by the Sales Convention. The special requirements imposed by article 79(2) increase the obstacles confronting a party claiming exemption, so that it is important to know when it applies. A key issue, in this regard, is the meaning of the phrase "a third person whom he [i.e., the party claiming exemption] has engaged to perform the whole or a part of the contract." Several cases have addressed the question whether a supplier to whom the seller looks to procure or produce the goods is covered by the phrase, so that a seller who claims exemption because of a default by such a supplier would have to satisfy article 79(2). In one decision, a regional appeals court held that a manufacturer from whom the seller ordered vine wax to be shipped directly to the buyer was not within the scope of article 79(2), and the seller's exemption claim was governed exclusively by article 79(1). On appeal, a higher court avoided the issue, suggesting that the seller did not qualify for exemption under either article 79(1) or 79(2). An arbitral tribunal has suggested that article 79(2) applies when the seller claims exemption because of a default by a "sub-contractor" or the seller's "own staff", but not when the third party is a "manufacturer or sub-supplier." On the other hand, an arbitral tribunal has assumed that a fertilizer manufacturer with whom a seller contracted to supply the goods and to whom the buyer was instructed to send specified types of bags for shipping the goods was covered by article 79(2). It has also been suggested that a carrier whom the seller engaged to transport the goods is the kind of third party that falls within the scope of article 79(2).
22. Article 79(5) of the Convention specifies that a successful claim to exemption shields a party from liability for damages, but it does not preclude the other party from "exercising any right other than to claim damages". Claims against a party for damages have been denied in those cases in which the party qualified for an exemption under article 79. A seller's claim to interest on the unpaid part of the contract price has also been denied on the basis that the buyer had an exemption for its failure to pay. In one decision it appears that both the buyer's claim to damages and its right to avoid the contract were rejected because the seller's delivery of non-conforming goods "was due to an impediment beyond its control", although the court permitted the buyer to reduce the price in order to account for the lack of conformity.
23. Article 79 is not excepted from the rule in article 6 empowering the parties to "derogate from or vary the effect of" provisions of the Convention. Decisions have construed article 79 in tandem with force majeure clauses in the parties' contract. One decision found that a seller was not exempt for failing to deliver the goods under either article 79 or under a contractual force majeure clause, thus suggesting that the parties had not pre-empted article 79 by agreeing to the contractual provision. Another decision denied a buyer's claim to exemption where the circumstances that the buyer argued constituted a force majeure were not found in an exhaustive listing of force majeure situations included in the parties' contract.
1. [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)]. For further discussion of article 79(4), see para 7 of the Digest for Section II of Part III, Chapter V, and the Digest for article 74, para 13.
2. [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)]; [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)] (suggesting that a seller can be exempt from liability for failure to deliver only if suitable goods were no longer available in the market); [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)]. But see [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)] where the court implied that the standard for claiming exemption under article 79 is more lenient than impossibility: it held that the buyer was exempt from interest for a delayed payment of the price, even though timely payment was clearly possible although not reasonably to be expected in the circumstances, according to the court.
3. [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)].
4. [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)] (see full text of the decision).
5. [GERMANY Landgericht Aachen 14 May 1993 (Electronic hearing aid case)] (see full text of the decision).
6. [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)].
7. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)].
8. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
9. See [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)] (discussing application of article 79, the tribunal asserts "[o]nly the apportionment of the risk in the contract is relevant here") (see full text of the decision); [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)] ("The possibility of exemption under CISG article 79 does not change the allocation of the contractual risk"). For other cases suggesting or implying that the question of exemption under article 79 is fundamentally an inquiry into the allocation of risk under the contract, see [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)]; [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)]; [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)]; [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)]; [GERMANY Landgericht Alsfeld 12 May 1995 (Flagstone tiles case)]; [FRANCE Cour d'appel de Colmar 12 June 2001 (Polyurethane foam covers for air conditioners case)] (denying buyer an exemption when buyer's customer significantly reduced the price it would pay for products that incorporated the goods in question as a component; the court noted that in a long term contract like the one between the buyer and the seller such a development was foreseeable, and it concluded that it was thus up to the [buyer], a professional experienced in international market practice, to lay down guarantees of performance of obligations to the [seller] or to stipulate arrangements for revising those obligations. As it failed to do so, it has to bear the risk associated with non-compliance.).
10. See [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)] (see full text of the decision).
11. [HUNGARY Arbitration Court attached to the Hungarian Chamber of and Industry 10 December 1996 (Caviar case)].
12. [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)].
13. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)] (seller was granted exemption from damages for delivery of non-conforming goods, although the court ordered the seller to give the buyer a partial refund); [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (seller found exempt from damages for late delivery of goods).
14. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)]; [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)]; [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)]; [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)], affirming (on somewhat different reasoning); [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)]; [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)]; [GERMANY Landgericht Ellwangen 21 August 1995 (Paprika case)]. See also [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)] (tribunal applies Yugoslav national doctrines, but also indicates that exemption would have been denied under article 79).
15. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)] (buyer that had paid price for goods granted exemption for damages caused by its failure to take delivery); [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)] (buyer granted exemption from liability for interest and damages due to late payment).
16. [RUSSIA Arbitration-Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 123/1992 of 17 October 1995 (Equipment / automatic diffractameter case)]; [RUSSIA Information Letter No. 29 of the High Arbitration Court of the Russian Federation, 16 February 1998 (Onions case)]; [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)]; [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)]; [GERMANY Landgericht Alsfeld 12 May 1995 (Flagstone tiles case)]; [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)]; [FRANCE Cour d'appel de Colmar 12 June 2001 (Polyurethane foam covers for air conditioners case)].
17. [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)].
18. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)].
19. [GERMANY Bundesgerichtshof 9 January 2002 (Powdered milk case)].
20. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
21. [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)].
22. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
23. [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)].
24. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)].
25. [RUSSIA Arbitration-Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 123/1992 of 17 October 1995 (Equipment / automatic diffractameter case)]; [RUSSIA Information Letter No. 29 of the High Arbitration Court of the Russian Federation, 16 February 1998 (Onions case)]; [HUNGARY Arbitration Court attached to the Hungarian Chamber of and Industry 10 December 1996 (Caviar case)]; [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)]; [GERMANY Landgericht Alsfeld 12 May 1995 (Flagstone tiles case)].
26. [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)]; [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)].
27. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)]; [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)]; [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)]; [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)]; [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)].
28. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)]; [GERMANY Landgericht Ellwangen 21 August 1995 (Paprika case)];. See also [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)] (denying exemption for seller who could not acquire conforming goods and for this reason failed to deliver).
29. [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)] (see full text of the decision).
30. [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)]. The court nevertheless denied the seller's claim of exemption on the facts of the particular case.
31. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)]. For further discussion of the question whether a seller can claim exemption under article 79 for delivery of non-conforming goods, see supra para. 8.
33. [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)]. The seller also claimed exemption for failing to deliver the goods (coal) because of a strike by coal miners, but the court denied the claim because the seller was already in default when the strike occurred.
34. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)].
35. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
36. [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (see full text of the decision).
37. [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)].
38.[RUSSIA Arbitration-Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 123/1992 of 17 October 1995 (Equipment / automatic diffractameter case)].
39. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)].
40. [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)].
41. [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)].
42. [RUSSIA Information Letter No. 29 of the High Arbitration Court of the Russian Federation, 16 February 1998 (Onions case)].
43. [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)].
44. [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)].
45. [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)].
46. [HUNGARY Arbitration Court attached to the Hungarian Chamber of and Industry 10 December 1996 (Caviar case)] (see full text of the decision).
47. [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)].
48. [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)].
49. [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)].
50. [GERMANY Landgericht Alsfeld 12 May 1995 (Flagstone tiles case)].
51. [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)] (see full text of the decision).
52. [GERMANY Landgericht Ellwangen 21 August 1995 (Paprika case)]. An arbitral panel has noted that, under domestic Yugoslavian law, a 13.16 per cent rise in the cost of steel -- which the tribunal found was a predictable development -- would not exempt the seller from liability for failing to deliver the steel, and suggested that the domestic Yugoslavian law was consistent with article 79. See [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)] (see full text of the decision).
53. [FRANCE Cour d'appel de Colmar 12 June 2001 (Polyurethane foam covers for air conditioners case)].
54. This situation also raises issues concerning the applicability of article 79(2)a topic that is discussed infra, para. 21.
55. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)]; [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)].
56. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)]; [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
57. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)]. In another case, the seller claimed that chemical contamination of the goods was not the result of the seller's own processing of the goods, but the court declared that the source of the contamination was irrelevant for purposes of article 79. See [GERMANY Landgericht Ellwangen 21 August 1995 (Paprika case)].
58. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)] (see full text of the decision).
59. The lower court opinion is [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)]. Another case also suggested that a seller's opportunity to discover a lack of conformity by pre-delivery inspection was relevant in determining the seller's entitlement to exemption under article 79. See [GERMANY Landgericht Ellwangen 21 August 1995 (Paprika case)].
60. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)]. For discussion of the requirement that an impediment be beyond a party's control as applied to situations in which a seller's failure of performance is due to a default by its supplier, see para. 16 infra.
61. [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)]; [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)]. See also [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)] (seller argued that article 79 exempted it from liability for non-delivery where the market price of the goods rose remarkably and unforeseeably after the contract was concluded).
62. [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)]; [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)].
63. See [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)]; [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)]; [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)]; [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)].
64. [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)].
65. [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)].
66. [ITALY Tribunale Civile di Monza 14 January 1993 (Ferrochrome case)] (see full text of the decision).
67. [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)].
68. [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)]; [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)]. See also [FRANCE Cour d'appel de Colmar 12 June 2001 (Polyurethane foam covers for air conditioners case)] (denying buyer an exemption when buyer's customer significantly reduced the price it would pay for products that incorporated the goods in question as a component; the court noted that in a long term contract like the one between the buyer and the seller such a development was foreseeable, and it concluded that it was thus "up to the [buyer], a professional experienced in international market practice, to lay down guarantees of performance of obligations to the seller] or to stipulate arrangements for revising those obligations. As it failed to do so, it has to bear the risk associated with non-compliance.").
69. [RUSSIA Information Letter No. 29 of the High Arbitration Court of the Russian Federation, 16 February 1998 (Onions case)].
70. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)].
71. [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)] (denying an exemption because the impediment was foreseeable at the time of the conclusion of the contract).
72. For further discussion of the application of article 79 to situations in which the seller's failure of performance was caused by a supplier's default, see supra para. 14, and infra paras. 17, 18 and 21.
73. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
74. [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)].
75. [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)].
76. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)]. A tribunal that finds a party exempt under article 79 presumably is satisfied that there was an impediment beyond the control of the party, even if the tribunal does not expressly discuss this requirement. The following decisions fall into this category: [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (seller found exempt from damages for late delivery of goods); [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)] (buyer granted exemption from liability for interest and damages due to late payment).
77. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)]. For further discussion of the application of article 79 to situations in which the seller's failure of performance was caused by a suppliers default, see supra paras. 14 and 16, and infra paras. 18 and 21.
78. [NETHERLANDS Rechtbank 's-Hertogenbosch 2 October 1998 (Powdered milk case)].
79. [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)].
80. [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)] (see full text of the decision).
81. [BELGIUM Rechtbank van Koophandel, Hasselt 2 May 1995 (Frozen raspberries case)] (a significant drop in the world market price of frozen raspberries was foreseeable in international trade and the resulting losses were included in the normal risk of commercial activities; thus buyer's claim of exemption was denied); [BULGARIA Arbitration before the Bulgarian Chamber of Commerce and Industry, 12 February 1998 (Steel ropes case)] (negative developments in the market for the goods were to be considered part of the buyer's commercial risk and were to be reasonably expected by the buyer upon conclusion of the contract); [ICC International Court of Arbitration, Award 6281 of 26 August 1989 (Steel bars case)] (when the contract was concluded a 13.16 per cent rise in steel prices in approximately three months was predictable because market prices were known to fluctuate and had begun to rise at the time the contract was formed; although decided on the basis of domestic law, the court indicated that the seller would also have been denied an exemption under article 79) (see full text of the decision); [FRANCE Cour d'appel de Colmar 12 June 2001 (Polyurethane foam covers for air conditioners case)] (denying buyer an exemption when buyer's customer significantly reduced the price it would pay for products that incorporated the goods in question as a component; the court noted that in a long term contract like the one between the buyer and the seller such a development was foreseeable, and it concluded that it was thus up to the [buyer], a professional experienced in international market practice, to lay down guarantees of performance of obligations to the [seller] or to stipulate arrangements for revising those obligations. As it failed to do so, it has to bear the risk associated with non-compliance.).
A tribunal that finds a party is exempt under article 79 presumably believes that the party could not reasonably have taken the impediment at issue into account when entering into the contract, whether or not the tribunal expressly discusses that requirement. The following decisions fall into this category: [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (seller found exempt from liability for damages for late delivery of goods); [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)] (buyer granted exemption from liability for interest and damages due to late payment); [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)] (seller granted exemption from liability for damages for delivery of non-conforming goods, although the court ordered the seller to give the buyer a partial refund); [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)] (buyer that had paid price for goods granted exemption from liability for damages caused by its failure to take delivery).
82. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)], affirming (on somewhat different reasoning) [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)]. In [the Vine wax case, the Federal Supreme Court of Germany] generalized that a supplier's breach is normally something that, for purposes of article 79, the seller must avoid or overcome.
83. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)].
84. For further discussion of the application of article 79 to situations in which the seller's failure of performance was caused by a suppliers default, see supra paras. 14, 16 and 17, and infra para. 21.
85. [GERMANY Oberlandesgericht Hamburg 4 July 1997 (Tomato concentrate case)]. A tribunal that finds a party exempt under article 79 presumably believes that the party could not reasonably be expected to have avoided an impediment or to have overcome it or its consequences, whether or not the tribunal expressly discusses these requirements. The following decisions fall into this category: [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (seller found exempt from liability for damages for late delivery of goods); [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)] (buyer granted exemption from liability for interest and damages due to late payment); [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)] (seller granted exemption from liability for damages for delivery of non-conforming goods, although the court ordered the seller to give the buyer a partial refund); [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)] (buyer that had paid price for goods granted exemption from liability for damages caused by its failure to take delivery).
86. [GERMANY Oberlandesgericht Zweibrücken 2 February 2004 (Milling equipment case)] (see full text of the decision).
87. [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)] (see full text of the decision). See also [BULGARIA Arbitration Award 56/1995 of the Bulgarska turgosko-promishlena palata 24 April 1996 (Coal case)] (seller's argument that a miners' strike should exempt it from liability for damages for failure to deliver coal rejected because at the time of the strike seller was already in default).
88. [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)].
89. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)].
90. [GERMANY Oberlandesgericht Zweibrücken 2 February 2004 (Milling equipment case)] (see full text of the decision).
91. [ITALY Tribunale di Vigevano 12 July 2000 (Sheets of vulcanized rubber used in manufacture of shoe soles case)]; [GERMANY Bundesgerichtshof 9 January 2002 (Powdered milk case)]. The latter case, however, distinguishes the question of the effect on the burden of proof of an extra-judicial admission of liability, viewing this matter as beyond the scope of the Convention and subject to the forums procedural law.
92. [ITALY Tribunale di Vigevano 12 July 2000 (Sheets of vulcanized rubber used in manufacture of shoe soles case)]; [GERMANY Bundesgerichtshof 9 January 2002 (Powdered milk case)]; [ITALY Tribunale di Pavia 29 December 1999 (High fashion textiles case)] (see full text of the decision).
93. [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 155/1994 of 16 March 1995 (Metallic sodium case)] (denying the seller's claim to exemption because seller was unable to prove the required facts); [ICC International Court of Arbitration, Award 7197 of 1992 (Failure to open letter of credit and penalty clause case)] (denying the buyer's exemption claim because buyer did not prove that its failure to perform was caused by the impediment); [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)] (employing language suggesting that the seller, who claimed exemption, had to submit facts to substantiate the claim).
94. The application of the requirements of article 79(1) to situations in which a seller claims exemption because its supplier defaulted on its own obligations to the seller is discussed supra paras. 14, 16, 17 and 18.
95. [GERMANY Oberlandesgericht Zweibrücken 31 March 1998 (Vine wax case)].
96. [GERMANY Bundesgerichtshof 24 March 1999 (Vine wax case)].
97. [GERMANY Arbitration-Schiedsgericht der Handelskammer Hamburg, 21 March 1996 and 21 June 1996 (Chinese goods case)] (see full text of the decision).
98. [ICC International Court of Arbitration, Award 8128 of 1995 (Chemical fertilizer case)].
99. [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)].
100. [SWITZERLAND Handelsgericht des Kantons Zürich 10 February 1999 (Art books case)] (see full text of the decision); [RUSSIA Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce, Award 155/1996 of 22 January 1997 (Butter case)].
101. [GERMANY Amtsgericht Charlottenburg 4 May 1994 (Shoes case)].
102. [FRANCE Tribunal de commerce de Besançon 19 January 1998 (Sports clothes / judo suits case)].
103. [GERMANY Oberlandesgericht Hamburg 28 February 1997 (Iron molybdenum case)].
104. [RUSSIA Arbitration-Tribunal of International Commercial Arbitration at the Russian Federation Chamber of Commerce and Industry, Award 123/1992 of 17 October 1995 (Equipment / automatic diffractameter case)]; [RUSSIA Information Letter No. 29 of the High Arbitration Court of the Russian Federation, 16 February 1998 (Onions case)].
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\chapter{Long Gamma-Ray Burst Host Galaxies and their Environments}
\section{Introduction}
Host galaxies have played an important role in determining the nature of
Gamma-Ray Burst (GRB) progenitors even before the first optical afterglows were
detected. After the first detections of host galaxies, their properties
provided strong evidence that long-duration GRBs were associated with massive
stellar deaths and hence the concept of using long-duration GRBs to probe the
evolution of the cosmic star-formation rate was conceived.
In this chapter we first briefly discuss some basic observational issues
related to what a GRB host galaxy is (whether they are operationally well
defined as a class) and sample completeness. We then describe some of the early
studies of GRB hosts starting with statistical studies of upper limits done
prior to the first detections, the first host detection after the
\textit{BeppoSAX} breakthrough and leading up to the current {\it Swift} era.
Finally, we discuss the status of efforts to construct a more complete sample
of GRBs based on {\it Swift} and end with an outlook. We only consider the host
galaxies of long-duration GRBs. For short GRBs we refer to Berger (2009). The
study of GRB host galaxies has previously been reviewed by van Paradijs et al.\
(2000) and Djorgovski et al.\ (2003).
\section{Early results based on GRB host galaxy studies}
\subsection{Pre-afterglow host galaxy studies}
Prior to the first afterglow detections a few gamma-ray bursts
with relatively small uncertainties on their derived position were studied
based on the Interplanetary Network (Hurley et al.\ 1993; see also Chapter 2). Limits
on host galaxy magnitudes in such boxes coupled with distance estimates based
on $\log{N}$ vs. $\log{S}$ arguments (see also Chapter 3) suggested that GRB host galaxies
predominantly had to be subluminous (Fenimore et al.\ 1993). However, such
limits depended strongly on the assumed GRB luminosity function. As argued by
Woods \& Loeb (1995), above a certain width of the GRB luminosity function the
probability of detection of the then possibly very distant GRBs with apparently
faint hosts would be considerable (see also Larson 1997 and Wijers et al.\
1998). In fact both explanations turned out to contain part of the truth.
\subsection{The first host galaxy detections}
Obviously, the detection of the first optical afterglows (van Paradijs et al.\
1997) fundamentally changed the study of host galaxies. The first detection of
an extended source at the position of a GRB afterglow was for GRB\,970228 (Sahu
et al.\ 1997) -- the first GRB with a detected X-ray and optical afterglow
(Costa et al.\ 1997, van Paradijs et al.\
1997). At the time, it was not firmly proven that this extended source actually was the
host galaxy so the distance scale of GRBs was yet to be established. The next
GRB with a detected optical afterglow was GRB\,970508. This burst was found to have
a redshift of 0.695 (Metzger et al.\ 1997) and hence
the extragalactic nature of (the majority of) long-duration GRBs was
established. GRB hosts were subsequently soon found to be
predominantly blue, star-forming galaxies, suggesting a young population origin
for the bursts (Paczy{\'n}ski 1998, Hogg et al. 1999, Christensen et al.\ 2004). Another important point
was clear after the detection of GRB\,970508, namely that GRBs allow the
detection of distant (star-forming) dwarf galaxies that are very difficult to
detect with other methods (Natarajan et al.\ 1997).
A few months later
relatively deep, early limits were obtained on the magnitude of the optical afterglow of
GRB\,970828. The non-detection of this optical afterglow suggested that some
GRBs occur along sightlines with substantial dust extinction in the observed
optical band (Groot et al. 1998). The third\footnote{Much later, in 2003, it
was established that an optical afterglow was also detected for GRB\,970815
(Soderberg et al. 2004).} GRB to have its optical afterglow detected was
GRB\,971214 for which a redshift of $z=3.42$ was established from the likely
host galaxy (Kulkarni et al.\ 1998). Hence, it was immediately clear that GRBs
allow us to probe ongoing star-formation throughout the observable Universe
(e.g., Wijers et al.\ 1998).
In late March 1998 two further optical afterglows
were detected and then in April, GRB\,980425 was found to be associated with a
hyperluminous type Ic SN in a nearby dwarf galaxy at redshift $z=0.0085$
(Galama et al.\ 1998). The intrinsic fluence of this burst was
about 4 orders of magnitude fainter than the GRBs with optical afterglows
studied before and hence this discovery started discussions both on
low-luminosity bursts and the issue of chance projection. It is remarkable that
roughly within a year after the first
detected optical afterglow some of the most important conclusions were already
reached: GRBs are related to massive stellar deaths, are located predominantly
in star-forming dwarf galaxies, are detected throughout most of the observable
Universe, are sometimes hidden by dust, and there seems to be a
population of low-luminosity GRBs only detectable in the relatively
local Universe.
\section{Operational issues related to GRB host galaxy studies}
After having established that (at least the majority of) GRBs have hosts we
make a short interlude discussing operational issues related to GRB host
galaxies as a class. It is prudent to keep these points in mind whenever
considering conclusions made about GRB hosts and their relation to other
classes of in particular high-redshift galaxies.
\subsection{Dark bursts and incomplete samples}
A crucial issue when discussing the nature of GRB host galaxies and the
implication thereof on the nature of GRB progenitors is sample completeness.
The detection of the GRB itself is of course limited by the sensitivity of the
gamma-ray detector and the GRB sample from a given mission will hence be
representative of a smaller and smaller part of the (possibly evolving) GRB
fluence distribution as one moves to higher and higher redshifts. However, in
terms of observational selection bias, the
GRB detection should not be affected by
host galaxy properties. In contrast, detection of the longer
wavelength afterglow emission, which is crucial for obtaining the precise
localization as well as measuring redshifts (see, e.g., Fiore et al.\ 2007), is
strongly dependent on the dust column density along the line-of-sight in the
host galaxy.
In the samples of GRBs detected with satellites prior to the currently
operating {\it Swift} satellite the fraction of GRBs with detected optical
afterglows was only about 30\% (Fynbo et al.\ 2001, Lazzati et al.\ 2002).
Much of this incompleteness was caused by random factors affecting ground-based
optical observations, such as weather or
unfortunate celestial positions of the bursts, but some remained undetected
despite both early and deep observations. It is possible that some of these so called
``dark bursts'' could be caused by GRBs in very dusty environments (Groot et
al.\ 1998) and hence the sample of GRBs with detected optical afterglows could
very well be systematically biased against dust obscured hosts (see also
Jakobsson et al.\ 2004a, Rol et al.\ 2005, 2007, Jaunsen et al.\
2008, Tanvir et al.\ 2008 for more recent discussions of the dark bursts).
Other causes for having a very faint optical afterglow are high redshifts (e.g. Greiner
et al.\ 2009, Ruiz-Velasco et al. 2007) or -- but only to some extent -- intrinsically hard spectra
(e.g., Pedersen et al.\ 2006).
In any case, such a high incompleteness imposes a large uncertainty on
statistical studies based on GRB host galaxies derived from these early
missions.
It should be stressed that the conclusions based on these samples may
only be relevant for a minority of all GRBs and consequently a biased subsample
of the GRB host population. Galaxies hosting GRBs located in high-metallicity and hence
more dusty environments, (and we know already that such systems exist), will be
systematically underrepresented. Due to the much more precise and rapid X-ray
localization capability of {\it Swift} it is possible to build a much more
complete sample of GRBs from this mission as we will discuss later in this
chapter.
\subsection{Contamination from chance projection and Galactic transients}
An important question to ask is: are GRB host galaxies operationally
well-defined as a class? The answer may seem to be trivially ``yes'', but
reality is more complex. If we define the host galaxy of a particular burst to
be the galaxy nearest to the line-of-sight, we need to worry about chance
projection (Sahu et al.\ 1997, Band \& Hartmann 1998, Campisi \& Li 2008, Cobb \& Baylin 2008). In
the majority of cases where an optical afterglow has been detected and
localized with subarcsecond accuracy and where the field has been observed to
deep limits, a galaxy has been detected with an impact parameter less than 1
arcsec (see, e.g., Bloom et al.\ 2002, Fruchter et al.\ 2006 and
Fig.\,\ref{021004host} for an example). The probability for this to happen by
chance depends on the magnitude (and angular size) of the galaxy. The number of
galaxies per
arcmin$^2$ has been well determined to deep limits in the various Hubble deep
fields. To limits of $R=24$, 26 and 28 there are about 20, 80 and 400 galaxies
arcmin$^{-2}$. Hence, the probability to find an $R=24$ galaxy by chance in an
error circle with radius 0.5 arcsec is about 4$\times$10$^{-3}$. For an $R=28$
galaxy the probability is about 8\%. If the error circle is
defined only by the X-ray afterglow (with a radius of 2 arcsec in the best cases)
then we expect a random $R=24$ and $R=28$ galaxy in 6\% and all of the error
circles. For a sample of a few hundred GRBs chance projection should hence not
be a serious concern for GRBs localized to subsarcsecond precision, but for
error-circles with radius of a few arcseconds we expect many chance
projections. In some cases it may be possible to eliminate the chance
projections, e.g., based on conflicting redshift information from the afterglow
and proposed host (e.g., Jakobsson et al.\ 2004b); without such extra
information this is impossible.
Finally, it is worth noting that some events triggering the gamma-ray detectors
are Galactic high-energy transients rather than extragalactic GRBs. Examples
are GRB\,070610 (Kasliwal et al.\ 2008, Castro-Tirado et al. 2008) later renamed to SWIFT J195509.6+261406
and GRB\,060602B, which was found to be a low-mass X-ray binary (Wijnands et al.\
2009).
Judged from the high energy properties alone these bursts looked just
like GRBs, so in principle such events can contaminate GRB samples. Presumably,
most of these events will, like these two examples, be located at low
Galactic latitude and hence most can be rejected based on a Galactic latitude
cut in the sample selection.
\begin{figure}
\begin{center}
\epsfxsize=7cm \epsfbox{fig021004.ps}
\caption{
The {\it HST} 1$\times$1 arcsec$^2$ field around the host galaxy of GRB\,021004 at $z=2.33$ found with {\it HETE-2} (from Fynbo et al.\
2005). The GRB went off near the center of the galaxy. The position
of the GRB is marked with a cross and an error circle and in coincides with the
centroid of the galaxy to within a few hundredths of an arcsec. In cases like
this there is no problem in identifying the correct host galaxy. However, in
cases of bursts localized to only a few arcsec accuracy, chance projection
needs to be considered.
}
\label{021004host}
\end{center}
\end{figure}
\section{Status prior to the \textit{Swift} mission}
\subsection{GRBs as probes of star formation}
As outlined above, the properties of the first handful of detected hosts showed
a clear link to star-formation. Moreover, Mao \& Mo (1998) made a model
incorporating a power-law shaped luminosity function of GRBs and the assumption
that the GRB rate is proportional to the cosmic star-formation density. From
this model, Mao \& Mo (1998) were able to get remarkably good agreement with the
properties of observed hosts suggesting that GRBs were close to being good
tracers of star-formation. Hogg \& Fruchter (1999) reached a similar
conclusion.
\subsection{Biased tracers?}
\begin{figure}
\begin{center}
\epsfxsize=10cm \epsfbox{r_k_color_z.eps}
\caption{
Observed $R-K$ colours versus redshift for the sample of GRB host galaxies
selected by Le Floc'h et al. (2003) ({\it filled diamonds}). The point of
the figure is that GRB host galaxies are bluer than other starburst galaxies
studied at similar redshifts.
For comparison the authors also plot the
colours and redshifts for optically-selected field sources ({\it dots}) and of
ISO sources from the Hubble Deep Field ({\it open squares}) and of SCUBA
galaxies with confirmed redshifts ({\it filled squares}). See Le Floc'h et al.\
(2003) for more details and references. In addition, the authors found
that the $K$-band luminosities of GRB hosts were substantially fainter than,
e.g., the ISO galaxies which at redshifts around one are believed to
dominate the total star-formation activity.
Solid curves
indicate the observed colours of local E, Sc, Scd and Irr galaxies if they were
moved back to increasing redshifts.
}
\label{Emeric}
\end{center}
\end{figure}
However, as the sample size grew evidence started to collect suggesting that
GRBs may be related only to massive stars with metallicity below a certain
threshold. As discussed in Chapter 10, such a metallicity dependence is expected
in the collapsar model. The first empirical evidence for this came with the
realization that the GRB hosts were fainter and bluer than expected according to
certain models about the nature of the galaxies dominating the integrated
(over all galaxies)
star-formation rate density (Le Floc'h et al.\ 2003, see also Fig.~\ref{Emeric}).
Also, SCUBA imaging of GRB hosts in the sub-mm range produced only a few rather
tentative detections, again seemingly at odds with the expectations if GRB hosts
were selected in an unbiased way from all star-forming galaxies (Tanvir et al.\
2004). The analysis is complicated, and it has been pointed out by
Priddey et al.\ (2006) that ``there is sufficient uncertainty in models and
underlying assumptions, as yet poorly constrained by observation (e.g., the
adopted dust temperature) that a correlation between massive, dust-enshrouded
star formation and GRB production cannot be firmly ruled out.'' The issue of
dust temperature has been discussed in detail by Micha\l{}owski et al. (2008;
see also Fig.~\ref{fg:broadband}).
They find that the few GRB hosts that have been tentatively detected in the
sub-mm range have hotter dust and lower masses than typical sub-mm detected
galaxies.
Further circumstantial evidence for a preference towards low metallicity came
from the observation that Lyman-$\alpha$ (Ly$\alpha$) emission seemed to be
ubiquitous from GRBs hosts (Fynbo et al. 2003a, Jakobsson et al. 2005b). At this
point, around 2003, redshifts had been measured for ten $z>2$ GRBs.
Ly$\alpha$ emission was detected for 5 of these and for the remaining 5 it was
not yet
searched for to sufficient depth to allow detection of even a large equivalent
width emission line. As only 25\% of continuum selected starbursts at
similar redshifts are Ly$\alpha$ emitters and as Ly$\alpha$ emission on
theoretical grounds should be more common for metal poor starbursts (Charlot \&
Fall 1993; but see also Mas-Hesse et al.\ 2003) this suggested that there could
be a low metallicity bias in making GRBs. However, the reason could also be an
observational bias against dusty (and hence likely higher metallicity) GRBs or
simply that the majority of the star formation is associated with
low-luminosity galaxies, which tend to be metal poor in accordance with the
luminosity-metallicity relation. The broad-band luminosity distribution of
$z>2$ GRB hosts was found by Jakobsson et al. (2005b) to be consistent with the
assumption that GRBs are selected from the rest-frame UV selection function
based on the total UV emission per luminosity bin. Assuming that the rest-frame
UV emission is proportional to the star formation rate this suggested that GRBs
cannot be strongly biased towards low metallicity.
\begin{figure}
\includegraphics[width=\textwidth]{broadband_SED_Michalowski.eps}
\begin{center}
\vskip -0.2cm
\caption{Broad-band spectral energy distribution of four GRB hosts with firm
submillimetric or radio detections, demonstrating the role played by observations
in various wavebands (from Micha\l{}owski et al. 2008). Dotted lines show the
rescaled SED of the prototypical ultraluminous infrared galaxy Arp\,220,
while solid lines show synthetic best-fit models based on the GRASIL code
(Silva et al. 1998).\label{fg:broadband}}
\end{center}
\end{figure}
A very important result is that GRBs and core-collapse (CC) SNe are found in
different environments (Fruchter et al.\ 2006 and Fig.~\ref{andyhosts}). GRBs
are significantly more concentrated on the very brightest regions of their host galaxies
than CC SNe. The same study also found that GRB host galaxies at
$z<1$ are fainter and more irregular than the host galaxies of CC
SNe. Fruchter et al.\ (2006) suggest that these results may imply that long-duration
GRBs are associated with the most extremely massive stars and may be restricted
to galaxies of limited chemical evolution. This would directly imply that
long-duration GRBs are relatively rare in galaxies such as our own Milky Way.
This study is also based on incomplete pre-{\it Swift} samples, but as the SN
samples are, if anything, more biased against dusty regions than GRBs, the
fact that GRB hosts have lower luminosities than CC SN hosts
does seem to provide substantial evidence that GRBs are biased towards
massive stars with relatively low metallicity. Regarding the size of the
effect, Wolf \& Podsiadlowski (2007) find, based on an analysis of the Fruchter
et al.\ (2006) data, that the metallicity threshold cannot be significantly
below half the solar metallicity. Concerning the different environments of
CC SNe and GRBs it has recently been found that type Ic SNe have
similar positions relative to their host galaxy light profiles as GRBs,
whereas all other SN types have (similar) distributions less centred on their
host light than GRBs and SN Ic (Kelly et al.\ 2007).
Larsson et al.\ (2007),
modeling the distribution of young star clusters and total light
in the local starburst NGC 4038/39 (the Antennae),
find that the different distributions of GRBs and CC SNe relative
to their
host light can be naturally explained by assuming different mass ranges for the
typical progenitor stars: $>8~M_{\odot}$ for typical CC SNe and
$>20~M_{\odot}$ for GRB progenitors. The picture is complicated by the
finding that type Ic SNe typically are found in substantially more metal rich
environments than GRBs (Modjaz et al.\ 2008). It is well established that
WR stars become more frequent with {\it increasing} metallicity - opposite to
GRBs that, if anything, are biased towards low metallicity. Taken together, these
results suggest that progenitors of GRBs and ``normal'' type Ic SNe are two
different subsets of the $>20~M_{\odot}$ stars. For a thorough discussion of
the relation between WR stars, SN Ic and GRBs we refer to Crowther (2007). In
conclusion, there seems to be a low metallicity preference for GRBs, but the
metallicity threshold cannot be much below half solar and we need a more
unbiased and uniform sample to estimate the severity of the effect (see also Chapter 10).
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.99\textwidth]{expl_fig1.ps}
\vskip -0.2cm
\caption{
A mosaic of GRB host galaxies imaged with {\it HST} (from Fruchter et al.\ 2006). Each
individual image corresponds to a square region on the sky $3.75$ arcsec on a
side. These images were taken with the {\it HST}. In cases where the location of the GRB on the host
is known to better than $0.15$ arcsec, the position of the GRB is shown by a green
mark. If the positional error is smaller than the point spread function of the
image ($0.07$ arcsec for STIS and ACS, $0.13$ arcsec for WFPC2) the position is
marked by a cross-hair, otherwise the positional error is indicated by a
circle.
Due to the redshifts of the hosts, these images generally correspond
to blue or ultra-violet images of the hosts in their rest frame, and thus
detect light largely produced by the massive stars in the hosts.
}
\label{andyhosts}
\end{center}
\end{figure}
\subsection{The galactic environment of GRB hosts}
The galactic environments of GRBs have so far not been studied much. At low
redshifts Foley et al.\ (2006) studied the field of the GRB\,980425
host galaxy,
which was reported to be a member of a group. However, based on
redshift measurements of the proposed group members, Foley et al.\ (2006) could
establish that the host of GRB\,980425 is an isolated dwarf
galaxy\footnote{H$\alpha$ imaging of the host has revealed a very faint
companion about 1 arcmin NE of the host (L. Christensen, private
communication).}. Levan et al.\ (2006) also proposed GRB\,030115 to be
connected to a cluster around $z \sim 2.5$ based on photometric redshifts. At
redshifts $z\gtrsim2$ a few GRB fields have been studied using narrow band
Ly$\alpha$ imaging (Fynbo et al.\ 2002, Fynbo et al.\ 2003a, Jakobsson et al.\
2005b). In all cases several other galaxies at the same redshift as the GRB host
were identified, but it is not certain whether the galaxy densities in these
fields are higher than in blank fields as no blank field studies have been
carried out at similar redshifts. However, the density of Ly$\alpha$ emitters
was found to be as high as in the fields around powerful radio sources that
have been proposed to be forming proto-clusters (Kurk et al.\ 2000), which
would suggest that GRBs could reside in overdense fields at $z\gtrsim$ 2.
Bornancini et al.\ (2004), on the other hand, argue for a low galaxy density in
GRB host galaxy environments. In conclusion, the evidence is currently too
sparse to establish whether GRB hosts are located in special environments.
\subsection{GRB host absorption line studies}
GRB host galaxies have the unique advantage over galaxies selected on the
base of their emission that spectroscopy of their afterglows (X-ray, optical,
and other bands) can reveal detailed information about the gas in the host
galaxy ranging from the circumstellar material to halo gas.
Two interesting examples of what GRB afterglow spectroscopy can teach us about
GRB hosts are the cases of GRB\,030323 at redshift $z=3.372$ (Vreeswijk et al.\
2004) and GRB\,080607 at $z=3.036$ (Prochaska et al.\ 2009). As seen in
Fig.~\ref{andyhosts} the host galaxy of GRB\,030323 is among the faintest
detected (it has a magnitude of about $R=28$). The spectrum of the optical
afterglow of the event is shown in Fig.~\ref{030323}. The dominant feature in
the spectrum is a very large damped Ly$\alpha$ absorption (DLA) line
corresponding to a redshift of $z=3.37$ (shown in more detail in the inset in
the upper left corner). A DLA is a hydrogen absorption line with column density
above $2\times10^{20}$ cm$^{-2}$, where the total equivalent width is dominated by
the broad damping wings. Also seen are numerous low- and high-ionization lines
at a redshift $z=3.3718\pm0.0005$. The inferred neutral hydrogen column
density, $\log{(N_{\rm HI}/{\rm cm}^{-2})} = 21.90\pm0.07$, is very large -- higher than
in DLAs seen against the light of QSOs (Wolfe et al.\ 2005). From an analysis of
the metal line
strengths compared to the hydrogen column density, a sulphur abundance of
about 0.05 solar is determined.
An upper limit to the H$_2$
molecular fraction of
$2N_{{\rm H}_2}/(2N_{{\rm H}_2}+N_{\rm HI}) < 10^{-6}$
is derived from an
analysis of the absence of molecular lines.
In the DLA trough, a
Ly$\alpha$ emission line is detected, which corresponds to a star formation
rate (not corrected for dust extinction) of roughly $1~M_{\odot}$ yr$^{-1}$.
All these results are consistent with the host galaxy of GRB\,030323 containing
low metallicity gas with a low dust content. In addition, fine-structure lines
of silicon, Si II*, are detected in the spectrum. These have not been clearly detected
in QSO-DLAs suggesting that these lines are produced in the vicinity of the GRB
explosion site. The optical spectrum of GRB\,080607 displays an even larger HI
column density of $\log{(N_{\rm HI}/{\rm cm}^{-2})} = 22.7$, but contrary to GRB\,030323, the
GRB\,080607 host is responsible for a very metal rich absorption system (close to solar
metallicity) with significant dust extinction and a clear detection of both
H$_2$ and CO molecular absorption (Fig.~\ref{fig:080607}). At the time of
writing (June 2010) the host galaxy has not been detected.
Dozens of similar quality spectra have been obtained, some with
high resolution spectrographs (e.g., Chen et al.\ 2005, Starling et al.\ 2005,
Vreeswijk et al.\ 2007, D'Elia et al.\ 2007). The
properties of the GRB absorbers as a class are discussed in Savaglio (2006),
Fynbo et al.\ (2006) and Prochaska et al.\ (2007). The GRB absorbers are
characterized by HI column densities spanning five orders of magnitude from
$\sim$10$^{17}$ cm$^{-2}$ to $\sim$10$^{23}$ cm$^{-2}$ and metallicities
spanning two orders of magnitude from 1/100 to nearly solar (see Fig.~\ref{Z}).
Calura et al.\ (2009) have taken the first steps towards modelling the
abundance ratios in GRB hosts using models for chemical evolution in galaxies.
The majority of the GRB absorbers have metallicities exceeding the cosmic mean
metallicity of atomic gas at $z>2$ as determined from QSO-DLAs. The difference
in abundances between the QSO DLAs and GRB absorbers can be reconciled in a
simple model where the two populations are drawn randomly from the distribution
of star-forming galaxies according to their star formation rate and HI cross
section, respectively (Fynbo et al.\ 2008). However, it should be noted that
these results are based on a small sample of GRBs in which most have bright
optical afterglows.
\begin{figure}[h]
\begin{center}
\includegraphics[width=4.0in,angle=90]{f1.ps}
\caption{
The histograms show the cumulative distribution of QSO-DLA and GRB-DLA
metallicities in the statistical samples compiled by Prochaska et al.\ (2003)
and
Prochaska et al.\ (2007). The GRB-DLA metallicities are systematically
higher than the QSO-DLA metallicities. Based on a KS test, the probability that the two observed distributions
are drawn from the same parent distribution is 2.1\% (see also Fynbo et al.\ 2008 for
a simple model of these two distributions).
}
\label{Z}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.00\textwidth]{grb030323.f3.ps}
\caption{
The spectrum of the optical afterglow of GRB\,030323 (from Vreeswijk et
al.\ 2004). The insets show close-ups on the DLA (with Ly$\alpha$ in emission) and of the C IV doublet, showing the presence of at least two distinct
components.
}
\label{030323}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.75\textwidth,angle=90]{fig080607.eps}
\caption{
The spectrum of the optical afterglow of GRB\,080607 (from Prochaska et
al.\ 2009).
}
\label{fig:080607}
\end{center}
\end{figure}
\section{GRB host galaxies in the \textit{Swift} era}
The currently operating {\it Swift} satellite (Gehrels et al.\ 2004
and Chapter 5) has
revolutionized GRB research with its frequent, rapid, and precise localization
of both long and short duration GRBs. The breakthrough enabled by {\it Swift} was the ability to build a much more complete sample of localized GRBs and hence of GRB host galaxies. By complete we here mean unbiased in
terms of the optical properties of the GRB afterglows.
\subsection{Building a complete sample of \textit{Swift} GRBs}
The {\it Swift} satellite has been
superior to previous GRB missions due to the combination
of several factors: {\it i)} it detects GRBs at a rate of about two bursts per
week, about an order of magnitude larger than the previous successful \textit{BeppoSAX}
and {\it HETE-2} missions; {\it ii)} with its X-Ray Telescope (XRT) it localizes the
bursts with a precision of about 5 arcsec (often later refined to less than 2
arcsec), also orders of magnitude better than previous missions; {\it iii)} it
has a much shorter reaction time, allowing the study of the evolution of the
afterglows literally seconds after the burst, sometimes while the prompt
$\gamma$-ray emission is still being emitted. For a detailed description of the {\it Swift} era see also Chapter 5.
We will here discuss the status of an ongoing effort to build such a complete
sample (see, e.g., Jakobsson et al.\ 2006a for earlier descriptions of this
work). Rather than including all {\it Swift} detected GRBs only
those GRB afterglows with favourable observing conditions are included,
in particular those
fulfilling the following selection criteria:
\begin{enumerate}
\item{XRT afterglow detected within 12 hr;}
\item{small foreground Galactic extinction: $A_V<0.5$ mag;}
\item{favourable declination: $-70^\circ < \delta < +70^\circ$;}
\item{Sun-to-field angular distance larger than 55$^\circ$.}
\end{enumerate}
By introducing these constraints the sample is not biased towards GRBs with
optically bright afterglows, but will contain bursts for
which useful follow-up observations are likely to be secured.
The fact that 6--10 m class telescopes have made tremendous efforts to secure
redshifts means that this sample has a much higher redshift completion than for
pre-{\it Swift} samples (see Fig.~\ref{zdist}). Still, it is clear that we will not
get redshifts for all bursts from spectroscopy of the afterglows for multiple
reasons. In a small fraction of the cases where a spectrum of the afterglow is
secured no redshift can be measured. This is either due to
lack of significantly detected absorption lines or
because the spectrum is simply too noisy. For these
bursts the only way to measure the redshift is via spectroscopy of the host
galaxy, but this is a challenging task due to the faintness of most GRB hosts
(see below).
\subsection{The redshift distribution of \textit{Swift} GRBs: current status}
The first conclusion from Fig.~\ref{zdist} is that most {\it Swift} GRBs are
very distant. {\it Swift} GRBs are more distant than GRBs from previous
missions due to the higher sensitivity of the satellite to the lower energies
prevalent in the more distant events (Fiore et al.\ 2007). The faster reaction
time of the satellite probably also contributes to shifting the median
redshift upwards. The median and mean
redshift are now both around 2, while for previous missions it was closer to 1
(Jakobsson et al.\ 2006a). The record holders are GRB\,080913 at $z=6.7$
(Greiner et al.\ 2009; Pattel et al.\ 2010) and GRB\,090423 at $z=8.2$
(Tanvir et al.\ 2010; Salvaterra et al.\ 2010).
It is striking how events at redshifts as large as $\sim 8$ can be detected within
such a small sample. For comparison, only a few QSOs are detected at $z \sim 6$
(and none at $z \sim 8$!) out of a sample of hundred thousand QSOs.
Given the current redshift completeness level, the data are consistent with a
broad range of redshift distributions (see, e.g., Jakobsson et al.\ 2006a).
\begin{figure}[h]
\begin{center}
\includegraphics[width=5.3in]{zdist.ps}
\caption{
Redshift distribution (up to March 2009) of {\it Swift} GRBs localized
with the X-ray telescope and with low foreground extinction $A_V\le0.5$.
Bursts for which only an upper limit on the redshift could be established so
far are indicated by arrows. The histogram at
the right indicates the bursts for which no optical/$J$/$H$ afterglow was
detected and hence no redshift constraint could be inferred (see Ruiz-Velasco
et al.\ 2007 for a full discussion of an earlier version of this plot).
}
\label{zdist}
\end{center}
\end{figure}
\subsection{HI column densities}
The HI column density distribution for GRB sightlines is extremely broad. It
covers a range of about 5 orders of magnitude from $\sim10^{17}$ cm$^{-2}$
(Fig.~\ref{LyC}, Chen et al.\ 2007) to nearly $10^{23}$ cm$^{-2}$
(Jakobsson et al.\ 2006b). It still remains to be understood if this distribution
is representative of the intrinsic distribution of HI column densities
towards massive stars in galaxies or if the distribution is rather controlled
by the ionizing emission from the afterglows themselves. In any case, as
pointed out by Chen et al. (2007), the HI column density distribution provides
an upper limit to the escape fraction of Lyman continuum emission from
star-forming galaxies. This is crucial for the issue of determining the sources
for the ionising photons in the metagalactic UV background.
\begin{figure}
\begin{center}
\includegraphics[width=5in]{050908.eps}
\caption{
The {\it VLT}\/FORS2 spectrum of the afterglow of GRB\,050908 (Fynbo et al.\ 2009)
Plotted is the flux-calibrated 1-dimensional
spectrum against observed wavelength. The vertical
dashed line shows the position of the Lyman limit at the
redshift of the GRB ($z=3.343$). As seen, there
is clear excess flux blueward of the Lyman limit.
}
\label{LyC}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{070802.eps}
\caption{
The {\it VLT}\/FORS2 spectrum of the afterglow of GRB\,070802 (El\'iasd\'ottir
et al.\ 2009).
Plotted is the flux-calibrated spectrum
against observed wavelength. Metal lines at the host redshift are
marked with solid lines whereas the features from two intervening systems
are marked with dotted lines. The broad depression centred around 7500 \AA \
is caused by the 2175 \AA \ extinction bump in the host system
at $z_{\rm abs}=2.4549$.
}
\label{bump}
\end{center}
\end{figure}
\subsection{Extinction}
In addition to HI column densities, metal and molecular abundances and
kinematics, the afterglow spectra also provide information about the extinction
curves. The intrinsic spectrum of the afterglow is predicted from theory to be
a power-law and therefore any curvature or other broad features in the spectrum
can be interpreted as being due to the extinction curve shape. So far,
almost all the extinction curves derived for GRB sightlines have
been consistent with an extinction curve similar to that of the SMC
(e.g., Starling et al.\ 2007, Schady et al.\ 2010). Recently,
a clear detection of the 2175 \AA \ bump known from the
Milky Way was found in a $z=2.45$ GRB sightline (El\'iasd\'ottir et al.\ 2009,
Kr\"uhler et al.\ 2008 and
Fig.~\ref{bump}). This GRB absorber also has unusually strong metal lines
suggesting that the presence of the 2175 {\AA} extinction bump is related to
high metallicity (as expected from sightlines in the local group). However, we
have examples of GRBs with nearly solar
metallicity for which the bump is not seen, so it seems that metallicity is not
the only parameter controlling the presence of the 2175 {\AA} extinction bump.
Concerning the amount of extinction, the GRB sightlines vary from no extinction
(e.g., GRB\,050908, Fig.~\ref{LyC}) to $A_V > 5$ mag (e.g., GRB\,070306, Jaunsen et
al.\ 2008).
\subsection{The host sample}\label{sc:sample}
\begin{figure}
\includegraphics[width=\columnwidth]{Fig11.ps}
\caption{All-sky map (Mollweide projection) of the 238 \textit{Swift} GRBs
occurred between 2005 March and 2007 August. Empty circles: GRBs with no
\textit{Swift}/XRT detection. Filled circles: GRBs with \textit{Swift}/XRT
detection. Filled, encircled circles: GRBs obeying all selection
criteria of the Hjorth et al. (2010) sample. Squares: GRBs classified as short. Crosses: nontriggered
GRBs. Overplotted are the declination cuts ($-70$ and $+27^\circ$, dashed
lines) and the region with Galactic latitude $|b| > 20^\circ$ (dotted curves),
which roughly corresponds to the sample selection criterion $A_V < 0.5$
mag.\label{fg:map}}
\end{figure}
Hjorth and collaborators have been working on building up a sample of {\it
Swift} GRB host galaxies observed with the ESO {\it VLT}. The main science driver for this
work is to build a representative, unbiased sample of GRB host galaxies than
can be used to firmly establish the statistical properties of GRB hosts. Similar to
the philosophy of the afterglow sample discussed above, this work
focusses on the systems with the best observability, which also
have the best available information. GRBs included in the sample fullfil the
following criteria:
\begin{enumerate}
\item triggered by BAT;
\item belong to the long-duration class;
\item trigger time between March 2005 and August 2007;
\item low Galactic extinction ($A_V \le 0.5$ mag);
\item prompt XRT localization ($< 12$~hr);
\item good observability from the {\it VLT} ($-70^\circ < \delta < +27^\circ$);
\item small position error ($\mbox{radius} \le 2''$) from based on the X-ray
afterglow astrometry.
\end{enumerate}
Note that criteria (iii)--(vi) do not introduce selection effects in the sample,
since they are not based on intrinsic properties of the GRBs. Only criterion
(vii) may in principle bias against faint events (which will have on the average
worse localizations), however in practice it only excludes a few events (3\%
of the total). On the other hand, a burst satisfying all the above criteria
must have been on the average better studied and characterized, since its
afterglow was well observable from the ground.
There are 69 GRBs fulfilling the above criteria. For 52 (75\%) of these the
optical/NIR afterglow has been detected and for 38 (55\%) the redshift has been
measured (spanning the range $0.033 < z < 6.295$). Figure~\ref{fg:map} shows
the distribution of this sample in the sky. For more details about the
program, we refer to Hjorth et al.\ (2012).
\subsubsection{Results}\label{sc:results}
\textbf{Magnitudes.} The primary objective in this study is the search and
localization of the hosts. The applied strategy is a moderately deep $R$-band
exposure (30 min), followed by deeper imaging ($\sim 2$~hr) in case of no
detection. The survey was quite successful, with a detection rate of 80\% of
the hosts. The success rate drops significantly at large redshifts, with a
recovery fraction of 40\% for GRBs with a measured redshift larger than 3.
Figure~\ref{fg:mags} shows the distribution of the observed and absolute
magnitudes. To compute the $k$-corrections, a spectrum $F_\nu \propto \nu^{-1}$
was assumed. As can be seen, most GRB hosts are subluminous, at a
level $(0.01 \mbox{--} 1) \times L^*$. This is in line with the previous
findings mentioned above based on smaller and less complete samples (Le Floc'h
et al.\ 2003, Fruchter et al.\ 2006). Hence, this conclusion is not a
result of a bias against dusty GRB hosts, but it is an intrinsic property
of the GRB host population.
\begin{figure}
\includegraphics[width=0.49\columnwidth]{Fig12_top.ps}\hspace{0.02\columnwidth}%
\includegraphics[width=0.49\columnwidth]{Fig12_bottom.ps}
\caption{Top: observed $R$-band magnitudes of the hosts in the Hjorth et al.\ (2012) sample as a
function of redshift. The left panel shows objects with unknown redshift (the
$x$-axis value is arbitrary). The dashed line shows the magnitude of an $L^*$
galaxy ($M_B = -21$) as a function of redshift. Bottom: absolute luminosity of
hosts as a function of redshift. The dashed line shows the level of $L^*$,
while the dotted curve indicates the effective survey limit ($R \sim
27$).\label{fg:mags}}
\end{figure}
\textbf{Colours.} In addition to $R$-band imaging the Hjorth et al.\ (2012) survey also
obtained $K$-band imaging of all the fields. The detection rate at NIR
wavelengths is significantly lower than in the optical. Hosts were detected in
only about 40\% of the systems in the sample. Overall, colours are in the range $2 <
R-K < 4.5$, with two possible examples of extremely red objects (EROs)
with $R-K \approx 5$. These GRBs had no reported optical afterglow. While the lack of
optical emission is consistent with the presence of dust (and reddening), a
chance association between the GRB and the galaxy cannot be excluded.
Figure~\ref{fg:colours}
shows the distribution of observed colours for bursts with and without an
optical afterglow. Note that the two groups have a comparable distribution of
colours. Overall, even considering bursts with no detected optical afterglow,
the earlier findings of Le Floc'h et al.\ (2003) that GRB hosts have mostly
blue colours are confirmed (though a few cases of red systems have been found
(Levan et al.\ 2006, Berger et al.\ 2007). This does not exclude that dust is
present in these objects (e.g. Jaunsen et al.\ 2008, Tanvir et al.\ 2008), but
is probably confined only in a fraction of the volume occupied by the young
stars (Micha\l{}owski et al.\ 2008).
\begin{figure}
\includegraphics[width=\columnwidth]{Fig13.ps}
\caption{$R-K$ colour of the hosts in the Hjorth et al.\ (2012) sample as a function
of $z$. Filled symbols are for GRBs with detected optical afterglows and open
symbols for GRBs with no detected optical afterglow. Only the galaxies with
optical detections are shown. The left panel shows systems with no known
redshift (the $x$-axis value is arbitrary). The horizontal line marks the
boundary of extremely red objects (EROs; $R - K >
5$).\label{fg:colours}}
\end{figure}
\textbf{Redshifts.}
Hosts without known redshift were observed with a variety of
spectroscopic setups (Jakobsson et al.\ 2012, Kr{\"u}hler et al.\ 2012).
In several cases, only upper and/or lower limits could be placed on the redshift due to the lack of prominent
features. Many GRBs are bound to be in the so-called redshift desert ($1
\lesssim z \lesssim 2$), where the most prominent nebular lines are shifted
into wavelength regions difficult to observe. In some cases,
this hypothesis was confirmed thanks to the use of red grisms sensitive up to 10,000~\AA,
which probes redshifts up to $z \approx 1.7$ through the [O II] emission line.
Overall, 15 new redshifts were determined from host galaxy spectroscopy, and,
surprisingly, a few of the redshifts reported in the literature were found to
be inconsistent with the redshift derived from the likely host. This is most likely
not due to a wrong host identification as these are bursts with detected
optical afterglows (and for some the reported wrong redshifts were based on
emission lines). For an additional three systems the redshift could be
constrained in the range $1 \lesssim z \lesssim 2$. Surprisingly, only three
of the targeted systems have a redshift larger than 2. While this is partly due
to the selection of the brightest systems for spectroscopy, it also shows that
many dark or optically faint GRBs probably lie at moderate redshifts.
Fig.~\ref{fg:redshift} shows the redshift distribution of the sample, outlining
the contribution from the Hjorth et al.\ (2012) program. We caution that the analysis
of the spectroscopic data is not yet complete. The most noticeable effect is
the reduction of the ``gap'' at $z \sim 1.7$, which was likely due to the lack
of prominent features in the observed spectral range, for both GRB afterglows and hosts (Fiore et al.\ 2007).
\textbf{Ly$\alpha$ emisson.} For all hosts with a known redshift in the range
$2 < z < 4.5$ (where the Ly$\alpha$ falls in a favorable wavelength range), a
spectrum was obtained with the aim of looking for Ly$\alpha$ in emission
(Milvang-Jensen et al.\ 2012). The
presence of a host galaxy detection in the optical was not required as
Ly$\alpha$ can easily be detected from galaxies that are very faint in the
continuum (e.g., Fynbo et al.\ 2003b). These spectra also provided a way to
double check some of the redshifts reported in the literature (leading to the
aforementioned discovery of a few likely wrong redshifts). The recovery fraction for these \textit{Swift} GRB
hosts is lower than in earlier cases (with moderately deep exposures of 1.5--4
hr). Ly$\alpha$ is detected in about 35\% of the cases. As mentioned above,
pre-\textit{Swift} studies provided five detections out of five studied cases
(Fynbo et al.\ 2003a). The peak of the Ly$\alpha$ line is observed to be redshifted by a
few hundred km~s$^{-1}$ with respect to the absorption-line redshift inferred
from afterglow spectroscopy. This has been already observed in Lyman break
galaxies (e.g., Adelberger et al.\ 2003).
In summary, the Hjorth et al. (2012) study of {\it Swift} hosts has established that
many of the conclusions reached from the smaller and more biased pre-{\it Swift}
samples still hold: GRB hosts are all star forming and they are predominantly
(but not all) subluminous and blue.
\begin{figure}
\includegraphics[width=0.49\columnwidth]{Fig14_top.ps}\hspace{0.02\columnwidth}
\includegraphics[width=0.49\columnwidth]{Fig14_bottom.ps}
\caption{Redshift distribution of GRBs in the Hjorth et al.\ (2012) sample. Top
panel: redshifts taken from the literature. Bottom panel: including the results
from host spectroscopy in the program.\label{fg:redshift}}
\end{figure}
\section{Simulations of GRB host galaxies}
In this final section we will briefly discuss the work that has been done on
simulating and modeling GRB host galaxies. The modeling of star-forming
galaxies in their cosmological context is a field in rapid progress. Whereas
the treatment of growth of structure in the dark matter component seems to be
very well understood it is still far from trivial to include the baryonic
physics. In particular star formation and its feedback on the interstellar
medium are difficult to include in the simulations. Different ``recipes'' exist,
but it is difficult objectively to establish if they provide an adequate
description of reality rather than providing, e.g., a fitting parameter that
helps reproducing a set of observations. GRB host galaxies are interesting for
testing simulations of galaxies due to their nature as star-formation selected
galaxies. Hence, it is relatively easy to predict from a given simulation what
the properties (e.g., luminosities, metallicities, environments, etc.) of a
sample of GRB host galaxies should be under the assumptions behind the
simulation. This work has only started recently and there is certainly
potential for a lot of development in this area. The first pioneering work
was done by Courty et al.\ (2004, 2007) who basically established that galaxies
similar to the GRB hosts exist in their simulations. Nuza et al.\ (2007)
predicted the properties of GRB hosts under the assumption that GRBs are only
formed by low-metallicity stars. The study used a rather small simulated volume
with a box length of only 10 Mpc, but it was still an important step forward.
Most lately, Nagamine et al.\ (2008) used a similar simulation to examine the HI absorption
of GRB host absorption systems as a function of redshift.
\section{Conclusions and outlook}
About a decade after the first detection of host galaxies we have progressed
tremendously. GRBs host galaxies have been found to be predominantly
young, actively star-forming and subluminous. This conclusion seems to
reflect the intrinsic properties of the host populations and is not only a result of selection effects. The study of the GRB host absorption
systems is an entire new emerging field providing an interesting link
between the study of QSO-DLAs and star-forming galaxies selected in
emission.
There are still unclarified issues. One of the most important questions to
answer is whether GRBs trace all star-formation or only a limited,
low-metallicity segment. The evidence seems to point in different
directions. Most likely the road to further progress will be by building
yet more complete samples with more detailed information for all
GRBs and their hosts (metallicities, luminosities, dust masses, etc.).
Bursts like GRB\,080607 (Prochaska et al.\ 2009) and GRB\,070306 (Jaunsen
et al.\ 2008) suggest that we do preferentially lose GRBs in
high metallicity, dusty environments. We need to establish the frequency
of such systems in the host population to determine where GRBs fit
into the big picture. Also, more work is needed on improving the predictions
of how the properties of GRB hosts should be under various assumptions
about the link between GRBs and star-formation.
Another important area for future work is the use of GRBs to probe galaxies at
the epoch of re-ionization. It is now established that GRBs allow us to move
further back in time than what is currently possible with QSOs (Greiner et al.\
2009, Tanvir et al.\ 2010, Salvaterra et al.\ 2010). The discovery of bursts
like GRB\,090423 also allows us to study in what
type of galaxies most of the star formation happened at $z > 8$, and what was the
nature of the sources responsible for the re-ionization. There is evidence
that the bright $z > 6$ galaxies discovered using colour-colour (drop-out)
selection or more advanced photometric redshifts are too rare to provide the
total star formation rate as well as to have done the re-ionization (e.g.,
Bouwens et al.\ 2007). GRB measurements provide the tool to find the
more typical galaxies responsible for the bulk of the production of ionizing
photons (Ruiz-Velasco et al.\ 2007), and will allow further study of these
galaxies in the future (e.g., with {\it JWST} or 30m ground-based telescopes).
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\cleardoublepage
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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10 Chip Brands Ranked From Worst To Best
10 Ways Costco Has Been AFFECTED by Covid-19 / Coronavirus Pandemic (Part 2)
Sandra Lesniak
Chips are one of the most popular snacks in the world. The salty, crunchy, snacks that come in a variety of delicious flavors are a go-to for many. However, there is a huge disparity when it comes to chip brands. Some are great and others…well, maybe not so great. So here are the top 10 Chip Brands Ranked From Worst To Best.
10. Old Dutch
Old Dutch chips are old news and aren't as popular as they once were. Old Dutch was founded in 1934. Today their product line includes Old Dutch Potato Chips, Dutch Crunch, Ripples, Cheese Pleesers, and Restaurante Style Tortilla Chips. The Old Dutch chips come in standard flavors like original, BBQ, Sour Cream and Onion and Ketchup (for the Canadians, of course), along with a few unique flavors like Smokey Bacon and Onion & Garlic. The problem with the Old Dutch brand is that the chips aren't tasty enough. Their quality rating used to be a lot higher. Until their competitors began showing up on shelves next to them, they were wildly popular with kids. Old Dutch just failed to keep up with the times. One commenter on a review site perfectly encapsulated their decline, saying they "used to be my favorite, but the new kettle type chips sure put these under the table". The commenter went on to say how they wished the chips themselves were better, that they were really thin and can get oversaturated with oil as a result. The Old Dutch BBQ chips have more of a tangier flavor, whereas most prefer a sweeter barbecue taste on potato chips. The brand has been around for many years, so they have picked up some loyal chip fans, but they are no longer a popular favorite like some of the relatively new kids on the block.
9. Sun Chips
First off, Sun Chips are made with whole grains instead of potatoes so these may qualify as a "healthier" chip. But if they aren't made with potatoes can they even be considered chips? Sun Chips are fried, rippled, multigrain chips produced by Frito-Lay. We have to give them credit though. They wanted to be different so they created what they call a "one-of-a-kind chip". They do have some pretty good flavors like Harvest Cheddar, French Onion, Sweet and Spicy BBQ, and Garden Salsa. Although the flavors are yummy, some chip connoisseurs don't enjoy the multi-grain taste these chips serve up. But in the late 2000s, Frito-Lay decided to make their bags compostable. While this was a step in the right direction for the environment, unexpected backlash occurred. The newly formulated compostable bags made way more noise then a standard chip bag. Yes, this was an actual complaint, and many people disliked the extra noise the bag made while snacking on Sun Chips. The sales of Sun Chips actually experienced an 11 percent drop around that same time period! That sprung execs into action to create a, you guessed it, quieter Sun Chips bag. But maybe they should have focused on upgrading their chips as well as their bags.
8. Tostitos
Tostitos are merely shovels used to dig up copious amounts of salsa, cheese, and guacamole. It's very uncommon for someone to grab a bag of Tostitos chips if they don't plan on heavily dipping them in some sort of sauce. Don't take this the wrong way – they are a good chip for these types of dips, but overall they lack flavor and excitement. That's why they come in scoops or massive sized hexagons and triangles, the makers intended on them being used as a dipping chip and not one that is enjoyed all on its own. This makes Tostitos a mediocre chip when standing next to other stand-alone chips. You'll find a bag of these on the table at most parties and almost every Super Bowl get-together but it is not necessarily a pantry staple. The restaurant-style Tostitos are ideal for making nachos! The popular tortilla chips also make a great sidekick to Mexican dishes especially with their hint of lime flavor. According to a few blogs and university polls, Tostitos don't even make it to the top of the list for best tortilla chips! Sorry Tostitos, but the more popular choices are Calidad, Late July, Santitas and Mission tortilla chips.
Pringles are an American brand of stackable snack chips owned by Kellogg's. The chips have been around for ages and even though it feels like we are eating chips from a tennis ball container, no one cares because they are that awesome. These are potato and wheat-based chips and offer a good crunch with the perfect wave shape in a variety of flavors. They are made from a dough made of dried potatoes, wheat starch, potato flour, corn flour, and rice flour. Pringles look like they were made to survive anything in those cylindrical containers. Their commercials are good at convincing us that once you pop the top you can't stop. This is definitely true – we've all been there – staring down into that empty Pringles tube, wondering what happened. Still, Pringles remains a hit-or-miss with many chip lovers. Some snack aficionados would probably prefer some of the other brands on this list. But why would someone omit a chip that pops with fun? Well, one reason might be because Pringles contain 2.5 times more saturated fat per serving, which is one of the worst types of fat you can consume. Maybe this is why they came out with a reduced-fat version. One of the coolest aspects of eating Pringles is the vast array of flavors they come in! Things start to get really crazy when you look into international, local flavors, but even here in America there's plenty of flavors abound, though often available for a limited time only. But unique flavors like Mozzarella Sticks, Pecan Pie and Seaweed flavored Pringles are actually the norm for this delicious chip brand!
6. Ruffles
Ruffles is another chip company that puts a little spin on the average potato chip. It's the ridges that make this chip stand out. The name is a direct reference to the "ruffle" made when folds are created. The chips were sold using the slogan "RRRuffles Have Ridges!" The idea of these ruffled chips was for them to not only be a sturdier and crunchier potato chip but also less likely to break in the bag. This might be a great concept, but the bottom of the bag still ends up with broken pieces. An extra bonus feature when it comes to Ruffles is that they are the perfect chip for standing up to stiffer dips. Ruffles are best known for their All Dressed chip flavor. Like Drake, these used to be exclusive to Canada, but now they can be found in other places too. Ruffles All Dressed are the perfect combination of the holy trinity of chip flavors: ketchup, salt and vinegar, and BBQ. Let's not forget the other awesome flavors Ruffles has like original, which is crunchy and has just the right amount of saltiness. They also have sour cream and onion, cheddar and sour cream, and unique flavors like poutine and beer-battered onion rings. The ridges, and their classic taste, are what gives this chip a leg up on some of the other brands out there.
5. Miss Vickie's
Next on the list is a fine kettle chip if there ever was one. There are so many intriguing flavors in this chip family. Miss Vickie's doesn't get any better than their Salt & Vinegar chips, however they do have several other flavors that stand out. Each flavor is as delicious and crunchy as the next. There is a fan for every chip Miss Vickie's has come out with, especially Jalapeno! Dare we say that there is a possible tie for top spot when it comes to Miss Vickie's chips? "Kettle" chips have become widely popular and loved by many. They come with a delicious kettle-cooked crunch. Some online forum contributors claim Miss Vickie's are a healthier and better-for-you option because you can eat more Miss Vickie's chips than you can other chips for the same amount of calories. But who's counting calories when snacking on chips anyway? Miss Vickie's delivers a chip with the perfect amount of seasoning. Once you open a bag, you don't put it down until it's empty. Miss Vickie's has very high reviews for their kettle chips; a lot of people love the extreme crunchiness of this brand. The chips themselves are medium and small-sized instead of the big chunky chips that can't fit in your mouth with one bite. So, when it comes to kettle chips, the crunchier the better!
4. Cheetos
While maybe not technically being an actual chip, Cheetos have been a long time favorite. They are addictive. One reviewer commented how they love the orange cheesiness and the crunch that comes with Cheetos. Doesn't everyone? They are great to serve at parties because no dip is required. Cheetos are a much-loved cheesy treat that is fun for everyone. Kids can't eat Cheetos without licking the signature "cheetle" off their fingers once done. It might even be the best part! Cheetos came out with a flaming hot crunchy version everyone has been ranting and raving about. Whether snacking on them in the car during traffic, or on the couch binge-watching Netflix, these Flamin' Hot Cheetos have also been a hit on social media. Flamin' Hot Cheetos have many 5-star reviews with people claiming they are addicted to them because they have a great "kick" in addition to the crunch. A highly recommended option for those who like extra cheese or some heat! Katy Perry dressed up as a Flamin' Hot Cheeto for Halloween back in 2014, complete with a Cheetos handbag. Even she's letting us know they will light up your taste buds like fireworks. And of course, they do have one of the coolest mascots around in Chester Cheetah.
Now these are common tortilla chips you can find at most parties and friendly gatherings. They are not only flavorful but their perfect crunch and shape is ideal for salsa dipping and more. There are so many Doritos flavors including Zesty Cheese, Bold BBQ, Jalapeno, Spicy Nacho, Sweet Chili Heat, and Flaming Hot Nacho. Some flavors taste similar to others but who's complaining when they are all equally tasty? Who doesn't love a little spice or extra cheesy flavor on their chips! Doritos Nacho Cheese is the OG in terms of tortilla chip popularity! The original Doritos were not flavored. They then released a taco flavor but they didn't stop there. In 1972, their third flavor came out: Doritos Nacho Cheese. Today it's better and cheesier than ever. Every crunchy bite is packed with a burst of bold and cheesy flavor. It's impossible to eat just one or two, whether it be Cool Ranch, Cheese Supreme, or Salsa Verde! Even Avengers stars Chris Evans & Chris Hemsworth are big fans of Doritos, and claim to prefer the Cool Ranch flavor, which seems questionable, Nacho Cheese all the way!
2. Loads of Flavor by President's Choice
Literally, like the name says, there is loads of flavor in every chip in every bag you open. The Loads of Flavor chips by President's Choice brand — also known as PC — are a Canadian private label owned by Loblaws Companies Limited. Their chips can be found in many different stores besides Loblaws. Sadly, for all those outside of Canada you might have to do a little ordering online to try all the different flavors but trust us when we say it's worth it. The PC brand has high ratings online for their taste and the quality of the chips. You can find a variety of PC loaded flavors: including loads of ketchup, loads of all dressed, loads of barbeque, loads of sour cream and onion, and loads of creamy dill pickle. President's Choice describes their chips as "seasoned to the max for extreme snacking satisfaction, these delicious rippled chips are loaded with flavor and pack a mighty tasty crunch". Those who have tried these chips and then can't get their hands on them get very frustrated – and with good reason. A fan craving their ketchup chips commented how they grew up in Canada and now live in the States and can't get these anymore… Any time friends or family come down from Canada they always make sure to bring a couple of bags. Like, we said before, these chips are worth the wait.
1. Lay's
Lay's is arguably the most popular brand of potato chips out there. They sell their chips all over the world in almost every country in countless flavors. They are thinly sliced to perfection and have wonderful flavor. There are Lay's fans globally with one loyal fan stating that they have loved Lays for 50 years and the chips were always of a good size and color and great tasting. There never was a better chip. Lay's has become a staple in many cupboards across the world. Their classic original flavor is one of the most popular and well-loved. A classic plain chip that you cannot go wrong with. Lay's original are not too salty and perfect to eat alone or with a dip. Like their slogan says… betcha can't eat just one! Lay's chips have also become a popular side for sandwiches, burgers or hot dogs. People enjoy an added crunch and it's usually the original flavor that wins out when served with a hoagie or sub. Lay's always gets creative with funky, fun and unique flavors people worldwide love to try! Like Thai Sweet Chili, Deep Dish Pizza, Chili Con Queso, Cajun Spice and more! Lay's also knows how to get healthy eaters' attention. They launched their own style of baked chips and these have also done very well since they can't seem to stay on store shelves for long.
Next Next: 10 Unhealthy Foods You Probably Eat Every Day
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Top 10 Fast Food Items That Totally FAILED in America (Part 2)
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10 Fast Food Chains that are by Far the WORST in the Country! (Part 2)
Top 10 Secrets Of The Costco Bakery You'll Wish You Knew Sooner
Top 10 Dunkin' Donuts Secret Menu Items That Will Make Starbucks Jealous
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,471
|
NS_ASSUME_NONNULL_BEGIN
@interface NSUserDefaults (DefaultValues)
- (nullable id)objectForKey:(NSString *)defaultName defaultValue:(nullable NSString *)value;
- (nullable NSString *)stringForKey:(NSString *)defaultName defaultValue:(nullable NSString *)value;
- (nullable NSArray *)arrayForKey:(NSString *)defaultName defaultValue:(nullable NSArray *)value;
- (nullable NSDictionary<NSString *, id> *)dictionaryForKey:(NSString *)defaultName defaultValue:(nullable NSDictionary<NSString *, id> *)value;
- (nullable NSData *)dataForKey:(NSString *)defaultName defaultValue:(nullable NSData *)value;
- (nullable NSArray<NSString *> *)stringArrayForKey:(NSString *)defaultName defaultValue:(nullable NSArray<NSString *> *)value;
- (NSInteger)integerForKey:(NSString *)defaultName defaultValue:(NSInteger)value;
- (float)floatForKey:(NSString *)defaultName defaultValue:(float)value;;
- (double)doubleForKey:(NSString *)defaultName defaultValue:(double)value;;
- (BOOL)boolForKey:(NSString *)defaultName defaultValue:(BOOL)value;;
- (nullable NSURL *)URLForKey:(NSString *)defaultName defaultValue:(nullable NSURL *)value;
@end
NS_ASSUME_NONNULL_END
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,326
|
package org.jboss.weld.tests.alternatives;
public class Foo {
private final String name;
public Foo(String name) {
this.name = name;
}
public String getName() {
return name;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,914
|
{"url":"https:\/\/codereview.stackexchange.com\/questions\/172941\/how-to-avoid-duplicate-code-due-to-the-impossibility-of-using-multiple-inheritan","text":"# How to avoid duplicate code due to the impossibility of using multiple inheritance\n\nI'm working on a Spring - Hibernate App, and I have a question about how to correctly avoid duplicate code and using Hibernate, due to the impossibility of using use multiple inheritance (I usually work with Python so this is not a \"problem\").\n\nMy UML:\n\nhttp:\/\/i.imgur.com\/9GD1sjV.png\n\nMy class, Periodico, for example, the same for Livro or Prototexto extends GenericEntity:\n\n@Entity\n@Table(name = \"periodico\")\npublic class Periodico extends GenericEntity {\n\n}\n\n\nMy question is: \u00bfwhat is the way to implement inheritance in this case of two classes?\n\nAt this moment I'm doing this and it works.\n\n\/**\n* Created by hlorenzo on 03\/08\/2017.\n*\/\n@Entity\n@Table(name = \"prototexto\")\npublic class Prototexto extends GenericEntity {\n\nprivate String titulo;\n\n\/*\nC\u00f3digo alfanum\u00e9rico composto por n\u00ba de clase + tipo (3 iniciais) + n\u00ba Id (3 cifras) + data_ano\n*\/\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"numerasao\")\nprivate String numerasao;\n\n\/*\nLa utilizaci\u00f3n del prefijo nacimiento es para que funcione con el componente fecha.component.js de forma autom\u00e1tica.\n*\/\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"publicacion_dia\")\nprivate Integer nacimientoDia;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"publicacion_mes\")\nprivate Integer nacimientoMes;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"publicacion_ano\")\nprivate Integer nacimientoAno;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"primeira_linha\")\nprivate String primeiraLinha;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"ultima_linha\")\nprivate String ultimaLinha;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"numero_paginas\")\nprivate Long numeroPaginas;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"descrisao\")\nprivate String descrisao;\n\n@JsonView(JsonViews.DetailedList.class)\n@Column(name = \"localizasao\")\nprivate String localizasao;\n\n@Enumerated(value = EnumType.STRING)\nprivate TipoPrototexto tipo;\n\nprivate boolean concluido;\n\n@JsonView(JsonViews.DetailedList.class)\n@ManyToOne\n@JoinColumn(name=\"pais_id\")\nprivate Pais pais;\n\n@JsonView(JsonViews.DetailedList.class)\n@ManyToOne\n[...]\n}\n\n\nMy GenericEntity:\n\n@MappedSuperclass\npublic abstract class GenericEntity implements Serializable {\n\n@Id\n@GeneratedValue(strategy = GenerationType.IDENTITY)\n@JsonView(JsonViews.List.class)\nprivate Long id;\n\npublic Long getId() {\nreturn id;\n}\n\npublic void setId(Long id) {\nthis.id = id;\n}\n\n}\n\n\nThe problem is that my classes extends GenericEntity and now I don't know how to implement the second relationship and use hibernate. Maybe using interfaces? or should I implement another way.\n\nI've had a lot of occasions to face similar problems, but must admit that there is no ideal solution. It is difficult to avoid code duplications in JPA entities definitions.\n\nHowever, there are some things that I can recommend.\n\nBoth Fasterxml and JPA annotations work on fields and on methods, which makes possible to extract common things to interfaces.\n\nFor example, you have several entities that contain titulo and descricao fields like the one of your example.\n\nYou can create interfaces to manage JSON representations of these fields, for example:\n\npublic interface Intitulado {\n\n@JsonView(JsonViews.List.class)\n@JsonProperty(\"titulo\")\nString getTitulo();\n\n}\n\n\nBy analogy, there can be Descrito interface.\n\nI also recommend to always specify @JsonProperty annotation: if the name of the field changes some day, the JSON API will not be broken.\n\nIf Lombok is used in the project, this approach allows to reduce dozens of lines of code.\n\nNow, the initial class will resemble the following:\n\n@Data\npublic class Prototexto implements Intitulado, Descrito {\n\nprivate String titulo;\n\nprivate String descricao;\n\n}\n\n\nBut if you need to change the JSON view for this object only, you'll have to override the getter, which brings back the verbosity.\n\nSimilar approach can be used for JPA annotations if the relationship between entities is not very complex, but there are many pitfalls with them anyway and duplications will not be avoidable in many cases.\n\nOther remarks:\n\n1) Why do you use nacimientoDia, nacimientoMes and nacimientoAno fields instead of a single nacimiento field typed as LocalDate?\n\n2) Long might be too much for numeroPaginas field. Integer is hugely enough.\n\n\u2022 Thanks for your time @Antot. 1) Because it is an article cataloger, where you can know the year, but not the month. Or the month and year, but not the day. 2) Certain collections of hundreds of thousands of books can result in many pages. Probably got there by the existing limit anyway. But all the cases of groupings are unknown, I must opt for the highest ranking. \u2013\u00a0Hugo L.M Sep 23 '17 at 11:16\n\u2022 I finally opted to use the legacy in Java and leave the database as it was. As soon as I finish it, I'll put the code in my github. \u2013\u00a0Hugo L.M Sep 23 '17 at 11:18","date":"2019-06-15 23:54:05","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\": 0.24615244567394257, \"perplexity\": 5776.184910558113}, \"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-2019-26\/segments\/1560627997501.61\/warc\/CC-MAIN-20190615222657-20190616004657-00545.warc.gz\"}"}
| null | null |
Duval County je název dvou okresů ve Spojených státech amerických:
Duval County (Florida) – okres státu Florida
Duval County (Texas) – okres státu Texas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,260
|
by vietnamshipbuildingnews June 14, 20143:54 pm June 15, 2014
Building more vessels for Fisheries Surveillance Force
The Prime Minister has decided to invest US$200 million in building four more large-scale vessels for the Fisheries Surveillance Force to allow them to better carry out their law enforcement missions in the country's waters and supporting Vietnamese fishermen.
Prime Minister Nguyen Tan Dung made the investment decision at a June 4 working session with key officials of ministries, agencies and the Shipbuilding Industry Corporation (SBIC), following an inspection of a new large-sized ship built by Ha Long Shipyard for the force.
The ship, named KN-781, has a displacement of more than 2,000 tonnes, and is one of the largest patrol ships in the country. It will be handed over to the Fisheries Surveillance Force this month.
The Ha Long Shipyard is also putting final touch on another similar vessel for the force, which will be transferred in July.
The PM also agreed to the building of an additional 15 medium-scale ships, bringing the expected number of fisheries surveillance vessels to more than 50 in total, thus allowing the force to better perform its law enforcement and and rescue missions at sea.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,798
|
Q: websocket socket.io dir nodejs I have a question regarding socket.io and dir
I have an api to validate login / jwt and etc
I'm starting to create a websocket server to create a joken po game,
one question i already have this api folder, should i create my websocket server inside this src? Or is it better to create another folder for this separate websocket server and it will consume my login api and others dates
and about having to pass html directory on nodejs to use socketio ex:
webApp.get ('/', (req, res, next) => {
res.sendFile (__ dirname + 'game.html'
});
How would I do if I create in a separate folder from my front end how would I be able to communicate?
ex: my front end in front end folder
my back end in backend folder
A: You can use your existing express server to host your socket.io instance. You can reuse all the same infrastructure in place and continue to use jwt tokens for your authentication with socket.io. Look at the socketio-jwt package on npm.
const socketioJwt = require('socketio-jwt');
const express = require('express');
const app = express();
// rest of your express config
...
const server = app.listen(PORT,
console.log(`Server running in ${process.env.NODE_ENV} mode on port ${PORT}`)
);
const io = require('socket.io')(server);
io.use(socketioJwt.authorize({
secret: 'your secret or public key',
handshake: true
}));
io.on('connection', (socket) => {
console.log('hello!', socket.decoded_token.name);
socket.emit('hello', {message: 'hello world'});
socket.on('myevent', myhandler);
});
Ideally, you can simply use socket.io for all your backend communication.
In react you would simply import the socket.io client and then connect as follows:
let socket = null;
function App() {
useEffect(() => {
async function connect () {
socket = io.connect('http://myserver');
}
connect();
}, []);
return (
<div>
Hello
</div>
);
}
export default App;
There is a lot to cover and it is a broad question, but you have some starting points now to continue your research.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,402
|
868 Lova este o planetă minoră ce orbitează Soarele, descoperită pe 26 aprilie 1917.
Legături externe
Planete minore
Planete minore
Centura de asteroizi
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,020
|
\section{Introduction}
In a series of papers, Nekrasov (2007, 2008 a,b, and 2009 a,b), a general
theory for electromagnetic compressible streaming instabilities in
multicomponent rotating magnetized objects such as accretion disks and
molecular clouds has been developed. For accretion disks, the different
equilibrium velocities of different species (electrons, ions, dust grains,
and neutrals) have been found from the momentum equations taking into
account anisotropic thermal pressure and collisions of charged species with
neutrals (Nekrasov 2008 a,b, 2009 b). For molecular clouds, their cores, and
elephant trunks, the different equilibrium velocities of charged species
have been supposed due to the action of the background magnetic field
(Nekrasov 2009 a). It has been shown that differences of equilibrium
velocities generate compressible streaming instabilities having growth rates
much larger than the rotation frequencies. New fast instabilities found in
these papers have been suggested to be a source of electromagnetic
turbulence in accretion disks and molecular clouds.
Streaming instabilities can also be generated in neutral media. Recently,
the streaming instability originated due to the difference between
equilibrium velocities of small neutral solids and gas has been suggested as
a possible source that could contribute to the planetesimal formation
(Youdin \& Goodman 2005). These authors have numerically treated this
hydrodynamic instability in the Keplerian disk for the interpenetrating
streams coupled via drag forces. The growth rates of instability have been
found to be much smaller than the dynamical timescales. The particle density
perturbations generated by this instability could seed the planetesimal
formation without self-gravity.
In papers by Nekrasov devoted to electromagnetic streaming instabilities,
the viscosity has not been considered. However, numerical simulations of the
magnetorotational instability show that this effect can influence the
magnitude of the saturated amplitudes of perturbations and, correspondingly,
the turbulent transport of the angular momentum (e.g., Pessah \& Chan 2008;
Masada \& Sano 2008). Interstellar medium, molecular clouds, and accretion
disks around young stars are weakly ionized objects, where collisional
effects play the dominant role. The ratio of the viscosity to the
resistivity (the magnetic Prandtl number) for astrophysical objects takes a
wide range of values. In particular, in accretion disks around compact $X$%
-ray sources and active galactic nuclei, the magnetic Prandtl number varies
by several orders of magnitude across the entire disk (Balbus \& Henry
2008). The viscosity of astrophysical objects is studied mainly in the
framework of the one-fluid magnetohydrodynamics. Therefore, one needs to
consider this effect using a multicomponent fluid approach for studying
streaming instabilities in real multicomponent media.
In the present paper, we investigate electromagnetic streaming
instabilities\ in rotating astrophysical objects, such as, magnetized
accretion disks, molecular clouds, their cores, elephant trunks, and so on,
taking into account effects of collisions, thermal pressure and viscosity.
We consider a weakly ionized multicomponent plasma consisting of electrons,
ions, dust grains, and neutrals. The charged species are supposed to be
magnetized, i.e., their cyclotron frequencies are larger than their orbiting
frequencies and collisional frequencies with neutrals. We take into account
the effect of perturbation of collisional frequencies due to density
perturbations of species, which takes place at different background
velocities of species. We find the general expressions for the perturbed\
velocities of species for perturbations which are perpendicular\ to the
background magnetic field. (The analogous investigation of perturbations
along the magnetic field has been performed in (Nekrasov 2009 c).) These
expressions for any species also contain the perturbed velocity of other
species due to collisions. The dispersion relation is derived and
investigated for axisymmetric perturbations. The cold and thermal regimes
are considered when the pressure force is negligible or dominates the
inertia. The role of the viscosity of neutrals and charged species is
analyzed. The growth rates due to different azimuthal velocities of charged
species are found.
The paper is organized as follows. In Section 2 the basic equations are
given. In Section 3 we shortly discuss the equilibrium state. General
solutions for the perturbed velocities of species for the transverse
perturbations having the radial as well as azimuthal wave number are
obtained in Section 4. These expressions for axisymmetric (radial)
perturbations and magnetized charged species are given in Section 5. In
Section 6 the dispersion relation is derived and its solutions are found.
Discussion of the obtained results and their applicability are given in
Section 7. The main points of the paper are summarized in Section 8.
\section{Basic equations}
In the present paper, we investigate weakly ionized astrophysical objects or
their definite regions, where all the charged species are magnetized, i.e.,
when, in particular, their cyclotron frequencies are larger than their
collisional frequencies with neutrals. Then the momentum equations for
charged species and neutrals including the viscous terms have the form
(Braginskii 1965),
\begin{eqnarray}
\frac{\partial \mathbf{v}_{j\perp }}{\partial t}+\mathbf{v}_{j}\cdot \mathbf{%
\nabla v}_{j\perp } &=&-\mathbf{\nabla }U-\frac{\mathbf{\nabla }P_{j}}{%
m_{j}n_{j}}+\frac{q_{j}}{m_{j}}\mathbf{E}_{\perp }\mathbf{+}\frac{q_{j}}{%
m_{j}c}\left( \mathbf{v}_{j}\times \mathbf{B}\right) _{\perp }\mathbf{-}\nu
_{jn}\left( \mathbf{v}_{j\perp }-\mathbf{v}_{n\perp }\right) \\
&&+\mu _{j1}\mathbf{\nabla \nabla \cdot v}_{j}+\mu _{j2}\mathbf{\nabla }^{2}%
\mathbf{v}_{j}\mathbf{\times z+}\mu _{j3}\mathbf{\nabla }^{2}\mathbf{v}%
_{j\perp }, \nonumber
\end{eqnarray}%
\begin{equation}
\frac{\partial v_{jz}}{\partial t}+\mathbf{v}_{j}\cdot \mathbf{\nabla }%
v_{jz}=\frac{q_{j}}{m_{j}}E_{z}\mathbf{+}\frac{q_{j}}{m_{j}c}\left( \mathbf{v%
}_{j}\times \mathbf{B}\right) _{z}\mathbf{-}\nu _{jn}\left(
v_{jz}-v_{nz}\right) +4\mu _{j3}\mathbf{\nabla }^{2}v_{jz},
\end{equation}%
\begin{equation}
\frac{\partial \mathbf{v}_{n}}{\partial t}+\mathbf{v}_{n}\cdot \mathbf{%
\nabla v}_{n}=-\mathbf{\nabla }U-\frac{\mathbf{\nabla }P_{n}}{m_{n}n_{n}}%
-\sum_{j}\nu _{nj}\left( \mathbf{v}_{n}-\mathbf{v}_{j}\right) +\mu
_{n}\left( \mathbf{\nabla }^{2}\mathbf{v}_{n}+\frac{1}{3}\mathbf{\nabla
\nabla \cdot v}_{n}\right) ,
\end{equation}
\noindent where the index $j=e,i,d$ denotes the electrons, ions, and dust
grains, respectively, and the index $n$ denotes the neutrals. In Equations
(1)-(3), $q_{j}$ and $m_{j,n}$ are the charge and mass of species $j$ and
neutrals, $v_{j,n}$ is the hydrodynamic velocity, $n_{j,n}$ is the number
density, $P_{j,n}=n_{j,n}T_{j,n}$ is the thermal pressure, $T_{j,n}$ is the
temperature, $\nu _{jn}$ $=\gamma _{jn}m_{n}n_{n}$ ($\nu _{nj}=\gamma
_{jn}m_{j}n_{j}$) is the collisional frequency of species $j$ (neutrals)
with neutrals (species $j$), where $\gamma _{jn}=<\sigma
v>_{jn}/(m_{j}+m_{n})$ ($<\sigma v>_{jn}$is the rate coefficient for
momentum transfer). The coefficients of the kinematic viscosity are $\mu
_{j1}=c_{j1}\mu _{j}/3$, $\mu _{j2}=c_{j2}\mu _{j}/\omega _{cj}\tau _{jn}$,
and $\mu _{j3}=c_{j3}\mu _{j}/\omega _{cj}^{2}\tau _{jn}^{2}$. Here $\mu
_{j,n}=v_{Tj,n}^{2}/\nu _{jn,nn}$ ($\nu _{nn}$ is the neutral-neutral
collisional frequency), $\omega _{cj}=q_{j}B_{0}/m_{j}c$ is the cyclotron
frequency, $\tau _{jn}=\nu _{jn}^{-1}$, and $v_{Tj,n}=(T_{j,n}/m_{j,n})^{1/2}
$ is the thermal velocity. Numerical coefficients for the electrons and ions
are $c_{e1}=0.73$, $c_{i1}=0.96$, $c_{j2}=0.5$, $c_{e3}=0.51$, and $%
c_{i3}=0.3$ (Braginskii 1965). Further, $U=-GM/R$\ is the gravitational
potential of the central object having mass $M$ (when it presents), $%
R=(r^{2}+z^{2})^{1/2}$, $G$ is the gravitational constant, $E$\ and $B$ are
the electric and magnetic fields, and $c$ is the speed of light in vacuum.
The magnetic field $B$ includes the external magnetic field $B_{0ext}$ of
the central object and/or interstellar medium, the magnetic field $B_{0cur}$
of the stationary current in a steady state, and the perturbed magnetic
field. We use the cylindrical coordinate system $(r,\theta ,z),$ where $r$
is the distance from the symmetry axis $z$, and $\theta $ is the angle in
the azimuthal direction. The background magnetic field is assumed to be
directed along the $z$ axis, $B_{0}=B_{0zext}+B_{0zcur}$, the sign $\perp $
denotes the direction across the $z$ axis. In Equations (1) and (2), the
condition $\omega _{cj}\gg \nu _{jn}$ is satisfied in the viscous terms. For
unmagnetized charged particles of species $j,$ $\omega _{cj}\ll \nu _{jn},$
the viscosity coefficient has the same form as that for neutrals. We adopt
the adiabatic model for the temperature evolution when $P_{j,n}\sim
n_{j,n}^{\gamma _{a}}$, where $\gamma _{a}$ is the adiabatic constant. In
general, we consider the two-dimensional case in which $\nabla =(\partial
/\partial r,\partial /r\partial \theta ,0).$
The other basic equations are the continuity equation,
\begin{equation}
\frac{\partial n_{j,n}}{\partial t}+\mathbf{\nabla \cdot }n_{j,n}\mathbf{v}%
_{j,n}=0,
\end{equation}
\noindent Faraday`s equation,
\begin{equation}
\mathbf{\nabla \times E=-}\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t},
\end{equation}%
and Ampere`s law,
\bigskip
\begin{equation}
\nabla \times B=\frac{4\pi }{c}j,
\end{equation}
\noindent where $j=\sum_{j}q_{j}n_{j}v_{j}.$ We consider the wave processes
with typical time scales much larger than the time the light spends to cover
the wavelength of perturbations. In this case, one can neglect the
displacement current in Equation (6) that results in quasineutrality both
for the electromagnetic and purely electrostatic perturbations.
\section{Equilibrium}
We suppose that in equilibrium the electrons, ions, dust grains, and
neutrals rotate in the azimuthal direction of the astrophysical object
(accretion disk, molecular cloud, its cores, elephant trunk, and so on) with
different, in general, velocities $v_{j,n0}$. These velocities can depend on
the radial coordinate. The stationary dynamics of charged species is
undergone by the action of the background magnetic field and collisions with
neutrals. In their turn, the neutrals also experience a collisional coupling
with charged species influencing their equilibrium velocity. Some specific
cases of equilibrium have been investigated in papers by Nekrasov (2007,
2008 a,b, 2009 b).\ Stationary velocities of charged species in the
background electric field and gravitational field of the central mass in the
absence of collisions have been used in (Nekrasov 2007). The case of weak
collisional coupling of neutrals with light charged species (electrons and
ions) and weak and strong collisional coupling between neutrals and heavy
dust grains has been considered in (Nekrasov 2008 a). Equilibrium where the
neutrals have a strong coupling with light charged species as well as with
heavy dust grains has been investigated in (Nekrasov 2008 b and 2009 b).
In papers cited above, it has been shown that the different charged species
have different stationary velocities. Due to this effect, the electric
currents exist in the equilibrium state, which generate their magnetic
fields.
\section{Linear approximation: General expressions}
In the present paper, we do not treat electromagnetic perturbations
connected with the background pressure gradients. Thus, we exclude the drift
waves from our consideration. We take into account the induced reaction of
neutrals on the perturbed motion of charged species. The neutrals can be
involved in electromagnetic perturbations, if the ionization degree of
medium is sufficiently high. We also include the effect of perturbation on
the collisional frequencies due to density perturbations of charged species
and neutrals. This effect emerges when there are different background
velocities of species. Then the momentum equations (1)-(3) in the linear
approximation take the form,
\begin{eqnarray}
\frac{\partial \mathbf{v}_{j1\perp }}{\partial t}+\mathbf{v}_{j0}\cdot
\mathbf{\nabla v}_{j1\perp } &=&-c_{sj}^{2}\frac{\mathbf{\nabla }n_{j1}}{%
n_{j0}}+\frac{q_{j}}{m_{j}}\mathbf{E}_{1\perp }\mathbf{+}\frac{q_{j}}{m_{j}c}%
\left( \mathbf{v}_{j0}\times \mathbf{B}_{1}\right) _{\perp }\mathbf{+}\frac{%
q_{j}}{m_{j}c}\left( \mathbf{v}_{j1}\times \mathbf{B}_{0}\right) _{\perp } \\
&&\mathbf{-}\nu _{jn}^{0}\left( \mathbf{v}_{j1\perp }-\mathbf{v}_{n1\perp
}\right) \mathbf{-}\nu _{jn}^{0}\frac{n_{n1}}{n_{n0}}\left( \mathbf{v}_{j0}-%
\mathbf{v}_{n0}\right) +\mu _{j1}\mathbf{\nabla \nabla \cdot v}_{j1}
\nonumber \\
&&+\mu _{j2}\mathbf{\nabla }^{2}\mathbf{v}_{j1}\mathbf{\times z+}\mu _{j3}%
\mathbf{\nabla }^{2}\mathbf{v}_{j1\perp }, \nonumber
\end{eqnarray}%
\begin{equation}
\frac{\partial v_{j1z}}{\partial t}+\mathbf{v}_{j0}\cdot \mathbf{\nabla }%
v_{j1z}=\frac{q_{j}}{m_{j}}E_{1z}\mathbf{+}\frac{q_{j}}{m_{j}c}\left(
\mathbf{v}_{j0}\times \mathbf{B}_{1}\right) _{z}\mathbf{-}\nu
_{jn}^{0}\left( v_{j1z}-v_{n1z}\right) +4\mu _{j3}\mathbf{\nabla }%
^{2}v_{j1z},
\end{equation}%
\begin{eqnarray}
\frac{\partial \mathbf{v}_{n1}}{\partial t}+\mathbf{v}_{n0}\cdot \mathbf{%
\nabla v}_{n1} &=&-c_{sn}^{2}\frac{\mathbf{\nabla }n_{n1}}{n_{n0}}%
-\sum_{j}\nu _{nj}^{0}\left( \mathbf{v}_{n1}-\mathbf{v}_{j1}\right)
-\sum_{j}\nu _{nj}^{0}\frac{n_{j1}}{n_{j0}}\left( \mathbf{v}_{n0}-\mathbf{v}%
_{j0}\right) \\
&&+\mu _{n}\left( \mathbf{\nabla }^{2}\mathbf{v}_{n1}+\frac{1}{3}\mathbf{%
\nabla \nabla \cdot v}_{n1}\right) , \nonumber
\end{eqnarray}
\noindent where $c_{sj,n}=(\gamma _{a}T_{j,n0}/m_{j,n})^{1/2}$ is the sound
velocity ($\gamma _{a}=2(3)$ for the two(one)-dimensional perturbations), $%
\nu _{jn}^{0}=\gamma _{jn}m_{n}n_{n0}$, $\nu _{nj}^{0}=\gamma
_{jn}m_{j}n_{j0}$. The terms proportional to $n_{n1}$ in Equation (7) and $%
n_{j1}$ in Equation (9) describe the effect of perturbation of the
collisional frequencies $\nu _{jn}^{1}=\nu _{jn}^{0}(n_{n1}/n_{n0})$ and $%
\nu _{nj}^{1}=\nu _{nj}^{0}(n_{j1}/n_{j0})$ due to number density
perturbations. The index $1$ denotes quantities of the first order of
magnitude. The neutrals participate in the electromagnetic dynamics only due
to collisional coupling with the charged species. On the left hand-sides of
Equations (7)-(9), we do not take into account the terms of the form $%
v_{j,n1}\cdot \nabla v_{j,n0}$. These terms without neutral dynamics have
been involved in papers by Nekrasov (2007 and 2008 a). The neutral dynamics
has been included in papers (Nekrasov 2008 b and 2009 b). In all specific
cases that were considered the terms mentioned above were negligible. From
general expressions for perturbed velocities given in (Nekrasov (2008 b and
2009 b) it can be seen that these Coriolis terms can be neglected under
sufficient condition $\omega _{j,nz}\gg 2\Omega _{j,n}$, where $\omega
_{j,nz}$\ in the present case are given below and $\Omega _{j,n}$\ are the
rotation frequencies of species in equilibrium.
The continuity Equation (4) in the linear regime is the following:
\begin{equation}
\frac{\partial n_{j,n1}}{\partial t}+\mathbf{v}_{j,n0}\cdot \mathbf{\nabla }%
n_{j,n1}+n_{j,n0}\mathbf{\nabla \cdot v}_{j,n1}=0.
\end{equation}
We further apply the Fourier transform to Equations (7)-(10), supposing
perturbations of the form $\exp (ik_{r}r+im\theta -i\omega t).$ Below, we
find expressions for the perturbed velocities of neutrals and charged
species.
\subsection{Perturbed velocity and number density of neutrals}
Using the $z$ component of Equation (9), we find the Fourier amplitude of
the induced longitudinal velocity of neutrals $v_{n1zk}$,
\begin{equation}
-i\omega _{nz}v_{n1zk}=\sum_{j}\nu _{nj}v_{j1zk},
\end{equation}
\noindent where
\begin{eqnarray*}
\omega _{nz} &=&\omega _{n}+i\nu _{n}+i\mu _{n}k_{\perp }^{2}, \\
\omega _{n} &=&\omega -\mathbf{k\cdot v}_{n0}, \\
\nu _{n} &=&\sum_{j}\nu _{nj},
\end{eqnarray*}
\noindent $k=\{k,\omega \}=\{k_{r},k_{\theta }=m/r,\omega \}$, $k_{\perp
}^{2}=k_{r}^{2}+k_{\theta }^{2}$. Here and below, we omit, for simplicity,
the index $0$ for $\nu _{jn}^{0}$ and $\nu _{nj}^{0}$.
From Equations (9) and (10), we obtain the induced transverse velocity and
induced perturbed number density of neutrals,\
\begin{equation}
-i\omega _{nz}\mathbf{v}_{n1\perp k}=\sum_{j}\nu _{nj}\frac{n_{j1k}}{n_{j0}}%
\left[ \mathbf{k}\eta _{n}\frac{\omega _{n}}{\omega _{n\perp }}-\left(
\mathbf{v}_{n0}-\mathbf{v}_{j0}\right) \right] +\sum_{j}\nu _{nj}\mathbf{v}%
_{j1\perp k},
\end{equation}%
\begin{equation}
-i\omega _{n\perp }\frac{n_{n1k}}{n_{n0}}=\sum_{j}\nu _{nj}\frac{n_{j1k}}{%
n_{j0}},
\end{equation}
\noindent where
\begin{eqnarray*}
\omega _{n\perp } &=&\omega _{nz}-k_{\perp }^{2}\eta _{n}, \\
\eta _{n} &=&\frac{c_{sn}^{2}}{\omega _{n}}-i\frac{1}{3}\mu _{n}.
\end{eqnarray*}
\noindent We see from Equation (12) that the perturbed velocity of neutrals
is induced by the perturbed number density and velocity of charged species.
Equation (13) shows that the density perturbation of neutrals is defined by
the density perturbations of charged species. When $\omega _{n\perp }\sim
\nu _{nj}$, we obtain $n_{n1k}/n_{n0}\sim n_{j1k}/n_{j0}$. Note that to
obtain the correct expressions (12) and (13) it is necessary to take into
account perturbation on the collisional frequency of neutrals with charged
species.
\
\subsection{The longitudinal perturbed velocity of charged species}
Let us now find the longitudinal perturbed velocity of charged species $%
v_{j1z}$. From Equations (5) and (8) in the linear approximation and by
using Equation (11) we obtain,
\begin{equation}
\omega _{nz}\omega _{jz}v_{j1zk}+\nu _{jn}\sum_{l}\nu _{nl}v_{l1zk}=i\omega
_{nz}F_{j1zk},
\end{equation}
\noindent where
\begin{eqnarray*}
\omega _{jz} &=&\omega _{j}+i\nu _{jn}+i4\mu _{j3}k_{\perp }^{2}, \\
\omega _{j} &=&\omega -\mathbf{k\cdot v}_{j0}, \\
F_{j1zk} &=&\frac{q_{j}}{m_{j}}\frac{\omega _{j}}{\omega }E_{1zk}.
\end{eqnarray*}
Equation (14) shows that due to collisions of neutrals with charged species
and charged species with neutrals, the charged species experience the
feedback on their perturbations. If neutrals collide with two charged
species only, for example, with electrons and ions, the solutions of
Equation (14) for $e$ and $i$ have the form,
\begin{eqnarray}
D_{z}v_{e1zk} &=&i\alpha _{iz}F_{e1zk}-i\nu _{en}\nu _{ni}F_{i1zk}, \\
D_{z}v_{i1zk} &=&i\alpha _{ez}F_{i1zk}-i\nu _{in}\nu _{ne}F_{e1zk},
\nonumber
\end{eqnarray}
\noindent where
\[
\alpha _{jz}=\omega _{nz}\omega _{jz}+\nu _{jn}\nu _{nj},
\]%
\[
D_{z}=\omega _{nz}\omega _{iz}\omega _{ez}+\omega _{iz}\nu _{en}\nu
_{ne}+\omega _{ez}\nu _{in}\nu _{ni}.
\]
\noindent Substituting $F_{j1zk}$ in Equations (15), we obtain,
\begin{equation}
D_{z}v_{j1zk}=i\left( a_{jz}-n_{\theta }b_{jz}\right) E_{1zk},
\end{equation}
\noindent where $n_{\theta }=k_{\theta }c/\omega $ and
\begin{eqnarray*}
a_{ez} &=&\alpha _{iz}\frac{q_{e}}{m_{e}}-\nu _{en}\nu _{ni}\frac{q_{i}}{%
m_{i}}, \\
b_{ez} &=&\alpha _{iz}\frac{q_{e}}{m_{e}}\frac{v_{e0}}{c}-\nu _{en}\nu _{ni}%
\frac{q_{i}}{m_{i}}\frac{v_{i0}}{c}, \\
a_{iz} &=&\alpha _{ez}\frac{q_{i}}{m_{i}}-\nu _{in}\nu _{ne}\frac{q_{e}}{%
m_{e}}, \\
b_{iz} &=&\alpha _{ez}\frac{q_{i}}{m_{i}}\frac{v_{i0}}{c}-\nu _{in}\nu _{ne}%
\frac{q_{e}}{m_{e}}\frac{v_{e0}}{c}.
\end{eqnarray*}
If neutrals collide with ions and dust grains, the index $e$ in Equations
(15) and (16) must be substituted by the index $d$.
\subsection{The transverse perturbed velocity and perturbed number
density of charged species}
We now find the transverse perturbed velocity of charged species $v_{j1k}$.
From Equation (7), we obtain two equations for components $v_{j1rk}$ and $%
v_{j1\theta k},$
\begin{eqnarray}
-i\omega _{j\perp }v_{j1rk} &=&-ik_{r}c_{sj}^{\ast 2}\frac{n_{j1k}}{n_{j0}}%
\mathbf{+}\omega _{cj}^{\ast }v_{j1\theta k}+G_{j1rk}, \\
-i\omega _{j\perp }v_{j1\theta k} &=&-ik_{\theta }c_{sj}^{\ast 2}\frac{%
n_{j1k}}{n_{j0}}\mathbf{-}\omega _{cj}^{\ast }v_{j1rk}+G_{j1\theta k}.
\nonumber
\end{eqnarray}
\noindent Here, the following notations are introduced:
\begin{eqnarray*}
\omega _{j\perp } &=&\omega _{j}+i\nu _{jn}+i\mu _{j3}k_{\perp }^{2}, \\
\omega _{cj}^{\ast } &=&\omega _{cj}-\mu _{j2}k_{\perp }^{2}, \\
c_{sj}^{\ast 2} &=&c_{sj}^{2}-i\omega _{j}\mu _{j1}, \\
G_{j1r,\theta k} &=&F_{j1r,\theta k}+Q_{j1r,\theta k}, \\
F_{j1rk} &=&\frac{q_{j}}{m_{j}}\left[ E_{1rk}\mathbf{+}\frac{v_{j0}}{c}%
\left( n_{r}E_{1\theta k}-n_{\theta }E_{1rk}\right) \right] , \\
F_{j1\theta k} &=&\frac{q_{j}}{m_{j}}E_{1\theta k}, \\
Q_{j1rk} &=&\nu _{jn}v_{n1rk}, \\
Q_{j1\theta k} &=&\nu _{jn}v_{n1\theta k}\mathbf{-}\nu _{jn}\frac{n_{n1k}}{%
n_{n0}}\left( v_{j0}-v_{n0}\right) .
\end{eqnarray*}
\noindent In the expression for $F_{j1rk}$, we have used Equation (5).
Solutions of Equations (17) have the form,
\begin{eqnarray}
D_{j\perp }v_{j1rk} &=&i\omega _{j\perp }\left( 1-\delta _{j\theta \theta
}\right) G_{j1rk}+i\left( i\omega _{cj}^{\ast }+\omega _{j\perp }\delta
_{jr\theta }\right) G_{j1\theta k}, \\
D_{j\perp }v_{j1\theta k} &=&i\omega _{j\perp }\left( 1-\delta _{jrr}\right)
G_{j1\theta k}+i\left( -i\omega _{cj}^{\ast }+\omega _{j\perp }\delta
_{jr\theta }\right) G_{j1rk}, \nonumber
\end{eqnarray}
\noindent where
\[
D_{j\perp }=\omega _{j\perp }^{2}\left( 1-\delta _{j\perp }\right) -\omega
_{cj}^{\ast 2},
\]
\noindent and $\delta _{jlm}=k_{l}k_{m}c_{sj}^{\ast 2}/\omega _{j}\omega
_{j\perp }$, $l,m=r,\theta $, $\delta _{j\perp }=\delta _{jrr}+\delta
_{j\theta \theta }$. The number density perturbation, $n_{j1k}=n_{j0}k\cdot
v_{j1k}/\omega _{j}$, is the following:
\begin{equation}
\omega _{j}D_{j\perp }\frac{n_{j1k}}{n_{j0}}=\left( ik_{r}\omega _{j\perp
}+k_{\theta }\omega _{cj}^{\ast }\right) G_{j1rk}+\left( ik_{\theta }\omega
_{j\perp }-k_{r}\omega _{cj}^{\ast }\right) G_{j1\theta k}.
\end{equation}
\noindent In the absence of the viscosity, expressions (18) and (19)
coincide with the corresponding expressions in the paper by Nekrasov (2009
b).
\section{Axisymmetric perturbations}
Below, we shall consider axisymmetric perturbations with $k_{r}\neq 0$, $%
k_{\theta }=0$. In this case, Equations (18) and (19) take the form,
\begin{eqnarray}
D_{j\perp }v_{j1rk} &=&i\omega _{j\perp }\frac{q_{j}}{m_{j}}E_{1rk}+\frac{%
q_{j}}{m_{j}}\left( i\omega _{j\perp }\frac{k_{r}v_{j0}}{\omega }-\omega
_{cj}^{\ast }\right) E_{1\theta k}+i\omega _{j\perp }\nu _{jn}v_{n1rk} \\
&&-\nu _{jn}\omega _{cj}^{\ast }\left[ v_{n1\theta k}\mathbf{-}\frac{n_{n1k}%
}{n_{n0}}\left( v_{j0}-v_{n0}\right) \right] , \nonumber \\
D_{j\perp }v_{j1\theta k} &=&\left[ i\omega _{j\perp }\left( 1-\delta
_{jrr}\right) +\omega _{cj}^{\ast }\frac{k_{r}v_{j0}}{\omega }\right] \frac{%
q_{j}}{m_{j}}E_{1\theta k}+\omega _{cj}^{\ast }\frac{q_{j}}{m_{j}}%
E_{1rk}+\omega _{cj}^{\ast }\nu _{jn}v_{n1rk} \nonumber \\
&&+i\omega _{j\perp }\nu _{jn}\left( 1-\delta _{jrr}\right) \left[
v_{n1\theta k}\mathbf{-}\frac{n_{n1k}}{n_{n0}}\left( v_{j0}-v_{n0}\right) %
\right] . \nonumber
\end{eqnarray}%
\begin{equation}
\frac{n_{j1k}}{n_{j0}}=\frac{k_{r}v_{j1rk}}{\omega }.
\end{equation}
\noindent From Equation (12), we can find components of the perturbed
velocity of neutrals in the present case,
\begin{eqnarray}
v_{n1rk} &=&i\frac{1}{\omega _{n\perp }}\sum_{j}\nu _{nj}v_{j1rk}, \\
v_{n1\theta k} &=&-i\frac{1}{\omega _{nz}}\sum_{j}\nu _{nj}\frac{n_{j1k}}{%
n_{j0}}\left( v_{n0}-v_{j0}\right) +i\frac{1}{\omega _{nz}}\sum_{j}\nu
_{nj}v_{j1\theta k}. \nonumber
\end{eqnarray}%
Substituting these expressions in Equations (20) and using Equations (13)
and (21), we derive equations that contain only perturbed velocities of
charged species,
\begin{eqnarray}
D_{j\perp }v_{j1rk} &=&i\omega _{j\perp }\frac{q_{j}}{m_{j}}E_{1rk}+\frac{%
q_{j}}{m_{j}}\left( i\omega _{j\perp }\frac{k_{r}v_{j0}}{\omega }-\omega
_{cj}^{\ast }\right) E_{1\theta k}-\frac{\omega _{j\perp }\nu _{jn}}{\omega
_{n\perp }}\sum_{l}\nu _{nl}v_{l1rk} \\
&&-i\frac{\nu _{jn}\omega _{cj}^{\ast }}{\omega _{nz}}\sum_{l}\nu
_{nl}v_{l1\theta k}+i\frac{\nu _{jn}\omega _{cj}^{\ast }}{\omega }%
\sum_{l}\nu _{nl}v_{l1rk}\left[ \frac{k_{r}\left( v_{n0}-v_{l0}\right) }{%
\omega _{nz}}+\frac{k_{r}\left( v_{j0}-v_{n0}\right) }{\omega _{n\perp }}%
\right] , \nonumber \\
D_{j\perp }v_{j1\theta k} &=&\left[ i\omega _{j\perp }\left( 1-\delta
_{jrr}\right) +\omega _{cj}^{\ast }\frac{k_{r}v_{j0}}{\omega }\right] \frac{%
q_{j}}{m_{j}}E_{1\theta k}+\omega _{cj}^{\ast }\frac{q_{j}}{m_{j}}E_{1rk}
\nonumber \\
&&+i\frac{\omega _{cj}^{\ast }\nu _{jn}}{\omega _{n\perp }}\sum_{l}\nu
_{nl}v_{l1rk}-\frac{\omega _{j\perp }\nu _{jn}}{\omega _{nz}}\left( 1-\delta
_{jrr}\right) \sum_{l}\nu _{nl}v_{l1\theta k} \nonumber \\
&&+\frac{\omega _{j\perp }\nu _{jn}}{\omega }\left( 1-\delta _{jrr}\right)
\sum_{l}\nu _{nl}v_{l1rk}\left[ \frac{k_{r}\left( v_{n0}-v_{l0}\right) }{%
\omega _{nz}}+\frac{k_{r}\left( v_{j0}-v_{n0}\right) }{\omega _{n\perp }}%
\right] . \nonumber
\end{eqnarray}
We shall find solutions of Equations (23) for the case in which charged
species are magnetized, i.e., when the Lorentz force dominates inertia,
thermal, and drag forces. More specifically, we suppose that
\begin{equation}
\frac{\omega _{j\perp }}{\omega _{cj}}\left( 1-\delta _{jrr}\right) \ll
\frac{k_{r}v_{j0}}{\omega }\ll \frac{\omega _{cj}}{\omega _{j\perp }}.
\end{equation}%
It follows from these inequalities that the condition $\omega _{cj}^{2}\gg
\omega _{j\perp }^{2}\left( 1-\delta _{jrr}\right) $ must be satisfied. From
the latter, we obtain that $\omega _{cj}^{2}\gg \omega _{j\perp }^{2}$ and $%
\omega /\omega _{j\perp }\gg k_{r}^{2}\rho _{j}^{2}$, where $\rho
_{j}=c_{sj}/\omega _{cj}$ is the Larmor radius. Thus, the viscosity of
charged species will be negligible for our consideration given below.
Besides, $c_{sj}^{\ast 2}\simeq c_{sj}^{2}$ because we consider that $\omega
\ll \nu _{jn}$. In the case (24), the main perturbed velocities of charged
species (denoted by the upper index $0$) are the electric and magnetic (due
to the Lorentz force) drifts, \
\begin{eqnarray}
v_{j1rk}^{0} &=&\frac{q_{j}}{m_{j}\omega _{cj}}E_{1\theta k}, \\
v_{j1\theta k}^{0} &=&-\frac{q_{j}}{m_{j}\omega _{cj}}\left( E_{1rk}+\frac{%
k_{r}v_{j0}}{\omega }E_{1\theta k}\right) . \nonumber
\end{eqnarray}
\noindent Substituting expressions (25) in the collisional terms in
Equations (23), we obtain the following expressions for the transverse
velocities of charged species:
\begin{eqnarray}
D_{j\perp }v_{j1rk} &=&i\frac{q_{j}}{m_{j}}\sigma _{jrr}E_{1rk}+\frac{q_{j}}{%
m_{j}}\left( -\sigma _{jr\theta 1}+i\sigma _{jr\theta 2}\frac{k_{r}v_{j0}}{%
\omega }+i\sigma _{jr\theta 3}\frac{k_{r}v_{n0}}{\omega }\right) E_{1\theta
k}, \\
D_{j\perp }v_{j1\theta k} &=&\frac{q_{j}}{m_{j}}\sigma _{j\theta r}E_{1rk}
\nonumber \\
&&+\frac{q_{j}}{m_{j}}\left[ i\sigma _{jr\theta 2}-i\omega _{j\perp }\delta
_{jrr}+\left( \sigma _{jr\theta 1}-\sigma _{j\theta \theta 1}\right) \frac{%
k_{r}v_{j0}}{\omega }+\sigma _{j\theta \theta 2}\frac{k_{r}v_{n0}}{\omega }%
\right] E_{1\theta k}, \nonumber
\end{eqnarray}
\noindent where notations are introduced,
\begin{eqnarray*}
\sigma _{jrr} &=&\omega _{j\perp }+\frac{\nu _{jn}\nu _{n}}{\omega _{nz}}%
,\sigma _{jr\theta 1}=\omega _{cj}+\frac{\omega _{j\perp }\nu _{jn}\nu _{n}}{%
\omega _{n\perp }\omega _{cj}},\sigma _{jr\theta 2}=\omega _{j\perp }+\frac{%
\nu _{jn}\nu _{n}}{\omega _{n\perp }}, \\
\sigma _{jr\theta 3} &=&\nu _{jn}\nu _{n}\left( \frac{1}{\omega _{nz}}-\frac{%
1}{\omega _{n\perp }}\right) ,\sigma _{j\theta r}=\omega _{cj}+\frac{\omega
_{j\perp }\nu _{jn}\nu _{n}}{\omega _{nz}\omega _{cj}}\left( 1-\delta
_{jrr}\right) , \\
\sigma _{j\theta \theta 1} &=&\frac{\omega _{j\perp }\nu _{jn}\nu _{n}}{%
\omega _{n\perp }\omega _{cj}}\delta _{jrr},\sigma _{j\theta \theta 2}=\frac{%
\omega _{j\perp }\nu _{jn}\nu _{n}}{\omega _{cj}}\left( 1-\delta
_{jrr}\right) \left( \frac{1}{\omega _{nz}}-\frac{1}{\omega _{n\perp }}%
\right) .
\end{eqnarray*}
\noindent Below, we shall derive the dispersion relation and find its
solutions.
\section{Dispersion relation}
From Faraday's and Ampere's laws (5) and (6), we find the following
equations:
\begin{eqnarray}
j_{1rk} &=&0, \\
n_{r}^{2}E_{1\theta k} &=&\frac{4\pi i}{\omega }j_{1\theta k}, \nonumber \\
n_{r}^{2}E_{1zk} &=&\frac{4\pi i}{\omega }j_{1zk}, \nonumber
\end{eqnarray}
\noindent where $j_{1r,zk}=\sum_{j}q_{j}n_{j0}v_{j1r,zk}$, $j_{1\theta
k}=\sum_{j}q_{j}n_{j0}v_{j1\theta k}+\sum_{j}q_{j}n_{j1k}v_{j0}$, and $%
n_{r}^{2}=k_{r}^{2}c^{2}/\omega ^{2}$. We see that perturbations with
polarizations of the electric field along and across the background magnetic
field are split. We do not consider perturbations with longitudinal
polarization because the current $j_{1zk}$ does not contain the equilibrium
velocities in the axisymmetric case (see Equation (16)). Using expressions
(26), we find the transverse electric currents,
\begin{eqnarray}
\frac{4\pi i}{\omega }j_{1rk} &=&-\varepsilon _{rr}E_{1rk}-\varepsilon
_{r\theta }E_{1\theta k}, \\
\frac{4\pi i}{\omega }j_{1\theta k} &=&\varepsilon _{\theta
r}E_{1rk}+\varepsilon _{\theta \theta }E_{1\theta k}. \nonumber
\end{eqnarray}
\noindent Here,
\begin{eqnarray*}
\varepsilon _{rr} &=&\sum_{j}\frac{\omega _{pj}^{2}}{\omega D_{j\perp }}%
\sigma _{jrr},\varepsilon _{r\theta }=\sum_{j}\frac{\omega _{pj}^{2}}{\omega
D_{j\perp }}\left( i\sigma _{jr\theta 1}+\sigma _{jr\theta 2}\frac{%
k_{r}v_{j0}}{\omega }+\sigma _{jr\theta 3}\frac{k_{r}v_{n0}}{\omega }\right)
, \\
\varepsilon _{\theta r} &=&\sum_{j}\frac{\omega _{pj}^{2}}{\omega D_{j\perp }%
}\left( i\sigma _{j\theta r}-\sigma _{jrr}\frac{k_{r}v_{j0}}{\omega }\right)
, \\
\varepsilon _{\theta \theta } &=&\sum_{j}\frac{\omega _{pj}^{2}}{\omega
D_{j\perp }}\left[ -\sigma _{jr\theta 2}\left( 1+\frac{k_{r}^{2}v_{j0}^{2}}{%
\omega ^{2}}\right) +\omega _{j\perp }\delta _{jrr}+\left( i\sigma _{j\theta
\theta 2}-\sigma _{jr\theta 3}\frac{k_{r}v_{j0}}{\omega }\right) \frac{%
k_{r}v_{n0}}{\omega }\right] .
\end{eqnarray*}%
where $\omega _{pj}=\left( 4\pi n_{j0}q_{j}^{2}/m_{j}\right) ^{1/2}$ is the
plasma frequency.
From Equations (27) and (28), we obtain the dispersion relation,
\begin{equation}
\varepsilon _{rr}n_{r}^{2}=\varepsilon _{rr}\varepsilon _{\theta \theta
}-\varepsilon _{r\theta }\varepsilon _{\theta r}.
\end{equation}
\noindent Below, we will find solutions of Equation (29) in limiting cases.
\subsection{The case of cold species}
We at first consider the case in which neutrals and charged species are
sufficiently cold, $\omega ^{2}\gg k_{r}^{2}c_{sn}^{2}$ and $\omega \nu
_{jn}\gg k_{r}^{2}c_{sj}^{2}$ (it is obvious that the first condition is the
main one). We also suppose that $\nu _{jn}\gg \nu _{n}\gg \omega $. Under
these conditions the values $\sigma $\ have a simple form, $\sigma
_{jrr}=\sigma _{jr\theta 2}=\omega \nu _{jn}/\nu _{n}$,\ $\sigma _{j\theta
r}=\sigma _{jr\theta 1}=\omega _{cj}+\nu _{jn}^{2}/\omega _{cj}+i\omega \nu
_{jn}^{2}/\omega _{cj}\nu _{n}$, and $\sigma _{jr\theta 3}=\sigma _{j\theta
\theta 2}=0$. When calculating the value $\varepsilon _{rr}\varepsilon
_{\theta \theta }-\varepsilon _{r\theta }\varepsilon _{\theta r}$, we have
carried out symmetrization, used the condition of quasineutrality, $%
\sum_{j}q_{j}n_{j0}=0$, and kept the main terms. Then the dispersion
relation (29) takes the form,
\begin{equation}
\omega ^{2}\sum_{jl}\frac{\omega _{pj}^{2}\nu _{jn}}{\omega _{cj}^{2}}\frac{%
\omega _{pl}^{2}\nu _{ln}}{\omega _{cl}^{2}}=\nu _{n}\sum_{j}\frac{\omega
_{pj}^{2}\nu _{jn}}{\omega _{cj}^{2}}k_{r}^{2}c^{2}-\frac{1}{2}\sum_{jl}%
\frac{\omega _{pj}^{2}\nu _{jn}}{\omega _{cj}^{2}}\frac{\omega _{pl}^{2}\nu
_{ln}}{\omega _{cl}^{2}}k_{r}^{2}\left( v_{j0}-v_{l0}\right) ^{2}.
\end{equation}
\noindent We see that the streaming instability is possible when the
difference of the equilibrium velocities of charged species is sufficiently
large to exceed the threshold defined by the first term on the right-hand
side of Equation (30).
Let us consider, for example, Equation (30) for the electron-ion plasma
(without dust grains). Then this equation will be the following:%
\begin{equation}
\omega ^{2}=k_{r}^{2}c_{A}^{2}-\frac{m_{e}\nu _{en}}{m_{i}\nu _{in}}%
k_{r}^{2}\left( v_{e0}-v_{i0}\right) ^{2},
\end{equation}%
where $c_{A}=\left( B_{0}^{2}/4\pi m_{n}n_{n0}\right) ^{1/2}$ is the Alfv%
\'{e}n velocity. When obtaining this equation, we have used that $m_{i}\nu
_{in}\gg m_{e}\nu _{en}$ (see e.g. Section 7). The first term on the right
hand-side of Equation (31) describes the magnetosonic waves in a weakly
ionized plasma. The second term can generate the streaming instability of
the hydrodynamic kind. This term does not change the order of the dispersion
relation for perturbations under consideration that results in appearance of
the threshold for the absolute instability to develop (see also Nekrasov
2007).
\subsection{The case of thermal species}
In the case of thermal species, we suppose conditions $\mu _{n}k_{r}^{2}\gg
\omega $ and $k_{r}^{2}c_{sj}^{2}\gg \omega \nu _{jn}$ to be satisfied. Then
we have that $k_{r}^{2}c_{sn}^{2}\gg \omega ^{2}$ (the intermediate case for
neutrals, $\mu _{n}k_{r}^{2}\ll \omega \ll k_{r}c_{sn}$, we will not
consider). We also adopt, as above, that $\nu _{jn}\gg \nu _{n}$. However,
the condition $\nu _{n}\gg \omega $ is now not necessary. Then the values $%
\sigma $\ take the form,
\begin{eqnarray*}
\sigma _{jrr} &=&i\kappa \nu _{jn}\frac{\mu _{n}k_{r}^{2}}{\nu _{n}},\sigma
_{jr\theta 1}=\omega _{cj},\sigma _{jr\theta 2}=i\nu _{jn},\sigma _{jr\theta
3}=-i\kappa \nu _{jn}, \\
\sigma _{j\theta r} &=&\omega _{cj}+\sigma _{j\theta \theta 2},\sigma
_{j\theta \theta 2}=i\kappa \nu _{jn}d_{j},
\end{eqnarray*}%
where $\kappa =\nu _{n}/\left( \nu _{n}+\mu _{n}k_{r}^{2}\right) $\ and $%
d_{j}=k_{r}^{2}c_{sj}^{2}/\omega \omega _{cj}$. Substituting these values in
$\varepsilon _{rr,\theta }$ and $\varepsilon _{\theta r,\theta }$, we find
expression $\varepsilon _{rr}\varepsilon _{\theta \theta }-\varepsilon
_{r\theta }\varepsilon _{\theta r}$,
\begin{equation}
\varepsilon _{rr}\varepsilon _{\theta \theta }-\varepsilon _{r\theta
}\varepsilon _{\theta r}=i\kappa \frac{\mu _{n}k^{2}}{\nu _{n}}%
\sum_{jl}\lambda _{j}\lambda _{l}d_{j}+\kappa \frac{\mu _{n}k^{2}}{\nu _{n}}%
\sum_{jl}\lambda _{j}\lambda _{l}\frac{k_{r}^{2}\left( v_{j0}-v_{l0}\right)
^{2}}{2\omega ^{2}},
\end{equation}%
where $\lambda _{j}=-\omega _{pj}^{2}\nu _{jn}/\omega \omega _{cj}^{2}$.
When deriving Equation (32), we have used that $\sum_{j}\omega
_{pj}^{2}\omega _{cj}/\omega D_{j\perp }=-i\sum_{j}\lambda _{j}d_{j}$ due to
condition of quasineutrality. The first term on the right hand-side of this
equation is the main one according to conditions at hand among other terms
that are independent of background velocities. We see from Equation (32)
that the viscosity of neutrals plays an important role in the case under
consideration.
The dispersion relation (29) takes the form,%
\begin{equation}
\omega \left[ 2\sum_{j}\frac{\omega _{pj}^{2}\nu _{jn}}{\omega _{cj}^{2}}%
c^{2}+\sum_{jl}\frac{\omega _{pj}^{2}}{\omega _{cj}^{2}}\frac{\omega
_{pl}^{2}}{\omega _{cl}^{2}}\left( \nu _{ln}c_{sj}^{2}+\nu
_{jn}c_{sl}^{2}\right) \right] =i\sum_{jl}\frac{\omega _{pj}^{2}\nu _{jn}}{%
\omega _{cj}^{2}}\frac{\omega _{pl}^{2}\nu _{ln}}{\omega _{cl}^{2}}\left(
v_{j0}-v_{l0}\right) ^{2}
\end{equation}
\noindent Thus, the streaming instability exists for the sufficiently short
wavelengths of perturbations when the viscosity of neutrals and thermal
pressure of species play a role.
We now consider Equation (33) for the electron-ion plasma (as above). In
this case, we obtain,%
\begin{equation}
\omega =i\frac{m_{e}}{m_{i}}\nu _{en}\frac{\left( v_{e0}-v_{i0}\right) ^{2}}{%
\left( c_{Ai}^{2}+c_{s}^{2}\right) },
\end{equation}%
where $c_{Ai}=\left( B_{0}^{2}/4\pi m_{i}n_{i0}\right) ^{1/2}$ is the ion
Alfv\'{e}n velocity and $c_{s}=\left[ \gamma _{a}\left( T_{e}+T_{i}\right)
/m_{i}\right] ^{1/2}$ is the ion sound velocity. We see from Equation (34)
that the growth rate $\gamma =$Im $\omega $ can be sufficiently large in
comparison to the rotation frequency in media where $c_{Ai}^{2}+c_{s}^{2}$
is not too large as compared to $\left( v_{e0}-v_{i0}\right) ^{2}$.
The instability described by Equation (33) belongs to the class of
dissipative instabilities because the dispersion relation contains the
corresponding imaginary term which is proportional to the collisional
frequency. Without additional damping mechanisms, the threshold of this
dissipative-streaming instability is absent.
\section{Discussion}
Let us discuss some consequences that follow from the results obtained
above. From Equation (13), we see that the relative number density
perturbation of neutrals is of the order of that for charged species, if $%
\omega _{n\perp }\sim \nu _{n}$. However, in the case $k^{2}c_{sn}^{2}\gg
\omega \nu _{n}$, a neutral fluid can be considered as incompressible. In
the latter case, the perturbed radial velocity of neutrals in the
axisymmetric perturbations is also negligible (see Equation (22)). At the
same time, the perturbed azimuthal velocity of neutrals is of the order of
that for charged species for small viscosity of neutrals, $\nu _{n}\gg \mu
_{n}k_{r}^{2}$ $(\nu _{n}\gg \omega )$, and is negligible for large
viscosity of neutrals, $\nu _{n}\ll \mu _{n}k_{r}^{2}$ (see Equation (22)).
Thus, the neutrals are immobile in perturbations when their viscosity is
large.
As a specific example, we consider the case of the dense regions of
molecular cloud cores where the number density of neutrals is $n_{n}\sim
10^{5}$ cm$^{-3}$ and $n_{i}/n_{n}\sim 10^{-7}$ (Caselli et al. 1998; Ruffle
et al. 1998). For simplicity, we omit the index $0$ by $n_{j0}$. The ratio
of mass densities of dust grains, $\rho _{d}=m_{d}n_{d}$, and neutrals, $%
\rho _{n}=m_{n}n_{n}$, in the interstellar medium is of the order of $10^{-2}
$ (e.g. Abergel et al. 2005). From these relations, we obtain that $n_{d}\ll
n_{i}$ at $m_{d}\gg 10^{5}m_{n}$. For usually adopted $m_{n}=2.33m_{p}$,
where $m_{p}$ is the proton mass, and for $\sigma _{d}=1$ g cm$^{-3}$, where
$\sigma _{d}$ is the density of grain material, we obtain that $m_{d}\gg
10^{5}m_{n}=3.9\times 10^{-19}$ g is satisfied at $r_{d}$ $\gg 4.53\times
10^{-3}$ $\mu $m, where $r_{d}$ is the grain radius. The typical values of
grain radius are $r_{d}$ $\sim 0.01-1$ $\mu $m (Mendis \& Rosenberg 1994;
Wardle \& Ng 1999). However, in spite of that $n_{d}\ll n_{i}$ the mass
density of dust grains $\rho _{d}\sim 7.77\times 10^{3}\rho _{i}$ for $%
m_{i}=30m_{p}$.
Further, we take the magnetic field $B_{0}=150$ $\mu $G and the radius and
charge of dust grains $r_{d}=0.01$ $\mu m$ and $q_{d}=q_{e}$ (we consider
one typical type of dust grains). In this case, $m_{d}=4.19\times 10^{-18}$
g and $n_{d}=0.93\times 10^{-3}$ cm$^{-3}$. Then we obtain the following
values for the plasma and cyclotron frequencies of charged species: $\omega
_{pe}=5.64\times 10^{3}$ s$^{-1}$, $\omega _{ce}(>0)=2.64\times 10^{3}$ s$%
^{-1}$, $\omega _{pi}=24$ s$^{-1}$, $\omega _{ci}=4.78\times 10^{-2}$ s$^{-1}
$ ($q_{i}=-q_{e}$), $\omega _{pd}=2.53\times 10^{-2}$ s$^{-1}$, and $\omega
_{cd}(>0)=5.75\times 10^{-7}$ s$^{-1}$. The rate coefficients for momentum
transfer by elastic scattering of electrons and ions with neutrals are $%
<\sigma \nu >_{en}=4.5\times 10^{-9}(T_{e}/30$ K$)^{1/2}$ cm$^{3}$ s$^{-1}$
and $<\sigma \nu >_{in}=1.9\times 10^{-9}$ cm$^{3}$ s$^{-1}$ (Draine et al.
1983). We take $T_{e}=300$ K. Then we obtain $\nu _{en}$ $=1.42\times 10^{-3}
$ s$^{-1}$ and $\nu _{in}$ $=1.37\times 10^{-5}$ s$^{-1}$. The collisional
frequency of dust grains with neutrals has the form $\nu _{dn}\simeq 6.7\rho
_{n}r_{d}^{2}v_{Tn}/m_{d}$ ($T_{n}\sim T_{e}$) (e.g., Wardle \& Ng 1999).
Using parameters given above, we obtain $\nu _{dn}=6.24\times 10^{-8}$ s$%
^{-1}$. The collisional frequency of neutrals with charged species is $\nu
_{n}\simeq \nu _{nd}=6.24\times 10^{-10}$ s$^{-1}$.
The values $\omega _{pj}^{2}\nu _{jn}/\omega _{cj}^{2}$ which are contained
in Equations (30) and (33) are under conditions at hand the following: $%
\omega _{pe}^{2}\nu _{en}/\omega _{ce}^{2}=6.4\times 10^{-3}$ s$^{-1}$, $%
\omega _{pi}^{2}\nu _{in}/\omega _{ci}^{2}=3.44$ s$^{-1}$, $\omega
_{pd}^{2}\nu _{dn}/\omega _{cd}^{2}=120.8$ s$^{-1}$. Thus, we can write
Equation (30) in the form,
\[
\omega ^{2}=\frac{\nu _{nd}}{\nu _{dn}}\left[ 1-\frac{\nu _{in}}{\nu _{nd}}%
\frac{\left( v_{i0}-v_{d0}\right) ^{2}}{c_{Ai}^{2}}\right]
k_{r}^{2}c_{Ad}^{2},
\]
\noindent where $c_{Ad}=c\omega _{cd}/\omega _{pd}$ is the dust Alfv\'{e}n
velocity. Supposing that $|v_{i0}-v_{d0}|\gg (\nu _{nd}/\nu
_{in})^{1/2}c_{Ai}$ (the sign $|$ $|$ denotes an absolute value), the growth
rate of instability $\gamma $ will be equal to
\begin{equation}
\gamma =\left( \frac{\nu _{in}}{\nu _{dn}}\right) ^{1/2}\frac{c_{Ad}}{c_{Ai}}%
k_{r}\left\vert v_{i0}-v_{d0}\right\vert .
\end{equation}
\noindent This solution exists if
\[
\frac{\nu _{n}}{k_{r}}\left( \frac{\nu _{dn}}{\nu _{in}}\right) ^{1/2}\frac{%
c_{Ai}}{c_{Ad}}>\left\vert v_{i0}-v_{d0}\right\vert >c_{sn}\left( \frac{\nu
_{dn}}{\nu _{in}}\right) ^{1/2}\frac{c_{Ai}}{c_{Ad}}.
\]
\noindent From these inequalities, it follows that $\lambda _{r}\gg 2\pi
c_{sn}/\nu _{n}$, where $\lambda _{r}=2\pi /k_{r}$ is the wavelength of
perturbations. For parameters given above, we obtain $c_{sn}=1.73$ km s$^{-1}
$ $(\gamma _{a}=3)$, $c_{Ai}=600$ km s$^{-1}$, $c_{Ad}=6.8$ km s$^{-1}$, $%
\left\vert v_{i0}-v_{d0}\right\vert >10$ km s$^{-1}$, and $\lambda _{r}\gg
1.73\times 10^{10}$ km. The estimation of the growth rate (35) at $%
\left\vert v_{i0}-v_{d0}\right\vert =15$ km s$^{-1}$ and $\lambda
_{r}=5\times 10^{10}$ km is the following: $\gamma =3\times 10^{-10}$ s$^{-1}
$. The condition of magnetization (24) is satisfied for ions as well as for
dust grains. The case of unmagnetized dust grains has been considered in
(Nekrasov 2009 a)
In the case of thermal species and/or short wavelength perturbations, $\mu
_{n}k_{r}^{2}\gg \omega $ and $k_{r}^{2}c_{sj}^{2}\gg \omega \nu _{jn}$, we
consider the electron-ion plasma because conditions $k_{r}^{2}c_{sd}^{2}\gg
\omega \nu _{dn}$ and $k_{r}v_{d0}/\omega \ll \omega _{cd}/\nu _{dn}$ (see
inequalities (24)) are incompatible for solution (33) and parameters given
above. In the absence of dust grains, Equation (33) has the form (34). The
condition $k_{r}^{2}c_{si}^{2}\gg \gamma \nu _{in}$ and the right inequality
(24) for ions $k_{r}v_{i0}/\gamma \ll \omega _{ci}/\nu _{in}$ (for electrons
these conditions are weaker) are compatible if
\[
\frac{m_{i}\nu _{in}}{m_{e}\nu _{en}}\ll \frac{\beta }{2\left( 1+\beta
\right) }\frac{\omega _{ci}^{2}}{\nu _{in}^{2}},
\]
\noindent where $\beta =12\pi n_{i}\left( T_{e}+T_{i}\right) /B_{0}^{2}$.
For number densities $n_{i}$ and $n_{n}$ used above, this inequality is also
not satisfied. At $\beta \ll 1$\ it can be written in the form,%
\[
\frac{\omega _{pi}}{\nu _{in}}\frac{c_{s}}{c}\gg 1.41\left( \frac{m_{i}\nu
_{in}}{m_{e}\nu _{en}}\right) ^{1/2}.
\]%
For $n_{n}\sim $\ $10^{4}$\ cm$^{-3}$\ and $n_{i}/n_{n}\sim 10^{-6}$,\ this
condition is satisfied.
We have excluded drift waves in our study. This can be done if the frequency
spectra of these waves and perturbations under consideration are different.
As is known, the frequency of drift waves, for example, in the electron-ion
plasma, is equal to $\omega =\left( T_{e}/m_{i}\omega _{ci}n_{i}\right) %
\left[ \mathbf{k\times \nabla }n_{i}\right] _{z}$. For axisymmetric
perturbations, this frequency is equal to zero, if the number density is
uniform in the azimuthal direction. If $\partial n_{i}/n_{i}r\partial \theta
\sim L_{\theta }^{-1}\neq 0$\ in any region, then $\omega \sim $\ $\omega
_{ci}k_{r}\rho _{i}^{2}L_{\theta }^{-1}$. Comparing this expression, for
example, with the growth rate (34), we see that under condition
\[
\left( \frac{\nu _{en}}{\omega _{ce}}\right) ^{1/2}\frac{|v_{e0}-v_{i0}|}{%
c_{Ai}}\gg \left( k_{r}L_{\theta }\right) ^{1/2}\frac{\rho _{i}}{L_{\theta }}
\]%
($\beta \ll 1$), the growth rate is larger than the drift frequency. The
last condition can easily be satisfied because $\rho _{i}$ is much smaller
than $L_{\theta }$.
\section{Conclusion}
In the present paper, we have studied electromagnetic streaming
instabilities\ in the thermal viscous regions of rotating astrophysical
objects, such as magnetized accretion disks, molecular clouds, their cores,
and elephant trunks. However, the results obtained can be applied to any
regions of interstellar medium, where different background velocities
between charged species can arise.
We have considered a weakly ionized multicomponent plasma consisting of
electrons, ions, dust grains, and neutrals. The cyclotron frequencies of
charged species have been supposed to be larger than their collisional
frequencies with neutrals. The axisymmetric perturbations across the
background magnetic field have been investigated. We have taken into account
the effect of perturbation of collisional frequencies due to density
perturbations of species. New compressible instabilities generated by the
different equilibrium velocities of species have been found in the cold and
thermal limits either when the viscosity of neutrals can be neglected or
when it is important. For the perturbations considered, the viscosity of
magnetized charged species is negligible.
In dense accretion disks, the ions are unmagnetized while the electrons
remain magnetized (e.g., Wardle \& Ng 1999). In this case, the viscosity of
ions can play the same role as that for neutrals. However, our present model
does not describe such objects.
The electromagnetic streaming instabilities studied in the present paper can
be a source of turbulence in weakly ionized magnetized astrophysical objects
in regions where thermal and viscous effects can play a role.
\bigskip
The insightful and constructive comments and suggestions of the anonymous
referee are gratefully acknowledged.
\bigskip
\paragraph{\noindent References\\}
\smallskip
\noindent \\
\noindent Abergel, A., Verstraete, L., Joblin, C., Laureijs, R., \&
Miville-Desch\^{e}nes, M.-A. 2005, Space Sci. Rev., 119, 247
\noindent Balbus, S. A., \& Henri, P. 2008, ApJ, 674, 408
\noindent Braginskii, S. I. 1965, Rev. Plasma Phys., 1, 205
\noindent Caselli, P., Walmsley, C. M., Terzieva, R., \& Herbst, E. 1998,
ApJ, 499, 234
\noindent Draine, B. T., Roberge, W. G., \& Dalgarno, A. 1983, ApJ, 264, 485
\noindent Masada, Y., \& Sano, T. 2008, ApJ, 689, 1234
\noindent Mendis, D. A., \& Rosenberg, M. 1994, Annu. Rev. Astron.
Astrophys., 32, 419
\noindent Nekrasov, A. K. 2007, Phys. Plasmas, 14, 062107
\noindent Nekrasov, A. K. 2008 a, Phys. Plasmas, 15, 032907
\noindent Nekrasov, A. K. 2008 b, Phys. Plasmas, 15, 102903
\noindent Nekrasov, A. K. 2009 a, Phys.Plasmas, 16, 032902
\noindent Nekrasov, A. K. 2009 b, ApJ, 695, 46
\noindent Nekrasov, A. K. 2009 c, ApJ, 704, 80
Pessah, M. E., \& Chan, C. 2008, ApJ, 684, 498
\noindent Ruffle, D. P., Hartquist, T. W., Rawlings, J. M. C., \& Williams,
D. A. 1998, A\&A, 334, 678
\noindent Wardle, M., \& Ng, C. 1999, MNRAS, 303, 239
\noindent Youdin, A. N., \& Goodman, J. 2005, ApJ, 620, 459
\end{document}
|
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| 6,623
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National help
Oregon – Law Offices of Bob Mionske
California – Emison Cooper & Cooper LLP
About Bob Mionske
About Bicycle Accidents
Defective Cycling Products
How to Handle a Traffic Ticket
How to Avoid Car-on-Bike Accidents
About Bike Theft
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Nolo
Bicycling and The Law
Driver Who Killed Man Arrested
BicycleLaw January 11, 2010 May 8th, 2016
The Morgan Hill Times: Driver who killed man arrested
By Michael Moore
The driver of a vehicle that dealt fatal injuries to Rory Tomasello, 22, as he was riding his bicycle through a crosswalk has been arrested on suspicion of manslaughter.
Morgan Hill resident Sandra Arlia, 66, turned herself in to police Dec. 28 after a $7,000 warrant was issued for her arrest. She was booked on suspicion of manslaughter at the San Jose Police Department and released, according to Morgan Hill Sgt. Jerry Neumayer.
The Santa Clara County District Attorney's Office charged Arlia with a misdemeanor count of vehicular manslaughter without gross negligence, according to spokeswoman Amy Cornell. If convicted, Arlia faces a maximum sentence of one year in county jail.
On Oct. 23, 2009, Arlia was driving a 2004 Cadillac Escalade west on West Edmundson Avenue, when the vehicle struck Tomasello in the crosswalk, just west of the intersection of Monterey Road. Police responded to the 5 p.m. accident and found Tomasello lying in the middle of the outside westbound lane. Arlia was still present at the scene.
According to police, the bicyclist was conscious at the scene, but he appeared disoriented and had redness to the back of his neck and stomach. He complained of an injury to his leg, and was transported by ambulance to Saint Louise Regional Hospital. On the way there, he was rerouted to San Jose Regional Hospital. Tomasello died Nov. 2 as a result of injuries suffered in the accident. He was not wearing a helmet at the time of the accident, Neumayer said.
Witnesses who saw the accident said while Tomasello was traversing the crosswalk on his bicycle, traveling north, a motorist stopped in the inside westbound lane to allow him to cross. Tomasello made it across the two eastbound lanes, and continued through the crosswalk and his bicycle hit the front driver's side fender of the Cadillac, which was traveling through the crosswalk at the same time, Neumayer said.
The impact caused Tomasello to fall off his bicycle and land in the middle of the outside westbound lane.
No one was ticketed or arrested at the scene of the accident, which remains under investigation, Neumayer said.
The crosswalk in which the accident occurred, located just east of the Centennial Recreation Center, connects a paved pedestrian and bike path that parallels West Little Llagas Creek across West Edmundson Avenue.
Tomasello suffered a fractured skull and brain swelling in the accident, according to his mother Kathee Tomasello. As soon as he was placed in the ambulance at the scene of the accident, he passed out and did not regain consciousness. He was declared brain dead Nov. 2.
Tomasello's mother and a close friend described him as an aspiring playwright with a creative mind. He was a graduate of Live Oak High School, and a former classmate said he began taking philosophy classes at De Anza College in Cupertino shortly before the accident.
On his MySpace page, Tomasello described himself as "creative but not annoyingly eccentric." His blogs included diatribes on gay marriage and religion, New World Order conspiracy theories and the top 10 reasons why the federal drug war should end. For heroes, he listed his father, professional musician Tom Tomasello, first, followed by beat poets including Allen Ginsberg, philosophers like Jean-Paul Sartre and comedians such as Bill Hicks and David Cross.
The accident led city staff to begin a reassessment of the safety of all its mid-block crosswalks. The crosswalk in which Tomasello was killed is not intended to be ridden across on a bicycle, and in fact city ordinance demands cyclists dismount to cross the street. Where the bike path meets West Edmundson Avenue on the south side of the road is a narrow opening in a chain link fence with about a 3-foot post in the middle of it, a design intended to encourage cyclists to dismount.
Arlia will be arraigned in the South County Courthouse Jan. 19.
The accident occurred just more than a year after the last cyclist was killed on Morgan Hill roads. On Oct. 8, 2008, Bruce Finch of Gilroy was killed when he was cycling on Uvas Road when he struck a 1997 Honda Civic at the intersection of Little Uvas Road. The motorist who was driving the Honda, Rita Campos of Morgan Hill, faces the same misdemeanor charge that Arlia faces. Campos' case is scheduled for a jury trial to begin Jan. 25.
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| 3,023
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Q: How to pass the option value from the drop down list to other pages using php? I am working on a website that allows the user to initially select his destination and then show results of four categories on the basis of the preferences that he selects from the check boxes. I am unable to post the value of the option selected in the drop down list using session variable to another webpages. Without pressing any submit button I want to pass the value selected in the drop down list to several pages. I have seen many previous posts related to the issue but found none of them helpful. My code with the drop down list is below:
index.php
<div class="dropdownstay">
<select name="city" class="option3" id="dropdown">
<option value="1" id="lhr" style="font-size:20px; font-family:Monotype Corsiva;" >Lahore</option>
<option value="2" id="dub" style="font-size:20px; font-family:Monotype Corsiva;">Dubai </option>
<option value="3" id="new" style="font-size:20px; font-family:Monotype Corsiva;">Newyork</option>
<option value="4" id="can"style="font-size:20px; font-family:Monotype Corsiva;">Canberra</option>
<option value="5" id="kl" style="font-size:20px; font-family:Monotype Corsiva;">Kuala Lampur</option>
<br>
</select>
</form>
<?php
session_start();
if(isset($_POST['city']))
$selected_city = $_POST['city'];
?>
<!--end for drop down -->
</div>
</div>
</li>
</ul>
</div>
<div class="reservation">
<ul>
<li class="span1_of_1">
<h5>What you want in hotel?</h5>
<br>
<form action="checkbox_value.php" method="post">
<section title="preferences">
<input type="checkbox" value="is_pool" id="pool" name="check_list[]" checked />
<text style="font-size:20px; font-family: Times New Roman;"> Pool </text>
<br/>
<input type="checkbox" value="is_gym" id="gym" name="check_list[]" checked />
<text style="font-size:20px; font-family: Times New Roman;"> Gym </text>
<br/>
<input type="checkbox" value="is_beach" id="beach" name="check_list[]" />
<text style="font-size:20px; font-family: Times New Roman;"> Beach </text>
<br/>
<input type="checkbox" value="is_spa" id="spa" name="check_list[]" />
<text style="font-size:20px; font-family: Times New Roman;"> Spa </text>
<br/>
<input type="checkbox" value="is_wifi" id="wifi" name="check_list[]" checked />
<text style="font-size:20px; font-family: Times New Roman;"> Wifi </text>
<br/>
<input type="checkbox" value="is_familyoriented" id="family" name="check_list[]"/>
<text style="font-size:20px; font-family: Times New Roman;"> Family </text>
<br/>
<input type="checkbox" value="is_economical" id="economical" name="check_list[]" />
<text style="font-size:20px; font-family: Times New Roman;"> Economical </text>
<br>
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<br>
<br>
</section>
<div>
<input type="submit" name="submit" value="Submit" style="color: orange;" />
</div>
<?php include 'checkbox_value.php';?>
</form>
</li>
<!--
The code that retrieves data for each category is "checkbox_value.php". I want to access the value of the variable "city" on total 4 pages like this page.
Checkbox_value.php
<?php
class MyDB extends SQLite3
{
function __construct()
{
$this->open('mytrip.db');
}
}
$db = new MyDB();
if(!$db){
echo $db->lastErrorMsg();
} else {
}
$hotelOptions = array('is_pool', 'is_gym', 'is_spa', 'is_wifi', 'is_beach', 'is_familyoriented', 'is_economical');
$countOptions = array(
'is_pool' => 'pool_count',
'is_gym' => 'gym_count',
'is_spa' => 'spa_count',
'is_wifi' => 'wifi_count',
'is_beach' => 'beach_count',
'is_familyoriented' => 'family_count',
'is_economical' => 'econo_count',
);
//$cities = array(1 => 'Dubai');
if (isset($_POST['submit'])) {
// $selected_city= $_POST['city'];
if (!empty($_POST['check_list']) && is_array($_POST['check_list'])) {
$profpic = "images/Yellow.jpg";
echo "<p> Destination: ".$selected_city ."</p>";
// Counting number of checked checkboxes.
$checked_count = count($_POST['check_list']);
echo "You have selected following ".$checked_count." option(s): <br/>";
// Loop to store and display values of individual checked checkbox.
$where = '';
$order = '';
foreach($_POST['check_list'] as $selected) {
echo "<p>".$selected ."</p>";
if (array_search($selected, $hotelOptions) !== false) {
$where .= " AND {$selected} = 1";
$order .= " {$countOptions[$selected]} DESC,";
}
}
$where = substr($where, 5, strlen($where));
$order = substr($order, 0, strlen($order) - 1);
//echo "<p>".$where ."</p>";
//echo "<p>".$order ."</p>";
session_start();
$id=$_SESSION['Id'];
$city=$_SESSION['selected_city'];
if (isset($city)) {
$sql = "SELECT hotel_name FROM ".$city." WHERE ".$where." ORDER BY ".$order.";";
// echo "<p>".$sql ."</p>";
$ret = $db->query($sql);
session_start();
$id=$_SESSION['Id'];
$ar= array();
$i=0;
while($row = $ret->fetchArray(SQLITE3_ASSOC) ){
echo "<p> <br /></p>\n";
echo "\n". $row['hotel_name'] . "\n";
$ar[$i]= $row['hotel_name'];
$i++;
}
if (is_array($ar))
{
foreach ($ar as $value)
{
echo "<p> Array is " .$value. "<p>";
$sql= "INSERT INTO Search (Id, History)
VALUES ('$id','$value')";
$ret = $db->query($sql);
}
}
}
} else {
echo "<b>Please Select Atleast One Option.</b>";
}
$db->close();
}
?>
<html>
<head>
<style type="text/css">
body {
background-image: url('<?php echo $profpic;?>');
}
</style>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
</head>
<body>
</body>
</html>
How can I pass the value selected in the drop down list and be able to access it on multiple pages without submitting it using session variable?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,353
|
\section{Introduction}
\label{intro}
\indent Star-forming galaxies harbour large-scale populations of energetic particles accelerated in violent phenomena associated with stellar evolution, such as supernova explosions, colliding-wind binaries, or pulsar wind nebulae. These galactic cosmic rays (hereafter CRs) interact with the various components of the interstellar medium (ISM): gas, turbulence, radiation, and magnetic fields. These interactions determine the transport of CRs from sources to intergalactic medium, and are responsible for their confinement and accumulation in the galactic volume over a large number of acceleration episodes (of the order of $\sim10^5$ for the Milky Way). The existence of such large-scale populations is attested by various extended emissions, notably in radio through synchrotron emission, but also in $\gamma$-rays\ through inverse-Compton scattering, Bremsstrahlung, and inelastic nuclear collisions. These emissions are signatures of the combined processes of CR acceleration and transport, and their interpretation in the frame of other astronomical data can provide us with a better picture of the life cycle of CRs \citep[see][for a review]{Strong:2007}.
\indent Understanding CRs is relevant to many fields beyond high-energy astrophysics. This is mostly because they are more than a simple side effect of the end stages of stellar evolution. Galactic cosmic rays interact with the ISM and can significantly modify the environment in which they are flowing. They can heat and ionise the gas, especially at the centre of dense molecular clouds, thereby impacting star formation conditions \citep{Papadopoulos:2010}; through their diffusion, CRs can alter the turbulence of the interstellar medium, with consequences on their own transport \citep{Ptuskin:2006}; they can contribute to large-scale outflows, an important feedback effect on star formation over cosmic times \citep{Hanasz:2013}; CRs may play a role in shaping large-scale galactic magnetic fields through a dynamo process \citep{Hanasz:2004}; and last, CRs from classical astrophysical objects can hide the signatures of more exotic components of the Universe such as dark matter \citep{Delahaye:2010}. For all these reasons, understanding the non-thermal content and emission of star-forming galaxies is crucial.
\indent The current generation of high-energy (HE) and very-high-energy (VHE) $\gamma$-ray\ telescopes has enabled us to detect several external galaxies that shine at photon energies $>100$\,MeV\ as a result of CRs interacting with their ISM (and not because of a central black hole activity, as is the case for the vast majority of detected $\gamma$-ray-emitting external galaxies). The {\em Fermi}/LAT space-borne pair-creation telescope has permitted the detection of five external galaxies at GeV energies (SMC, LMC, M31, M82, and NGC253), with two other systems being possible candidates \citep[NGC4945 and NGC1068, whose emission may be contaminated by central black hole activity; see][]{Abdo:2009l,Abdo:2010d,Abdo:2010e,Abdo:2010f,Ackermann:2012}. In the same time, the HESS and VERITAS ground-based Cherenkov telescopes have permitted the detection at TeV energies of NGC253 and M82, respectively \citep{Acero:2009,Abramowski:2012,Acciari:2009}. Star-forming galaxies can now be considered as an emerging class of $\gamma$-ray\ sources, with a notable increase of the number of detected objects from the previous generation of instruments \citep[only the LMC was detected by CGRO/EGRET; see][]{Sreekumar:1992}.
\indent Even with such a limited sample, it is tempting to perform a population study to find clues about the main parameters regulating CR populations. From the {\em Fermi}/LAT sample of five objects plus an estimate for the global emission of the Milky Way and an upper limit for M33, the work of \citet{Abdo:2010f} showed first hints for a slightly non-linear correlation between $\gamma$-ray\ luminosity and star formation rate (SFR). A more systematic study by \citet{Ackermann:2012}, including upper limits for about sixty luminous and ultra-luminous infrared galaxies, found further evidence for a quasi-linear correlation between $\gamma$-ray\ luminosity and tracers of the star formation activity. Such a relationship can be expected to first order, because the production of CRs is expected to scale with the rate of massive star formation; but it is surprising that the correlation is so close to linear, over such a wide range of galactic properties, from dwarfs like the SMC to interacting systems like Arp 220.
\indent The interpretation of this correlation therefore raised the question of the scaling laws that can be expected for the diffuse $\gamma$-ray\ emission as a function of global galactic properties. Clarifying this at the theoretical level is required to establish whether the current experimental data can be constraining in any way. This is all the more needed because the sample of detected objects is small, the coverage over the GeV-TeV range is uneven, and the spectral characterization is still rather limited. In addition, the determination of global properties of external systems, such as total mass or star formation rate, is in itself a challenge, and there is most likely some intrinsic scatter in the actual physical correlations.
\indent That star-forming galaxies might become a class of $\gamma$-ray\ sources was anticipated early, well before the launch of the \textit{Fermi} satellite \citep[see for instance][]{Volk:1996, Pavlidou:2001,Torres:2004}. The interest in the subject extended beyond the study of nearby objects, because this emission cumulated over cosmic time might account for a significant fraction of the isotropic diffuse $\gamma$-ray\ background \citep{Pavlidou:2002,Thompson:2007, Ackermann:2012}. Before and after the launch of the \textit{Fermi} satellite, the $\gamma$-ray\ emission from individual starbursts was studied by several authors, with the focus being mainly on M82 and NGC253 \citep{Persic:2008,deCeadelPozo:2009,Lacki:2011,Paglione:2012}.
\indent The purpose of this paper is to provide a more global picture of the evolution of the $\gamma$-ray\ interstellar emission from star-forming galaxies, in the context of the recent population study conducted on the basis of {\em Fermi}/LAT observations. Instead of trying to fit a model to a particular object, I defined a global framework by fixing as many cosmic-ray-related parameters as possible from Milky Way studies, leaving only a few global quantities as independent variables. Here, I present a generic model for the transport of CRs and the associated non-thermal interstellar emissions in star-forming galaxies. Simulating systems with various sizes and gas distributions thus allows investigating the impact of global galactic parameters on the $\gamma$-ray\ output. In a first part, I introduce the different components and related assumptions of the model, together with the set of synthetic galaxies that was considered. Then, the $\gamma$-ray\ emission over 100\,keV-100\,TeV\ for this series of synthetic galaxies is presented, and its evolution with global properties is analysed. In a subsequent part, the scaling of $\gamma$-ray\ luminosity with global galactic properties is derived for soft, high-energy, and very-high-energy $\gamma$-ray\ ranges, and is compared with observations in the case of the high-energy band. Last, the impact of the model parameters is discussed, and the connection to the far-infrared - radio correlation is addressed.
\section{Model}
\label{model}
\indent The model is based on a modified version of the GALPROP public code\footnote{GALPROP is a software package for modelling the propagation of CRs in the Galaxy and their interactions in the ISM. It allows simultaneous and coherent predictions of the CR population in the interplanetary and interstellar medium, gamma-ray emission, and recently also synchrotron radiation intensity and polarization. See http://galprop.stanford.edu/ and http://sourceforge.net/projects/galprop/ for the latest developments.}, originally designed for the Milky Way and adapted here to model any star-forming galaxy. For the most part, this was achieved by implementing general models for the interstellar medium components, as described below. This generic model uses as input and for calibration some characteristics of a typical GALPROP modelling of the Milky Way in the plain diffusion case: Model z04LMPDS in Table 1 of \citet{Strong:2010}, referred to in the following as the \textit{reference GALPROP run}.
\indent All the synthetic galaxies discussed in this work are two-dimensional in space with cylindrical symmetry, and are defined by a disk radius $R_{\textrm{max}}$ and a halo height $z_{\textrm{max}}$. The model is therefore better suited to simulate grand-design spiral galaxies than strongly irregular or interacting galaxies, but it can provide insights into the effects of galaxy size and star formation conditions on non-thermal emissions.
\subsection{Gas}
\label{model_gas}
\indent The interstellar gas distribution was modelled through simple analytical functions of the galactocentric radius $R$ and height above the galactic plane $z$. I considered only atomic and molecular gas, and neglected ionised gas. A ratio of He/H=0.11 was assumed for the interstellar gas.
\indent Three types of gas distributions are discussed in the following: a disk of atomic hydrogen with exponential density profile in radius and height (three parameters: scale radius, scale height, and peak atomic gas density); a central core of molecular gas with uniform density (three parameters: core radius, core height, and molecular gas density); a ring of molecular gas with uniform density (four parameters: ring inner and outer radii, ring height, and molecular gas density). These profiles are intended to reproduce a typical star-forming galaxy, where star formation can operate at a low surface density in a disk, and/or be concentrated in more compact regions like central cores (such as in M82 and NGC253), or in spiral arms or rings (such as in the Milky Way or M31). The different models and their parameters are summarised below.
\indent An important aspect of the model, as becomes obvious in the following, is that the gas content and distribution drives many other components, such as the magnetic field model, the infrared energy density of the interstellar radiation field, or the CR source distribution.
\subsection{Magnetic field}
\label{model_mag}
\indent The magnetic field strength is assumed to depend at each point on the vertical gas column density, according to
\begin{equation}
\label{eq_bfield}
B(R,z)=B_0 \, \left(\frac{\Sigma_{\textrm{gas}}(R)}{\Sigma_{0}} \right)^a \, e^\frac{-|z|}{z_{B}} ,
\end{equation}
where $\Sigma_{\textrm{gas}}$ is the gas surface density. Such a scaling of the mid-plane magnetic field value can be justified by different theoretical expectations, from hydrostatic equilibrium to turbulent magnetic field amplification \citep[see][]{Schleicher:2013}. The vertical profile is assumed to be exponential, and I adopted a magnetic field scale height equal to the halo size \citep[following][and quite similar to the \textit{reference GALPROP run}; the influence of this assumption is examined in Sect. \ref{res_params}]{Sun:2008}. The exponent $a$ was set to a value of 0.6 to match the observed far-infrared - radio continuum correlation, as discussed in Sect. \ref{res_radio}. The normalisation doublet $(B_0, \Sigma_{0})$ was set to (5\,$\mu$G, 4\,M$_{\odot}$\,pc$^{-2}$), from a comparison with the \textit{reference GALPROP run}, as explained in Sect. \ref{model_calib}.
\indent I did not consider a global topology model for the magnetic field, or a separation into regular and turbulent components, because the diffusion of CRs is treated in a homogeneous and isotropic way. A recent GALPROP modelling using a more complete magnetic field model can be found in \citet{Orlando:2013}. Theoretically, anisotropic diffusion along field lines can be expected, depending on the ratio of turbulent field to regular field strength, and lead for instance to channelling and clustering of particles preferentially along spiral arms \citep{Dosch:2011}. But, the reality and impact of this compared with the homogeneous and isotropic approximation is poorly constrained at the experimental level \citep[see the references and discussion in][]{Jaffe:2013}.
\subsection{Radiation field}
\label{model_isrf}
\indent The interstellar radiation field model has three main components: the cosmic microwave background (CMB), the infrared dust emission (IR), and the stellar emission (UVO). The first contribution is implemented straightforwardly as a blackbody with a temperature of $T_{\textrm{CMB}}$=2.7\,K. The other two components are dependent on the particular galactic setting and ideally result from a complex radiation transfer problem involving a variety of stellar sources and a mixture of dust grains, each with specific spatial distributions.
\indent The interstellar radiation field model (ISRF) is based on the work by \citet{Draine:2007a}. Interstellar dust is irradiated by a distribution of starlight intensities with the $I_{\textrm{M83}}(\lambda)$ spectrum determined by \citet{Mathis:1983} for the local Galactic environment and scaled by a dimensionless factor $U$. The resulting dust emission $I_{\textrm{dust}}(\lambda)$ in the 1-10000\,$\mu$m wavelength band and normalized per hydrogen atom is available for various combinations of the model parameters (see the appendix).
\indent The infrared intensity distribution over the galactic volume ideally results from integrating at each point the contribution of all sources, possibly affected by absorption, reemission, and scattering \citep[see for instance][]{Ackermann:2012b}. As a simplification, the energy density of the infrared dust component in the ISRF model was computed as
\begin{align}
\label{eq_ir}
u_{\textrm{IR}}(R,z)= \frac{4 \pi}{c} \, E_{\textrm{dust}} \, \Sigma_{\textrm{thres}} \, \left( \frac{\Sigma_{\textrm{gas}}(R)}{\Sigma_{\textrm{thres}}} \right)^N \, e^\frac{-|z|}{z_{\textrm{IR}}} ,\\
\textrm{where } E_{\textrm{dust}} = \int I_{\textrm{dust}}(\lambda) d\lambda ,
\end{align}
where $\Sigma_{\textrm{thres}}$ is a threshold gas surface density for star formation and $N$ the index of the Schmidt-Kennicutt relation (see the definitions below). The quantity $c$ is the speed of light. The intensity at each point $(R,z)$ in the volume is thus assumed to be dominated by dust emission from the same galactocentric radius $R$, and the energy density is taken to decrease over a scale height $z_{\textrm{IR}}$=2\,kpc above and below the galactic plane. The latter value was set by comparison with the \textit{reference GALPROP run}, as explained in Sect. \ref{model_calib}. The approximation adopted implements an IR energy density in the galactic plane that has a minimum value for the threshold of star formation, and then scales with the star formation rate according to the term in parentheses\footnote{With the adopted parameterization for the infrared dust emission, the conversion factor between star formation rate and infrared luminosity in the 8-1000\,$\mu$m band is 1.3\,10$^{-10}$\,M$_{\odot}$\,yr$^{-1}$/\,L$_{\odot}$. This can be compared with 1.7\,10$^{-10}$\,M$_{\odot}$\,yr$^{-1}$/\,L$_{\odot}$\ in the original formulation by \citet{Kennicutt:1998} using a Salpeter initial mass function \citep{Salpeter:1955}, or with 1.3\,10$^{-10}$\,M$_{\odot}$\,yr$^{-1}$/\,L$_{\odot}$\ when using a Chabrier initial mass function \citep{Chabrier:2003}.}. I neglected the cirrus emission arising from evolved star heating, which can increase the IR energy density by up to a factor of 2 \citep{Kennicutt:2009}, but I tested the impact of varying the IR field intensity, as discussed in Sect. \ref{res_params}.
\indent The mid-plane value for the starlight energy density distribution was assumed to be the $I_{\textrm{M83}}$ field scaled by the dust-weighted mean starlight intensity $\langle U \rangle$,
\begin{align}
\label{eq_uvo}
u_{\textrm{UVO}}(R,z)= \frac{4 \pi}{c} \, \langle U \rangle \, E_{\textrm{M83}} \, e^\frac{-|z|}{z_{\textrm{UVO}}} ,\\
\textrm{where } E_{\textrm{M83}} = \int I_{\textrm{M83}}(\lambda) d\lambda
\end{align}
With the parameters adopted (see the appendix), this mean starlight intensity is $\langle U \rangle$=3.4, so it is about 3 times stronger everywhere in the galactic disk than in the local Galactic environment. Getting the same field everywhere in the galactic disk is probably not realistic. The stellar photon field experienced by CRs is expected to be higher in regions of intense star formation, especially because this is where they are released into the ISM, but these effects occur on scales that are similar to or below the typical spatial resolution of the model. The energy density of the starlight component is taken to decrease over a scale height $z_{\textrm{UVO}}$=1\,kpc above and below the galactic plane, according to a comparison with the \textit{reference GALPROP run} (see Sect. \ref{model_calib}).
\subsection{Cosmic-ray source}
\label{model_src}
\indent The main sources of CRs are thought to be supernova remnants, possibly with some contribution of other objects such as pulsars, pulsar wind nebula, or colliding-wind binaries. Since most of these are end products of massive star evolution with lifetimes shorter than that of their progenitors, the CR source distribution was assumed by default to follow the star formation distribution. The latter is determined from the gas distribution using the empirical Schmidt-Kennicutt relation
\begin{equation}
\label{eq_sk}
\Sigma_{\textrm{SFR}}=A\,\Sigma_{\textrm{gas}}^{N} \quad \textrm{ for } \Sigma_{\textrm{gas}} > \Sigma_{\textrm{thres}} ,
\end{equation}
where $\Sigma_{\textrm{SFR}}$ and $\Sigma_{\textrm{gas}}$ are the star formation rate and atomic+molecular gas surface density (in M$_{\odot}$\,yr$^{-1}$\,kpc$^{-2}$ and M$_{\odot}$\,pc$^{-2}$ units), and with $A=2.5\,10^{-4}$ and $N=1.4$ the canonical values \citep{Kennicutt:1998}. I used a threshold at $\Sigma_{\textrm{thres}} = $4\,M$_{\odot}$\,pc$^{-2}$, which is a lower limit from spatially resolved observations of galaxies in \citet{Kennicutt:1998}. The relation is observed to hold from galactic averages down to kpc averages. At each galactocentric radius, the vertical distribution of the star formation rate was taken to be proportional to the gas density.
\indent The total CR source luminosity was assumed by default to be proportional to the total star formation rate, using as normalisation the \textit{reference GALPROP run}. The following CR input power integrated over 100\,MeV-100\,GeV\ was used:
\begin{align}
\label{eq_crpower}
Q_{CRp,e} &= Q_{MW} \, \frac{\textrm{SFR}}{1.9 \, \textrm{M}_{\odot}\,\textrm{yr}^{-1}} , \\
\textrm{where } Q_{MW} &= 7.10\,10^{40} \, \textrm{erg\,s}^{-1} \, \textrm{ for CRp } \notag \\
\textrm{ } Q_{MW} &= 1.05\,10^{39} \, \textrm{erg\,s}^{-1} \, \textrm{ for CRe } \notag ,
\end{align}
where CRp stands for cosmic-ray hydrogen and helium nuclei, and CRe stands for cosmic-ray electrons. The SFR normalisation was obtained by integrating Eq. \ref{eq_sk} over the Galaxy given the gas distribution of the original GALPROP model; the result lies in the range of current estimates \citep{Diehl:2006}, especially if one considers that the scatter in the Schmidt-Kennicutt relation can reach an order of magnitude \citep{Kennicutt:1998}. The relation between total star formation rate and CR luminosity is probably far from straightforward because of a possible dependence of the stellar initial mass function or particle acceleration process on the interstellar conditions, but taking this into account is beyond the scope of the present work.
\indent The CR injection spectra used in this work were consistently taken from the \textit{reference GALPROP run}. Particles at source have a power-law distribution in rigidity with a break, following
\begin{align}
\label{eq_crspec}
\frac{dQ_{CRp,e}}{dR} &= Q_{0} \, \left( \frac{R}{R_{0}} \right)^{-q} , \\
\textrm{with } q &= 1.80 \textrm{ for } R \leq R_{b} \notag \\
\textrm{ } q &= 2.25 \textrm{ for } R > R_{b} \notag \\
\textrm{ } R_{b} &= 9\,\textrm{GV} \, \textrm{ for CRp } \notag \\
\textrm{ } R_{b} &= 4\,\textrm{GV} \, \textrm{ for CRe } , \notag
\end{align}
This means that the same electron-to-proton ratio at injection was used for all synthetic galaxies. This may introduce errors because acceleration can be expected to proceed differently in environments where magnetic and radiation energy densities are a factor of 100-1000 higher (as is the case in the cores of starburst galaxies). But these differences are currently poorly constrained at the experimental level, and it is interesting enough to assess the effects of global galactic properties for a given CR injection spectrum. Moreover, the assumption that particle acceleration in other galaxies works similarly to what we think takes place in the Milky Way can be tested by comparison with the emerging class of $\gamma$-ray\ star-forming galaxies.
\subsection{Transport}
\label{model_trans}
\indent At galactic scales, the spatial transport of CRs in the ISM is currently understood as being ruled by two main processes: advection in large-scale outflows such as galactic winds or superbubbles breaking out into the halo, and diffusion on interstellar magnetohydrodynamic turbulence \citep{Strong:2007}. While the former implies adiabatic energy losses, the latter may be associated with reacceleration of the CRs. Recent studies seem to dismiss the need for reacceleration, and show that a large set of cosmic-ray-related observations can be reasonably well accounted for from a pure diffusion scheme with isotropic and homogeneous properties \citep[][note, however, that the good fit to the data is obtained by simultaneously tuning the source and transport parameters]{Strong:2011}. Several discrepancies between such predictions and observations remain, such as the so-called gradient problem, connected with the longitude distribution of the diffuse interstellar $\gamma$-ray\ emission, and the large-scale cosmic-ray anisotropy, which is observed to be smaller than expected at high energies. Several authors have shown that these can be resolved by adopting inhomogeneous and anisotropic diffusion schemes \citep{Evoli:2012}.
\indent For simplicity and as a starting point, the spatial transport of CRs in the present model of star-forming galaxy was assumed to occur by energy-dependent diffusion in a homogeneous and isotropic way. For the diffusion coefficient, I used the value from the \textit{reference GALPROP run}
\begin{align}
D_{xx}=3.4 \,10^{28} \, \beta \left( \frac{R}{4\,\textrm{GV}} \right)^{0.5} \, \textrm{cm}^{2} \, \textrm{s}^{-1} \quad \textrm{for} \quad R \geq 4\,\textrm{GV} \notag \\
D_{xx}=3.4 \,10^{28} \, \textrm{cm}^{2} \, \textrm{s}^{-1} \quad \textrm{for} \quad R < 4\,\textrm{GV} ,
\end{align}
where $R$ is the particle rigidity and $\beta=v/c$, the ratio of particle velocity to speed of light.
\indent The spatial transport of CRs in other galaxies than the Milky Way may take place in a different way. In systems harbouring a large starburst region, the interstellar turbulence may be strongly enhanced and result in different diffusion conditions, or strong outflows out of the disk may be generated and advect particles away. I discuss these two possibilities in Sect. \ref{res_params}.
\begin{table}[t!]
\caption{List of the galaxy configurations in the initial sample}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\rule{0pt}{2.8ex} $R_{\textrm{max}}$ & $z_{\textrm{max}}$ & HI & H$_2$ \rule[-1.4ex]{0pt}{0pt} \\
\hline
2.0 & 1.0 & disk with $R_{\textrm{H}}$=1.0 & core with $R_{\textrm{core}}$=0.1 \\
\hline
- & - & - & core with $R_{\textrm{core}}$=0.2 \\
\hline
5.0 & 1.0 & disk with $R_{\textrm{H}}$=2.5 & core with $R_{\textrm{core}}$=0.2 \\
\hline
- & - & - & core with $R_{\textrm{core}}$=0.5 \\
\hline
- & - & - & core with $R_{\textrm{core}}$=1.0 \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(1.5,2.0)$ \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(2.0,3.0)$ \\
\hline
10.0 & 2.0 & disk with $R_{\textrm{H}}$=5.0 & core with $R_{\textrm{core}}$=0.5 \\
\hline
- & - & - & core with $R_{\textrm{core}}$=1.0 \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(2.0,3.0)$ \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(4.0,6.0)$ \\
\hline
20.0 & 4.0 & disk with $R_{\textrm{H}}$=10.0 & core with $R_{\textrm{core}}$=1.0 \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(4.0,6.0)$ \\
\hline
- & - & - & ring with $(R_{i}, R_{o})=(8.0,12.0)$ \\
\hline
\end{tabular}
\end{center}
Note to the Table: The first two columns are the maximum radius and half-height of the galactic volume, respectively. The third column gives the scale radius of the exponential distribution of atomic gas (the vertical distribution is exponential with a scale height of 100\,pc). The fourth column indicates whether the molecular gas distribution is a uniform core or uniform ring, and gives the corresponding core radius or inner/outer radii (the vertical distribution is uniform with a half-height of 100\,pc). All lengths are in kpc units. For each model, 7 molecular gas densities were tested: $n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, 500\,H$_2$\,cm$^{-3}$, for the same peak atomic gas density of $n_{\textrm{H}}=$2\,H\,cm$^{-3}$.
\label{tab_modellist}
\end{table}
\subsection{Test and calibration}
\label{model_calib}
\indent The star-forming galaxy model described above has a total of five undetermined parameters. One of these is the index of the magnetic field model, and it was set using the constraint of the far-infrared - radio correlation (see Sect. \ref{res_radio}). The other four parameters are the normalisation doublet $(B_0,\Sigma_0)$ for the magnetic field model, and the scale heights $z_{\textrm{IR}}$ and $z_{\textrm{UVO}}$ for the ISRF model. These parameters were set by a comparison with the reference GALPROP modelling of the Milky Way, which is basically what the code was optimised for. From the built-in axisymmetric atomic, molecular, and ionised gas distributions, models for the ISRF, magnetic field, and source distribution were derived using the assumptions detailed above. A typical run was then performed with these modified conditions, and its results were compared with those of the original GALPROP calculation.
\indent The comparison was first made on the ISM conditions, using source-weighted and volume averages for the energy densities of the radiation and magnetic ISM components (these averages are relevant to the conditions experienced by CRs at injection and over their long-term transport, respectively). From this, the four undefined parameters listed above were set. Applied to the Milky Way, the generic model for star-forming galaxies results in radiation (respectively magnetic) energy densities with volume and source-weighted averages that are different by about $\sim$20-30\% (respectively $\sim$5-10\%) from those of the original GALPROP setup. Then, the total Galactic non-thermal outputs were compared and were found to agree very well, without any need for further tuning of the model. The radio emission over 1\,MHz-1\,THz differs at most by 6\% at low frequencies\footnote{Note that we compare the output of two different models for the unabsorbed synchrotron emission over a very large frequency range. At the observational level, free-free absorption causes a turnover at low frequencies, and free-free and thermal dust emission dominate at high frequencies. Eventually, unabsorbed non-thermal emission dominates in the range $\sim$100\,MHz-10\,GHz.}, while the $\gamma$-ray\ emission over 100\,keV-100\,TeV\ shows a maximum deviation of 5\%. When integrating the emission over $>$100\,MeV, the differences in total luminosity are even smaller. The similarity of the results to that of the original GALPROP calculation applies not only to the total emission but also to its components: inverse-Compton, Bremsstrahlung, and pion decay.
\subsection{Setup}
\label{model_setup}
\indent All the models discussed in this work are two-dimensional in space with cylindrical symmetry, and are defined by a disk radius $R_{\textrm{max}}$ and a halo half-height $z_{\textrm{max}}$, with a cell size of 50\,pc in both dimensions. The particle energy grid runs from 100\,MeV\ to 1\,PeV\ with ten bins per decade. The transport of CRs arises from diffusion for the spatial part, and includes ionisation, synchrotron, Bremsstrahlung, inverse-Compton scattering, and hadronic interaction losses for the momentum part. For the latter, the production of secondary electrons and positrons is taken into account and included in the complete transport and radiation calculation. All solutions correspond to a steady state. For more details about the implementation of the physical processes and the inner workings of the code, see the GALPROP Explanatory Supplement available at the GALPROP website.
\indent Four different sizes were considered for the synthetic star-forming disk galaxies: ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(2\,kpc, 1\,kpc), (5\,kpc, 1\,kpc), (10\,kpc, 2\,kpc), and (20\,kpc, 4\,kpc). The larger size corresponds to the dimensions of the Milky Way in the reference GALPROP run, and I kept the same aspect ratio for the smaller sizes, except for the smaller model where it would have resulted in a too small halo. The size of the halo for the Milky Way and how it evolves with galaxy size are unsettled questions, therefore this assumption of a nearly constant aspect ratio in galaxy dimensions is challenged below (see Sects. \ref{res_params}, \ref{res_fermilat}, and \ref{res_radio}).
\indent For each size, the galaxy consisted of a disk of atomic gas with an exponential distribution in $R$ and $z$, with scale lengths of $R_{\textrm{H}}=R_{\textrm{max}}/2$ and $z_{\textrm{H}}=$100\,pc respectively, and a peak gas density of $n_{\textrm{H}}=$2\,H\,cm$^{-3}$. On top of that, different molecular gas distributions were implemented: cores of radius $R_{\textrm{core}}=$100, 200, 500\,pc, and 1\,kpc, and two rings with inner and outer radius $(R_{i}, R_{o})=(0.2 \, R_{\textrm{max}},0.3 \, R_{\textrm{max}})$ and $(0.4 \, R_{\textrm{max}},0.6 \, R_{\textrm{max}})$, all distributions extending 100\,pc on either side of the galactic plane (it was verified that using a smaller molecular gas scale height of 50\,pc does not alter the trends in the evolution of the $\gamma$-ray\ luminosity with galactic properties). Not all combinations of molecular gas distributions and galaxy sizes were tested, see the list of models summarised in Table \ref{tab_modellist}. Then, for each molecular gas profile, seven gas densities were tested: $n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, and 500\,H$_2$\,cm$^{-3}$. The initial sample of models includes $14 \times 7=88$ synthetic star-forming galaxies, not counting variations of some parameters to assess their importance. While some configurations may be considered as good models for existing systems such as the Milky Way or M82, others may be less realistic, such as the ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(20\,kpc, 4\,kpc) model with a large ring of $n_{\textrm{H}_2}=$500\,H$_2$\,cm$^{-3}$ gas, yielding an SFR of $\sim10^4$\,M$_{\odot}$\,yr$^{-1}$. These were included to illustrate the effects of changing galactic parameters.
\subsection{Time scales}
\label{model_timescales}
\indent To facilitate the interpretation of the results presented in the following, I provide the typical time scales for the various processes involved in the transport of CRs. These hold for the assumptions used in the star-forming galaxy model (such as the spatial diffusion coefficient).
\begin{align}
\tau_{\textrm{diff}} &= 2.8\,10^{14} \, \left( \frac{E}{4\,\textrm{GeV}} \right)^{-0.5} \, \left( \frac{z}{1\,\textrm{kpc}} \right)^2 \, \textrm{s} \\
\tau_{\textrm{adv}} &= 3.1\,10^{14} \, \left( \frac{z}{1\,\textrm{kpc}} \right) \, \left( \frac{V}{100\,\textrm{km}\,\textrm{s}^{-1}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{pp}} &= 2.2\,10^{15} \, \left( \frac{n_{\textrm{H}}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{Br}} &= 8.9\,10^{14} \, \left( \frac{n_{\textrm{H}}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{IC}} &= 9.9\,10^{15} \, \left( \frac{E}{1\,\textrm{GeV}} \right)^{-1} \, \left( \frac{U_{\textrm{rad}}}{1\,\textrm{eV}\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{Synch}} &= 9.9\,10^{15} \, \left( \frac{E}{1\,\textrm{GeV}} \right)^{-1} \, \left( \frac{U_{\textrm{mag}}}{1\,\textrm{eV}\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{ion}} &= 4.8\,10^{15} \, \left( \frac{E}{1\,\textrm{GeV}} \right) \, \left( \frac{n_{\textrm{H}}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s}
\end{align}
The time scales are those of the diffusion, advection, hadronic interaction, Bremsstrahlung, inverse-Compton, synchrotron, and ionisation processes. For the spatial transport processes, the typical length was taken at 1\,kpc, which is the order of magnitude for the galactic halo size. The dependence on energy for the diffusion time scale applies only for $E > 4$\,GeV. The hadronic interaction time scale was computed for an energy of 1\,GeV, with a cross-section of 30\,mbarn and an inelasticity parameter of 0.5 \citep[note that the cross-section increases slightly to reach 40\,mbarn at 1\,TeV, see][]{Kelner:2006}. The Bremsstrahlung-loss time scale corresponds to an electron energy of 1\,GeV, and typically has a logarithmic dependence on energy that was neglected in the above formula \citep{Schlickeiser:2002}. The inverse-Compton-loss time scale was computed in the Thomson approximation, with $U_{\textrm{rad}}$ corresponding to the radiation energy density. The synchrotron-loss time scale is the same formula, with $U_{\textrm{mag}}$ corresponding to the magnetic energy density. The ionisation-loss time scale was computed for protons in the ISM, using the lowest value of the loss rate (which is obtained for a Lorentz factor of $\sim3$) and neglecting its energy dependence (which leads to variation by a factor of at most 3 over the 100\,MeV-10\,GeV\ range).
\indent Because the ISRF and magnetic field components are deduced from the gas distribution in my model, it is useful to recast some of the above formulae as a function of molecular gas density. For a given gas distribution, such as a molecular core or a ring of a given extent on top of the atomic disk, the molecular gas density is the independent variable that drives the non-thermal properties. Accordingly, from Eqs. \ref{eq_bfield} and \ref{eq_ir}, and using a thickness of 200\,pc for the molecular gas, one obtains
\begin{align}
\tau_{\textrm{pp}} &= 1.1\,10^{15} \, \left( \frac{n_{\textrm{H}_2}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{Br}} &= 4.5\,10^{14} \, \left( \frac{n_{\textrm{H}_2}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} \\
\tau_{\textrm{IC}} &= 1.8\,10^{16} \, \left( \frac{E}{1\,\textrm{GeV}} \right)^{-1} \, \left( \frac{n_{\textrm{H}_2}}{1\,\textrm{cm}^{-3}} \right)^{-1.4} \, \textrm{s} \\
\tau_{\textrm{Synch}} &= 5.0\,10^{15} \, \left( \frac{E}{1\,\textrm{GeV}} \right)^{-1} \, \left( \frac{n_{\textrm{H}_2}}{1\,\textrm{cm}^{-3}} \right)^{-1.2} \, \textrm{s} \\
\tau_{\textrm{ion}} &= 2.4\,10^{15} \, \left( \frac{E}{1\,\textrm{GeV}} \right) \, \left( \frac{n_{\textrm{H}_2}}{1\,\textrm{cm}^{-3}} \right)^{-1} \, \textrm{s} ,
\end{align}
where only the infrared component of the ISRF was considered in the inverse-Compton time scale (it is by far the dominant component above $n_{\textrm{H}_2}=$10-20\,H$_2$\,cm$^{-3}$).
\section{Results}
\label{res}
\indent In this section, I present the results for the synthetic star-forming galaxies introduced above. To allow the connection with observations, the $\gamma$-ray\ luminosities are often given in three spectral bands: 100\,keV-10\,MeV\ (band I, soft gamma-rays), 100\,MeV-100\,GeV\ (band II, high-energy gamma-rays), 100\,GeV-100\,TeV\ (band III, very-high-energy gamma-rays). Examples of current instruments covering these bands are INTEGRAL/SPI, Fermi/LAT, and HESS, respectively.
\indent I first focus on a subset of the results to illustrate how gas distribution and density affect the $\gamma$-ray\ emission, and how the latter scales with global parameters. This is followed by a discussion of the impact of changing some parameters of the model. I then discuss the complete set of results in the context of the population study performed from \textit{Fermi}/LAT observations, and of the far-infrared - radio correlation.
\subsection{Gamma-ray luminosities and spectra}
\label{res_gamma}
\indent To illustrate the effects of the gas distribution and density on $\gamma$-ray\ emission in a star-forming galaxy, I restrict the discussion to the case of a disk galaxy with $R_{\textrm{max}}=$10\,kpc for the four molecular gas profiles and seven molecular gas densities introduced above. To allow a clear comparison, the runs were performed for the same input CR luminosity (which is that of the Milky Way, see Sect. \ref{model_src}). The plots of Fig. \ref{fig_lumgamma_bands_noscale} show the evolution of the $\gamma$-ray\ luminosity in three different bands for this particular setup, while the plots of Fig. \ref{fig_distribgamma_bands_noscale} show the distribution of the emission in terms of physical processes and emitting particles (primary electrons versus secondary electrons and positrons, hereafter primaries and secondaries for short).
\indent These plots do not only show the effect of increasing gas density, and correspondingly of the enhanced energy losses, but also include an effect of the distribution of CR sources. Indeed, from the smaller core to the larger ring, and with increasing density, the ratio of star formation occurring in dense molecular regions is higher than that distributed in the lower-density atomic disk. This is why for a same increase in molecular gas density, the 500\,pc core configuration exhibits the highest contrasts between the low-density case ($n_{\textrm{H}_2}=5$\,H$_2$\,cm$^{-3}$) and the high-density case ($n_{\textrm{H}_2}=500$\,H$_2$\,cm$^{-3}$): the luminosities increase because of a higher conversion efficiency of particle energy to radiation, which arises from a higher gas density and from a larger proportion of CRs being injected in a higher-density medium.
\indent The evolution of the luminosity in all three $\gamma$-ray\ bands, for an increase of a factor 100 of the molecular gas density, shows the following trends (I recall that this discussion holds for the same input CR luminosity):
\indent Band I (100\,keV-10\,MeV): The emission is always dominated by IC, with Bremsstrahlung contributing at the 15-25\% level depending on gas density. The IC from primaries is almost constant, which is consistent with primaries being IC loss-dominated (the increase in the radiation field density is balanced by a decrease in the steady-state primaries population). The IC from secondaries increases by a factor of about 10, which is due to a rise of the production rate of secondaries (the production rate increases not as much as the gas density because low-energy CRp progressively move from a diffusion-dominated regime to a loss-dominated regime). The net result is that emission in band I gradually becomes dominated by secondaries, at a level of 70-80\% for the highest gas density \citep[for observations and interpretation of the diffuse soft $\gamma$-ray\ emission from the Milky Way, see][]{Porter:2008,Bouchet:2011}. Overall, this contribution of secondaries increases from the very-high-energy band to the soft-gamma-ray band (band III to I), which reflects mainly that secondaries are the particles with the steepest spectrum.
\indent Band II (100\,MeV-100\,GeV): The emission is always dominated by pion decay, while leptonic processes contribute at the 20-30\% level, with an increasing fraction of it coming from secondaries (eventually reaching 75\% for the highest gas densities). The evolution of the pion decay and Bremsstrahlung emission with gas density flattens, and this is due to a transition in the transport regime. At the lowest gas densities, diffusion is an important process at intermediate particle energies $\sim$1-10\,GeV, so the steady-state particle population is not strongly affected by gas density and the luminosity of gas-related components increases with gas density, although slower than linearly. At the highest gas densities, Bremsstrahlung losses (for CR leptons) and hadronic losses (for CR nuclei) seriously compete with diffusion, therefore the luminosity of gas-related components rises more weakly with gas density as the transport is moving towards a loss-dominated regime. Note that the shape of this flattening probably depends on the diffusion conditions (a higher diffusion coefficient at low energies would imply a flattening occurring later, at a higher density).
\indent Band III (100\,GeV-100\,TeV): The emission is dominated by pion decay, except for the lowest average gas densities where it is either dominated by IC or where IC is within a factor of 2-3. Inverse-Compton emission is here always dominated by primaries, and this contribution is constant with gas density because the steady-state primary CRe population is loss-dominated. In contrast, the contribution from secondaries will rise with gas density, following the increase of the production rate of secondaries from high-energy CRp. But because secondaries have a softer injection spectrum, and because the parent CRp population gradually suffers from losses, secondaries never dominate primaries at these high energies, and the increase of the total IC emission remains moderate, only by a factor of 2-3. In contrast, pion decay emission increases faster with gas density.
\newpage
\begin{figure}[H]
\begin{center}
\includegraphics[width= \columnwidth]{Lum-by-run_1_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= \columnwidth]{Lum-by-run_2_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= \columnwidth]{Lum-by-run_3_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\caption{Luminosity in 3 $\gamma$-ray\ bands for a star-forming disk galaxy with ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(10\,kpc, 2\,kpc), 4 profiles of molecular gas (given in abscissa), and 7 molecular gas densities for each profile ($n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, 500\,H$_2$\,cm$^{-3}$). To illustrate the effects of the gas distribution, the $\gamma$-ray\ luminosities are given for the same input cosmic-ray luminosity.}
\label{fig_lumgamma_bands_noscale}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[width= \columnwidth]{Lum-distrib_1_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= \columnwidth]{Lum-distrib_2_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= \columnwidth]{Lum-distrib_3_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\caption{Distribution of the luminosity in 3 $\gamma$-ray\ bands, in terms of physical processes and emitting particles, for a star-forming disk galaxy with ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(10\,kpc, 2\,kpc), 4 profiles of molecular gas (given in abscissa), and 7 molecular gas densities for each profile ($n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, 500\,H$_2$\,cm$^{-3}$).}
\label{fig_distribgamma_bands_noscale}
\end{center}
\end{figure}
\newpage
\begin{figure}[H]
\begin{center}
\includegraphics[width= 8.25cm]{Spectra_R10-ring4-6-n5-50-500_Pi.eps}
\includegraphics[width= 8.25cm]{Spectra_R10-ring4-6-n5-50-500_Br.eps}
\includegraphics[width= 8.25cm]{Spectra_R10-ring4-6-n5-50-500_IC.eps}
\caption{Spectra of pion decay (top), Bremsstrahlung (middle), and inverse-Compton (bottom) emission for a star-forming disk galaxy with ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(10\,kpc, 2\,kpc) and a 4-6\,kpc ring distribution of molecular gas. The solid, dashed, and dot-dashed curves correspond to $n_{\textrm{H}_2}$=5, 50, and 500\,H$_2$\,cm$^{-3}$, respectively. For the leptonic processes, the contribution from primary electrons and secondary positrons are shown. In all plots, the total $\gamma$-ray\ emission from all processes and all particles is shown in black.}
\label{fig_specgamma}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[width= 8.25cm]{Gammaidx-by-sfr_1_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.25cm]{Gammaidx-by-sfr_2_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.25cm]{Gammaidx-by-sfr_3_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\caption{Photon spectral indices over 3 $\gamma$-ray\ bands for a star-forming disk galaxy with ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(10\,kpc, 2\,kpc), 4 profiles of molecular gas, and 7 molecular gas densities for each profile ($n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, 500\,H$_2$\,cm$^{-3}$). Blue, red, green, and purple curves correspond to the 500\,pc core, 1\,kpc core, 2-3\,kpc ring, and 4-6\,kpc ring gas profiles, respectively. Note that the energy bands differ from those used for the luminosities.}
\label{fig_idxgamma}
\end{center}
\end{figure}
\newpage
For the lowest gas densities, the increase is almost linear because high-energy CRp emitting in this band are close to diffusion-dominated. This trend flattens with further increase of the gas density, however, because pion decay losses become strong enough to reduce the steady-state CRp population. The effect on the pion decay emission extends well into the 100\,GeV-1\,TeV\ range.
\indent The plots of Fig. \ref{fig_specgamma} show the $\gamma$-ray\ spectra of each emission process independently, for the large ring distribution with gas densities $n_{\textrm{H}_2} =$5, 50, and 500\,cm${-3}$, and still for the same input CR power. The respective contribution of primaries and secondaries is given. These plots illustrate some of the statements made above, in the discussion of the luminosity evolution in the three $\gamma$-ray\ bands. One can see the flattening of the pion decay emission in the 100\,MeV-1\,TeV\ band as gas density increases, the relative steadiness of the primary leptonic components, and the growth of the secondary leptonic contribution.
\indent The plots of Fig. \ref{fig_idxgamma} show the photon spectral indices over three spectral bands that differ from those used before for the luminosities (choices made so that a power-law approximation can hold). These are given for the disk galaxy model with $R_{\textrm{max}}=$10\,kpc, for the four molecular gas profiles and seven molecular gas densities, as a function of SFR. The flattening of the spectrum with increasing gas density is clearly apparent for band II. The emission in band III exhibits that same feature but also flattens at low SFRs due to the predominance of inverse-Compton emission.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width= 8.25cm]{Lum-by-sfr_1_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.25cm]{Lum-by-sfr_2_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.25cm]{Lum-by-sfr_3_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\caption{Luminosity in 3 $\gamma$-ray\ bands as a function of star formation rate and cosmic-ray power, for a star-forming disk galaxy with ($R_{\textrm{max}}$, $z_{\textrm{max}}$)=(10\,kpc, 2\,kpc), 4 profiles of molecular gas (given in abscissa), and 7 molecular gas densities for each profile ($n_{\textrm{H}_2}=$5, 10, 20, 50, 100, 200, 500\,H$_2$\,cm$^{-3}$). The cosmic-ray input power generated by the galaxy is assumed to scale with total star formation rate, using the Milky Way as normalisation.}
\label{fig_lumgamma_bands_scalesfr}
\end{center}
\end{figure}
\subsection{Scaling of luminosities with global properties}
\label{res_scaling}
\indent Under the assumption that the CR power generated by a star-forming galaxy is only proportional to its total SFR, and that the latter can be described from the Schmidt-Kennicutt relation, total luminosities were computed for all models in the sample. In the following, I present the theoretical scaling of the luminosity in the three $\gamma$-ray\ spectral bands in a qualitative and quantitative way. For clarity, the plots only illustrate the results for the disk galaxy with $R_{\textrm{max}}=$10\,kpc for the four molecular gas profiles and seven molecular gas densities introduced previously. The complete sample is shown for band II in a following subsection, because it is the only band where a population study was recently performed.
\indent The plots of Fig. \ref{fig_lumgamma_bands_scalesfr} show the evolution of $\gamma$-ray\ luminosity in the three spectral bands as a function of SFR and CR power. For each gas distribution (the two cores and two rings), the parameters of a scaling law were computed from the high gas density cases, under the hypothesis of a scaling of the form
\begin{equation}
\log L_{\gamma} = \beta \log P_{\textrm{CR}} + \delta
\end{equation}
Most of the discussion below focuses on the index $\beta$ because this contains most of the physics of the transport, while the offset $\delta$ is more related to normalisation considerations.
\indent Band I (100\,keV-10\,MeV): The luminosity at high gas densities evolves with an index $\delta \sim$1.2-1.3. At lower gas densities, the luminosity decreases faster than that because an increasing fraction of the CR power is released into the low-density atomic disk (the effect is stronger for the small-core model). The more-than-linear increase is ultimately due to the contribution of secondaries to the IC emission, which causes an increase of the luminosity by a factor of $\sim$5-7 on top of the increase of the CR power by a factor of about $\sim$150-550 (depending on the molecular gas distribution). The variation of luminosity for a given CR power or SFR can reach a factor of about 3, depending on the layout of the gas.
\indent Band II (100\,MeV-100\,GeV): The luminosity at high gas densities evolves with an index $\delta \sim$1.1-1.2. Compared with band I, the luminosity exhibits larger deviations from this scaling with decreasing gas density, and not only for the small-core case. The scaling index at low gas densities is in the range $\delta \sim$1.4-1.7 depending on the molecular gas distribution. This behaviour is connected with the shift between diffusion-dominated and loss-dominated regimes for the steady-state CR nuclei population (see Sect. \ref{res_gamma}). In the present setup, the transport is not fully loss-dominated over the energy range of particles emitting in the 100\,MeV-100\,GeV\ band, even for the highest molecular gas density, and diffusion remains significant (the calorimetric efficiency is at most $\sim$30\%, see Sect. \ref{res_calo}). As a consequence, luminosity still increases modestly with gas density, on top of the increase arising from enhanced star formation.
\indent Band III (100\,GeV-100\,TeV): The luminosity evolves with an index $\delta \sim$1.4-1.5 at high gas densities, and with an index $\delta \sim$1.5-1.7 at low gas densities, depending on the molecular gas distribution. The reason for the higher index than in band II is that the steady-state CR nuclei population emitting in band III is closer to the diffusion-dominated side, hence a higher increase of the luminosity for a given increase in gas density. Then, at low gas densities, the effect of an increased fraction of CR power released in the low-density atomic disk is compensated for by the IC contribution, so the decrease in luminosity is more shallow. The variation of luminosity for a given CR power or SFR and depending on the layout of the gas is the strongest over all three bands. This is because particles emitting in this band are significantly affected by diffusion, a process that depends on the distribution of the sources.
\indent The scaling of the $\gamma$-ray\ luminosity for the other star-forming disk galaxy models with $R_{\textrm{max}}=$5 and 20\,kpc follows the same trends. The indices are similar to the values given above for each of the three $\gamma$-ray\ bands, with differences $<0.1$. The variations of the luminosity for a given CR power or SFR and depending on the distribution of the molecular gas are also similar to those given above for $R_{\textrm{max}}=$10\,kpc.
\begin{figure}[!t]
\begin{center}
\includegraphics[width= \columnwidth]{Calorimetric-pion_R10-2cores-2rings-rev3-thres4_scaleSFR.eps}
\caption{Calorimetric efficiency for cosmic-ray nuclei, for a star-forming disk galaxy with $R_{\textrm{max}}=$10\,kpc, 4 profiles of molecular gas distribution, and 7 molecular gas densities. The energy conversion efficiency was computed for hadronic interactions only.}
\label{fig_calo}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width= \columnwidth]{ISM-energydens_R10-2cores-2rings-rev3-thres4-gas-avg_scaleSFR.eps}
\caption{Cosmic-ray energy densities for a star-forming disk galaxy with $R_{\textrm{max}}=$10\,kpc, 4 profiles of molecular gas distribution, and 7 molecular gas densities. The energy densities are source-weighted averages, and they are given for particles with energies $\geq$1\,GeV\ and $\geq$1\,TeV\ (solid and dashed lines, respectively).}
\label{fig_nrjdens}
\end{center}
\end{figure}
\subsection{Cosmic-ray calorimetry and energy densities}
\label{res_calo}
\indent The anticipation and then detection of star-forming galaxies as HE and VHE $\gamma$-ray\ sources raised the question of how much of the cosmic-ray energy is converted into $\gamma$-rays, and how this fraction would evolve with galactic properties. A more general question is that of the calorimetric efficiency of these galaxies, the fraction of cosmic-ray energy that is lost in the ISM, whatever the process and resulting energy type (radiative or thermal). Nuclei are the dominant CR component, and most of their energy lies above a few 100\,MeV, in a range where hadronic interactions are the main energy-loss process\footnote{Adiabatic losses in a galactic wind are irrelevant since I am interested here in gas-related $\gamma$-ray\ emission and cosmic-ray energy densities in the galactic plane.}. The cosmic-ray calorimetric efficiency can therefore be approximated by the fraction of energy lost to hadronic interactions. Assuming that charged pions are produced at the same rate as neutral pions, the calorimetric efficiency was computed for each galaxy simulated in this work.
\indent The results are presented in Fig. \ref{fig_calo} for the $R_{\textrm{max}}=$10\,kpc models. The calorimetric efficiencies are given as a function of a source-weighted average gas surface density to illustrate the primary dependence on gas density. Overall, the calorimetric efficiency moves from 2-4\% at the lowest gas densities to 25-30\% at the highest gas densities, which is a rise by one order of magnitude in efficiency for an increase by two orders of magnitude in gas density. These high calorimetric efficiencies are about 10-20 times higher than that inferred for the Milky Way \citep{Strong:2010}. This shows that even at volume-averaged gas densities of about 1000\,H\,cm$^{-3}$, most CR energy flows out of the system. This has at least two reasons: first, CR nuclei with energies of just about 10\,GeV\ or more can diffuse out of a layer of 100\,pc faster than they lose energy through hadronic interactions; then, a fraction of the lower-energy CRs can also diffuse out without losing too much energy when they are released close to the edges of the dense molecular regions.
\indent Connected with the latter point, another interesting feature of the plot in Fig. \ref{fig_calo} is the vertical scatter in calorimetric efficiency for a given average gas density. This arises from the spatial distribution of gas and cosmic-ray sources. The lowest efficiencies are obtained for the 500\,pc core model, because the latter molecular gas distribution has the highest surface-to-volume ratio. This ratio controls the importance of diffusion (proportional to surface) over energy losses (proportional to volume). This is further demonstrated by the fact that the 1\,kpc core model and the 2-3\,kpc ring model have the same surface-to-volume ratio and nearly identical calorimetric efficiencies, while the 4-6\,kpc ring model has the lowest surface-to-volume ratio of all $R_{\textrm{max}}=$10\,kpc models and the highest calorimetric efficiencies.
\indent The total (nuclei and leptons) CR energy densities resulting from increasing gas densities are shown in Fig. \ref{fig_nrjdens} for the $R_{\textrm{max}}=$10\,kpc models. In the context of pion-decay emission from CR nuclei in the HE and VHE $\gamma$-ray\ bands, the CR energy densities are given as gas-weighted averages for CR energies above 1\,GeV\ and 1\,TeV\ (note that the CR energy density above 100\,MeV\ and 1\,GeV\ are almost identical). The CR energy density $\geq$1\,GeV\ is of the order of 0.1-1\,eV\,cm$^{-3}$ at low average gas densities and rises to 50-80\,eV\,cm$^{-3}$ at high average gas densities (depending on the molecular gas distribution). Because of the importance of energy losses in the transport of particles at high gas densities, the increase in CR energy density does not follow the increase in star formation rate. The CR energy density $\geq$1\,TeV\ is two orders of magnitude lower at low average gas densities, but this difference shrinks to just one order of magnitude at high average gas densities. This occurs because higher-energy particles are closer to a diffusion-dominated regime, so in this range, the CR energy density rise follows more closely the increase in star formation rate (hence cosmic-ray input luminosity).
\indent For comparison, the energy density of radiation and magnetic field rises to $\sim$3300\,eV\,cm$^{-3}$ each for the highest gas densities ($n_{\textrm{H}_2}=500$\,H$_2$\,cm$^{-3}$). Cosmic rays are therefore a largely subdominant contribution to the ISM energy density in the densest and most active star-forming galaxies. Their share in the ISM energy density budget increases with decreasing gas density, until it becomes a factor of a few below that of radiation or magnetic field.
\subsection{Impact of model parameters}
\label{res_params}
\indent A subset of models was run with different parameters to assess the extent to which the above discussion depends either on simplifying assumptions of the model or on poorly known aspects of the phenomenon. In the following, these are discussed in turn, with emphasis on the luminosities and their scaling in the three $\gamma$-ray\ bands. The configurations used for these tests are the $R_{\textrm{max}}=$10\,kpc model with the 2-3\,kpc ring distribution and the $R_{\textrm{max}}=$5\,kpc model with the 500\,pc core distribution.
\indent Halo height: The size of the halo for the Milky Way and other galaxies remain a poorly known parameter of the problem. For the Milky Way, a halo size from 4 to 10\,kpc is allowed by measurements of unstable secondary cosmic rays \citep[][depending on propagation model]{Strong:2007}, and $\gamma$-ray\ observations of the outer Galaxy favour a larger than 4\,kpc halo \citep[][among other possibilities]{Ackermann:2012}. A three times larger halo height was therefore tested. This resulted in higher luminosities in all bands since CRs are confined for a longer time and can thus lose a larger part of their energy to radiation. In bands I and II, differences are of the order of 40-50\% at low star formation rates and shrink as molecular gas density increases and transport evolves to an increasingly loss-dominated regime. In band III, differences are of the order of 20-30\% and are almost constant with the star formation rate. This is so because the higher-energy particles emitting in band III are closer to the diffusion-dominated regime over the whole range of gas densities.
\indent Infrared field: The infrared dust emission is a major component in the energy density of the ISM, and dominates starlight for modest molecular gas densities and is as important as the magnetic field for the highest gas densities. It contributes to setting the steady-state lepton populations, therefore I tested cases where it is lower by a factor of $\sim2$. The main effect is on the luminosities in band I, which are lower by about 50\%; the lowest average gas surface densities are affected even more. The reason for this is that decreasing the IR energy density by about 2 reduces the energy losses for leptons, although by less than 2 since synchrotron and Bremsstrahlung are also contributing to the losses. Eventually, the steady-state leptons are more numerous but not to such an extent that it compensates for the reduction in photon number density for inverse-Compton scattering. The luminosity in band II is almost unaffected, which is expected since it is of hadronic origin. The same argument holds for the luminosity in band III except for the lowest average gas surface densities, which are dominated by inverse-Compton emission and thus undergo the same decrease as in band I, by 50\% or more. As mentioned in Sect. \ref{model_isrf}, no radiation transfer calculation was performed and the IR energy density was simply assumed to decrease vertically, away from the galactic plane. Since CRs can spend a significant amount of time in the halo, the actual extent of the IR radiation field may be relevant. A scale height $z_{\textrm{IR}}$ of 20 instead of 2\,kpc was therefore tested. The resulting differences are negligible in all bands, which indicates that regarding inverse-Compton emission everything occurs in the vicinity of the galactic plane .
\indent Diffusion coefficient: The spatial diffusion coefficient was assumed to be the one inferred for the Milky Way from a large set of data interpreted in the framework of a fairly complete GALPROP modelling. But this parameter very likely has some dependence on the actual interstellar conditions in other star-forming galaxies. The tremendous density of star formation in starbursts' cores may result in very strong turbulence such that CRs are expected to be confined more efficiently, diffuse less easily, and lose more energy to radiation. I tested the case of a diffusion coefficient that was ten times lower than the base case value, which yielded $\gamma$-ray\ luminosities higher than those presented previously, although with variations over bands and configurations. In band I, luminosities are almost the same at high densities $n_{\textrm{H}_2} = $500\,H$_2$\,cm$^{-3}$, because the transport is loss-dominated in these conditions, but they increasingly differ as the gas density decreases to eventually be a factor 2-3 higher at low densities $n_{\textrm{H}_2} = $5\,H$_2$\,cm$^{-3}$. This difference arises mostly from the larger population of secondaries that contribute to the inverse-Compton emission. This is connected to the higher pion-decay luminosity observable in band II. Again, the situation is almost the same at high densities but deviates significantly at low densities, by factors of up to 3-4. Importantly, this low diffusion coefficient moves almost all points above the correlation inferred from {\em Fermi}/LAT observations (see Sect. \ref{res_fermilat}). Last, in band III, the differences are even larger and range between about 2 and 5 because nuclei emitting in this band are more affected by diffusion. With this low diffusion coefficient, inverse-Compton never dominates the emission at very high energies.
\indent Galactic wind: Galactic winds are very common in active star-forming galaxies and can be traced primarily through the expelled gas and dust, but also by the entrained cosmic rays and magnetic fields \citep{Veilleux:2005}. They constitute an energy-independent spatial transport process for CRs, which differs fundamentally from diffusion and is thus likely to alter particle populations and emission spectra. Galactic wind properties are expected to scale with star formation rate surface density, and evidence for this has been observed \citep{McCormick:2013}. Winds are thus expected to be more relevant to star-forming galaxies where most of the star formation rate is confined to small, dense regions. The code used here does not allow the simulation of radially dependent advection, such as a wind emanating only from the inner core of a starburst. I tested the case of a wind with vertical velocity of about 500\,km\,s$^{-1}$\ at the level of the galactic plane that accelerates by 100\,km\,s$^{-1}$/kpc above the plane. Focusing on the highest gas density and star formation rate cases, which are the most likely to generate such a strong outflow, the wind effect is almost negligible in all $\gamma$-ray\ bands (it is only of a few \%). This can be understood from the time-scale equations given in Sect. \ref{model_timescales}. When considering the time needed to escape a gas layer of 100\,pc thickness, diffusion with the properties assumed here prevails over a 500\,km\,s$^{-1}$\ wind over the entire energy range. Time scales become similar (factor of $\sim$2-3) only at energies below a few GeV, but in this range energy losses also occur faster than advection.
\indent Schmidt-Kennicutt relation: The model used in this work relies on the Schmidt-Kennicutt relation for the determination of the star formation rate and its distribution, which in turn affects the CR input luminosity and source profile and the infrared radiation field density. I tested the impact of the index of the Schmidt-Kennicutt relation by varying its value within the uncertainty range and using 1.6 and 1.2 instead of 1.4. Keeping the same normalisation for the dependence of the CR input luminosity on SFR for a given galaxy model with a given gas distribution, a higher (lower) index causes the global SFR to be higher (lower) and more (less) concentrated in the high-density regions, and it increases (decreases) the infrared radiation field density. Apart from the shift in SFR and CR input power, the effects are the following: in band I, the evolution of the luminosity at high gas densities is faster (flatter) in the higher (lower) index case, which arises from variations in the infrared radiation field density; in band II, which is dominated by pion decay, the luminosities are simply increased as a consequence of the higher CR input luminosity, and the scaling of the luminosity with SFR is left essentially unchanged with variations of less than 50\% for a given SFR; in band III, the luminosity evolution with SFR is flattened in the higher index case as a result of a higher contribution of inverse-Compton scattering at low average gas densities (and the opposite for the lower index case). Overall, variations in the index of the Schmidt-Kennicutt relation introduces some scatter about the trend in band II and alters the trend in bands I and III as one moves from high to low SFRs.
\indent The magnetic field model and the impact of its parameters are discussed in Sect. \ref{res_radio} in the context of the constraint set by the observed far-infrared - radio continuum correlation.
\newpage
\begin{figure}[H]
\begin{center}
\includegraphics[width= 8.6cm]{Scaling-fermi_R2-5-10-20-4cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.6cm]{Scaling-fermi_R2-5-10-20-4cores-2rings-rev3-thres4-largerhalo-irabs_scaleSFR.eps}
\includegraphics[width= 8.6cm]{Scaling-fermi_R2-5-10-20-4cores-2rings-rev3-thres4-largerhalo-irabs-ratioplots_scaleSFR.eps}
\caption{Luminosity in the 100\,MeV-100\,GeV\ band as a function of total infrared luminosity and star formation rate for the complete sample of models. The orange region is the uncertainty range of the correlation determined experimentally. The top panel shows the result for the basic model, and the middle panel shows the result obtained with larger halos for the small galaxies and using a correction for starlight leakage. The bottom panel shows the ratio of 100\,MeV-100\,GeV\ to total infrared luminosities.}
\label{fig_lumgamma_all_vs_fermilat}
\end{center}
\end{figure}
\subsection{Comparison with the Fermi/LAT population study}
\label{res_fermilat}
\indent From observations of about 60 star-forming galaxies using the {\em Fermi}/LAT space telescope, \citet{Ackermann:2012} found evidence for a quasi-linear correlation between $\gamma$-ray\ luminosity and tracers of the star formation activity, although not with high significance (conservative P-values $\leq$ 0.05, for a power-law scaling with index close to 1). Since such a correlation cannot be considered as firmly established, the model introduced above was not tuned to match it. Instead, a generic model was defined, and the corresponding predictions are compared and discussed in the context of the {\em Fermi}/LAT population study. The situation is different for the far-infrared - radio correlation, which is a firmly established empirical correlation with which consistency was sought by defining the magnetic field scaling (see Sect. \ref{res_radio}).
\indent Figure \ref{fig_lumgamma_all_vs_fermilat} shows the luminosity in band II as a function of total infrared luminosity and star formation rate for the complete sample of models. This is compared with the scaling inferred from {\em Fermi}/LAT observations. I used the scaling relationship between $L_{8-1000\mu\textrm{m}}$ and $L_{0.1-100\textrm{GeV}}$, computed using the expectation-maximisation method and excluding galaxies hosting {\em Swift}/BAT-detected AGN, which corresponds to a power-law index of 1.09$\pm$0.10. This does not compare exactly the same things here because the {\em Fermi}/LAT luminosities were derived from power-law fits to the data, whereas the model luminosities correspond to the integration of physical spectra.
\indent The agreement between models and observations is rather good over the nearly 6 orders of magnitude in SFR covered by the models, especially considering that no parameter was tuned to reproduce the inferred correlation. At SFRs of $\sim$1-10\,M$_{\odot}$\,yr$^{-1}$, the models and the correlation uncertainty range clearly overlap. Some models are above or beyond the uncertainty range by factors of up to a few, but they remain in the estimated intrinsic dispersion \citep[not shown here, see][]{Ackermann:2012}. I show below that these outliers can easily be brought to a better agreement with the correlation. At higher SFRs, all relevant models converge with a decreasing scatter towards a scaling that completely agrees with the one inferred from observations (as anticipated from the discussion in Sect. \ref{res_scaling}).
\indent At the lowest densities and SFRs, the luminosities of all models decrease and start deviating from the observed correlation, which contrasts with the fact that the observed correlation appears to hold down to SFRs of 10$^{-2}$\,M$_{\odot}$\,yr$^{-1}$, with objects such as the Small and Large Magellanic Clouds being firmly detected by the {\em Fermi}/LAT. There are, however, two caveats to be considered about the behaviour of the predicted $\gamma$-ray\ luminosity versus SFR at the low end.
\begin{itemize}
\item The galactic halo of the small galaxies may be larger than assumed here (by keeping a nearly constant aspect ratio of the galactic volume between disk size and halo height).
\item The infrared luminosity may be lower than estimated here for the low gas surface densities, because a fraction of the UV/optical stellar light can escape the galaxy without being converted to infrared.
\end{itemize}
The first effect was assessed using larger halos with $z_{\textrm{max}}$=2\,kpc for the small galaxies with $R_{\textrm{max}}$=2 and 5\,kpc, and the second effect was tested using a correction based on the total gas surface density and an average extinction coefficient for the starlight of $\kappa(1\,\mu\textrm{m})=1.995\,10^{-22}$\,cm$^{2}$\,H$^{-1}$ \citep[][1\,$\mu$m is approximately the peak of the optical radiation field]{Draine:2003}. Combined, they significantly improved the match and reduced the scatter even more, as illustrated in the middle panel of Fig. \ref{fig_lumgamma_all_vs_fermilat}, although a deviation still remains.
\indent For the problem of deviation at low SFRs and/or for small galaxies, one should note that the correlation inferred from {\em Fermi}/LAT observations is not exactly of the same type as the one modelled in this work: only the interstellar $\gamma$-ray\ emission is modelled here, while the total $\gamma$-ray\ emission is used to determine the observed correlation. Most detected objects of the {\em Fermi}/LAT sample are point-like, and for the resolved ones, the Magellanic Clouds, the exact origin of the emission is unclear. For the SMC, it was estimated that the galactic pulsar population may account for a significant fraction of the signal \citep{Abdo:2010e}. A larger sample of detected galaxies with low SFRs would help to clarify the situation. The best prospects here are the detection of the Triangulum Galaxy M33 and a better spatial resolving of the emission from the Magellanic Clouds. At the other end, at high SFRs, doubling the {\em Fermi}/LAT exposure time from five to ten years will lower the constraints but probably not challenge the picture by much, considering that the observed and expected scaling is so close to linear \citep[see][for quantitative estimates of further detections]{Ackermann:2012}.
\indent Overall, the observed correlation can be accounted for to a large extent without major modifications to the global scheme inferred for CRs in the Milky Way. In terms of CR transport, the present results suggest that using a single diffusion coefficient and the assumption that CRs experience large-scale volume-averaged interstellar conditions is enough to account for the population constraints available today \citep[for a recent theoretical investigation of CR sampling of the ISM, see][]{Boettcher:2013}. Interestingly, the same scheme also seems to be able to explain the far-infrared - radio correlation, as illustrated in the following section. The bottom panel of Fig. \ref{fig_lumgamma_all_vs_fermilat} shows the ratio of 100\,MeV-100\,GeV\ to total infrared luminosities for the complete sample of models and with the corrections mentioned above. It reveals that the global scaling does not result from a universal relation, but rather from an ensemble of relations that depend to some degree on galactic global properties and are stretched over many orders of magnitude in luminosities.
\indent Last, when exploring possible correlations between the $\gamma$-ray\ luminosity of star-forming galaxies and their global properties both theoretically and experimentally, it was put forward that the luminosity in band II might scale with the product of SFR and total gas mass \citep{Abdo:2009l}. The basic reasoning behind this is that the former quantity traces the input CR power, while the latter quantity traces the amount of target material available for hadronic interactions. Checking that possibility against the models presented here shows that there is indeed a correlation, although with a scatter that is twice as large as that obtained with SFR alone.
\newpage
\begin{figure}[H]
\begin{center}
\includegraphics[width= 8.6cm]{Radio-by-sfr_R2-5-10-20-4cores-2rings-rev3-thres4_scaleSFR.eps}
\includegraphics[width= 8.6cm]{Radio-by-sfr_R2-5-10-20-4cores-2rings-rev3-thres4-largerhalo-irabs_scaleSFR.eps}
\includegraphics[width= 8.6cm]{Radio-by-sfr_R2-5-10-20-4cores-2rings-rev3-thres4-largerhalo-irabs-ratioplots_scaleSFR.eps}
\caption{Radio luminosity at 1.4\,GHz as a function of far-infrared luminosity for the complete sample of models. The orange region is the uncertainty range of the linear scaling derived from the observed far-infrared - radio correlation. The top panel shows the result for the basic model, and the middle panel shows the result obtained with larger halos for the small galaxies and using a correction for starlight leakage. The bottom panel shows the ratio of 1.4\,GHz to far-infrared luminosities.}
\label{fig_lumradio_all_vs_firradio}
\end{center}
\end{figure}
\subsection{Comparison with the far-infrared-radio correlation}
\label{res_radio}
\indent Before it was possible to probe the $\gamma$-ray\ emission from external star-forming galaxies, mostly thanks to the {\em Fermi}/LAT, the non-thermal radiation from galactic populations of CRs interacting with the ISM had been deeply explored at $\sim$100\,MHz-10\,GHz radio frequencies from the synchrotron emission of CR leptons propagating in the galactic magnetic field. The number of star-forming galaxies detected as a result of this process is orders of magnitude larger than the number of those now accessible through $\gamma$-rays. Population studies can thus lead to much more significant results, and among these is the observed far-infrared - radio correlation.
\indent Because of the high significance of this correlation, the star-forming galaxy model presented was checked against it before any discussion of the $\gamma$-ray\ counterpart. The definition of the far-infrared/radio flux ratio $q$ was taken from \citet{Helou:1985}: $q=\log_{10}(S_{\textrm{FIR}}/3.75 \times 10^{12}\,\textrm{Hz})-\log_{10}(S_{1.4\,\textrm{GHz}})$, where $S_{\textrm{FIR}}$ (W\,m$^{-2}$) is the FIR flux from 42.5 to 122.5\,$\mu$m and $S_{1.4\,\textrm{GHz}}$ (W\,m$^{-2}$\,Hz$^{-1}$) is the radio flux at 1.4\,GHz. The best-fit value $q=2.34 \pm0.26$ was adopted from \citet{Yun:2001}. Interestingly, estimates for $q$ based on a GALPROP modelling of the Milky Way are in the range 2.26-2.69 \citep[2.45-2.69 for plain diffusion models, see][]{Strong:2010}. The far-infrared and 1.4\,GHz luminosities are readily available for each synthetic galaxy and can thus be compared with the expected values using the measured proportionality factor $q$. The result is shown in the top panel of Fig. \ref{fig_lumradio_all_vs_firradio}.
\indent The index $a$ of the magnetic field model determines the slope of the 1.4\,GHz luminosity evolution for high enough average gas surface densities, while the doublet $(B_0,\Sigma_0)$ sets the normalisation of the 1.4\,GHz luminosities (see Eq. \ref{eq_bfield}). Note that the latter is dependent on the ISRF assumptions because radiation and magnetic field jointly determine the steady-state lepton population. In the star-forming galaxy model used here, index $a$ of the magnetic field prescription was selected so that the predicted radio luminosity matched the far-infrared - radio correlation (although not by a formal fit, but by testing values in the range 0.3-0.8 with steps of 0.1), while the normalisation of the magnetic field was set to reproduce the conditions in the Milky Way (see Sect. \ref{model_calib}). This means that, formally, there was only one degree of freedom to improve the fit to the far-infrared - radio correlation. Despite this, the model results in a fairly good agreement over more than four orders of magnitude, with a remarkably small scatter. Only a few points at high SFRs are marginally out of the uncertainty range of the observed correlation. Deviations appear at the lowest far-infrared luminosities, and especially for the smallest galaxy sizes. This arises because of the low conversion efficiency of CR leptons energy into radio emission, as a result from low magnetic fields (low average gas surface densities) and small confinement volume (small galaxy haloes).
\indent The same arguments put forward to explain the downturn of the 100\,MeV-100\,GeV\ luminosity at low densities and SFRs can be invoked here. In addition, in the context of synchrotron emission, one should note that the magnetic field evolution with gas surface density may differ from the power law assumed here, and may be flatter at low densities for instance. Using larger halos for the small galaxies and a correction for starlight leakage significantly improved the match, as illustrated in the middle panel of Fig. \ref{fig_lumradio_all_vs_firradio}. The resulting scatter in radio luminosity at any given infrared luminosity is relatively small, at most $\pm30$\% and at least twice as small as that of the of 100\,MeV-100\,GeV\ luminosity. But although the effects listed above would contribute to bring outliers closer to a single proportionality relation, it is interesting to note that a deviation from a pure linear scaling at low luminosities was observed and discussed \citep{Yun:2001}.
\indent The bottom panel of Fig. \ref{fig_lumradio_all_vs_firradio} shows the ratio of radio-to-infrared luminosities for the complete sample of models and with the corrections mentioned above. Although most models are consistent with the measured value, there is a slight trend in the radio-to-luminosity ratio, corresponding to a variation by a factor of a few, which is concealed by the stretching over 6 orders of magnitude in luminosity. I emphasise here again that only one parameter was adjusted to reproduce the far-infrared - radio correlation, and not through a formal fit. A flatter trend in the radio-to-luminosity ratio may therefore be obtained through a fine-tuning of the model components (magnetic field and ISRF, mainly).
\indent Addressing the latter two problems is beyond the scope of this work. In a forthcoming paper, I will examine in more detail the synchrotron emission as a function of galactic properties as predicted in the framework of the star-forming galaxy model presented here, and connect it with the observed far-infrared - radio correlation.
\section{Conclusion}
\label{conclu}
\indent The interaction of galactic cosmic rays (CRs) with the interstellar medium (ISM) produces $\gamma$-rays\ through hadronic interactions, inverse-Compton scattering, and Bremsstrahlung. This interstellar emission has been observed in the Milky Way for decades and provided a way to indirectly probe CRs outside the solar system, especially the dominant nuclei component. The current generation of high-energy (HE) and very-high-energy (VHE) $\gamma$-ray\ instruments has enabled us to extend this kind of study to several other star-forming galaxies. Five external systems, possibly seven, were recently detected at GeV energies with the \textit{Fermi}/LAT. Combined with upper limits on about sixty other star-forming galaxies, evidence was found for a quasi-linear correlation between $\gamma$-ray\ luminosity and tracers of the star formation activity. This raised the question of the scaling laws that can be expected for the interstellar $\gamma$-ray\ emission as a function of global galactic properties, with the aim of knowing whether this early population study can constrain the origin and transport of CRs.
\indent In the present work, I introduced a model for the non-thermal emissions from steady-state CR populations interacting with the ISM in star-forming disk galaxies. The model is two-dimensional and includes realistic CR source distributions as well as a complete treatment of their transport in the galactic disk and halo (in the diffusion approximation). This contrasts with most previous works, which relied on a one-zone approach and typical time-scale estimates. Prescriptions were used for the gas distribution, the large-scale magnetic field, and the interstellar radiation field. Cosmic-ray-related parameters such as injection rates and spectra or diffusion properties were taken from Milky Way studies. The model is set to reproduce the average interstellar conditions of the Galaxy, its global non-thermal emission, and the far-infrared - radio correlation. A series of star-forming galaxies with different sizes, masses, and gas distributions was simulated to assess the impact of global galactic parameters on the $\gamma$-ray\ output. The link between gas content and star formation was made from the Schmidt-Kennicutt relation. The full set covered almost six orders of magnitude in star formation rate (SFR). The emission was examined over 100\,keV-100\,TeV, with dedicated discussions for instrumental bands: 100\,keV-10\,MeV\ (band I, soft $\gamma$-rays), 100\,MeV-100\,GeV\ (band II, HE $\gamma$-rays), and 100\,GeV-100\,TeV\ (band III, VHE $\gamma$-rays).
\indent In band I, the emission is dominated by inverse-Compton emission, with Bremsstrahlung contributing at the $\sim20$\% level. With increasing average gas surface density, the luminosity rises more than linearly, with a power-law index on SFR $\sim$1.2-1.3, depending on the gas distribution. This is the result of the growing contribution of secondary leptons. The latter eventually accounts for 70-80\% of the emission.
\indent In band II, the emission is dominated by pion-decay emission, with Bremsstrahlung contributing at the $\sim20$\% level, and inverse-Compton being equally important for small galaxies with low average gas surface density. With increasing average gas surface density, the luminosity rises more than linearly, with a power-law index on SFR initially in the range $\sim$1.4-1.7 and then decreasing to $\sim$1.1-1.2. This behaviour is the result of the progressive shift of the CR nuclei population from a diffusion-dominated regime towards a loss-dominated regime.
\indent In band III, the emission is dominated by inverse-Compton emission for small galaxies with low average gas surface density. For larger and denser systems, pion decay is the main emission mechanism, accounting for more than 95\% of the luminosity for the densest ones. With increasing average gas surface density, the luminosity rises more than linearly, with a power-law index on SFR in the range $\sim$1.4-1.5. The behaviour is the same as in band II, with emitting particles being closer to diffusion-dominated, and with inverse-Compton becoming increasingly important at the low end.
\indent Without any specific tuning, the model was able to account for the normalisation and trend inferred from the {\em Fermi}/LAT population study over almost the entire range of SFRs tested here. This suggests that the \textit{Fermi}/LAT population study does not call for major modifications of the scheme inferred for CRs in the Milky Way when extrapolated to other systems, probably because uncertainties are still too large. A good match can be obtained with a plain diffusion scheme, a single diffusion coefficient, and the assumption that CRs experience large-scale volume-averaged interstellar conditions. There is no need for a strong dependence of the diffusion properties on interstellar conditions, and no impact of strong galactic winds in the most active star-forming systems. The only requirement in the framework used here is that small galaxies with a disk radius of a few kpc have halos of at least 2\,kpc in half height. There is, however, no universal relation between high-energy $\gamma$-ray\ luminosity and star formation activity, as illustrated by the downturn in $\gamma$-ray\ emission at low SFR values and the scatter introduced by different galactic global properties. Varying the galaxy size, gas distribution, and gas density introduces a variation in the 100\,MeV-100\,GeV\ luminosity at a given SFR by up to a factor of 3, while varying the SFR for a given gas layout was found to add another 50\% of scatter. Interestingly, the same model accounted pretty well for the far-infrared - radio correlation over most of the range of galactic properties tested here, with a remarkably small scatter of $\leq 30$\% at any infrared luminosity and a decrease in synchrotron emission at the low end.
\indent Progress on this kind of population study is expected from the doubling of the {\em Fermi}/LAT exposure, especially at low SFRs with the detection of M33 and the resolving of the Magellanic Clouds. An additional test of our understanding of interstellar $\gamma$-ray\ emission from star-forming galaxies may come from the 100\,GeV-100\,TeV\ range. In this band, the scaling of luminosity with SFR has a higher non-linearity favourable to distant but more active objects, and the next-generation {\em Cherenkov Telescope Array} with its ten-fold improvement in sensitivity is expected to perform constraining observations.
\begin{acknowledgement}
I acknowledge support from the European Community via contract ERC-StG-200911 for part of this work. I wish to thank Olivier Bern\'e and D\'eborah Paradis for their assistance on interstellar radiation field models, and Andy Strong for his help with the GALPROP package. I also thank Keith Bechtol and Guillaume Dubus for reading and commenting on a draft version of the manuscript.
\end{acknowledgement}
\bibliographystyle{aa}
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{
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Home Tech Study Reveals 'Digital Divide,' But Not Necessarily The One You'd Think It Is
by Joe Mandese @mp_joemandese, October 19, 2004
Given the rapid proliferation of new media of all kinds, the term "digital divide" appears to have been dropped from most industry, or even political discussions. But a new report on consumer media technologies reveals economic, racial and other demographic gaps continue to influence the adoption of digital media technologies, but they are not necessarily the ones you might think they are.
The report, the latest installment of The Home Technology Monitor, being released today by Knowledge Networks/SRI, does find a surprisingly wide gap in the penetration of seemingly ubiquitous digital media technologies such as personal computers and broadband access, but it also reveals that some newer media, including digital TV and cell phone services are accelerating more rapidly among lower or niche socio-economic groups.
In a comparison of white, African American and Hispanic households, for example, KN/SRI found that ownership of DVD players was essentially even, while African Americans were almost twice as likely to subscribe to digital cable TV services.
"It's not like there's one digital marketplace. We all tend to look at it that way, but it's really a diverse array of digital products and penetration levels," says David Tice, vice president of client service at KN/SRI and chief analyst on the Home Technology Monitor.
Percent Ownership Among Various Demographic Groups
Digital Cable DVD Player Home PC Broadband
Earning <$30,000 11% 38% 41% 8%
Earning $30-49,000 16% 63% 71% 16%
Earning $50,000+ 27% 74% 89% 39%
No Children 16% 47%
59% 16%
Any Children 22% 71% 76% 29%
White 17% 57% 69% 23%
26% 55% 56% 15%
Hispanic 17% 55% 48% 14%
Source: Knowledge Networks/SRI's Home Technology Monitor 2004.
While politicians, economists and industry analysts had projected that economics would lead to a digital divide that split America into classes of digital "haves" and "have notes," Tice says different patterns are emerging and that lower income households are even showing signs of adopting some new digital media at faster rates. He said cell phones with Internet access, for example, are gaining more rapidly among lower income households and theorized that might be because they cannot afford the expense of a personal computer and conventional Internet access.
In fact, economics are a significant factor in the adoption of those latter technologies. Households earning $50,000 or more a year are more than twice as likely to have a personal computer than those earning less than $30,000 a year. They are also more than four times more likely to have broadband Internet access.
Tice says these patterns are evolving rapidly and likely will continue to shift over the next several years as the cost of digital technologies and services come down, but they still are crucial for marketers and ad agencies planning to use digital media to understand.
Indeed, despite their relatively low penetration, digital video recorders have been giving Madison Avenue fits due to their ability to easily bypass TV commercials, but the problem may be most severe among the highest end consumer targets. While the Home Technology Monitor estimates that only 4 percent of U.S. households have DVRs nationwide, the penetration is twice that for households earning $50,000-plus, and 2.5 times that for households earning $75,000-plus. Conversely, DVRs are present in only 1.5 percent of households earning less than $50,000.
Digital Video Recorder Penetration
DVR Penetration
Households Earning <$50,000 1.5%
Households Earning $50,000+
Households Earning $75,000+ 10.0%
All U.S. Households 4.0%
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{"url":"https:\/\/benediktehinger.de\/blog\/science\/page\/3\/","text":"## Scatterplots, regression lines and the first principal component\n\nI made some graphs that show the relation between X1~X2 (X2 predicts X1), X2~X1 (X1 predicts X2) and the first principal component (direction with highest variance, also called total least squares).\n\nThe line you fit with a principal component is not the same line as in a regression (either predicting X2 by X1 [X2~X1] or X1 by X2 [X1~X2]. This is quite well known (see references below).\n\nWith regression one predicts X2 based on X1 (X2~X1 in R-Formula writing) or vice versa. With principal component (or total least squares) one tries to quantify the relation between the two. To completely understand the difference, image what quantity is reduced in the three cases.\n\nIn regression, we reduce the residuals in direction of the dependent variable. With principal components, we find the line, that has the smallest error orthogonal to the regression line. See the following image for a visual illustration.\n\nFor me it becomes interesting if you plot a scatter plot of two independent variables, i.e. you would usually report the correlation coefficient. The \u2018correct\u2019 line accompaniyng the correlation coefficient would be the principal component (\u2018correct\u2019 as it is also agnostic to the order of the signals).\n\n#### Further information:\n\nHow to draw the line, eLife 2013\nGelman, Hill \u2013 Data Analysis using Regression, p.58\nAlso check out the nice blogpost from Martin Johnsson doing practically the same thing but three years earlier \ud83d\ude09\n\n#### Source\n\nlibrary(ggplot2)\nlibrary(magrittr)\nlibrary(plyr)\n\nset.seed(2)\ncorrCoef = 0.5 # sample from a multivariate normal, 10 datapoints\ndat = MASS::mvrnorm(10,c(0,0),Sigma = matrix(c(1,corrCoef,2,corrCoef),2,2))\ndat[,1] = dat[,1] - mean(dat[,1]) # it makes life easier for the princomp\ndat[,2] = dat[,2] - mean(dat[,2])\n\ndat = data.frame(x1 = dat[,1],x2 = dat[,2])\n\n# Calculate the first principle component\n# see http:\/\/stats.stackexchange.com\/questions\/13152\/how-to-perform-orthogonal-regression-total-least-squares-via-pca\n\n### And that\u2019s how you do cluster permutation statistics.\n\nThanks to Tim C Kietzmann & Jos\u00e9 Ossandon for teaching me this some (long ;)) time ago. Thanks to Silja Timm for preparing the graphics and Anna Lisa Gert for comments on the text!\n\n3 of 4\n1234","date":"2019-02-21 08:52:15","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\": 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\": 0.73728346824646, \"perplexity\": 2652.101174651503}, \"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-2019-09\/segments\/1550247503249.58\/warc\/CC-MAIN-20190221071502-20190221093502-00312.warc.gz\"}"}
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\section{Introduction}\label{section_introduction}
An astronomical scaling relation of great historical merit is the Tully-Fisher relation \citep[TFR,][]{Tully1977}, an empirical scaling law between the absolute magnitude of disk galaxies and their edge-on spectral linewidths. This relation originates from a fundamental connection between baryon mass, roughly traced by luminosity, and circular velocity, imprinted in the Doppler-broadening of emission lines. Today, any power-law relation between a mass tracer and a proxy for the circular velocity of disk galaxies is called a TFR, such as the optical/infrared TFR \citep{Pizagno2007,Meyer2008}, the stellar mass TFR \citep{Kassin2007}, the neutral atomic hydrogen (H{\sc\,i}) TFR \citep{Chakraborti2011}, and the baryonic TFR \citep{McGaugh2000}. Such TFRs can be used to constrain the dark matter potential of galaxies \citep{Trachternach2009}, their gas-content \citep{McGaugh1997,Pfenniger2005}, their luminosity evolution \citep{Ziegler2002,Boehm2004}, the stellar initial mass function \citep{Bell2001}, the formation of central bars \citep{Courteau2003}, galactic outflows \citep{Dutton2012}, the transport of angular momentum between halos and disks \citep{Sommer-Larsen2001}, standard dark matter models \citep{Obreschkow2013b}, and alternative theories of gravity \citep{Chakraborti2011,McGaugh2012}. Adopting the TFR as a prior, observed H{\sc\,i}\ linewidths can be converted into absolute magnitudes, which, given apparent magnitudes, provide redshift-independent distances. Combined with spectroscopic redshifts ($z$), these can constrain the cosmic expansion rate \citep{Tully1977} and superposed peculiar motions \citep{Masters2006}, including the local bulk flow \citep{Nusser2011}.
Circular velocities of galaxies are normally derived from the width of the total 21~cm emission line of H{\sc\,i}, because most H{\sc\,i}\ lies beyond the stellar scale radius, hence better tracing the asymptotic circular velocity than optical emission lines. The apparent H{\sc\,i}\ linewidth must then be corrected for the orbital inclination $i\in[0,\pi/2]$, defined as the angle between the line-of-sight and the rotational axis of the disk. This inclination is inferred from optical images, because the H{\sc\,i}\ detections tend to be spatially unresolved. Even future interferometric H{\sc\,i}\ surveys with the Square Kilometer Array (SKA) and its pathfinders will only marginally resolve most H{\sc\,i}\ detections \citep[e.g.,][for ASKAP]{Duffy2012}. However, the need for optical inclinations leads to serious issues.
First, optical inclinations may be simply unavailable. For example, the optical counterparts of the H{\sc\,i}\ Parkes All Sky Survey \citep[HIPASS,][]{Barnes2001} have insufficient sensitivity to recover the inclinations of most disks with circular velocities below $50~{\rm km~s^{-1}}$ \citep{Obreschkow2013b} and dust extinction and stellar confusion in the galactic plane cause poor completeness \citep{Doyle2005}. Future high-$z$ H{\sc\,i}\ surveys with the SKA will need parallel sub-arcsecond imaging to even barely resolve typical $L^*$-galaxies (at $z\approx1$). The hence required active optics-assisted or space-based observations might yield insufficient sky coverage.
Second, optical inclinations are uncertain; they often represent the largest uncertainty in recovering circular velocities. Unless kinematic maps are available, inclinations are mostly obtained by fitting ellipses to the isophotes, giving an apparent axis ratio $q$. Upon assuming a spheroidal disk with an intrinsic axis ratio $q_0$, $i$ is then given by
\begin{equation}\label{eq_cosi}
\cos^2i=\frac{q^2-q_0^2}{1-q_0^2}\,.
\end{equation}
The measurement of $q$ is deteriorated by smearing effects \citep{Giovanelli1997b,Masters2003} and physical features such as disk asymmetries and differential dust absorption. The value of $q_0$ can only be approximated from local edge-on galaxies and its dependence on the Hubble type remains debated \citep{Tully2009}. At high $z$ the determination of $q_0$ is further complicated by the cosmic evolution of the disk thickness \citep{Bournaud2007} and the bulge-to-disk ratio \citep{Driver2013}.
Third, optical inclinations can differ significantly from H{\sc\,i}\ inclinations. Examples are galaxies with warped H{\sc\,i}\ disks \citep[e.g., NGC 4013,][]{Bottema1987}, which are more the rule than the exception \citep{Sancisi1976}.
The objective of this paper is to establish a new method to derive TFRs using limited or no information on inclinations. This method can recover the most likely TFR, assumed to be a power-law with Gaussian scatter, given a galaxy sample where the inclination of each object $k$ is described by a probability density function (PDF) $\rho_k(i)$. This method can then be applied to several cases: (1) a galaxy sample with approximate inclination measurements for every galaxy; (2) a galaxy sample without individual inclination measurements, but with a known prior PDF $\rho(i)$, obtained, for example, from deep imaging of a small subsample; and (3) a galaxy sample without any prior information on inclinations. In the latter case, the galaxy axes can be assumed isotropically oriented, that is $\rho(i)=\sin(i)$.
This paper first derives a generic maximum likelihood estimation (MLE) to obtain a TFR using limited or no knowledge on galaxy inclinations (Section \ref{section_method}). This new method is then tested and characterized in detail (Section \ref{section_simulation}) using idealized control samples of virtual galaxies. In particular, these tests allow a comparison of the classical approach, where slightly erroneous inclinations are treated as exact, against a version of the MLE that does not use any inclinations at all. In Section \ref{section_observation}, we then apply this new MLE approach to the local $K$-band and $B$-band TFRs studied in detail by \cite{Meyer2008}. The key findings are summarized in Section \ref{section_conclusion}.
\section{Analytical method}\label{section_method}
\subsection{Derivation of the maximum likelihood approach}\label{subsection_generic_mle}
This section introduces our generic method to estimate TFRs using uncertain/unknown inclinations. Here, we focus on the basic model of a linear TFR with Gaussian scatter (in log-space). Extensions to this model, e.g., to deal with non-linear TFRs and non-Gaussian scatter, are described in Appendix \ref{appendix_extensions}.
The basic TFR can be written as
\begin{equation}\label{eq_observable_tfr}
\tilde{m} = \alpha+\beta\tilde{v},
\end{equation}
where tildes denote logarithms, i.e.,~$\tilde{x}\equiv\log_{10}x$, $m$ is a tracer of the galaxy mass, e.g., an optical luminosity, a gas mass, the stellar mass, or the baryon mass, and $v$ is a tracer of the circular velocity. The parameters $\alpha$ (zero-point) and $\beta$ (slope) are scalar constants. The `true' values $\tilde{m}_k$ and $\tilde{v}_k$ of real galaxies scatter around \eq{observable_tfr} with an ensemble variance $\sigma^2\equiv\ensavg{(\alpha+\beta\tilde{v}_k-\tilde{m}_k)^2}$. Its square-root $\sigma$ is called the intrinsic scatter. Upon assuming that this scatter derives from a log-normal distribution, the conditional PDF per unit $\tilde{v}$ for a galaxy with mass $m$ to exhibit a circular velocity $v$ is given by
\begin{equation}\label{eq_rho_tfr}
\rho(\tilde{v}|\tilde{m},\mathbf{T}) = \frac{\beta}{\sqrt{2\pi}\sigma}\exp\Big[-\tfrac{(\tilde{m}-\alpha-\beta\tilde{v})^2}{2\sigma^2}\Big].
\end{equation}
In this expression and for the rest of this work
\begin{equation}\label{eq_def_tfr}
\mathbf{T}\equiv(\alpha,\beta,\sigma)
\end{equation}
is the vector grouping the three TFR parameters.
The circular velocity $v$ of a rotating disk cannot be measured directly, since the Doppler-effect only traces the line-of-sight projection
\begin{equation}\label{eq_wv}
w=v\sin i,
\end{equation}
where $i$ is the orbital inclination of the material that $v$ refers to. In a simplistic galaxy model with a H{\sc\,i}\ disk of constant circular velocity $v$ and zero dispersion, $w$ is given by the width $W$ of the H{\sc\,i}\ emission line via $w=W/2$. When dealing with real H{\sc\,i}\ emission lines with a dispersive component, $W$ is often substituted for $W_{20}$ \citep{McGaugh2000} or $W_{50}$ \citep{Kannappan2002}, where $W_p$ is the H{\sc\,i}\ linewidth measured at $p$ percent of the peak flux density. More elaborate equations can also account for cosmological and instrumental line broadening, as well as remove turbulent motion from the linewidth \citep[e.g.][]{Tully1985,Verheijen2001b,Springob2007,Meyer2008}. In this section, there is no need to specify the definition of $w$. We only assume that $w$ is defined and measured in some way and used with \eq{wv} to compute $v$.
Given a sample of $N$ galaxies $k=1,...,N$ and upon assuming no prior information on $\mathbf{T}$, the most likely $\mathbf{T}$ maximizes the global likelihood function $\mathcal{L}(\mathbf{T})$, conveniently written in log-form,
\begin{equation}\label{eq_likelihood}
\ln\mathcal{L}(\mathbf{T})=\sum_{k=1}^N g_k \ln\rho_k(\mathbf{T}),
\end{equation}
where $\rho_k(\mathbf{T})$ is the conditional PDF of observing a galaxy with the properties of galaxy $k$ given the Tully-Fisher parameters $\mathbf{T}$. The scalars $g_k$ are normalized weights, such that $\sum g_k=1$. If all galaxies are considered equally important, then $g_k=N^{-1}~\forall k$. To estimate the TFR of a sample that is complete inside a galaxy-dependent volume $V_k$, the weights could be chosen as $g_k\propto V_k^{-1}$.
If each galaxy had perfectly measured values $\{\tilde{m}_k,\tilde{w}_k\}$, $\rho_k(\mathbf{T})$ would be equal to the conditional PDF $\rho_k(\tilde{w}_k,\tilde{m}_k|\mathbf{T})$ of finding $\{\tilde{m}_k,\tilde{w}_k\}$ given the TFR with parameters $\mathbf{T}$. However, in practice, the true masses and projected velocities differ from the measured values $\tilde{m}_k$ and $\tilde{w}_k$ by measurement errors. The true values can only be described in terms of finite PDFs $\rho_k(\tilde{m})$ and $\rho_k(\tilde{w})$, respectively. In that case, $\rho_k(\mathbf{T})$ can generally be written as
\begin{equation}\label{eq_rhok_general}
\rho_k(\mathbf{T}) = \iint d\tilde{m}\,d\tilde{w}\,\rho_k(\tilde{m})\,\rho_k(\tilde{w})\,\rho_k(\tilde{w},\tilde{m}|\mathbf{T}).
\end{equation}
The remaining task is to express \eq{rhok_general} in a usable analytical form. The rule of joint probabilities implies $\rho_k(\tilde{m},\tilde{w}|\mathbf{T})=\rho_k(\tilde{w}|\tilde{m},\mathbf{T})\rho_k(\tilde{m}|\mathbf{T})$. Here, we assume no prior constraints on $\tilde{m}$, thus $\rho_k(\tilde{m}|\mathbf{T})$ can be taken as unity; hence, $\rho_k(\tilde{m},\tilde{w}|\mathbf{T})=\rho_k(\tilde{w}|\tilde{m},\mathbf{T})$. Summing over all possible inclinations $i$ then implies $\rho_k(\tilde{w}|\tilde{m},\mathbf{T})=\int_0^{\pi/2}\d i\,\rho_k(i)\,\rho_k(\tilde{v}|\tilde{m},\mathbf{T})$, where $v=w\,(\sin i)^{-1}$ and $\rho_k(i)$ is the normalized prior PDF of the inclination $i_k\in[0,\pi/2]$.
Using Equations~(\ref{eq_rho_tfr}) and (\ref{eq_wv}),
\begin{equation}\label{eq_main_rho1}
\begin{split}
\rho_k(\tilde{w},\tilde{m}|\mathbf{T})=\,&\frac{\beta}{\sqrt{2\pi}\sigma}\int_0^{\pi/2}\!\!\!\!\d i\,\rho_k(i) \times \\
&\exp\Big[-\tfrac{(\tilde{m}-\alpha-\beta\tilde{w}+\beta\log_{10}\sin i)^2}{2\sigma^2}\Big].
\end{split}
\end{equation}
This PDF is indexed with $k$, since in general different galaxies may have different inclination priors $\rho_k(i)$.
Upon assuming that the measurement uncertainties of $\tilde{m}_k$ and $\tilde{w}_k$ are Gaussian with galaxy-dependent standard deviations $\sigma_{\tilde{m}_k}$ and $\sigma_{\tilde{w}_k}$, \eq{rhok_general} can be solved\footnote{Using $\rho_k(\tilde{m})=\exp[-(\tilde{m}-\tilde{m}_k)^2/(2\sigma_{\tilde{m}_k}^2)]/(\sqrt{2\pi}\sigma_{\tilde{m}_k})$ and $\rho_k(\tilde{w})=\exp[-(\tilde{w}-\tilde{w}_k)^2/(2\sigma_{\tilde{w}_k}^2)]/(\sqrt{2\pi}\sigma_{\tilde{w}_k})$ in \eq{rhok_general} and applying the convolution theorem results in \eq{main_rho2}.},
\begin{equation}\label{eq_main_rho2}
\begin{split}
\rho_k(\mathbf{T}) =\,&\frac{\beta}{\sqrt{2\pi}\sigma_{tot,k}}\int_0^{\pi/2}\!\!\!\!\d i\,\rho_k(i) \times \\
&\exp\Big[-\tfrac{(\tilde{m}_k-\alpha-\beta\tilde{w}_k+\beta\log_{10}\sin i)^2}{2\sigma_{tot,k}^2}\Big],
\end{split}
\end{equation}
where the total scatter $\sigma_{tot,k}\equiv(\sigma^2+\sigma_{\tilde{m}_k}^2+\beta^2\sigma_{\tilde{w}_k}^2)^{1/2}$ is the combination of the intrinsic scatter $\sigma$ of the TFR and the observational scatter $(\sigma_{\tilde{m}_k}^2+\beta^2\sigma_{\tilde{w}_k}^2)^{1/2}$.
In summary, if every galaxy $k$ has measured values $\tilde{m}_k$ and $\tilde{w}_k$ with Gaussian uncertainties $\sigma_{\tilde{m}_k}$ and $\sigma_{\tilde{w}_k}$, as well as an inclination $i_k$ specified by the PDF $\rho_k(i)$, then the most likely TFR is found by maximizing \eq{likelihood} with $\rho_k(\mathbf{T})$ given in \eq{main_rho2}. If the measurement uncertainties are highly non-Gaussian, then the full integral of \eq{rhok_general} should be used instead of \eq{main_rho2}. If all $\tilde{m}_k$ and $\tilde{w}_k$ are measured perfectly ($\sigma_{\tilde{m}_k}=\sigma_{\tilde{w}_k}=0$), \eq{main_rho2} remains valid and $\sigma_{tot,k}$ reduces to the intrinsic scatter $\sigma$.
\subsection{Practical considerations}\label{subsection_real_data}
The formalism of \S\ref{subsection_generic_mle} assumes an idealized scenario with perfect data bar Gaussian measurement uncertainties. This section explains how to deal with two typical deviations from this scenario, namely (i) inclination-dependent extinction and (ii) non-Gaussian outliers in the data. More advanced extensions, dealing with complex detection limits, early-type galaxies, and non-linear TFRs, are sketched out in Appendix \ref{appendix_extensions}.
\textit{Inclination-dependent extinction}: The internal extinction of galaxies depends on their inclination $i$ \citep[e.g., ][]{Giovanelli1994}, meaning that the derived mass $\tilde{m}$ depends on the inclination $i$. Sometimes this extinction can be neglected -- typically for the H{\sc\,i}\ line and rest-frame near-infrared continuum \citep{Gordon2003}. In most cases, however, extinction is significant. If extinction is important and the inclination $i_k$ is not precisely known, then $\tilde{m}_k$ is not a well-defined value, but of a function $\tilde{m}_k(i)$,~$i\in[0,\pi/2]$, for every galaxy. In that case, it suffices to substitute $\tilde{m}_k$ in \eq{main_rho2} for $\tilde{m}_k(i)$. For example, let's consider $\tilde{m}_k$ to be a mass computed from the absolute magnitude $M_{k,\lambda}(i)$ at wavelength $\lambda$. This magnitude is estimated from the observed absolute magnitude $M_{k,\lambda}$ via $M_{k,\lambda}(i)=M_{k,\lambda}-\Delta M_{\lambda}(i)$, where $\Delta M_{\lambda}(i)$ is the extinction factor. Simple empirical models for $\Delta M_{\lambda}(i)$ in the $B$, $R$, $J$, $H$, and $K$ bands are summarized in \S3.1.4 of \cite{Meyer2008} \citep[based on][]{Tully1998,Masters2003}, and extended models were made available by \cite{Driver2008}, \cite{Maller2009}, and \cite{Masters2010}. Using those models, observed absolute magnitudes can be translated into $\tilde{m}_k(i)\propto M_{k,\lambda}-\Delta M_{\lambda}(i)$. This approach will be used in Section \ref{section_observation} when recovering the $K$-band and $B$-band TFR without using inclination measurements.
\textit{Non-Gaussian outliers}: Data with erroneous outliers, such as misidentifications between H{\sc\,i}\ sources (used to measure $w$) and optical counterparts (used to measure $m$), can adversely affect the estimation of a TFR. Of course, this applies to any method used to estimate a TFR, not just to the MLE introduced in this work. However, when dealing with galaxies of uncertain/unknown inclinations $i_k$ -- a particular strength of the MLE -- outliers are hard to spot. This is because without inclinations measurements the data can only be shown in the $(\tilde{w},\tilde{m})$-plane but not in the $(\tilde{v},\tilde{m})$-plane. If the dataset is likely to include some outliers, the likelihood function (\eq{likelihood}) should be maximized using a robust technique, which ignores the $fN$ galaxies with the lowest likelihood terms, where $f<1$ is the estimated upper bound for the fraction of outliers.
\subsection{Special case: completely unknown inclinations}
In the case where galaxy inclinations are completely unknown, the MLE of \S\ref{subsection_generic_mle} can be used to recover the most likely TFR by choosing $\rho_k(i)=\sin i$. The $\sin i$-distribution is that expected from an isotropic random orientation of galaxies, hence encoding our complete lack of knowledge on individual galaxy orientations. Hereafter, this method will be called the `inclination-free' maximum-likelihood estimation (MLE). If a better prior for $\rho_k(i)$ is available, for example from inclination measurements in a subsample of galaxies or from modelling the survey properties, then this prior can be used instead (see quantitative analysis in \S\ref{subsection_inclination_distribution_errors}).
\subsection{Special case: perfectly known inclinations}\label{subsection_MLE_with_inclinations}
Let us now investigate the opposite extreme, where all galaxy inclinations $i_k$ are precisely known -- the default assumption in most current literature on TFRs. In this case, the extinction-corrected masses $\tilde{m}_k$ and deprojected velocities $\tilde{v}_k=\tilde{w}_k-\log_{10}\sin i_k$ are known up to the observational uncertainties $\sigma_{\tilde{m}_k}$ and $\sigma_{\tilde{v}_k}=\sigma_{\tilde{w}_k}$.
It is an occasional misconception that the TFR can be accurately recovered by fitting a linear regression to the data in the $(\tilde{v},\tilde{m})$-plane. To show that regressions are not generally appropriate one can depart from \eq{main_rho2} assuming that the inclinations $i_k$ are precisely known; thus $\rho_k(i)=\delta(i_k-i)$, where $\delta$ is the Dirac-delta function. \eq{main_rho2} then reduces to
\begin{equation}\label{eq_rho_tfr2}
\rho_k(\mathbf{T}) = \frac{\beta}{\sqrt{2\pi}\sigma_{tot,k}}\exp\Big[-\tfrac{(\tilde{m}_k-\alpha-\beta\tilde{v}_k)^2}{2\sigma_{tot,k}^2}\Big].
\end{equation}
Hence the likelihood function $\mathcal{L}(\mathbf{T})$ of \eq{likelihood} is maximized by minimizing
\begin{equation}\label{eq_likelihood_fixedi}
\sum_{k=1}^N g_k\left[\ln\frac{\sigma_{tot,k}}{\beta}+\frac{(\tilde{m}_k-\alpha-\beta\tilde{v}_k)^2}{2\sigma_{tot,k}^2}\right].
\end{equation}
Only upon neglecting the contribution of $\ln(\sigma_{tot,k}/\beta)$ and assuming vanishing intrinsic scatter, $\sigma\rightarrow0$, \eq{likelihood_fixedi} is minimized by minimizing
\begin{equation}\label{eq_likelihood_fixedi_simple}
\chi^2\equiv\sum_{k=1}^N g_k \frac{(\tilde{m}_k-\alpha-\beta\tilde{v}_k)^2}{\sigma_{\tilde{m}_k}^2+\beta^2\sigma_{\tilde{v}_k}^2}\,.
\end{equation}
This minimization corresponds to a bivariate linear regression. It reduces to a standard linear regression, if all $\sigma_{\tilde{m}_k}$ are identical and all $\sigma_{\tilde{v}_k}=0$.
The difference between Equations (\ref{eq_likelihood_fixedi}) and (\ref{eq_likelihood_fixedi_simple}) shows that linear regressions, even bivariate ones, do not generally recover the most likely TFR. Instead, \eq{likelihood_fixedi} must be minimized by simultaneously varying all three parameters $\alpha$, $\beta$, and $\sigma$. In practice, the errors made by using regression techniques are often small -- perhaps smaller than the statistical uncertainties of the Tully-Fisher parameters $\mathbf{T}$. However, as TFR measurements become increasingly accurate, it seems worth noting that linear regressions techniques are not optimal (illustrations in \S\ref{subsection_convergence}).
\begin{figure}[t]
\includegraphics[trim=31 10 47 30,clip,width=\columnwidth]{fig_basic_example}
\caption{Illustration of a galaxy control sample (points) drawn from an input TFR shown as black dotted lines. The TFR has an intrinsic vertical scatter of 0.3 dex. In addition, each data point has been randomly perturbed by a random measurement error in both coordinates, represented by the 68\%-error bars shown for one point. The colored lines show the TFRs recovered using the different methods detailed in \S\ref{subsection_basic_example}. For each line type there are three lines representing the mean TFR ($\tilde{m}=\alpha+\beta\tilde{v}$) and the $1\sigma$-scatter intervals ($\tilde{m}=\alpha+\beta\tilde{v}\pm\sigma$). This figure only provides a qualitative illustration of the different methods, since the recovered TFRs depend on the random realization of the sample. A quantitative assessment of the different methods is given in \fig{errors}.\\[1ex](A color version of this figure is available in the online journal.)}
\label{fig_basic_example}
\end{figure}
\section{Verification using mock data}\label{section_simulation}
This section tests the accuracy and liability of the MLE to recover the TFR using idealized control samples of model-galaxies.
\subsection{Control sample}\label{subsection_control_sample}
The control samples consist of $N$ model-galaxies $k=1,...,N$ with identical weights $g_k=N^{-1}$ and random `observed' log-masses $\tilde{m}_k$ drawn from a normal distribution with mean $9$ and unit variance. Log-normal mass distributions roughly describe the empirical mass distributions in sensitivity limited surveys (e.g., absolute magnitude distributions in SDSS, see Figure 5 in \citealp{Montero-Dorta2009}; H{\sc\,i}\ mass distribution in HIPASS, see Figure 1 bottom in \citealp{Zwaan2003}). The low-mass drop-off is generally due to the sensitivity limit, thus small effective volume, while the high-mass drop-off is caused by the steepness of the galaxy mass function.
The `true' log-masses $\tilde{m}_k^t=\tilde{m}_k-\Delta \tilde{m}_k$ differ from the observed log-masses $\tilde{m}_k$ by a measurement error $\Delta \tilde{m}_k$, drawn from a normal distribution with $\sigma_{\tilde{m}}=0.2$. Given the true masses $m_k^t$, the corresponding circular velocities $v_k^t$ are drawn randomly from a TFR, i.e.~from the PDF of \eq{rho_tfr}, with input values $\mathbf{T}_0=(\alpha_0,\beta_0,\sigma_0)=(3,3,0.3)$. These values roughly coincide with those of the baryonic TFR, if $m_k$ and $v_k$ are given units of ${\rm M}_{\odot}$ and ${\rm km~s^{-1}}$ \citep{Gurovich2010}, but the precise choice of $\mathbf{T}_0$ does not matter here. Every galaxy is also assigned a random inclination $i_k$, drawn from a $\sin i$-distribution, unless otherwise specified. The $\sin i$-distribution is expected from galaxies having random orientations in three dimensions. The `true' line-of-sight projected velocities $w_k^t$ are then computed as $w_k^t=v_k^t \sin i_k$. Finally, measurement errors $\Delta \tilde{w}_k$ are drawn from a normal distribution with $\sigma_{\tilde{w}}=0.1$ to compute the observables $\tilde{w}_k=\tilde{w}_k^t+\Delta\tilde{w}_k$ and $\tilde{v}_k=\tilde{v}_k^t+\Delta\tilde{w}_k$. Inclination-dependent extinction is neglected here but will be considered in Section \ref{section_observation}.
\subsection{Basic example}\label{subsection_basic_example}
To \textit{qualitatively} illustrate the different methods to estimate a TFR, we use a single realization of a control sample (\S\ref{subsection_control_sample}) with $N=10^3$ galaxies, displayed in \fig{basic_example}. The four line-types represent four different TFRs and their 1$\sigma$-scatter: ($\cdots$) the input TFR with parameters $\mathbf{T}_0$ used to construct the data points; ($\cdot-$) the TFR recovered by fitting a bivariate linear regression to the $(\tilde{v}_k,\tilde{m}_k)$-pairs, i.e., $\alpha$ and $\beta$ minimize \eq{likelihood_fixedi_simple} and $\sigma^2=\avg{(\alpha+\beta\tilde{v}_k-\tilde{m}_k)^2}-\sigma_{\tilde{m}}^2-\beta^2\sigma_{\tilde{v}}^2$, where the over-line represents the average over all galaxies; ($--$) the TFR recovered by applying the MLE to all $(\tilde{v}_k,\tilde{m}_k)$-pairs, i.e., this TFR minimizes \eq{likelihood_fixedi}; and (---) the TFR recovered by applying the inclination-free MLE to \textit{all} $(\tilde{w}_k,\tilde{m}_k)$-pairs, i.e., this TFR maximizes \eq{likelihood}, where $\rho_k(\mathbf{T})$ is given in \eq{main_rho2} and $\rho_k(i)=\sin i$ expresses our lack of information on the inclinations $i_k$. We emphasize that the methods ($\cdot-$) and ($--$) implicitly use the inclinations $i_k$, since in practice the $v_k$-values are recovered from $w_k$ and $i_k$ via \eq{wv}. In turn, the inclination-free MLE (---) truly disregards the galaxy inclinations by exclusively relying on the $(\tilde{w}_k,\tilde{m}_k)$-pairs.
Qualitatively, all three methods recover the input TFR fairly well. Surprisingly, even the MLE (---) that only uses the projected velocities $\tilde{w}_k$ while ignoring the inclinations $i_k$ closely recovers the input TFR. This method even outperforms the bivariate linear regression. As expected, the MLE ($--$), which uses the deprojected velocities $\tilde{v}_k$, performs better than the inclination-free MLE (---). In the following, the performance of these methods will be properly quantified.
\begin{figure*}[t]
\centering
\begin{tabular}{cc}
\begin{overpic}[width=\columnwidth]{fig_convergence}\put(13.5,71.0){\normalsize\textbf{(a)}}\end{overpic} &
\begin{overpic}[width=\columnwidth]{fig_inclination_errors1}\put(14.5,70.0){\normalsize\textbf{(c)}}\end{overpic} \\
\begin{overpic}[width=\columnwidth]{fig_inclination_distribution_errors}\put(13.5,71.0){\normalsize\textbf{(b)}}\end{overpic} &
\begin{overpic}[width=\columnwidth]{fig_inclination_errors2}\put(14.5,70.0){\normalsize\textbf{(d)}}\end{overpic} \\
\end{tabular}
\caption{Mean relative errors $\ensavg{\Delta\mathbf{T}}\equiv(\ensavg{\Delta\alpha},\ensavg{\Delta\beta},\ensavg{\Delta\sigma})$ made in recovering the parameters $\alpha$, $\beta$, and $\sigma$ in the TFR $\tilde{m} = \alpha+\beta\tilde{v}~(\pm\sigma)$ of different control samples with $N$ galaxies. The four panels are described in detail in different sections: (a) in \ref{subsection_convergence}, (b) in \ref{subsection_inclination_distribution_errors}, (c) and (d) in \ref{subsection_inclination_errors}. Panel (b) uses the same line styles as \fig{inclination_distributions} and refers to the inclination distributions listed in \tab{inclination_distributions}.\\[1ex](A color version of this figure is available in the online journal.)}
\label{fig_errors}
\end{figure*}
\subsection{Ensemble of control samples}\label{subsection_ensemble}
We can quantify the goodness of a method to estimate the TFR in terms of the error vector $\abs{\mathbf{T}-\mathbf{T}_0}$, where $\mathbf{T}$ denotes the TFR parameters fitted to a control sample with input parameters $\mathbf{T}_0$. However, this error depends on the particular random realization of the control sample. Rather than considering a single control sample, we construct an ensemble of $10^3$ random control samples and compute the ensemble mean of the relative error,
\begin{equation}\label{eq_DT}
\ensavg{\Delta\mathbf{T}}=\left\langle\left|\frac{\mathbf{T}-\mathbf{T}_0}{\mathbf{T}_0}\right|\right\rangle.
\end{equation}
This mean error will be used in \S\ref{subsection_convergence} to \S\ref{subsection_inclination_errors}.
\subsection{Numerical convergence}\label{subsection_convergence}
Let us now \textit{quantify} the performance of the different TFR estimators illustrated in \S\ref{subsection_basic_example} and investigate their asymptotic convergence with the number of galaxies $N$. To this end, we consider 12 \textit{ensembles} of control samples (see \S\ref{subsection_ensemble}) with different $N$, spaced logarithmically between $N=10$ and $N=2\cdot10^4$. The latter corresponds to the largest TFR samples used today \citep[e.g.,][]{Mocz2012}. The mean relative error $\ensavg{\Delta\mathbf{T}}$ is plotted against $N$ in \fig{errors}(a) for the different methods.
As expected, $\ensavg{\Delta\mathbf{T}}$ decreases monotonically with $N$. However, the error of the bivariate linear regression levels out at a value larger than zero: no matter how many galaxies we use, the linear regression never recovers the input TFR. Formally, this is a direct consequence of the difference between the full likelihood function of \eq{likelihood_fixedi} and the reduced \eq{likelihood_fixedi_simple} used for the bivariate linear regression. Figuratively speaking, the linear regression tends to follow the average slope of the shape spanned in the $(\tilde{w},\tilde{m})$-plane. Depending on the selection function of the data -- here a normal distribution in $\tilde{m}$ -- this slope can differ from the actual slope of the input TFR. For mass-limited samples, the slope of the linear regression is likely to be somewhat shallower than the true TFR, but depending on the precise selection criteria, the slope error may also be inverted, or the linear regression may provide perfectly acceptable results. The important message is that the linear regression does not guarantee an optimal recovery of the TFR, although the results might suffice in many cases.
The mean relative errors of the MLEs scale numerically as $\ensavg{\Delta\mathbf{T}}\propto N^{-1/2}$ and tend to $0$ as $N\rightarrow\infty$, as expected for Poisson noise. In line with the qualitative conclusion from \fig{basic_example}, the MLE that uses inclinations performs best. However, the relative errors of the inclination-free MLE are only a factor $\sim\!1.5$ larger for any given $N$. Considering that the requirement for optical inclinations tends to reduce number of galaxies $N$ in the parent sample, the inclination-free MLE can typically make use of a larger number $N$ of galaxies than inclination-dependent methods. Hence the relative errors of the inclination-free method become even more competitive.
\subsection{Robustness against inclination distribution errors}\label{subsection_inclination_distribution_errors}
Sections \ref{subsection_basic_example} and \ref{subsection_convergence} demonstrated that the inclination-free MLE can reliably recover the TFR of a galaxy sample that, by construction, has an isotropic distribution of galaxy axes. Expressed alternatively, if the galaxy inclinations satisfy a $\sin i$-distribution, then the inclination-free MLE assuming a $\sin i$-distribution works well. But what if the galaxies in the sample do not obey a $\sin i$-distribution? To investigate this situation, we use control samples with random galaxy inclinations drawn from the modified PDFs shown in \fig{inclination_distributions} and listed in \tab{inclination_distributions}. The `HIPASS-fit' approximates the inclination distribution of the 3,618 tabulated optical counterparts \citep{Doyle2005} of HIPASS detections. These optical inclinations were calculated via \eq{cosi} with $q_0=0.2$. Edge-on galaxies are clearly underrepresented in this distribution, due to the difficulty of detecting the broadest H{\sc\,i}\ lines and due to optical extinction. The `square root' PDF well approximates the predicted inclination distribution of the WALLABY and DINGO HI blind surveys on the Australian SKA Pathfinder (ASKAP) telescope \citep[derived from][]{Duffy2012}.
\begin{table}[b]
\centering
\normalsize
\begin{tabular}{lllc}
\hline\hline \\ [-1.5ex]
Name & PDF $\rho(i)$~~~ & Quantile $Q(r)$ & $\ensavg{i}$ \\ [1.0ex]
\hline \\
HIPASS fit & $\frac{1-\cos 3i}{\pi/2+1/3}$ & $\approx\frac{\pi}{6}(2r^{0.3}+r^2)$~ & 0.98 \\ [2ex]
Sine & $\sin i$ & $\arccos r$ & 1.00 \\ [2ex]
Square root~~ & $\left(\frac{18i}{\pi^3}\right)^{1/2}$ & $\frac{\pi}{2}r^{2/3}$ & 0.94 \\ [2ex]
\hline
\hline
\end{tabular}
\caption{\upshape\raggedright Three examples of PDFs $\rho(i)$ for the inclinations $i$. In order to draw a random value $i$ from $\rho(i)$, one can draw a uniform random number $r\in[0,1]$ and set $i=Q(r)$, where the quantile function $Q(r)$ is the inverse function of the cumulative distribution function (CDF) $F(i)\equiv\int_0^i d i'\rho(i')$.}
\label{tab_inclination_distributions}
\end{table}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{fig_inclination_distributions}
\caption{PDFs used for choosing the random inclinations of the galaxy samples considered in \S\ref{subsection_inclination_distribution_errors}. The analytical expressions of these three functions are listed in \tab{inclination_distributions}.}
\label{fig_inclination_distributions}
\end{figure}
As in \S\ref{subsection_convergence}, we chose 12 logarithmically spaced values of $N=10,...,2\cdot10^4$. For any $N$ and each of the three PDFs in \fig{inclination_distributions} we construct ensembles of control samples (\S\ref{subsection_ensemble}). Their TFRs are then estimated using the inclination-free MLE, always assuming a $\sin i$-distribution, even if the actual input PDF is different.
The mean relative errors $\ensavg{\Delta\mathbf{T}}$ are plotted against $N$ in \fig{errors}(b), with the line styles matching those in \fig{inclination_distributions}. \fig{errors}(b) demonstrates the robustness of the inclination-free MLE against a systematic deviation of the galaxy inclinations from the assumed $\sin i$-distribution. The slope $\beta$ of the TFR remains virtually unaffected, since a systematic deviation of the inclinations affects galaxies of all masses in the same manner and thus merely shifts the TFR horizontally. Intriguingly, even the zero-point $\alpha$ is very robust, the asymptotic mean error being as small as 1-4\% for all PDFs considered here. The main reason for this robustness is the strong dependence of the inclination-free MLE on close-to-edge-on galaxies ($i\gtrsim75^\circ$, thus $w\approx v$): as long as some of these galaxies are present, the precise inclination PDF plays a subordinate role. An additional reason for the robustness of $\alpha$ are the similar means of the three PDFs (last column in \tab{inclination_distributions}). The recovery of intrinsic scatter $\sigma$ is most strongly affected by a systematic deviation of the galaxy inclinations from a $\sin i$-distribution, the mean errors being potentially around 10-20\%. By current observing standards this is nonetheless a small error on the scatter of the TFR.
In conclusion, the inclination-free MLE is robust against a mismatch between the assumed inclination distribution (here a $\sin i$-distribution) and the true inclinations of the galaxy sample. In principle, the true inclination PDF of a galaxy sample could be estimated, for example using a telescope simulator, and then used as the prior PDF in the inclination-free MLE rather than a $\sin i$-distribution. However, given the quantitative results of this section, such extra effort might be unnecessary.
\subsection{Robustness against inclination errors}\label{subsection_inclination_errors}
An advantage of the inclination-free MLE is its strict independence from inclination measurement errors, which deteriorate the TFRs recovered with inclination-dependent methods. We now quantify the effect of inclination errors based on ensembles of control samples of $N=10^3$ galaxies, where the individual inclinations are perturbed by random and systematic errors.
Random inclination errors are modeled by first drawing the true galaxy inclinations $i_k$ from a $\sin i$-distribution to compute the projected velocities $w_k=v_k\sin i_k$. The true inclinations $i_k$ are then perturbed with Gaussian random noise (mirrored at $i_k=0^\circ$ and $i_k=90^\circ$), resulting in erroneous inclinations $i_k'$ that mimic the observed values. For TFR estimations methods using inclinations, the observed velocities with inclination errors are given by $v_k'\equiv w_k|\sin i_k'|^{-1}$. An inclination cut $i_k'>45^\circ$ is also applied for these methods, since without such an inclination cut their errors $\Delta\mathbf{T}$ can become much larger. This cut reduces the number of galaxies available for these inclination-dependent methods to about $N\approx700$. By contrast, the inclination-free MLE can use all $N=10^3$ galaxies, since this method remains strictly unaffected by inclination measurement errors.
\fig{errors}(c) displays the mean relative error $\ensavg{\Delta\mathbf{T}}$ as a function of the standard deviation of the inclination errors. This figure reveals that for random inclination errors larger than about $10^\circ$, the TFR is more reliably recovered by the inclination-free MLE. The same result is found for other sample sizes $N$, not shown in the figure. Inclination errors of $10^\circ$ correspond to errors in the axis ratio $q$ of about $0.17, 0.20, 0.16$ at $i=45^\circ, 60^\circ, 75^\circ$. For inclination errors larger than $10^\circ$, it is better not to use any inclination data than to use the uncertain inclinations as exact. Of course, if the uncertainties of the inclinations are well-characterized in terms of probability distributions $\rho_k(i)$, then those distributions can be fed into our MLE to improve on the inclination-free MLE.
Secondly, we address systematic inclination errors arising when the true intrinsic axis ratio $q_0$ deviates uniformly from the assumed intrinsic ratio $q_0'=0.2$. To model this case we use unperturbed galaxy inclinations $i_k$, drawn from a $\sin i$-distribution, to compute the axis ratios $q_k=[q_0^2+(1-q_0^2)\cos^2i_k]^{1/2}$. This equation is then inverted (giving \eq{cosi}), while substituting the true intrinsic axis ratio $q_0$ for $q_0'=0.2$. This results in the erroneous inclinations $i_k'$, which serve to calculate the observed velocities with $q_0$-errors $v_k'=w_k|\sin i_k'|^{-1}$ used in estimating the TFR with the methods that use inclinations. As before, only galaxies with $i_k'>45^\circ$ are used with these methods.
\fig{errors}(d) shows the mean relative error $\ensavg{\Delta\mathbf{T}}$ as a function of the true $q_0$ (for fixed assumed $q_0'=0.2$). Although the error $\ensavg{\Delta\mathbf{T}}$ slightly increases with $\abs{q_0-q_0'}$, all methods turn out to be surprisingly robust against errors in the assumed intrinsic axis ratio. With hindsight this finding also justifies the use of a constant, and not well-measured, $q_0$ in many existing TFR studies.
\section{Application to observed data}\label{section_observation}
This section tests the inclination-free MLE to recover the TFRs of real galaxies. We re-analyze a well characterized sample of H{\sc\,i}\ selected galaxies, used by \cite{Meyer2008} to derive a $K$-band and $B$-band TFRs with minimal observational scatter. In selecting galaxies from the HIPASS catalogue for this purpose, \citeauthor{Meyer2008}\ included an inclination cut to nearly edge-on galaxies with $i>75^\circ$ (thus $\sin i\in[0.96,1]$), leading to very reliable circular velocities $v=w/\sin i$. We shall use the same samples, both with and without inclination cut, to compare a standard TFR estimation against the inclination-free MLE. These data allow us to test the inclination-free MLE in more challenging circumstances, including effects such as inclination-dependent extinction that will significantly impact $B$-band magnitudes.
\subsection{Reference Tully-Fisher sample}\label{subsection_empirical_data}
The study of the $K$-band and $B$-band TFRs by \cite{Meyer2008} relies on four data sets: the H{\sc\,i}\ Parkes All-Sky Survey (HIPASS) Catalogue (HICAT, \citealp{Meyer2004,Zwaan2004}) which provides the initial H{\sc\,i}\ selected sample with H{\sc\,i}\ linewidths; the HIPASS Optical Catalogue (HOPCAT, \citealp{Doyle2005}), which provides more accurate galaxy positions and aspect ratios $q$ used to calculate the inclinations $i$ via \eq{cosi} with $q_0=0.12$; the Two Micron All-Sky Survey (2MASS, \citealp{Jarrett2000}) providing the $M_K$ magnitudes; and the ESO Lauberts \& Valentijn catalogue (ESO-LV, \citealp{Lauberts1989}) providing the $M_B$ magnitudes. Those magnitudes have been corrected for redshift ($k$-correction), extinction by dust in the Milky Way, and intrinsic extinction by the source (details in \S3.1 of \citealp{Meyer2008}). The linewidths are taken as the H{\sc\,i}\ linewidths $W_{50}$, measured at the 50\% level of the peak flux density and corrected for relativistic broadening, instrumental broadening, and turbulent motion (\S3.2 of \citeauthor{Meyer2008}). To study the $K$-band and $B$-band TFRs, the log-mass $\tilde{m}$ of the generic TFR (\eq{observable_tfr}) is substituted for $M_K$ and $M_B$, and the projected velocities $w$ are taken as $w=W_{50}/2$.
Departing from the full sample of galaxies with sufficient data in HICAT, HOPCAT, 2MASS, and ESO-LV, \cite{Meyer2008} derived raw $K$-band $B$-band TFRs (see their Figure 3). They discovered that most of the scatter in these TFRs was observational, being correlated to observer-dependent quantities such as inclination, apparent size, and large-scale flows. Restricting the sample to inclinations $i>75^\circ$, effective semi-major axes $r_{max}>15''$, velocities $cz>1000~{\rm km~s^{-1}}$, and positions less than $20^\circ$ from the dipole equator of the cosmic microwave background (CMB), led to clean subsamples with tight $K$-band and $B$-band TFRs over six magnitudes; the bivariate regressions quoted by \citeauthor{Meyer2008} are $\beta_K=-9.38\pm0.19$, $\sigma_K=0.25\rm$, $\beta_B=-8.51$, $\sigma_B=0.33$.
\subsection{Recovery of the Tully-Fisher Relation}\label{subsection_KTFR}
The aforementioned clean subsamples of nearly edge-on ($i>75^\circ$) galaxies constructed by \cite{Meyer2008} contain 55 ($K$-band) and 36 ($B$-band) objects. They are displayed in \fig{real_tfr} in the $(\tilde{v},\tilde{m})$-plane (big dots). The red dashed lines represent the corresponding TFRs, fitted directly to the $(\tilde{v},\tilde{m})$-points using the MLE that simultaneously optimizes all three parameters $\mathbf{T}=(\alpha,\beta,\sigma)$ to minimize \eq{likelihood_fixedi}. The fitted TFR parameters (see \tab{real_tfr}) are consistent with those resulting from the bivariate regression applied by \citeauthor{Meyer2008}, although regressions are theoretically less exact (see \S\ref{subsection_MLE_with_inclinations}).
\begin{figure*}[t]
\centering
\begin{tabular}{cc}
\begin{overpic}[width=\columnwidth]{fig_real_tfrK}\put(13,70.0){\normalsize\textbf{~}}\end{overpic} &
\begin{overpic}[width=\columnwidth]{fig_real_tfrB}\put(14,70.0){\normalsize\textbf{~}}\end{overpic} \\
\end{tabular}
\caption{Absolute $K$-band (left) and $B$-band (right) magnitudes, plotted against $v$ (red big dots, $i>75^\circ$) and $w$ (blue small dots, all $i$). The lines represent the fitted TFRs with 1-$\sigma$ scatter. Red dashed lines show the TFRs obtained by applying the MLE using inclination measurements, i.e, these lines are the most likely fits to the red big dots. In turn, the blue solid lines are obtained without inclination data, by applying the inclination-free MLE, which assumes an isotropic prior on the inclinations, i.e., $\rho(i)=\sin i$. In other words, these fits are based purely on the scattered blue small dots. The numerical values of all TFRs are listed in \tab{real_tfr}.\\[1ex](A color version of this figure is available in the online journal.)}
\label{fig_real_tfr}
\end{figure*}
\begin{table}[t]
\centering
\normalsize
\begin{tabular}{lcccc}
\hline\hline \\ [-1.5ex]
Sample & $N$ & Offset $\alpha$ & Slope $\beta$ & Scatter $\sigma$ \\[1.0ex]
\hline \\[-1.0ex]
$K$, $i>75^\circ$ & 55 & $+2.5\pm 0.7$ & $-9.3\pm 0.3$ & $0.22\pm0.06$ \\
$K$, all $i$ & 351 & $+2.3\pm 0.6$ & $-9.4\pm 0.3$ & $0.24\pm0.06$ \\
$B$, $i>75^\circ$ & 36 & $+1.2\pm 0.8$ & $-8.7\pm 0.3$ & $0.34\pm0.06$ \\
$B$, all $i$ & 180 & $+1.4\pm 0.6$ & $-8.6\pm 0.3$ & $0.39\pm0.04$ \\
[1.0ex]\hline\hline
\end{tabular}
\caption{\upshape\raggedright Numerical values for the TFRs shown in \fig{real_tfr}. The TFRs of the edge-on samples with $i>75^\circ$ were obtained by applying the MLE using full information on inclinations, i.e., inclinations were used both to calculate the velocities $v$ from the linewidths and to correct to the magnitudes for internal extinction. By contrast, the TFRs fitted to the full samples (all $i$) use strictly no inclinations. These TFRs have been obtained by applying the inclination-free MLE that assume an isotropic distribution of the galaxies, i.e., $\rho(i)=\sin i$. Parameters are given with $68\%$ confidence intervals obtained by Bootstrapping the data.}
\label{tab_real_tfr}
\end{table}
To check the inclination-free MLE, we now drop the selection criterion $i>75^\circ$ and include galaxies of all inclinations. In practice no inclination data often means no resolved images, thus no values for the semi-major axes, and hence no possibility for an explicit size selection. We therefore couple the removal of the inclination cut with the removal of the size cut and instead add an absolute magnitude cut $M_K<-20$ and $M_B<-18$ to still suppress the gas-dominated dwarf galaxies that were excluded by the size cut ($r_{max}>15''$). This results in samples of 351 ($K$-band) and 180 ($B$-band) galaxies of all inclinations. \fig{real_tfr} shows this sample in the $(\tilde{w},\tilde{m})$-plane (small dots). Further, we also remove the inclination-dependent extinction correction from the data (by suppressing the term $A_i$ in Equation (1) of \citealp{Meyer2008}). These corrections have a significant (0.2~mag average) effect for the $B$-band magnitudes, but are negligible for $K$-band magnitudes.
To recover the TFRs we then feed the absolute magnitudes $\tilde{m}_k$ (without corrections for internal extinction) and the observed projected velocites $\tilde{w}_k$ into the MLE, strictly using no information on inclination. The likelihood function of \eq{likelihood} is maximized assuming random galaxy inclinations ($\rho(i)=\sin i$) and accounting for the unknown inclination-dependent extinction (see \S\ref{subsection_real_data}) $\tilde{m}_k(i)=\tilde{m}_k+A_i$ with the extinction factor $A_i$ defined as in \citealp{Meyer2008}. This results in the TFRs plotted as blue solid lines in \fig{real_tfr}.
\fig{real_tfr} demonstrates that the inclination-free MLE provides similar TFRs than the observationally more expensive approach, which uses inclinations. To quantify this similarity, both methods have been re-sampled $10^3$ times with random half-sized subsamples. This bootstrapping provides the 68\% confidence intervals listed in \tab{real_tfr}. All three TFR parameters $\alpha$, $\beta$, and $\sigma$ of the two methods are consistent in that they deviate by less than one standard deviation. Given the high co-variance of these three parameters, the same conclusion applies to the full vector $\mathbf{T}=(\alpha,\beta,\sigma)$. In summary, this illustration based on real data suggests that the inclination-free MLE can recover the TFR within errors comparable to the typical shot noise uncertainties.
\section{Conclusions}\label{section_conclusion}
This work introduced a mathematical framework for studying the Tully-Fisher relation (TFR) based on maximum likelihood estimation (MLE). This framework permits the estimation of the TFR in galaxy samples with uncertain inclinations, described by a probability distribution $\rho_k(i)$ for every galaxy $k$. This method consists of maximizing the likelihood function of \eq{likelihood}, where $\rho_k(\mathbf{T})$ is given in \eq{main_rho2} for Gaussian measurement uncertainties. This method also applies to the extreme situation, where no inclinations are known at all. In this case, the inclination priors can be replaced by an isotropic distribution, $\rho_k(i)=\sin i$ (unless a better prior is available). Our analysis of this `inclination-free MLE' has led to the following results:
\begin{itemize}
\item \textit{No galaxy inclinations are needed to recover the TFR.} If a galaxy sample obeys a TFR with normal scatter, the inclination-free MLE generally recovers the three TFR parameters (zero-point, slope, scatter) with statistical errors about 1.5-times larger than the best estimates based on perfectly known galaxy inclinations with zero uncertainty.
\item \textit{The inclination-free MLE is robust.} Its independence of galaxy inclinations makes it independent of inclination errors, be they measurement errors, model errors of $q_0$, or real misalignments between the disk used for linewidth measurements (e.g., H{\sc\,i}~disk) and the disk used for inclination measurements (e.g., optical disk). In a realistic scenario, where inclination errors are larger than $10^\circ$ and a typical inclination selection of $i>45^\circ$ must be applied for inclination-dependent methods, the inclination-free MLE performs in fact better than methods that use the measured inclinations while treating them as exact. Even if the assumed PDF $\rho(i)$ (typically a $\sin i$-distribution) of the galaxy inclinations is inaccurate, the errors of the TFR parameters remain small (\fig{errors}(b)).
\item \textit{The inclination-free MLE is flexible.} If desired, this method can account for a PDF $\rho(i)$ that differs from a $\sin i$-distribution. The method can also be extended in a straightforward way to deal with detection limits, complex measurement uncertainties, erroneous outliers in the data, elliptical/irregular galaxies, and inclination-dependent extinction.
\end{itemize}
The inclination-free MLE is potentially crucial for future surveys, namely H{\sc\,i}\ blind surveys with the Australian SKA Pathfinder (ASKAP), the South African SKA Pathfinder (MeerKAT), and the SKA. In these large surveys, ancillary optical data may not have sufficient spatial resolution for reliable inclination measurements. In this case, the inclination-free MLE can recover various TFRs, including the stellar mass and baryon mass TFRs. By removing the need for well-resolved spatial imaging, the inclination-free MLE also enables the TFR to be recovered at higher redshifts than might otherwise be possible, offering a new method of probing the cosmic evolution of the TFR. Should optical counterparts be completely unavailable, one might still use the inclination-free MLE to derive a H{\sc\,i}~mass TFR, as done for the HIPASS data in Figure 13a of \cite{Obreschkow2009b}.
Indirectly, the inclination-free MLE can be used to constrain the inclinations of galaxies with otherwise unknown inclinations. Once the TFR parameters $\mathbf{T}$ have been determined using the inclination-free MLE, every galaxy $k$ has a well-defined PDF $\rho_k(\tilde{v})$ for its circular velocity (\eq{rho_tfr}). Using $v_k=w_k/\sin i_k$, this PDF can be rewritten as a PDF for the inclinations $i_k$. When multiplied by the \textit{prior} PDF $\rho(i)=\sin i$, this results in a \textit{posterior} PDF $\rho^{post}_k(i)$. This posterior PDF\footnote{Explicitly, $\rho^{post}_k(i)=\rho(i)\exp[-[\log_{10}(\sin(i)/r)]^2/(2\sigma_{tot,k}'^2)]$, where $\sigma_{tot,k}'\equiv\sigma_{tot,k}/\beta$ is the horizontal scatter and $r\equiv w_k/\avg{v}_k$ with $\avg{v}_k=10^{(\tilde{m}_k-\alpha)/\beta}$ being the mean circular velocity predicted by the TFR. The width of $\rho^{post}_k(i)$ increases monotonically with $\sigma_{tot,k}'$, and as $\sigma_{tot,k}\rightarrow\infty$ we have $\rho^{post}_k(i)=\rho(i)$. This simply means that no constraints on inclinations could be found via the TFR, if the scatter of the latter were too large.} could be used, for instance, to study correlations between inclinations and other galaxy properties -- an idea to be explored in forthcoming studies.
This project was supported by the UWA Research Collaboration Award 2013 (PG12104401). We thank the anonymous referee for significant contributions.
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"redpajama_set_name": "RedPajamaArXiv"
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Gutch is a surname. Notable people by that name include:
Eliza Gutch (1840-1931), English author.
George Gutch, British architect.
John Gutch (clergyman) (1746–1831), Anglican clergyman and official of the University of Oxford.
John Gutch (colonial administrator), British colonial administrator.
John Mathew Gutch (1776-1861), English journalist and historian.
John Wheeley Gough Gutch (1809–1862), British surgeon and editor.
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"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/researchmap.jp\/read0102265\/misc\/4027142","text":"2003\u5e74\n\n# Convergence Properties of the Membership Set in the Presence of Disturbance and Parameter Uncertainty\n\n\u2022 KITAMURA Wataru\n\u2022 ,\n\u2022 FUJISAKI Yasumasa\n\n39\n4\n\n382\n\n387\n\nDOI\n10.9746\/sicetr1965.39.382\n\nThe Society of Instrument and Control Engineers\n\nThe size of the membership set is investigated in the presence of bounded disturbance and l2 bounded parameter uncertainty. A tight upper bound of the diameter of the membership set is derived in a deterministic setting, which indicates that the diameter does not converge to zero in general. Then, a probabilistic upper bound of the diameter is derived in a stochastic setting, where the disturbance and the parameter uncertainty are assumed to be random variables and to take a value near the worst ones with a probability. This probabilistic bound leads to the fact that the diameter converges to zero with probability one as the number of samples tends to infinity.\n\n\u30ea\u30f3\u30af\u60c5\u5831\nDOI\nhttps:\/\/doi.org\/10.9746\/sicetr1965.39.382\nCiNii Articles\nhttp:\/\/ci.nii.ac.jp\/naid\/130003791837\nURL\nhttps:\/\/jlc.jst.go.jp\/DN\/JALC\/00182539459?from=CiNii\nID\u60c5\u5831\n\u2022 DOI : 10.9746\/sicetr1965.39.382\n\u2022 ISSN : 0453-4654\n\u2022 CiNii Articles ID : 130003791837\n\n\u30a8\u30af\u30b9\u30dd\u30fc\u30c8","date":"2023-02-04 03:33:10","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\": 0.8727263808250427, \"perplexity\": 725.2471065319978}, \"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-2023-06\/segments\/1674764500080.82\/warc\/CC-MAIN-20230204012622-20230204042622-00174.warc.gz\"}"}
| null | null |
Єжо́в Валенти́н Іва́нович (7 липня 1927, , Івановська область, РРФСР, СРСР — 25 серпня 2010, Київ, Україна) — український радянський архітектор. Віце-президент Української академії архітектури, член Президії Національної спілки архітекторів України, почесний академік Національної академії мистецтв України, академік Академії будівництва та інженерних наук України, дійсний член Міжнародної академії архітектури, іноземний член Російської академії архітектури і будівельних наук. Професор Київського національного університету будівництва і архітектури (КНУБА).
У 1981–1987 — головний архітектор Києва.
Біографія
Народився 7 липня 1927 року в селі Кощєєвому (нині Івановська область, Росія). В 1949 рік у з відзнакою закінчив архітектурний факультет Київського інженерно-будівельного інституту — навчався у Каракіса Йосипа Юлійовича. Йосип Юлійович був керівником декількох його курсових проєктів і дипломного проєкту. Дипломний проєкт комісією був оцінений найвищим балом і багато років провисів на кафедрі в КІБІ. (Клара Георгіївна Демура вважає В. Єжова дуже здібним студентом І. Каракіса.)
Розпочав трудову діяльність архітектором у Севастополі.
У 1953–1963 — аспірант та науковий співробітник Академії архітектури УРСР.
З 1963 працював керівником сектору, відділу КиївЗНДІЕП, згодом став головним архітектором інституту.
У 1976–1981 — директор КиївНДІТІ (НДІТІАМ).
У 1981–1987 — головний архітектор Києва.
У 1987–2010 — завідувач кафедри основ архітектури і архітекутрного проєктування Київського національного університету будівництва і архітектури (КНУБА).
З 1991 — керівник АБ «Єжов».
Помер у Києві 25 серпня 2010. Похований на Байковому кладовищі.
Праці
Під керівництвом та за австорської участі В. І. Єжова розроблено понад 260 проєктів, за якими побудовано 80 будівель та комплексів різноманітного функціонального призначення в Києві, Кам'янському, Севастополі, Бердянську, Феодосії, Євпаторії, Прип'яті та інших містах України. Серед будівель:
Дім торгівлі;
шкільне містечко на 4000 учнів;
палац культури київського Авіазаводу ім. О. К. Антонова,
палац культури виробничого об'єднання ім. С. П. Корольова;
станції метро «Вокзальна» і «Дзержинська» (нині «Либідська»);
забудова житлових масивів Теремки-2, Вигурівщина-Троєщина, Північно-Броварський в Києві;
шкільний комплекс у Кам'янському;
Бердянський міський Палац культури ім. Т.Г. Шевченка;
житловий будинок по вулиці Івана Мазепи (2001) та інші.
Автор понад 200 наукових праць, серед яких монографії:
«Архитектура общественных зданий массового строительства» (1983);
«Архитектурно-конструктивные системы общественных зданий» (1981);
«Архитектурно-конструктивные системы гражданских зданий» (1998, у співавторстві);
«Полвека глазами архитектора» (2001);
«Эскизная графика архитектора» (2003);
«Архитектура общественных зданий и комплексов» (2006, у співавторстві);
«Архитектура южного жилища» (2012).
Серед художніх робіт: «Площа Богдана Хмельницького з видом на Софійський собор і житловий будинок, побудований за проєктом І. Каракіса», 1955 рік.
Відзнаки та нагороди
Почесний доктор НДІТІАМ, доктор аріхтектури, професор, має почесні звання «Заслужений архітектор України» (1994), «Народний архітектор України» (1999), удостоєний Державної премії України в галузі архітектури (2000).
Нагороджений орденами «Знак Пошани» (1981), «За заслуги» III ступеня (2004), Почесною грамотою Верховної Ради УРСР (1987), Премією Ради Міністрів СРСР (1989), медалями «За трудову доблесть» (1971), «В пам'ять 1500-річчя Києва» (1982), «Ветеран праці» (1984) та іншими.
Примітки
Посилання
Єжов Валентин Іванович на сайті who-is-who.com.ua
Джерела
Гасанова, Н. Зодчий и время: к 70-летию со дня рождения В. И. Ежова / Н. Гасанова, А. Пучков // Архитектура и престиж. — 1997. — № 1-2. — С. 29-30.
Організація комфортного середовища життєдіяльності міських поселень: зб. наук. пр. / під заг. ред. В. В. Куцевича ; Укр. зон. н.-д. і проект. ін-т по цив. буд-ву.— К., 2008 .
Єжов Валентин Іванович //
Єжов Валентин Іванович //
Валентин Иванович Ежов. Библиографический указатель построек, проектов и научных трудов: К семидесятилетию со дня рождения / Сост. А. А. Пучков. — К.: НИИТИАГ, 1997. — 64 с.: ил.
Валентин Иванович Ежов. Библиографический указатель построек, проектов и научных трудов: К семидесятипятилетию со дня рождения / Сост. и авт. вступит. ст. А. А. Пучков. — Изд. 2-е, перераб., исправ. и доп. — К.: НИИТИАГ, 2002. — 99 с.: ил.
Єжов Валентин Іванович // Імена України: Біограф. довідник. — : Фенікс, 2002. — С. 186.
Валентин Єжов // Українська академія архітектури: Персональний склад / За заг. ред. В. Г. Штолька. — : Вид. дім А+С, 2007. — С. 49.
Валентин Єжов // 100 найкращих будівельників та архітекторів України '2009. — , 2009.
Єжов Валентин Іванович: особова справа чл. НСА України (1974 — 25 серпня 2010) // Національна спілка архітекторів України.
Штолько В. Г. Єжов Валентин Іванович //
Випускники НАОМА
Радянські архітектори
Члени Національної спілки архітекторів України
Автори проєктів станцій Київського метрополітену
|
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<!DOCTYPE html>
<html lang="en">
<head>
<title>{{template "title" .}}</title>
<link href="{{asset "main.css"}}" media="all" rel="stylesheet" type="text/css" />
<link rel="apple-touch-icon" sizes="57x57" href="{{asset "favicons/apple-touch-icon-57x57.png"}}">
<link rel="apple-touch-icon" sizes="60x60" href="{{asset "favicons/apple-touch-icon-60x60.png"}}">
<link rel="apple-touch-icon" sizes="72x72" href="{{asset "favicons/apple-touch-icon-72x72.png"}}">
<link rel="apple-touch-icon" sizes="76x76" href="{{asset "favicons/apple-touch-icon-76x76.png"}}">
<link rel="apple-touch-icon" sizes="114x114" href="{{asset "favicons/apple-touch-icon-114x114.png"}}">
<link rel="apple-touch-icon" sizes="120x120" href="{{asset "favicons/apple-touch-icon-120x120.png"}}">
<link rel="apple-touch-icon" sizes="144x144" href="{{asset "favicons/apple-touch-icon-144x144.png"}}">
<link rel="apple-touch-icon" sizes="152x152" href="{{asset "favicons/apple-touch-icon-152x152.png"}}">
<link rel="apple-touch-icon" sizes="180x180" href="{{asset "favicons/apple-touch-icon-180x180.png"}}">
<link rel="icon" type="image/png" href="{{asset "favicons/favicon-32x32.png"}}" sizes="32x32">
<link rel="icon" type="image/png" href="{{asset "favicons/android-chrome-192x192.png"}}" sizes="192x192">
<link rel="icon" type="image/png" href="{{asset "favicons/favicon-96x96.png"}}" sizes="96x96">
<link rel="icon" type="image/png" href="{{asset "favicons/favicon-16x16.png"}}" sizes="16x16">
<link rel="manifest" href="{{asset "favicons/manifest.json"}}">
<meta name="msapplication-TileColor" content="#000000">
<meta name="msapplication-TileImage" content="{{asset "favicons/mstile-144x144.png"}}">
<meta name="theme-color" content="#000000">
</head>
<body class="pipelinesNav-enabled">
<div class="js-pipelinesNav">
<div class="nav-container">
<ul class="pipelines js-pipelinesNav-list"></ul>
</div>
<div class="page-to-slide">
<nav>
<ul class="js-groups groups">
<li class="main"><span class="js-pipelinesNav-toggle btn-hamburger" aria-label="Toggle List of Pipelines"><i class="fa fa-bars"></i></span></li><li class="main"><a href="/pipelines/{{.PipelineName}}" aria-label="Back to Pipeline View"><i class="fa fa-home"></i></a></li>
{{range .GroupStates}}
<li{{if .Enabled}} class="active"{{end}}><a href="/pipelines/{{$.PipelineName}}?groups={{.Name}}">{{.Name}}</a></li>
{{end}}
</ul>
<ul class="nav-right">
<li class="nav-item"><a href="/builds" aria-label="Builds List"><i class="fa fa-tasks"></i></a></li>
</ul>
</nav>
<div id="content">
{{template "body" .}}
</div>
</div>
</div>
</body>
</html>
|
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"redpajama_set_name": "RedPajamaGithub"
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Q: Can't find the syntax error in my SQL SELECT statement I keep getting a syntax error in my SQL SELECT statement. I know it is an issue with the DateDifffunction at the end of the statement, but I can't seem to find the correct format. I am sure it is simple but I am not finding it.
"SELECT
tblAllSchedules.[Schedule Name],
tblAllSchedules.[Arvl Sta],
(' & Format(DateValue([tblAllSchedules]![Arvl Local Date/Time]),'ddd') & ')' AS [Day],
DateValue([tblAllSchedules]![Arvl Local Date/Time]) AS [Date],
tblAllSchedules.Metal, tblAllSchedules.Type,
tblAllSchedules.[Key ID], tblAllSchedules.[LAA NB],
tblAllSchedules.[LAA WB], tblAllSchedules.[LUS NB],
tblAllSchedules.[LUS WB], tblAllSchedules.[Sub Fleet],
'Original', '" & csName & "', (' & DateDiff('d', '" & FSun & "', [Tables]![tblAllSchedules]![Arvl Local Date/Time])+ 1 & ')' AS CompSchedDay"
Thanks for the assistance.
A: Someone just had the winning answer posted but it went away before I could respond. Here is the corrected syntax.
SELECT tblAllSchedules.[Schedule Name], tblAllSchedules.[Arvl Sta],
Format(DateValue([tblAllSchedules]![Arvl Local Date/Time]),'ddd') AS [Day],
DateValue([tblAllSchedules]![Arvl Local Date/Time]) AS [Date],
tblAllSchedules.Metal, tblAllSchedules.Type, tblAllSchedules.[Key ID],
tblAllSchedules.[LAA NB], tblAllSchedules.[LAA WB], tblAllSchedules.[LUS NB],
tblAllSchedules.[LUS WB], tblAllSchedules.[Sub Fleet], 'Original', '" & csName & "',
DateDiff('d', '" & FSun & "' , [tblAllSchedules]![Arvl Local Date/Time])+ 1 AS CompSchedDay
I quess I had it too mudled and more complicated than it needed to be. Only issue now is correcting the new "aggrigated function" error. Thank you all for your quick and helpful comments.
A: FSun must be a valid date expression. So try:
Dim FSun As String
FSun = Format(YourFSunDateValue, "yyyy\/mm\/dd")
"SELECT
tblAllSchedules.[Schedule Name],
tblAllSchedules.[Arvl Sta],
Format([tblAllSchedules]![Arvl Local Date/Time]),'ddd') AS [Day],
DateValue([tblAllSchedules]![Arvl Local Date/Time]) AS [Date],
tblAllSchedules.Metal,
tblAllSchedules.Type,
tblAllSchedules.[Key ID],
tblAllSchedules.[LAA NB],
tblAllSchedules.[LAA WB],
tblAllSchedules.[LUS NB],
tblAllSchedules.[LUS WB],
tblAllSchedules.[Sub Fleet],
'Original',
'" & csName & "',
DateDiff('d', #" & FSun & "#, [tblAllSchedules]![Arvl Local Date/Time]) + 1 AS CompSchedDay
FROM
tblAllSchedules"
A: Please try the following...
SELECT [Schedule Name],
[Arvl Sta],
'(' & FORMAT( DATEVALUE( [Arvl Local Date/Time] ), 'ddd') & ')' AS [Day],
DATEVALUE( [Arvl Local Date/Time] ) AS [Date],
Metal,
[Type],
[Key ID],
[LAA NB],
[LAA WB],
[LUS NB],
[LUS WB],
[Sub Fleet],
'Original',
'''' & csName & '''',
'(' & ( DATEDIFF( 'd', FSun, DATEVALUE( [Arvl Local Date/Time] ) ) + 1 ) & ')' AS CompSchedDay
FROM tblAllSchedules;
Firstly, you have used tblAllSchedules. for some lines and tblAllSchedules! for others. Unless you have both a table and a form called tblAllSchedules, you should stick to the former to refer to fields in your table. The latter is used to refer to fields and controls in a form or report. For now, I am assuming that your statement refers only to fields in a table called tblAllSchedules.
Secondly, if all the fields are coming from the same table, namely the one specified in your FROM clause, then you do not need to specify the table's name against each field. If any are coming from a form, then you should use notation like frmAllSchedules!txtDate.
Thirdly, try to give each field its own line in the SELECT part of the statement. This makes it easier to find a targeted field as well as to recognise when you are referring to two separately selected fields or two fields being used in some manner to form another (both of which appear to be occurring on your final line).
Fourthly, you appear to be trying to place ( and ) around string representations of date values, e.g. (Wednesday) and (12). To do this you will need to place ' on both sides of the opening bracket as well as the closing bracket, and concatenate it with your value.
Fifthly, you appear to be trying to place a ' on either side of csName in the output. To do that you need to start with '' to signify the start and end of your string. In the middle of these place two more '' (giving you ''''), which is how you tell it that you want it to return a string with a single ' as its contents.
Sixthly, you should consider giving an alias to 'Original' and '''' & csName & '''' - Access will arbitrarily give them one if you don't.
If you have any questions or comments, then please feel free to post a comment accordingly.
|
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Q: Is there a 'ResponseInterceptor' (as its counterpart RequestInterceptor) component for Retrofit? I'm using Retrofit and most of its components like ErrorHandler, Callbacks and RequestInterceptor, etc. But I miss something like a 'ResponseInterceptor', something that would allow me to filter or read headers that came on the response in generic way, as a central point, instead of having to set a Callback to every service call. This is currently the only way I know how to achieve such a goal, but having to explicitly pass the callback object on very service call is not desirable.
What I want is to act upon any response that has a presence of a specific header, so it would be like a generic object (a 'ResponseInterceptor') that would analyse each response.
Is there such a thing? or any better idea on how to achieve this without having to set a Callback object on every service call?
Thanks in advance!
A: As Retrofit currently doesn't provide a standard way of doing it, I managed to make it work by decorating the retrofit.client.Client implementation that is given to the retrofit.RestAdapter, see it below:
1) define an interface for the interceptor
import retrofit.client.Response;
public interface ResponseInterceptor {
void intercept(Response response);
}
2) make an implementation of it.
public class MyResponseInterceptor implements ResponseInterceptor {
@Override
public void intercept(Response response) {
List<Header> headers = response.getHeaders();
//... read the headers the way you like
}
}
3) implement the decorator for the Client:
public final class ResponseInterceptorClient extends OkClient {
private final ResponseInterceptor responseInterceptor;
public ResponseInterceptorClient(OkHttpClient okHttpClient, ResponseInterceptor responseInterceptor) {
super(okHttpClient);
this.responseInterceptor = responseInterceptor;
}
@Override
public Response execute(Request request) throws IOException {
Response response = super.execute(request);
doIntercept(response);
return response;
}
private void doIntercept(Response response) {
if (responseInterceptor != null) {
responseInterceptor.intercept(response);
}
}
}
4) then pass it on to the RestAdapter:
RestAdapter.Builder restBuilder = new RestAdapter.Builder();
//...
restBuilder.setClient(new ResponseInterceptorClient(okHttpClient, new MyResponseInterceptor()));
//...
restAdapter = restBuilder.build();
It's very simple and it could be easily make available by the library, just like the RequestInterceptor, until it isn't, this is a way to do it. Enjoy!
UPDATE 03/2016: Since Retrofit 2.0 there's no need for making your own ResponseInterceptor, you can make use of the OkHttp interceptors that come integrated with it.
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Q: Not using the else I'm try to not use the else in the code and just the if. But when I run it and enter "Q" it say error of converting in the total += Convert.ToInt32(input);. Is there a way around this?
do
{
Console.Clear();
Console.WriteLine("Enter a number or Q to quit", input);
input = Console.ReadLine();
if (input.ToUpper() == "Q")
{
Console.WriteLine("Press any key to continue");
Console.ReadKey();
}
else
{
total += Convert.ToInt32(input);
numbersEntered++;
average = ((double)total / numbersEntered);
Console.WriteLine("Total: {0}\t Numbers Entered: {1}\t Average: {2}\t", total, numbersEntered, average);
Console.ReadKey();
}
}
while (input.ToUpper() != "Q");
A: Try using break; to stop the loop so you don't need the else.
do{
Console.Clear();
Console.WriteLine("Enter a number or Q to quit", input);
input = Console.ReadLine();
if (input.ToUpper() == "Q"){
Console.WriteLine("Press any key to continue");
Console.ReadKey();
break;
}
total += Convert.ToInt32(input);
numbersEntered++;
average = ((double)total / numbersEntered);
Console.WriteLine("Total: {0}\t Numbers Entered: {1}\t Average: {2}\t", total, numbersEntered, average);
Console.ReadKey();
} while (input.ToUpper() != "Q");
A: Use TryParse to test if the input is a numeric. Here's an example for int (you'll need to define the out variable yourself).
else if (Int32.TryParse(input,out inputVal) {
|
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"redpajama_set_name": "RedPajamaStackExchange"
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// <file>
// <copyright see="prj:///doc/copyright.txt"/>
// <license see="prj:///doc/license.txt"/>
// <owner name="David Srbecký" email="dsrbecky@gmail.com"/>
// <version>$Revision$</version>
// </file>
#pragma warning disable 108, 1591
namespace Debugger.Interop.CorDebug
{
using System;
public enum CorDebugMappingResult
{
// Fields
MAPPING_APPROXIMATE = 0x20,
MAPPING_EPILOG = 2,
MAPPING_EXACT = 0x10,
MAPPING_NO_INFO = 4,
MAPPING_PROLOG = 1,
MAPPING_UNMAPPED_ADDRESS = 8
}
}
#pragma warning restore 108, 1591
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,505
|
Performance Tracks/Downloads
The Taylors
Friday, May 6 @ 7:00PM Fri, May 6 @ 7:00PM Calvary Baptist Church Statham, GA Statham, GA
Saturday, May 14 @ 7:00PM Sat, May 14 @ 7:00PM South New Milford Baptist Church New Milford, PA New Milford, PA
Saturday, July 2 Sat, Jul 2 The Farm House Cafe & Bakery Newton Grove, NC Newton Grove, NC
Monday, January 30, 2023 — Friday, February 3, 2023 Mon, Jan 30, 2023 — Fri, Feb 3, 2023 Singing at Sea 2023 Port Canaveral, FL Port Canaveral, FL
Sunday, June 11, 2023 @ 6:00PM Sun, Jun 11, 2023 @ 6:00PM Music In The Park Lebanon, PA Lebanon, PA
Salvation's Song 3:57
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,044
|
package org.onlab.packet.ndp;
import org.junit.BeforeClass;
import org.junit.Test;
import org.onlab.packet.MacAddress;
import static org.hamcrest.Matchers.is;
import static org.junit.Assert.assertArrayEquals;
import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertThat;
import static org.junit.Assert.assertTrue;
/**
* Tests for class {@link Redirect}.
*/
public class RedirectTest {
private static final byte[] TARGET_ADDRESS = {
(byte) 0x20, (byte) 0x01, (byte) 0x0f, (byte) 0x18,
(byte) 0x01, (byte) 0x13, (byte) 0x02, (byte) 0x15,
(byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00,
(byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00
};
private static final byte[] DESTINATION_ADDRESS = {
(byte) 0x20, (byte) 0x01, (byte) 0x0f, (byte) 0x18,
(byte) 0x01, (byte) 0x13, (byte) 0x02, (byte) 0x15,
(byte) 0xca, (byte) 0x2a, (byte) 0x14, (byte) 0xff,
(byte) 0xfe, (byte) 0x35, (byte) 0x26, (byte) 0xce
};
private static final byte[] DESTINATION_ADDRESS2 = {
(byte) 0x20, (byte) 0x01, (byte) 0x0f, (byte) 0x18,
(byte) 0x01, (byte) 0x13, (byte) 0x02, (byte) 0x15,
(byte) 0xe6, (byte) 0xce, (byte) 0x8f, (byte) 0xff,
(byte) 0xfe, (byte) 0x54, (byte) 0x37, (byte) 0xc8
};
private static final MacAddress MAC_ADDRESS =
MacAddress.valueOf("11:22:33:44:55:66");
private static byte[] bytePacket;
@BeforeClass
public static void setUpBeforeClass() throws Exception {
byte[] byteHeader = {
(byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00,
(byte) 0x20, (byte) 0x01, (byte) 0x0f, (byte) 0x18,
(byte) 0x01, (byte) 0x13, (byte) 0x02, (byte) 0x15,
(byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00,
(byte) 0x00, (byte) 0x00, (byte) 0x00, (byte) 0x00,
(byte) 0x20, (byte) 0x01, (byte) 0x0f, (byte) 0x18,
(byte) 0x01, (byte) 0x13, (byte) 0x02, (byte) 0x15,
(byte) 0xca, (byte) 0x2a, (byte) 0x14, (byte) 0xff,
(byte) 0xfe, (byte) 0x35, (byte) 0x26, (byte) 0xce,
(byte) 0x02, (byte) 0x01, (byte) 0x11, (byte) 0x22,
(byte) 0x33, (byte) 0x44, (byte) 0x55, (byte) 0x66
};
bytePacket = new byte[byteHeader.length];
System.arraycopy(byteHeader, 0, bytePacket, 0, byteHeader.length);
}
/**
* Tests serialize and setters.
*/
@Test
public void testSerialize() {
Redirect rd = new Redirect();
rd.setTargetAddress(TARGET_ADDRESS);
rd.setDestinationAddress(DESTINATION_ADDRESS);
rd.addOption(NeighborDiscoveryOptions.TYPE_TARGET_LL_ADDRESS,
MAC_ADDRESS.toBytes());
assertArrayEquals(rd.serialize(), bytePacket);
}
/**
* Tests deserialize and getters.
*/
@Test
public void testDeserialize() {
Redirect rd = new Redirect();
rd.deserialize(bytePacket, 0, bytePacket.length);
assertArrayEquals(rd.getTargetAddress(), TARGET_ADDRESS);
assertArrayEquals(rd.getDestinationAddress(), DESTINATION_ADDRESS);
// Check the option(s)
assertThat(rd.getOptions().size(), is(1));
NeighborDiscoveryOptions.Option option = rd.getOptions().get(0);
assertThat(option.type(),
is(NeighborDiscoveryOptions.TYPE_TARGET_LL_ADDRESS));
assertArrayEquals(option.data(), MAC_ADDRESS.toBytes());
}
/**
* Tests comparator.
*/
@Test
public void testEqual() {
Redirect rd1 = new Redirect();
rd1.setTargetAddress(TARGET_ADDRESS);
rd1.setDestinationAddress(DESTINATION_ADDRESS);
rd1.addOption(NeighborDiscoveryOptions.TYPE_TARGET_LL_ADDRESS,
MAC_ADDRESS.toBytes());
Redirect rd2 = new Redirect();
rd2.setTargetAddress(TARGET_ADDRESS);
rd2.setDestinationAddress(DESTINATION_ADDRESS2);
rd2.addOption(NeighborDiscoveryOptions.TYPE_TARGET_LL_ADDRESS,
MAC_ADDRESS.toBytes());
assertTrue(rd1.equals(rd1));
assertFalse(rd1.equals(rd2));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,417
|
Aquaculture Research and Development Centre, Kajjansi (ARDC), is a national centre responsible for aquaculture research and development in Uganda. It is a branch of the National Fisheries Resources Institute (NAFIRRI).
The center undertakes the following studies:
Fish Feeds
Fish Health
Genetics
Hatchery Productivity
Mapping & Market
MSI Nile Perch Project
New species
Ornamentals
Production Systems.
The site attracts a number of bird species both Waterfowl species and land birds seen around the fish ponds and at the edges of the site. The centre undertakes research on 300 fish species that are extinct as well as the threatened ones like the riverine Ningu (Labeo victorianus), Kisinja (Barbus spp), Nkolongo (Synodontis spp) and Kasulu (Mormyrids).
Fish reared at the site
Nile Tilapia
Nile perch.
Mirror carp
African catfish
lung fish
Kisinja
References
External links
"How aquaculture can save Uganda's lakes"
"Uganda Promotes Aquaculture as Fish Stock in Lakes Dwindles "
"China-Uganda Aquaculture Project Launched"
"Uganda: Kajjansi Research Centre Renovated"
Research institutes in Uganda
Fish farming
Kumusha
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,382
|
\section*{Introduction} \label{sec_introduction}
\addcontentsline{toc}{section}{Introduction}
Generalized geometry \cite{Hitchin:2004ut, Gualtieri:2003dx, cavalcanti2005new} and Courant algebroids \cite{liu1997manin, 1999math.....10078R, Severa:2017oew} became one of the most useful tools in the understanding of geometry behind string theory. Courant algebroids can be used to describe current algebras of $\sigma$-models \cite{Alekseev:2004np}, various aspects of T-duality \cite{Baraglia:2013wua, Cavalcanti:2011wu, Garcia-Fernandez:2016ofz, Grana:2008yw}, geometrical approach to (exceptional, heterotic) supergravity \cite{Coimbra:2011nw, Coimbra:2012af, garcia2014torsion, Strickland-Constable:2019qti} and Poisson--Lie T-duality \cite{Severa:2015hta,Severa:2016prq, Severa:2017kcs, Severa:2018pag}. See also our contributions to the subject \cite{jurvco2016heterotic,Jurco:2016emw, Vysoky:2017epf, Jurco:2017gii, Jurco:2019tgt}.
On the other and, graded manifolds rose became a useful tool both in differential geometry and mathematical physics. They can be viewed as a modification of supergeometry, allowing for a grading of local coordinates governed by the abelian group $\mathbb{Z}$. Theory of graded manifolds is a crucial mathematical notion for BV \cite{batalin1983quantization,batalin1984gauge} and AKSZ \cite{alexandrov1997geometry} formalism in quantum field theory. There are excellent materials covering this topic, see e.g. \cite{qiu2011introduction,ikeda2017lectures,2011RvMaP..23..669C}. Graded manifolds provide an elegant description of Courant algebroids \cite{roytenberg2002structure}, they can be used to construct Drinfel'd doubles for Lie bialgebroids \cite{Voronov:2001qf}, Lie algebroids and their higher analogues \cite{Voronov:2010hj} or they are utilized for integration of Courant algebroids \cite{li2011integration, vsevera2015integration}. In our treatment of graded geometry, we follow the definitions and formalism established in \cite{vysoky2021global}. Let us point out that recently a similar treatment appeared in \cite{2021arXiv210813496K}. See also \cite{mehta2006supergroupoids, jubin2019differential, fairon2017introduction}.
This paper aims to develop a graded version of generalized geometry. In particular, we will study structures on the direct sum $T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$, where $\mathcal{M}$ is a general graded manifold and $T[\ell]\mathcal{M}$ and $T^{\ast}\mathcal{M}$ are its (degree shifted) tangent and cotangent bundles. More generally, we are going to introduce a concept of a graded Courant algebroid and provide non-trivial examples. In this way, we obtain a nice geometrical realization of (graded) Loday brackets introduced in \cite{Kosmann1996}.
The paper is organized as follows:
In \S 1, we summarize a basic knowledge of graded geometry necessary for this paper. In particular, we give a definition of a graded manifold and basic examples of a graded domain and a degree shifted vector bundle. Graded vector bundles are introduced together with an example of a tangent bundle to a graded manifold. We recall basic notions of graded vector bundle theory, e.g. dual graded vector bundles, degree shifted graded vector bundles, and vector bundle maps.
In \S 2, we introduce a graded version of an exterior algebra of differential forms. Rather then taking a usual path using local coordinates, we bring an equivalent global and fully algebraic description. We provide explicit expressions for the exterior differential, Lie derivative and interior product. A full set of Cartan relations is given. We introduce algebras of both skew-symmetric and symmetric multivector fields and show that they are equipped with a canonical Schouten-Nijenhuis bracket. We show that graded Poisson structures correspond to skew-symmetric (even case) or symmetric (odd case) bivector fields.
In \S 3, we define quadratic graded vector bundles and the main subject of this paper, graded Courant algebroids. Their basic properties following from the axioms are deduced. We prove that the generalized tangent bundle $T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$ can be equipped with a canonical structure of a graded Courant algebroid of degree $\ell$, generalizing a notion of a well-known Dorfman bracket (twisted by a closed $3$-form $H$). We show how to generalize the Ševera classification \cite{Severa:2017oew} of exact Courant algebroids to the graded case.
Dirac structures are one of the most important sub-structures of Courant algebroids. In \S 4, we define their graded analogue. This requires one to recall the notion of subbundles of graded vector bundles. We examine maximal isotropic subspaces of quadratic vector spaces and use this notion to define maximal isotropic subbundles. Finally, we define Dirac structures on graded Courant algebroids. We prove that a graph of a vector bundle map $\Pi^{\sharp}: T^{\ast}\mathcal{M} \rightarrow T[\ell]\mathcal{M}$ defines a Dirac structure with respect to the $H$-twisted Dorfman bracket, if and only if it corresponds to the unique $H$-twisted graded Poisson structure of degree $\ell$.
In \S 5, we show how generalized complex structures can be defined as certain involutive subbundles of complexified Courant algebroids. Equivalently, they correspond to orthogonal endomorphisms satisfying $\mathcal{J}^{2} = - \mathbbm{1}_{\mathcal{E}}$ whose eigenspaces form subbundles (this has to be assumed in the graded setting) with a vanishing Nijenhuis tensor. We show that a symplectic form $\omega$ of degree $-\ell$, viewed as a vector bundle map $\omega^{\flat}: T[\ell]\mathcal{M} \rightarrow T^{\ast}\mathcal{M}$, can be used to construct an example of a generalized complex structure. Similarly, so does a complex structure $J: T\mathcal{M} \rightarrow T\mathcal{M}$.
In the graded setting, a graded Courant algebroid $\mathcal{E}$ can be equipped with a degree $1$ differential $\Delta: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ forming a graded derivation of the bracket $[\cdot,\cdot]_{\mathcal{E}}$. It turns out that one has to impose a compatibility with other structures. We obtain a notion of differential graded Courant algebroids, discussed in \S 6. We show how so called ``homological sections'' of $\mathcal{E}$ can be used to produce $\Delta$. We classify all differentials on exact graded Courant algebroids. One can impose the $\Delta$-compatibility of Dirac structures and generalized complex structures. We show that this leads to a generalization of QP-manifolds and differential graded symplectic manifolds.
Finally, in \S 7, we give a definition of a graded Lie algebroid and show how it induces certain structures on its algebras of forms and multivector fields (we have actually moved the details to the appendix \ref{sec_appendixA}). This allows one to generalize a notion of Lie bialgebroids \cite{mackenzie1994, kosmann1995exact} to the graded setting. It turns out that every graded Lie bialgebroid can be used to produce a graded Courant algebroid, called its double. This is a generalization of the famous result \cite{liu1997manin}. We show how the graded Dorfman bracket can be viewed as a double of a Lie bialgebroid. It is argued how a graded Poisson manifold induces a graded Lie bialgebroid. Finally, graded Lie bialgebroids over a single point lead to the notion of graded Lie bialgebras. We show how this definition can be restated in terms of a Chevalley-Eilenberg cohomology of graded Lie algebras and give a non-trivial example of such structure.
\section{Elements of graded geometry}
In this section, we shall very briefly recall notions of graded geometry required in this paper. For a detailed exposition, we refer the reader to \cite{vysoky2021global}.
In general, we use the following somewhat unusual approach to graded algebra. Each graded object, say a \textbf{graded vector space} $V$, is always a sequence $V = \{ V_{j} \}_{j \in \mathbb{Z}}$, where $V_{j}$ is an ordinary vector space for each $j \in \mathbb{Z}$. If there a unique $j \in \mathbb{Z}$, such that $v \in V_{j}$, we write simply $v \in V$ and declare $|v| := j$. In particular, we do not consider inhomogeneous elements. Morphisms of graded objects can be described in a similar manner. For example, a \textbf{graded linear map $\varphi: V \rightarrow W$ of degree $|\varphi|$} is a sequence $\varphi = \{ \varphi_{j} \}_{j \in \mathbb{Z}}$, where $\varphi_{j}: V_{j} \rightarrow W_{j + |\varphi|}$ for every $j \in \mathbb{Z}$. We again use a simplified notation $\varphi(v)$ instead of $\varphi_{|v|}(v)$. Graded vector spaces together with graded linear maps of degree zero form a category $\gVect$. By a \textbf{graded dimension} of $V$, we mean a sequence $\gdim(V) := ( \dim(V_{j}))_{j \in \mathbb{Z}}$. We say that $V$ is finite-dimensional, if $\sum_{j \in \mathbb{Z}} \dim(V_{j}) < \infty$.
For our purposes, the most important category is the one of \textbf{graded commutative associative algebras}, denoted as $\gcAs$, where each object is a graded vector space $A = \{ A_{j} \}_{j \in \mathbb{Z}}$ equipped with an $\mathbb{R}$-bilinear product $\cdot: A \times A \rightarrow A$ having the following properties:
\begin{enumerate}[(i)]
\item It is of degree zero, that is $|a \cdot b| = |a| + |b|$.
\item It is associative, that is $a \cdot (b \cdot c) = (a \cdot b) \cdot c$.
\item There exists a (unique) unit element $1 \in A$ satisfying $a \cdot 1 = 1 \cdot a = a$. One has $|1| = 0$.
\item It is graded commutative, that is $a \cdot b = (-1)^{|a||b|} b \cdot a$.
\end{enumerate}
Axioms are assumed to hold for all elements of $A$. Morphisms in the category $\gcAs$ are degree zero graded linear maps preserving products and unit elements. For other examples appearing throughout this paper, we refer the reader to \S 1 of \cite{vysoky2021global}.
A \textbf{graded manifold} $\mathcal{M} = (M, \mathcal{C}^{\infty}_{\mathcal{M}})$ of a graded dimension $(n_{j})_{j \in \mathbb{Z}}$ is a second countable Hausdorff topological space $M$ together with a sheaf $\mathcal{C}^{\infty}_{\mathcal{M}}$ of graded commutative associative algebras over $M$. Moreover, it has to have the following properties:
\begin{enumerate}[1)]
\item $(M,\mathcal{C}^{\infty}_{\mathcal{M}})$ is a graded locally ringed space, that is each stalk $\mathcal{C}^{\infty}_{\mathcal{M},m}$ is a local graded ring.
\item There exists a \textbf{graded smooth atlas} $\mathcal{A} = \{ (U_{\alpha}, \varphi_{\alpha}) \}_{\alpha \in I}$, that is $M = \cup_{\alpha \in I} U_{\alpha}$, and for each $\alpha \in I$, $\varphi_{\alpha}: \mathcal{M}|_{U_{\alpha}} \rightarrow \hat{U}_{\alpha}^{(n_{j})}$ is an isomorphism of graded locally ringed spaces. By $\hat{U}_{\alpha}^{(n_{j})}$ we denote a graded domain over $\hat{U}_{\alpha} \subseteq \mathbb{R}^{n_{0}}$ of a graded dimension $(n_{j})_{j \in \mathbb{Z}}$, see the example below. Every such pair $(U,\varphi)$ is called a \textbf{graded chart} for $\mathcal{M}$ over $U$.
\end{enumerate}
Sections of the sheaf $\mathcal{C}^{\infty}_{\mathcal{M}}$ are usually called \textbf{functions on a graded manifold $\mathcal{M}$}.
\begin{example}
Let $(n_{j})_{j \in \mathbb{Z}}$ be a sequence of non-negative integers, such that $n := \sum_{j \in \mathbb{Z}} n_{j} < \infty$. Let $n_{\ast} = n - n_{0}$ and consider a collection of variables $\{ \xi_{\mu} \}_{\mu = 1}^{n_{\ast}}$ where each of them is assigned a degree $|\xi_{\mu}| \in \mathbb{Z}$. Suppose that $n_{j} = \#\{ \mu \in \{1, \dots, n_{\ast} \} \; | \; |\xi_{\mu}| = j \}$ for each $j \in \mathbb{Z} - \{0\}$. The variables are assumed to commute according to the rule
\begin{equation} \label{eq_commrelation}
\xi_{\mu} \xi_{\nu} = (-1)^{|\xi_{\mu}||\xi_{\nu}|} \xi_{\nu} \xi_{\mu}.
\end{equation}
Let $\hat{U} \in \Op(\mathbb{R}^{n_{0}})$ be an open subset of $\mathbb{R}^{n_{0}}$. Elements of $\mathcal{C}^{\infty}_{(n_{j})}(\hat{U}) \in \gcAs$ are formal power series in $\{ \xi_{\mu} \}_{\mu=1}^{n_{\ast}}$ with coefficients in the algebra $\mathcal{C}^{\infty}_{n_{0}}(\hat{U})$ of smooth functions on $\hat{U}$. In other words, one has $f \in \mathcal{C}^{\infty}_{(n_{j})}(\hat{U})$ of degree $|f|$, if
\begin{equation} \label{eq_formalpowseries}
f = \sum_{\mathbf{p} \in \mathbb{N}_{|f|}^{n_{\ast}}} f_{\mathbf{p}} \xi^{\mathbf{p}},
\end{equation}
where $N_{|f|}^{n_{\ast}} = \{ \mathbf{p} = (p_{1}, \dots, p_{n_{\ast}}) \in (\mathbb{N}_{0})^{n_{\ast}} \; | \; \sum_{\mu=1}^{n_{\ast}} p_{\mu} |\xi_{\mu}| = |f| \text{ and } p_{\mu} \in \{0,1\} \text{ for } |\xi_{\mu}| \text{ odd} \}$, $f_{\mathbf{p}} \in \mathcal{C}^{\infty}_{n_{0}}(\hat{U})$ and $\xi^{\mathbf{p}} = (\xi_{1})^{p_{1}} \dots (\xi_{n_{\ast}})^{p_{n_{\ast}}}$. We declare $(\xi_{\mu})^{0} = 1$. This is a unique representation of every formal power series of degree $|f|$ due to (\ref{eq_commrelation}) and the fact that $\xi_{\mathbf{p}} \neq 0$ for each $\mathbf{p} \in \mathbb{N}^{n_{\ast}}_{|f|}$.
Let $f,g \in \mathcal{C}^{\infty}_{(n_{j})}(\hat{U})$. Their product $f \cdot g$ is an element of degree $|f| + |g|$ given by the formula
\begin{equation}
f \cdot g := \hspace{-2mm} \sum_{\mathbf{p} \in \mathbb{N}^{n_{\ast}}_{|f|+|g|}} \hspace{-2mm} (f \cdot g)_{\mathbf{p}} \xi^{\mathbf{p}}, \text{ where } (f \cdot g)_{\mathbf{p}} := \sum_{\substack{\mathbf{q} \in \mathbb{N}^{n_{\ast}}_{|f|} \\ \mathbf{q} \leq \mathbf{p}}} \epsilon^{\mathbf{q},\mathbf{p}-\mathbf{q}} f_{\mathbf{q}} g_{\mathbf{p}-\mathbf{q}},
\end{equation}
and $\epsilon^{\mathbf{q},\mathbf{p}-\mathbf{q}} \in \{-1,1\}$ is the unique sign obtained by reordering $\xi^{\mathbf{q}} \xi^{\mathbf{p}-\mathbf{q}}$ into $\xi^{\mathbf{p}}$ in accordance with (\ref{eq_commrelation}). In other words, up to signs, $f \cdot g$ is a usual product of formal power series. It is not difficult to see that this makes $\mathcal{C}^{\infty}_{(n_{j})}(\hat{U})$ into a graded commutative associative algebra (with the unit). This algebra can be defined in a more abstract way, see \S 1.3 of \cite{vysoky2021global}.
Now, let $\hat{U} \in \Op(\mathbb{R}^{n_{0}})$ be fixed. For each $\hat{V} \in \Op(\hat{U})$, the assignment
\begin{equation}
\hat{V} \mapsto \mathcal{C}^{\infty}_{(n_{j})}(\hat{V})
\end{equation}
makes $\mathcal{C}^{\infty}_{(n_{j})}$ into a sheaf on $\hat{U}$ valued in $\gcAs$. Sheaf restrictions are given by restricting the coefficient functions in the expansion (\ref{eq_formalpowseries}). In fact, $\hat{U}^{(n_{j})} := (\hat{U}, \mathcal{C}^{\infty}_{(n_{j})})$ becomes a graded locally ringed space called a \textbf{graded domain of a graded dimension $(n_{j})_{j \in \mathbb{Z}}$}.
\end{example}
Let $\mathcal{M} = (M,\mathcal{C}^{\infty}_{\mathcal{M}})$ and $\mathcal{N} = (N, \mathcal{C}^{\infty}_{\mathcal{N}})$ be a pair of graded manifolds. We say that $\phi: \mathcal{M} \rightarrow \mathcal{N}$ is a \textbf{graded smooth map}, if $\phi = (\ul{\phi}, \phi^{\ast})$ for a continuous map $\ul{\phi}: M \rightarrow N$ and a sheaf morphism $\phi^{\ast}: \mathcal{C}^{\infty}_{\mathcal{N}} \rightarrow \ul{\phi}_{\ast} \mathcal{C}^{\infty}_{\mathcal{M}}$. In other words, for each $V \in \Op(N)$, we obtain a graded algebra morphism $\phi^{\ast}_{V}: \mathcal{C}^{\infty}_{\mathcal{N}}(V) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}( \ul{\phi}^{-1}(V))$ natural in $V$. Finally, the map $[g]_{\ul{\phi}(m)} \mapsto [\varphi^{\ast}_{V}(g)]_{m}$ must define a morphism of local graded rings for all $m \in M$, $V \in \Op_{\ul{\phi}(m)}(N)$ and $g \in \mathcal{C}^{\infty}_{\mathcal{N}}(V)$.
Hence in the above definition, $\varphi_{\alpha}: \mathcal{M}|_{U_{\alpha}} \rightarrow \hat{U}^{(n_{j})}_{\alpha}$ becomes a graded smooth map. It is easy to see that $\ul{\mathcal{A}} = \{ (U_{\alpha}, \ul{\varphi_{\alpha}}) \}_{\alpha \in I}$ defines an ordinary smooth atlas on $M$, hence making it into an $n_{0}$-dimensional smooth manifold. It then follows that for any graded smooth map $\phi: \mathcal{M} \rightarrow \mathcal{N}$, the underlying continuous map $\ul{\phi}: M \rightarrow N$ is smooth.
One can view $M$ as a (trivially) graded manifold. There exists a canonical graded smooth map $i: M \rightarrow \mathcal{M}$ over $\mathbbm{1}_{M}: M \rightarrow M$. For each $U \in \Op(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$, the ordinary smooth function $\ul{f} := i^{\ast}_{U}(f) \in \mathcal{C}^{\infty}_{M}(U)$ is called the \textbf{body of $f$}. For any $m \in U$, we then write $f(m) := \ul{f}(m)$. Note that $\ul{f} = 0$ whenever $|f| \neq 0$.
For $\mathcal{M} = \hat{U}^{(n_{j})}$, $\hat{V} \in \Op(\hat{U})$ and $f \in \mathcal{C}^{\infty}_{(n_{j})}(\hat{V})$ in the form (\ref{eq_formalpowseries}) with $|f| = 0$, one has $\ul{f} := f_{\mathbf{0}}$, where $\mathbf{0} := (0,\dots,0)$. For general $\mathcal{M}$, $\ul{f}$ is defined precisely to have this form when composed with graded charts and one only has to verify that this makes sense globally.
\begin{example}
Let $\pi: E \rightarrow M$ be a real vector bundle of a finite rank.
\begin{enumerate}[(i)]
\item Let $k \in \mathbb{Z}$ be a non-zero odd integer. Let us define a sheaf $\mathcal{C}^{\infty}_{E[k]}$ for each $U \in \Op(M)$ as
\begin{equation}
(\mathcal{C}^{\infty}_{E[k]}(U))_{q} := \left\{ \begin{array}{cl}
\Gamma_{\Lambda^{r} E^{\ast}}(U) & \text{ whenever } q = rk \text{ for some $r \in \mathbb{Z}$}, \\
0 & \text{ otherwise. }
\end{array} \right.
\end{equation}
There are thus non-zero functions on $E[k]$ only in integer multiples of $k$. Sheaf restrictions are inherited from the sheaf of sections $\Gamma_{\Lambda^{r}E^{\ast}}$ and the graded algebra product is obtained from the exterior product $\^$ on $\Gamma_{\Lambda^{\bullet}E^{\ast}}(U)$.
\item Let $k \in \mathbb{Z}$ be a non-zero even integer. Let us define a sheaf $\mathcal{C}^{\infty}_{E[k]}$ for each $U \in \Op(M)$ as
\begin{equation}
(\mathcal{C}^{\infty}_{E[k]}(U))_{q} := \left\{ \begin{array}{cc}
\Gamma_{S^{r} E^{\ast}}(U) & \text{ whenever } q = rk \text{ for some $r \in \mathbb{Z}$}, \\
0 & \text{ otherwise. }
\end{array} \right.
\end{equation}
This time restrictions are inherited from $\Gamma_{S^{r}E^{\ast}}$ and the graded algebra product is given by the symmetrized tensor product $\odot$.
\end{enumerate}
In both cases, $E[k] := (M, \mathcal{C}^{\infty}_{E[k]})$ becomes a graded locally ringed space. Let $(n_{j})_{j \in \mathbb{Z}}$ be a sequence given by $n_{0} := \dim(M)$, $n_{k} := \rk(E)$ and $n_{j} := 0$ for $j \notin \{0,k\}$. It is straightforward to obtain a graded smooth atlas $\mathcal{A}$ for $E[k]$ using the atlas for $M$ and a local trivialization for the vector bundle $E$. $E[k]$ thus becomes a graded manifold of a graded dimension $(n_{j})_{j \in \mathbb{Z}}$ called the \textbf{degree $k$ shift of a vector bundle $E$}. Note that functions of degree $0$ are ordinary functions on $M$ and functions of degree $k$ are sections of the dual vector bundle $E^{\ast}$.
\end{example}
Vector fields can be defined as sections of the sheaf of graded derivations of $\mathcal{C}^{\infty}_{\mathcal{M}}$. In other words, for each $U \in \Op(M)$ define
\begin{equation}
\mathfrak{X}_{\mathcal{M}}(U) := \gDer( \mathcal{C}^{\infty}_{\mathcal{M}}(U)).
\end{equation}
We say that $X \in \gDer(\mathcal{C}^{\infty}_{\mathcal{M}}(U))$ is a \textbf{vector field of degree $|X|$}. By definition, it is a graded linear map $X: \mathcal{C}^{\infty}_{\mathcal{M}}(U) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(U)$ of degree $|X|$ and for all $f,g \in \mathcal{C}^{\infty}_{\mathcal{M}}(U)$, one has
\begin{equation}
X(fg) = X(f)g + (-1)^{|X||f|} f X(g).
\end{equation}
$\mathfrak{X}_{\mathcal{M}}(U)$ has a natural structure of a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-module given by $(f \triangleright X)(g) := f X(g)$. There is a unique way how to make the assignment $U \mapsto \mathfrak{X}_{\mathcal{M}}(U)$ into a sheaf of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules, such that for all $V \subseteq U \subseteq M$, $X \in \mathfrak{X}_{\mathcal{M}}(U)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(U)$, one has
\begin{equation}
X|_{V}(f|_{V}) = X(f)|_{V}.
\end{equation}
Similarly to ordinary differential geometry, one utilizes the partitions of unity to define $X|_{V}$.
\begin{definice}[\textbf{Graded vector bundles over graded manifolds}]
Let $\mathcal{M} = (M,\mathcal{C}^{\infty}_{\mathcal{M}})$ be a graded manifold. Suppose we are given a sheaf of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules $\Gamma_{\mathcal{E}}$ that is locally freely and finitely generated of a finite graded rank $(r_{j})_{j \in \mathbb{Z}}$. In other words, $\Gamma_{\mathcal{E}}$ is a sheaf of graded vector spaces having the following properties:
\begin{enumerate}[(i)]
\item For each $U \in \Op(M)$, $\Gamma_{\mathcal{E}}(U)$ is a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-module, and restrictions are compatible with the action of $\mathcal{C}^{\infty}_{\mathcal{M}}$.
\item $r := \sum_{j \in \mathbb{Z}} r_{j} < \infty$, and for any $m \in M$, there exists $U \in \Op_{m}(M)$ and a collection $\{ \Phi_{\lambda} \}_{\lambda = 1}^{r}$ of sections in $\Gamma_{\mathcal{E}}(U)$, such that $r_{j} = \# \{ \lambda \in \{1,\dots,r\} \; | \; |\Phi_{\lambda}| = j \}$ for each $j \in \mathbb{Z}$. Moreover, for each $V \in \Op(U)$, every $\psi \in \Gamma_{\mathcal{E}}(V)$ can be uniquely decomposed as $\psi = \sum_{\lambda=1}^{r} f^{\lambda} \Phi_{\lambda}|_{V}$ for $f^{\lambda} \in \mathcal{C}^{\infty}_{\mathcal{M}}(V)$ satisfying $|f^{\lambda}| + |\Phi_{\lambda}| = |\psi|$.
\end{enumerate}
We say that $\Gamma_{\mathcal{E}}$ is a sheaf of sections of a \textbf{graded vector bundle} $\mathcal{E}$ over $\mathcal{M}$. Its \textbf{graded rank} is defined as $\grk(\mathcal{E}) := (r_{j})_{j \in \mathbb{Z}}$. $\{ \Phi_{\lambda} \}_{\lambda = 1}^{r}$ is called a \textbf{local frame for $\mathcal{E}$ over $U$}.
\end{definice}
\begin{rem} \label{rem_noSerreswan}
Since we define graded vector bundles rather algebraically, one may be tempted to define graded vector bundles as finitely generated projective graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-modules. However, although $\Gamma_{\mathcal{E}}(M)$ is always a finitely generated projective graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module, the converse is not true in the graded (or super) setting.
\end{rem}
\begin{example}
Let $K \in \gVect$ be a finite-dimensional graded vector space. For each $U \in \Op(M)$, let $\Gamma_{\mathcal{M} \times K}(U) := \mathcal{C}^{\infty}_{\mathcal{M}}(U) \otimes_{\mathbb{R}} K$. Suppose that $(r_{j})_{j \in \mathbb{Z}} := \gdim(K)$ and let $r := \sum_{j \in \mathbb{Z}} r_{j}$. Let $(\vartheta_{\lambda})_{\lambda=1}^{r}$ be a total basis for $K$. By definition $\# \{ \lambda \in \{1,\dots,r\} \; | \; |\vartheta_{\lambda}| = j \} = r_{j}$ for each $j \in \mathbb{Z}$. It follows that $\{ 1 \otimes \vartheta_{\lambda} \}_{\lambda =1}^{r}$ forms a local frame for $\mathcal{M} \times K$ over $M$. Hence $\mathcal{M} \times K$ forms a graded vector bundle, called a \textbf{trivial vector bundle}.
Note that every local frame for a graded vector bundle $\mathcal{E}$ over $U$ is equivalent to the $\mathcal{C}^{\infty}_{\mathcal{M}}|_{U}$-linear sheaf isomorphism $\Gamma_{\mathcal{E}}|_{U} \cong \Gamma_{\mathcal{M} \times \mathbb{R}^{(r_{j})}}|_{U}$. Recall that $(\mathbb{R}^{(r_{j})})_{k} := \mathbb{R}^{r_{k}}$ for all $k \in \mathbb{Z}$.
\end{example}
Let $(n_{j})_{j \in \mathbb{Z}}$ be a graded dimension of $\mathcal{M}$ and suppose $\mathcal{E}$ is a graded vector bundle over $\mathcal{M}$ of a graded rank $(r_{j})_{j \in \mathbb{Z}}$. One can show that there is a unique (up to a graded diffeomorphism) graded manifold $\mathcal{E} = (E, \mathcal{C}^{\infty}_{\mathcal{E}})$ of a graded dimension $(n_{j} + r_{-j})_{j \in \mathbb{Z}}$ together with a graded smooth map $\pi: \mathcal{E} \rightarrow \mathcal{M}$. $\mathcal{E}$ is called the \textbf{total space} of a graded vector bundle. One can recover the sheaf $\Gamma_{\mathcal{E}}$ from its sheaf of functions $\mathcal{C}^{\infty}_{\mathcal{E}}$. However, we will rarely actually use this graded manifold explicitly. Note that $\ul{\pi}: E \rightarrow M$ is an ordinary vector bundle of rank $r_{0}$.
\begin{example} \label{ex_tangent}
Let $\Gamma_{T\mathcal{M}} := \mathfrak{X}_{\mathcal{M}}$ and suppose $(n_{j})_{j \in \mathbb{Z}} = \gdim(\mathcal{M})$.
Let $(U,\varphi)$ be a graded local chart for $\mathcal{M}$ over $U$, that is $\varphi: \mathcal{M}|_{U} \rightarrow \hat{U}^{(n_{j})}$ is a graded diffeomorphism. If $n = \sum_{j \in \mathbb{Z}} n_{j}$ and $n_{\ast} = n - n_{0}$, we have coordinate functions $\{ x^{i} \}_{i=1}^{n_{0}} \subseteq \mathcal{C}^{\infty}_{(n_{j})}(\hat{U})$ of degree $0$ corresponding to standard coordinates in $\mathbb{R}^{n_{0}}$, and ``purely graded'' coordinate functions $\{ \xi_{\mu} \}_{\mu=1}^{n_{\ast}} \subseteq \mathcal{C}^{\infty}_{(n_{j})}(\hat{U})$. It is sometimes convenient to denote them altogether as $\{ \mathbbm{z}^{A} \}_{A=1}^{n}$. It is customary to use the same symbols to denote the corresponding functions $\mathbbm{z}^{A} := \varphi^{\ast}_{\hat{U}}(\mathbbm{z}^{A})$ in $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$.
It follows that for each $A \in \{1,\dots, n\}$, there is a vector field $\frac{\partial}{\partial \mathbbm{z}^{A}} \in \mathfrak{X}_{\mathcal{M}}(U)$ of degree $-|\mathbbm{z}^{A}|$ determined uniquely by the formula
\begin{equation}
\frac{\partial}{\partial \mathbbm{z}^{A}}(\mathbbm{z}^{B}) := \delta_{A}^{B},
\end{equation}
for every $B \in \{1,\dots,n\}$. One can show that $\{ \frac{\partial}{\partial \mathbbm{z}^{A}} \}_{A=1}^{n}$ forms a local frame for $T\mathcal{M}$ over $U$, thus making $T\mathcal{M}$ into a graded vector bundle of a graded rank $(n_{-j})_{j \in \mathbb{Z}}$. See \S 4.2 in \cite{vysoky2021global} for details. $T\mathcal{M}$ is called the \textbf{tangent bundle} of $\mathcal{M}$.
\end{example}
\begin{example} \label{ex_dual}
Let $\mathcal{E}$ be a graded vector bundle over $\mathcal{M}$ of a graded rank $(r_{j})_{j \in \mathbb{Z}}$. Let $\Gamma_{\mathcal{E}}$ be its sheaf of sections. For each $U \in \Op(M)$, let $\Gamma_{\mathcal{E}^{\ast}}(U)$ be a graded vector space of $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-linear maps from $\Gamma_{\mathcal{E}}(U)$ to $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$, that is $\alpha \in \Gamma_{\mathcal{E}^{\ast}}(U)$, if $\alpha: \Gamma_{\mathcal{E}}(U) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(U)$ is $\mathbb{R}$-linear of degree $|\alpha|$, and
\begin{equation}
\alpha(f \psi) = (-1)^{|\alpha||f|} f \alpha(\psi),
\end{equation}
for all $\psi \in \Gamma_{\mathcal{E}}(U)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(U)$. There is an obvious graded $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-module structure on $\Gamma_{\mathcal{E}^{\ast}}(U)$ and one can use partitions of unity to make $\Gamma_{\mathcal{E}^{\ast}}$ into a sheaf of $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules. If $\{ \Phi_{\lambda}\}_{\lambda=1}^{r}$ is a local frame for $\mathcal{E}$ over $U$, one can consider $\Phi^{\lambda} \in \Gamma_{\mathcal{E}^{\ast}}(U)$ determined uniquely by the requirement $\Phi^{\lambda}(\Phi_{\kappa}) := \delta^{\lambda}_{\kappa}$ for each $\kappa \in \{1,\dots,r\}$. It follows that $\{ \Phi^{\lambda} \}_{\lambda =1}^{r}$ is a local frame for $\mathcal{E}^{\ast}$ over $U$, making $\Gamma_{\mathcal{E}^{\ast}}$ into a sheaf of sections of the \textbf{graded vector bundle $\mathcal{E}^{\ast}$ dual to $\mathcal{E}$}. Note that $\grk(\mathcal{E}^{\ast}) = (r_{-j})_{j \in \mathbb{Z}}$.
\end{example}
\begin{example} \label{ex_degreeshift}
Let $\mathcal{E}$ be a graded vector bundle over $\mathcal{M}$ of a graded rank $(r_{j})_{j \in \mathbb{Z}}$. Let $\Gamma_{\mathcal{E}}$ be its sheaf of sections. Let $\ell \in \mathbb{Z}$. For each $U \in \Op(M)$, consider a graded vector space $\Gamma_{\mathcal{E}[\ell]}(U) := \Gamma_{\mathcal{E}}(U)[\ell]$. For each $j \in \mathbb{Z}$, we thus have $(\Gamma_{\mathcal{E}}(U)[\ell])_{j} := (\Gamma_{\mathcal{E}}(U))_{j + \ell}$.
A graded $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-module structure on $\Gamma_{\mathcal{E}[\ell]}(U)$ is for each $\psi \in \Gamma_{\mathcal{E}[\ell]}(U)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(U)$ given by $f \triangleright' \psi := (-1)^{|f|\ell} f \psi$. On the right-hand side, there is the original action on $\Gamma_{\mathcal{E}}(U)$. It is obvious that every local frame $(\Phi_{\lambda})_{\lambda=1}^{r}$ for $\mathcal{E}$ over $U$ can be viewed as a local frame for $\mathcal{E}[\ell]$ over $U$, whence $\Gamma_{\mathcal{E}[\ell]}$ becomes a sheaf of sections of a \textbf{degree $\ell$ shift $\mathcal{E}[\ell]$} of $\mathcal{E}$. Note that $\grk(\mathcal{E}[\ell]) = (r_{j+\ell})_{j \in \mathbb{Z}}$.
\end{example}
\begin{definice}
Let $\mathcal{E}$ and $\mathcal{E}'$ be two graded vector bundles over $\mathcal{M}$. By a \textbf{vector bundle map $F: \mathcal{E} \rightarrow \mathcal{E}'$ of degree $|F|$}, we mean a $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear map $F: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}'}(M)$ of degree $|F|$. In other words, it satisfies
\begin{enumerate}
\item $|F(\psi)| = |F| + |\psi|$ for all $\psi \in \Gamma_{\mathcal{E}}(M)$;
\item for each $\psi \in \Gamma_{\mathcal{E}}(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$, one has $F(f \psi) = (-1)^{|f||F|} f F(\psi)$.
\end{enumerate}
If the degree is not specified, we assume $|F| = 0$.
\end{definice}
\begin{rem}
A degree $|F|$ vector bundle map $F: \mathcal{E} \rightarrow \mathcal{E}'$ can be equivalently viewed as a degree $0$ vector bundle map $F: \mathcal{E} \rightarrow \mathcal{E}'[|F|]$ or $F: \mathcal{E}[-|F|] \rightarrow \mathcal{E}'$.
\end{rem}
\begin{rem} \label{rem_VBmorphisms}
In this paper, we consider only vector bundle maps over the identity $\mathbbm{1}_{\mathcal{M}}: \mathcal{M} \rightarrow \mathcal{M}$ for two graded vector bundles over the same graded manifold. One can show that $F$ can be uniquely extended to a $\mathcal{C}^{\infty}_{\mathcal{M}}$-linear sheaf morphism $F: \Gamma_{\mathcal{E}} \rightarrow \Gamma_{\mathcal{E}'}$. Note that there is a definition of vector bundle maps for any two graded vector bundles, formulated using their respective duals, see \S 5.1 of \cite{vysoky2021global}.
\end{rem}
\section{Exterior and mutltivector field algebras} \label{sec_extmutvectoralg}
In this section, let $\mathcal{M}$ be a fixed graded manifold. In graded geometry, there is a convenient way to introduce the exterior algebra sheaf $\Omega_{\mathcal{M}}$ as a sheaf of functions on the graded manifold $T[1+s]\mathcal{M}$. Here $s$ is a large enough non-negative even integer, and $T[1+s]\mathcal{M}$ is the total space corresponding to the degree shift of the tangent bundle $T\mathcal{M}$, see Example \ref{ex_tangent} and Example \ref{ex_degreeshift}. For an explanation of the strange additional degree shift $s$, see \S 6 of \cite{vysoky2021global}. In this paper, we give an equivalent more algebraic definition.
\begin{definice} \label{def_pform}
Let $p \in \mathbb{N}_{0}$. By a \textbf{$p$-form} $\omega$ on $\mathcal{M}$ of degree $|\omega|$, we mean a $p$-linear map
\begin{equation}
\omega: \mathfrak{X}_{\mathcal{M}}(M) \times \dots \times \mathfrak{X}_{\mathcal{M}}(M) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(M)
\end{equation}
satisfying $|\omega(X_{1},\dots,X_{p})| = |\omega| + |X_{1}| + \dots + |X_{p}|$ for all $X_{1},\dots,X_{p} \in \mathfrak{X}_{\mathcal{M}}(M)$, such that
\begin{equation}
\omega(f X_{1}, \dots, X_{p}) = (-1)^{|f||\omega|} f \omega(X_{1},\dots,X_{p}),
\end{equation}
\begin{equation} \label{eq_pform_skewsym}
\omega(X_{1}, \dots, X_{i},X_{i+1},\dots,X_{p}) = - (-1)^{|X_{i}||X_{i+1}|} \omega(X_{1}, \dots, X_{i+1},X_{i},\dots,X_{p}),
\end{equation}
for all $X_{1},\dots,X_{p} \in \mathfrak{X}_{\mathcal{M}}(M)$, $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $i \in \{1,\dots,p-1\}$. For each $p \in \mathbb{N}_{0}$, $p$-forms form a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module which we denote as $\Omega^{p}_{\mathcal{M}}(M)$. We declare $\Omega^{0}_{\mathcal{M}}(M) := \mathcal{C}^{\infty}_{\mathcal{M}}(M)$.
\end{definice}
For each $U \in \Op(M)$, one can declare $\Omega^{p}_{\mathcal{M}}(U) := \Omega^{p}_{\mathcal{M}|_{U}}(U)$. Using partitions of unity, one can make the assignment $U \mapsto \Omega^{p}_{\mathcal{M}}(U)$ into a sheaf of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules, so that
\begin{equation}
\omega|_{V}( X_{1}|_{V}, \dots, X_{p}|_{V}) = \omega(X_{1},\dots,X_{p})|_{V},
\end{equation}
for all $V \subseteq U \subseteq M$, $\omega \in \Omega^{1}_{\mathcal{M}}(U)$ and $X_{1},\dots,X_{p} \in \mathfrak{X}_{\mathcal{M}}(U)$. In this way we obtain a \textbf{sheaf $\Omega^{p}_{\mathcal{M}}$ of $p$-forms on $\mathcal{M}$}.
\begin{rem}
In \S 6.2 of \cite{vysoky2021global}, we have obtained $p$-forms as a certain subsheaf of $\mathcal{C}^{\infty}_{T[1+s]\mathcal{M}}$. The two sheaves are naturally isomorphic, except that one has to use an alternative $\mathbb{Z}$-grading different from the one coming with the sheaf $\mathcal{C}^{\infty}_{T[1+s]\mathcal{M}}$. See Proposition 6.4-$(ii)$ there.
\end{rem}
The (exterior) product of forms can be most easily obtained from the graded algebra product of the sheaf $\mathcal{C}^{\infty}_{T[1+s]\mathcal{M}}$. Since we will not actually use it in this paper, we only state the result. Note that the full exterior algebra $\Omega_{\mathcal{M}}(M)$ is \textit{not} a direct product $\prod_{p=0}^{\infty} \Omega^{p}_{\mathcal{M}}(M)$ with the $\mathbb{Z}$-grading we use here. For this reason, we will always talk exclusively about ``homogeneous'' forms.
\begin{tvrz} \label{tvrz_extproduct}
There is an $\mathbb{R}$-bilinear product $\^: \Omega^{p}_{\mathcal{M}}(M) \times \Omega^{q}_{\mathcal{M}}(M) \rightarrow \Omega^{p+q}_{\mathcal{M}}(M)$ for each $p,q \in \mathbb{N}_{0}$. It has the following properties:
\begin{enumerate}[(i)]
\item It is compatible with the grading introduced in Definition \ref{def_pform}, that is $|\omega \^ \tau| = |\omega| + |\tau|$ for all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$.
\item One has $\omega \^ \tau = (-1)^{(p + |\omega|)(q + |\tau|)} \tau \^ \omega$ for all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$.
\item It is distributive with respect to the addition: $(\omega + \omega') \^ \tau = \omega \^ \tau + \omega' \^ \tau$ for all $\omega,\omega' \in \Omega^{p}_{\mathcal{M}}(M)$ with $|\omega| = |\omega'|$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$.
\item For all $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M) \equiv \Omega^{0}_{\mathcal{M}}(M)$ and all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$, one has $f \^ \omega = f \omega$.
\item It satisfies $\omega \^ (\tau \^ \eta) = (\omega \^ \tau) \^ \eta$ for all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$, $\tau \in \Omega^{q}_{\mathcal{M}}(M)$ and $\eta \in \Omega^{r}_{\mathcal{M}}(M)$.
\item For each $U \in \Op(M)$, one has $(\omega \^ \tau)|_{U} = \omega|_{U} \^ \tau|_{U}$.
\end{enumerate}
$\^$ is usually called the \textbf{exterior product}.
\end{tvrz}
Explicit formulas for $\omega \^ \tau$ evaluated on a $(p+q)$-tuple of vector fields are in fact quite complicated. Should one actually need them, they can be obtained using the interior product which we will now introduce.
\begin{tvrz} \label{tvrz_intproduct}
Let $X \in \mathfrak{X}_{\mathcal{M}}(M)$ and $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ for $p > 0$. Define $i_{X}\omega \in \Omega^{p-1}_{\mathcal{M}}(M)$ by
\begin{equation} \label{eq_defintproduct}
(i_{X}\omega)(Y_{1},\dots,Y_{p-1}) := (-1)^{(|X|-1)|\omega|} \omega(X,Y_{1},\dots,Y_{p-1}),
\end{equation}
for all $Y_{1},\dots,Y_{p-1} \in \mathfrak{X}_{\mathcal{M}}(M)$. One declares $i_{X}\omega = 0$ for $p = 0$. Then $|i_{X}\omega| = |X| + |\omega|$, and
\begin{equation} \label{eq_iXonexteriorproduct}
i_{X}(\omega \^ \tau) = (i_{X}\omega) \^ \tau + (-1)^{(|X|-1)(|\omega|+p)} \omega \^ (i_{X}\tau),
\end{equation}
for any $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$. The operator $i_{X}: \Omega^{p}_{\mathcal{M}}(M) \rightarrow \Omega^{p-1}_{\mathcal{M}}(M)$ is called the \textbf{interior product (with $X$)}.
\end{tvrz}
The interior product can be used to evaluate $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ on $X_{1},\dots,X_{p} \in \mathfrak{X}_{\mathcal{M}}(M)$. One has
\begin{equation}
\omega(X_{1},\dots,X_{p}) = (-1)^{\sum_{r=1}^{p} (|\omega|+ |X_{1}| + \dots + |X_{r-1}|)(|X_{r}|-1)} i_{X_{p}} \dots i_{X_{1}} \omega.
\end{equation}
Together with (\ref{eq_iXonexteriorproduct}), one can use this to find explicit expressions for the exterior product.
\begin{example}
Let $\omega,\tau \in \Omega^{1}_{\mathcal{M}}(M)$ and let $X_{1},X_{2} \in \mathfrak{X}_{\mathcal{M}}(M)$. Then one has
\begin{equation}
\begin{split}
(\omega \^ \tau)(X_{1},X_{2}) = & \ (-1)^{(|\omega| + |\tau|)(|X_{1}|-1) + (|\omega| + |\tau| + |X_{1}|)(|X_{2}|-1)} i_{X_{2}} i_{X_{1}} ( \omega \^ \tau) \\
= & \ (-1)^{(|\omega|+|\tau|)(|X_{1}|-1) + |\tau|(|X_{2}|-1)} i_{X_{1}}\omega \^ i_{X_{2}}\tau \\
& + (-1)^{(1 + |\tau|)(|X_{1}|-1) + (|\omega| + |\tau| + |X_{1}|)(|X_{2}|-1)} i_{X_{2}}\omega \^ i_{X_{1}} \tau \\
= & \ (-1)^{|\tau|(|X_{1}|-1)} \omega(X_{1}) \tau(X_{2}) - (-1)^{|X_{1}||X_{2}| + |\tau|(|X_{2}|-1)} \omega(X_{2}) \tau(X_{1}).
\end{split}
\end{equation}
This expression may look a bit counter-intuitive, but it is easy to check that the right-hand side indeed defines a $2$-form on $\mathcal{M}$ of degree $|\omega| + |\tau|$.
\end{example}
Next, recall that $\mathfrak{X}_{\mathcal{M}}(M)$ has a natural structure of a graded Lie algebra (of degree zero) provided by the graded commutator, defined as
\begin{equation} \label{eq_gcommutator}
[X,Y](f) := X(Y(f)) - (-1)^{|X||Y|} Y(X(f)),
\end{equation}
for all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. It can be used to define the exterior differential on forms.
\begin{tvrz} \label{tvrz_formdifferential}
Let $\omega \in \Omega_{\mathcal{M}}^{p}(M)$ for some $p \in \mathbb{N}_{0}$. Define $\mathrm{d}{\omega} \in \Omega^{p+1}_{\mathcal{M}}(M)$ by the formula
\begin{equation}
\begin{split}
\mathrm{d}{\omega} & ( X_{1},\dots,X_{p+1}) = \\
& = \ \sum_{r=1}^{p+1} (-1)^{r+1 + |\omega|(|X_{r}|-1) + |X_{r}|(|X_{1}| + \dots + |X_{r-1}|)} X_{r}( \omega(X_{1},\dots,\hat{X}_{r},\dots,X_{p+1})) \\
& + \sum_{1 \leq i<j \leq p+1} (-1)^{|\omega|+i+j} \sigma_{ij}(|X_{1}|,\dots,|X_{p+1}|) \; \omega([X_{i},X_{j}],X_{2},\dots,\hat{X}_{i},\dots,\hat{X}_{j},\dots,X_{p+1}),
\end{split}
\end{equation}
for all $X_{1},\dots,X_{p+1} \in \mathfrak{X}_{\mathcal{M}}(M)$. $\hat{X}$ denotes an omitted element. For each tuple $(q_{1},\dots,q_{p+1})$ of integers, $\sigma_{ij}(q_{1},\dots,q_{p+1})$ is the unique sign obtained by reordering the term $v_{1} \dots v_{p+1}$ to $v_{i}v_{j} v_{2} \dots \hat{v}_{i} \dots \hat{v}_{j} \dots v_{p+1}$, where we assume that $v_{k}$'s are independent graded variables which commute in accordance with the rule $v_{k} v_{\ell} = (-1)^{q_{k}q_{\ell}} v_{\ell} v_{k}$.
Then $|\mathrm{d} \omega| = |\omega|$, and for all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$, one has
\begin{equation}
\mathrm{d}( \omega \^ \tau) = \mathrm{d} \omega \^ \tau + (-1)^{p + |\omega|} \omega \^ \mathrm{d}{\tau}.
\end{equation}
Finally, $\mathrm{d}$ squares to zero, $\mathrm{d}(\mathrm{d} \omega) = 0$ for any $\omega \in \Omega^{p}_{\mathcal{M}}(M)$. The operator $\mathrm{d}: \Omega^{p}_{\mathcal{M}}(M) \rightarrow \Omega^{p+1}_{\mathcal{M}}(M)$ is called the \textbf{exterior differential}.
\end{tvrz}
It remains to define the last of the operations on forms, the Lie derivative. Again, we only state the resulting explicit formula.
\begin{tvrz} \label{tvrz_Lieder}
Let $X \in \mathfrak{X}_{\mathcal{M}}(M)$ and $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ for $p \in \mathbb{N}_{0}$. Define $\Li{X}\omega \in \Omega^{p}_{\mathcal{M}}(M)$ by
\begin{equation}
\begin{split}
(\Li{X}\omega)(Y_{1},\dots,Y_{p}) := & \ X( \omega(Y_{1},\dots,Y_{p})) \\
& - \sum_{r=1}^{p} (-1)^{(|\omega| + |Y_{1}| + \dots + |Y_{r-1}|)|X|} \omega(Y_{1},\dots,[X,Y_{r}],\dots,Y_{p}),
\end{split}
\end{equation}
for all $Y_{1},\dots,Y_{p} \in \mathfrak{X}_{\mathcal{M}}(M)$. Then $|\Li{X}\omega| = |X| + |\omega|$, and
\begin{equation}
\Li{X}(\omega \^ \tau) = \Li{X}\omega \^ \tau + (-1)^{|X|(|\omega|+p)} \omega \^ \Li{X}\tau,
\end{equation}
for all $\omega \in \Omega^{p}_{\mathcal{M}}(M)$ and $\tau \in \Omega^{q}_{\mathcal{M}}(M)$. The operator $\Li{X}: \Omega^{p}_{\mathcal{M}}(M) \rightarrow \Omega^{p}_{\mathcal{M}}(M)$ is called the \textbf{Lie derivative (along $X$)}.
\end{tvrz}
\begin{theorem}[\textbf{Cartan relations}] \label{thm_Cartan}
For all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$, the above defined operators satisfy the following set of relations:
\begin{align}
\label{eq_Cartan1} \Li{X} = & \ i_{X} \circ \mathrm{d} + (-1)^{|X|} \mathrm{d} \circ i_{X}, \\
\label{eq_Cartan2} \Li{[X,Y]} = & \ \Li{X} \circ \Li{Y} - (-1)^{|X||Y|} \Li{Y} \circ \Li{X}, \\
\label{eq_Cartan3} i_{[X,Y]} = & \ \Li{X} \circ i_{Y} - (-1)^{(|X|(|Y|-1)} i_{Y} \circ \Li{X}, \\
\label{eq_Cartan4} 0 = & \ \Li{X} \circ \mathrm{d} - (-1)^{|X|} \mathrm{d} \circ \Li{X}, \\
\label{eq_Cartan5} 0 = & \ i_{X} \circ i_{Y} - (-1)^{(|X|-1)(|Y|-1)} i_{Y} \circ i_{X}.
\end{align}
\end{theorem}
\begin{rem} \label{rem_dualareoneforms}
It follows from Example \ref{ex_dual} and definitions that the sheaf $\Omega^{1}_{\mathcal{M}}$ is the sheaf of sections of the \textbf{cotangent bundle} $T^{\ast}\mathcal{M}$, defined as the graded vector bundle dual to the tangent bundle $T\mathcal{M}$, see Example \ref{ex_tangent}.
Similarly to the ordinary case, one can identify vector fields with the sections of the double dual. However, in the graded setting, the action of $X \in \mathfrak{X}_{\mathcal{M}}(M)$ on $\omega \in \Omega^{1}_{\mathcal{M}}(M)$ is given by the formula containing an additional sign, that is
\begin{equation} \label{eq_vfactionon1form}
X(\omega) := (-1)^{|X||\omega|} \omega(X).
\end{equation}
This is in accordance with the Koszul sign convention and one should not miss this subtle difference.
\end{rem}
This settles the discussion for $p$-forms. Let us now examine the dual picture. We will need both skew-symmetric and symmetric versions of $p$-vector fields.
\begin{definice} \label{def_pvector}
Let $p \in \mathbb{N}_{0}$. By a \textbf{skew-symmetric $p$-vector (field)} $X$ on $\mathcal{M}$ of degree $|X|$, we mean a $p$-linear map
\begin{equation}
X: \Omega^{1}_{\mathcal{M}}(M) \times \dots \times \Omega^{1}_{\mathcal{M}}(M) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(M)
\end{equation}
satisfying $|X(\xi_{1},\dots,\xi_{p})| = |X| + |\xi_{1}| + \dots + |\xi_{p}|$ for all $\xi_{1},\dots,\xi_{p} \in \Omega^{1}_{\mathcal{M}}(M)$, such that
\begin{equation}
X(f \xi_{1}, \dots, \xi_{p}) = (-1)^{|f||X|} f X(\xi_{1},\dots,\xi_{p}),
\end{equation}
\begin{equation} \label{eq_pvectorskewymmetry}
X(\xi_{1}, \dots, \xi_{i},\xi_{i+1},\dots,\xi_{p}) = - (-1)^{|\xi_{i}||\xi_{i+1}|} X(\xi_{1}, \dots, \xi_{i+1},\xi_{i},\dots,\xi_{p}),
\end{equation}
for all $\xi_{1},\dots,\xi_{p} \in \Omega^{1}_{\mathcal{M}}(M)$, $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $i \in \{1,\dots,p-1\}$. For each $p \in \mathbb{N}_{0}$, skew-symmetric $p$-vectors form a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module which we denote as $\mathfrak{X}^{p}_{\mathcal{M}}(M)$. We declare $\mathfrak{X}^{0}_{\mathcal{M}}(M) := \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. Note that $\mathfrak{X}^{1}_{\mathcal{M}}(M)$ can be identified with $\mathfrak{X}_{\mathcal{M}}(M)$ by means of the equation (\ref{eq_vfactionon1form}).
\end{definice}
Similarly to $p$-forms, skew-symmetric $p$-vectors form a sheaf of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules defined for each $U \in \Op(M)$ as $\mathfrak{X}^{p}_{\mathcal{M}}(U) := \mathfrak{X}^{p}_{\mathcal{M}|_{U}}(U)$. Restriction morphisms are defined using partitions of unity.
\begin{tvrz} \label{tvrz_extproduct2}
There is an $\mathbb{R}$-bilinear product $\^: \mathfrak{X}^{p}_{\mathcal{M}}(M) \times \mathfrak{X}^{q}_{\mathcal{M}}(M) \rightarrow \mathfrak{X}^{p+q}_{\mathcal{M}}(M)$ for each $p,q \in \mathbb{N}_{0}$. It has the same properties as in Proposition \ref{tvrz_extproduct}.
For each $\xi \in \Omega^{1}_{\mathcal{M}}(M)$, there is an operator $i_{\xi}: \mathfrak{X}^{p}_{\mathcal{M}}(M) \rightarrow \mathfrak{X}^{p-1}_{\mathcal{M}}(M)$ of degree $|\xi|$, defined using the same formula and having the same properties as the interior product in Proposition \ref{tvrz_intproduct}. Finally, for each $X \in \mathfrak{X}_{\mathcal{M}}(M)$, there is a Lie derivative operator $\Li{X}: \mathfrak{X}^{p}_{\mathcal{M}}(M) \rightarrow \mathfrak{X}^{p}_{\mathcal{M}}(M)$ of degree $|X|$, defined for all $Y \in \mathfrak{X}^{p}_{\mathcal{M}}(M)$ as
\begin{equation} \label{eq_LieXmultivector}
\begin{split}
(\Li{X}Y)(\xi_{1},\dots,\xi_{p}) := & \ X( Y(\xi_{1},\dots,\xi_{p})) \\
& - \sum_{r=1}^{p} (-1)^{(|Y| + |\xi_{1}| + \dots + |\xi_{r-1}|)|X|} Y(\xi_{1}, \dots, \Li{X}\xi_{r}, \dots, \xi_{p}),
\end{split}
\end{equation}
for all $\xi_{1},\dots,\xi_{p} \in \Omega^{1}_{\mathcal{M}}(M)$. It has the same properties as $\Li{X}$ in Proposition \ref{tvrz_Lieder}.
\end{tvrz}
\begin{rem}
$\mathfrak{X}^{p}_{\mathcal{M}}$ can be again viewed as a certain subsheaf of the sheaf $\mathcal{C}^{\infty}_{T^{\ast}[1+s']\mathcal{M}}$ of functions of the degree shifted cotangent bundle $T^{\ast}[1+s']\mathcal{M}$, where $s'$ is a large enough non-negative even integer, except that the $\mathbb{Z}$-grading is different.
\end{rem}
\begin{tvrz} \label{tvrz_SNB}
For each $p,q \in \mathbb{N}_{0}$, there is an $\mathbb{R}$-bilinear bracket $[\cdot,\cdot]_{S}: \mathfrak{X}^{p}_{\mathcal{M}}(M) \times \mathfrak{X}^{q}_{\mathcal{M}}(M) \rightarrow \mathfrak{X}^{p+q-1}_{\mathcal{M}}(M)$ having the following properties:
\begin{enumerate}[(i)]
\item One has $|[X,Y]_{S}| = |X| + |Y|$ for all $X \in \mathfrak{X}_{\mathcal{M}}^{p}(M)$ and $Y \in \mathfrak{X}_{\mathcal{M}}^{q}(M)$.
\item For every $f \in \mathfrak{X}^{0}_{\mathcal{M}}(M) \equiv \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ $X \in \mathfrak{X}^{1}_{\mathcal{M}}(M) \equiv \mathfrak{X}_{\mathcal{M}}(M)$ and $Y \in \mathfrak{X}^{q}_{\mathcal{M}}(M)$, one has
\begin{equation} \label{eq_SNB_special}
[f,Y]_{S} = (-1)^{|f|+1} i_{\mathrm{d}{f}} Y, \; \; [X,Y]_{S} = \Li{X}Y.
\end{equation}
\item For each $X \in \mathfrak{X}^{p}_{\mathcal{M}}(M)$ and $Y \in \mathfrak{X}^{q}_{\mathcal{M}}(M)$, one has
\begin{equation} \label{eq_SNB_gsymmetry}
[X,Y]_{S} = -(-1)^{(p + |X| - 1)(q + |Y| - 1)} [Y,X]_{S}.
\end{equation}
\item It satisfies the graded Leibniz rule with respect to the exterior product. For all $X \in \mathfrak{X}_{\mathcal{M}}^{p}(M)$, $Y \in \mathfrak{X}_{\mathcal{M}}^{q}(M)$ and $Z \in \mathfrak{X}_{\mathcal{M}}^{r}(M)$, one has
\begin{equation}
[X, Y \^ Z]_{S} = [X,Y]_{S} \^ Z + (-1)^{(p + |X| - 1)(q + |Y|)} Y \^ [X,Z]_{S}.
\end{equation}
\item It satisfies the graded Jacobi identity. For all $X \in \mathfrak{X}_{\mathcal{M}}^{p}(M)$, $Y \in \mathfrak{X}_{\mathcal{M}}^{q}(M)$ and $Z \in \mathfrak{X}_{\mathcal{M}}^{r}(M)$, one finds
\begin{equation} \label{eq_SNB_gJacobi}
[X, [Y,Z]_{S}]_{S} = [[X,Y]_{S}, Z]_{S} + (-1)^{(p + |X| -1)(q + |Y| - 1)}[Y, [X,Z]_{S}]_{S}.
\end{equation}
\end{enumerate}
$[\cdot,\cdot]_{S}$ is called the \textbf{Schouten-Nijenhuis bracket} of skew-symmetric multivector fields.
\end{tvrz}
Finally, we shall need the notion of symmetric multivector fields. As it differs only slightly from the skew-symmetric case, we shall be even more brief.
\begin{definice} \label{def_pvectorsym}
A \textbf{symmetric $p$-vector (field)} $X$ on $\mathcal{M}$ is defined as in Definition \ref{def_pvector}, except the requirement (\ref{eq_pvectorskewymmetry}) is replaced by
\begin{equation}
X(\xi_{1},\dots,\xi_{i},\xi_{i+1}, \dots, \xi_{p}) = (-1)^{|\xi_{i}||\xi_{i+1}|} X(\xi_{1}, \dots, \xi_{i+1},\xi_{i}, \dots, \xi_{p}).
\end{equation}
The module of symmetric $p$-vectors is denoted as $\~\mathfrak{X}^{p}_{\mathcal{M}}(M)$. Again $\~\mathfrak{X}^{0}_{\mathcal{M}}(M) := \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $\~\mathfrak{X}^{1}_{\mathcal{M}}(M)$ corresponds to $\mathfrak{X}_{\mathcal{M}}(M)$.
\end{definice}
\begin{tvrz} \label{tvrz_odotproduct}
There is an $\mathbb{R}$-bilinear product $\odot: \~\mathfrak{X}^{p}_{\mathcal{M}}(M) \times \~\mathfrak{X}^{q}_{\mathcal{M}}(M) \rightarrow \~\mathfrak{X}^{p+q}_{\mathcal{M}}(M)$ having the same properties as the exterior product, except that
\begin{equation}
X \odot Y = (-1)^{|X||Y|} Y \odot X,
\end{equation}
for all $X \in \~\mathfrak{X}^{p}_{\mathcal{M}}(M)$ and $Y \in \~\mathfrak{X}^{q}_{\mathcal{M}}(M)$.
For each $\xi \in \Omega^{1}_{\mathcal{M}}(M)$, one can define the interior product operator $j_{\xi}: \~\mathfrak{X}^{p}_{\mathcal{M}}(M) \rightarrow \~\mathfrak{X}^{p-1}_{\mathcal{M}}(M)$ of degree $|\xi|$ by formula
\begin{equation}
(j_{\xi} X)(\eta_{1},\dots,\eta_{p-1}) := (-1)^{|X||\xi|} X(\xi,\eta_{1},\dots,\eta_{p-1}),
\end{equation}
for all $X \in \~\mathfrak{X}^{p}_{\mathcal{M}}(M)$ and $\eta_{1},\dots,\eta_{p-1} \in \Omega^{1}_{\mathcal{M}}(M)$. Note the sign difference from (\ref{eq_defintproduct}). For each $X \in \~\mathfrak{X}_{\mathcal{M}}^{p}(M)$ and $Y \in \~\mathfrak{X}_{\mathcal{M}}^{q}(M)$, it satisfies $j_{\xi}(X \odot Y) = j_{\xi}X \odot Y + (-1)^{|\xi||X|} X \odot j_{\xi}Y$. Note that for $p = 1$, one has $j_{\xi} \neq i_{\xi}$. Finally, for each $X \in \mathfrak{X}_{\mathcal{M}}(M)$, there is a Lie derivative operator $\Li{X}: \~\mathfrak{X}^{p}_{\mathcal{M}}(M) \rightarrow \~\mathfrak{X}^{p}_{\mathcal{M}}(M)$ defined by the formula (\ref{eq_LieXmultivector}).
\end{tvrz}
We obtain a version of a Schouten-Nijenhuis bracket suitable for symmetric multivector fields.
\begin{tvrz} \label{tvrz_SNB2}
For each $p,q \in \mathbb{N}_{0}$, there is an $\mathbb{R}$-bilinear bracket $[\cdot,\cdot]_{S}: \~\mathfrak{X}^{p}_{\mathcal{M}}(M) \times \~\mathfrak{X}^{q}_{\mathcal{M}}(M) \rightarrow \~\mathfrak{X}^{p+q-1}_{\mathcal{M}}(M)$ having the following properties:
\begin{enumerate}[(i)]
\item One has $|[X,Y]_{S}| = |X| + |Y|$ for all $X \in \~\mathfrak{X}_{\mathcal{M}}^{p}(M)$ and $Y \in \~\mathfrak{X}_{\mathcal{M}}^{q}(M)$.
\item For every $f \in \~\mathfrak{X}^{0}_{\mathcal{M}}(M) \equiv \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ $X \in \~\mathfrak{X}^{1}_{\mathcal{M}}(M) \equiv \mathfrak{X}_{\mathcal{M}}(M)$ and $Y \in \~\mathfrak{X}^{q}_{\mathcal{M}}(M)$, one has
\begin{equation} \label{eq_SNB2_spexial}
[f,Y]_{S} = (-1)^{|f|+1} j_{\mathrm{d}{f}} Y, \; \; [X,Y]_{S} = \Li{X}Y.
\end{equation}
\item For each $X \in \~\mathfrak{X}^{p}_{\mathcal{M}}(M)$ and $Y \in \~\mathfrak{X}^{q}_{\mathcal{M}}(M)$, one has
\begin{equation} \label{eq_SNB2_gsymmetry}
[X,Y]_{S} = -(-1)^{|X||Y|} [Y,X]_{S}.
\end{equation}
\item It satisfies the graded Leibniz rule with respect to the symmetrized tensor product. For all $X \in \~\mathfrak{X}_{\mathcal{M}}^{p}(M)$, $Y \in \~\mathfrak{X}_{\mathcal{M}}^{q}(M)$ and $Z \in \~\mathfrak{X}_{\mathcal{M}}^{r}(M)$, one has
\begin{equation}
[X, Y \odot Z]_{S} = [X,Y]_{S} \odot Z + (-1)^{|X||Y|} Y \odot [X,Z]_{S}.
\end{equation}
\item It satisfies the graded Jacobi identity. For all $X \in \~\mathfrak{X}_{\mathcal{M}}^{p}(M)$, $Y \in \~\mathfrak{X}_{\mathcal{M}}^{q}(M)$ and $Z \in \~\mathfrak{X}_{\mathcal{M}}^{r}(M)$, one finds
\begin{equation} \label{eq_SNB2_gJacobi}
[X, [Y,Z]_{S}]_{S} = [[X,Y]_{S}, Z]_{S} + (-1)^{|X||Y|}[Y, [X,Z]_{S}]_{S}.
\end{equation}
\end{enumerate}
$[\cdot,\cdot]_{S}$ is called the \textbf{Schouten-Nijenhuis bracket} of symmetric multivector fields.
\end{tvrz}
\begin{example}[\textbf{Graded Poisson manifolds}] \label{ex_gPoisson} Let $\mathcal{M}$ be a graded manifold. By a \textbf{Poisson bracket of degree $\ell$} on $\mathcal{M}$, we mean an $\mathbb{R}$-bilinear bracket $\{ \cdot,\cdot\}: \mathcal{C}^{\infty}_{\mathcal{M}}(M) \times \mathcal{C}^{\infty}_{\mathcal{M}}(M) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ having the following properties:
\begin{enumerate}[(i)]
\item $|\{f,g\}| = |f| + |g| + \ell$ for all $f,g \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$;
\item \label{eq_PB_gsymmetry} $\{f,g\} = - (-1)^{(|f|+\ell)(|g|+\ell)} \{g,f\}$;
\item \label{eq_PB_Leibniz} $\{f, gh \} = \{f,g\}h + (-1)^{(|f|+\ell)|g|} g \{f,h\}$;
\item \label{eq_PB-gJacobi} $\{f, \{g,h\}\} = \{\{f,g\}, h\} + (-1)^{(|f|+\ell)(|g|+\ell)} \{g, \{f,h\}\}$.
\end{enumerate}
$(\mathcal{M},\{\cdot,\cdot\})$ is called a \textbf{graded Poisson manifold of degree $\ell$}. This is an example of a graded Poisson algebra \cite{cattaneo2018graded} of degree $\ell$.
As in ordinary Poisson geometry, there should be a skew-symmetric bivector $\Pi \in \mathfrak{X}^{2}_{\mathcal{M}}(M)$ of degree $\ell$, such that $\{\cdot,\cdot\}$ is obtained as a \textbf{derived bracket} from $\Pi$ using $[\cdot,\cdot]_{S}$, that is
\begin{equation} \label{eq_PBasderived}
\{f,g\} = [[f,\Pi]_{S},g]_{S}.
\end{equation}
However, using the properties (\ref{eq_SNB_gsymmetry}, \ref{eq_SNB_gJacobi}) one finds
\begin{equation}
[[f,\Pi]_{S},g]_{S} = [f, [\Pi,g]_{S}]_{S} = - (-1)^{(|f|+\ell)(|g|+\ell) + \ell} [[g,\Pi]_{S},f]_{S}
\end{equation}
We see that for odd $\ell$, the formula (\ref{eq_PBasderived}) does not define a graded Poisson bracket satisfying (\ref{eq_PB_gsymmetry}) of the above definition. This cannot be fixed by different sign choices in (\ref{eq_PBasderived}). Hence $\{\cdot,\cdot\}$ corresponds to a skew-symmetric bivector $\Pi \in \mathfrak{X}^{2}_{\mathcal{M}}(M)$ \textit{only for even $\ell$}.
It turns out that odd $\ell$, we have to assume that $\Pi \in \~\mathfrak{X}^{2}_{\mathcal{M}}(M)$. Indeed, one uses (\ref{eq_SNB2_gsymmetry}, \ref{eq_SNB2_gJacobi}) to find
\begin{equation}
[[f,\Pi]_{S},g]_{S} = [f, [\Pi,g]_{S}]_{S} = (-1)^{(|f|+\ell)(|g|+\ell) + \ell} [[g,\Pi]_{S},f]_{S}.
\end{equation}
This gives a correct sign in (\ref{eq_PB_gsymmetry}) when $\ell$ is odd.
In both cases, (\ref{eq_PB_Leibniz}) follows from the Leibniz rule for $[\cdot,\cdot]_{S}$, whereas (\ref{eq_PB-gJacobi}) is equivalent to the condition $[\Pi,\Pi]_{S} = 0$. Note that if $\Pi \in \mathfrak{X}^{2}_{\mathcal{M}}(M)$ has an odd degree, the condition $[\Pi,\Pi]_{S} = 0$ is trivial. The same happens for $\Pi \in \~\mathfrak{X}^{2}_{\mathcal{M}}(M)$ of even degree.
\end{example}
\begin{rem}
In fact, the bracket $[\cdot,\cdot]_{S}$ for for skew-symmetric multivectors can be viewed as a graded Poisson bracket on $T^{\ast}[1+s']\mathcal{M}$ of degree $-(1 + s')$. For symmetric multivectors, it defines a graded Poisson bracket on $T^{\ast}[s']\mathcal{M}$ of degree $-s'$.
\end{rem}
\begin{rem}
For all $p \in \mathbb{N}_{0}$, $\Omega^{p}_{\mathcal{M}}$, $\mathfrak{X}^{p}_{\mathcal{M}}$ and $\~\mathfrak{X}^{p}_{\mathcal{M}}$ form locally freely and finitely generated sheaves of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-modules, hence they can be viewed as sheaves of sections of graded vector bundles.
\end{rem}
\section{Graded Courant algebroids}
In this section, we shall generalize the concept of a Courant algebroid \cite{liu1997manin, 1999math.....10078R, Severa:2017oew}. We will then provide a key example generalizing the well-known Dorfman bracket \cite{dorfman1987dirac} of ordinary generalized geometry.
\begin{definice}
Let $\mathcal{E}$ be a graded vector bundle. By a \textbf{graded symmetric bilinear form on $\mathcal{E}$ of degree $\ell$}, we mean a vector bundle map $g_{\mathcal{E}}: \mathcal{E} \rightarrow \mathcal{E}^{\ast}$ of degree $\ell$, such that $\<\cdot,\cdot\>_{E}$, defined for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$ by the formula
\begin{equation} \label{eq_gEformErelation}
\<\psi,\psi'\>_{\mathcal{E}} := (-1)^{(|\psi|+\ell)\ell} [g_{\mathcal{E}}(\psi)](\psi'),
\end{equation}
is graded symmetric of degree $\ell$ in its inputs, that is
\begin{equation} \label{eq_gradedsymmetricform}
\< \psi,\psi'\>_{\mathcal{E}} = (-1)^{(|\psi|+\ell)(|\psi'|+\ell)} \<\psi',\psi\>_{\mathcal{E}},
\end{equation}
for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. We say that $g_{\mathcal{E}}$ is a \textbf{fiber-wise metric on $\mathcal{E}$ of degree $\ell$}, if it is a graded vector bundle isomorphism. In this case, we say that $(\mathcal{E},g_{\mathcal{E}})$ is a \textbf{quadratic graded vector bundle of degree $\ell$}.
\end{definice}
Let $\psi \in \Gamma_{\mathcal{E}}(M)$. Since $g_{\mathcal{E}}(\psi) \in \Gamma_{\mathcal{E}^{\ast}}(M)$ is $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear of degree $|\psi| + \ell$ , we find that for all $\psi' \in \Gamma_{\mathcal{E}}(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$, one has the rule
\begin{equation} \label{eq_metriclinearsecond}
\<\psi, f \psi'\>_{\mathcal{E}} = (-1)^{(|\psi|+\ell)|f|} f \<\psi,\psi'\>_{\mathcal{E}}.
\end{equation}
On the other hand, as $g_{\mathcal{E}}: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}^{\ast}}(M)$ is $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear of degree $\ell$, one gets
\begin{equation} \label{eq_metriclinearfirst}
\<f \psi, \psi'\>_{\mathcal{E}} = f \<\psi, \psi'\>_{\mathcal{E}}.
\end{equation}
Note that this rule is consistent with (\ref{eq_gradedsymmetricform}) and (\ref{eq_metriclinearsecond}). This also explains the signs in (\ref{eq_gEformErelation}).
\begin{rem} \label{rem_nondegen}
Since $g_{\mathcal{E}}$ can be equivalently viewed as a vector bundle map $g_{\mathcal{E}}: \mathcal{E} \rightarrow \mathcal{E}^{\ast}[\ell]$ of degree zero, it can be an isomorphism, only if $\grk(\mathcal{E}) = \grk(\mathcal{E}^{\ast}[\ell])$. It thus follows from Example \ref{ex_dual} and Example \ref{ex_degreeshift} that a fiber-wise metric on $\mathcal{E}$ of degree $\ell$ exists, only if $\grk(\mathcal{E}) = (r_{j})_{j \in \mathbb{Z}}$ satisfies the condition $r_{j} = r_{-(j+\ell)}$ for every $j \in \mathbb{Z}$.
\end{rem}
\begin{example} \label{ex_canpairing}
Let $\mathcal{E} = T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$. Each section in $\Gamma_{\mathcal{E}}(M)$ of degree $k \in \mathbb{Z}$ is a pair $(X,\xi)$, where $X \in \mathfrak{X}_{\mathcal{M}}(M)$ is a vector field of degree $|X| = k + \ell$ and $\xi \in \Omega_{\mathcal{M}}(M)$ is a $1$-form of degree $|\xi| = k$. In other words, we have $|(X,\xi)| = |X| - \ell = |\xi|$. The dual vector bundle $\mathcal{E}^{\ast}$ can be canonically identified with $T^{\ast}[-\ell]\mathcal{M} \oplus T\mathcal{M}$. One has to be a bit careful when doing this. Indeed, suppose $(\xi,X) \in \Omega^{1}_{\mathcal{M}}[-\ell](M) \oplus \mathfrak{X}_{\mathcal{M}}(M)$. This means that $\xi \in \Omega^{1}_{\mathcal{M}}(M)$ is a $1$-form of degree $|\xi| = |(\xi,X)| - \ell$ and $X \in \mathfrak{X}_{\mathcal{M}}(M)$ is a vector field of degree $|X| = |(\xi,X)|$. If we want to view $(\xi,X)$ as an element of $\Gamma_{\mathcal{E}^{\ast}}(M)$, we must declare its action on every element $(Y,\eta) \in \Gamma_{\mathcal{E}}(M)$. Set
\begin{equation} \label{eq_dualgentangentident}
[(\xi,X)](Y,\eta) := (-1)^{(|\xi|+\ell)\ell} \xi(Y) + (-1)^{|X||\eta|} \eta(X).
\end{equation}
Note that $|(Y,\eta)| = |Y| - \ell = |\eta|$. This must define a $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear map of degree $|(\xi,X)| = |\xi| + \ell = |X|$. We leave the verification to the reader - be aware that $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-actions are also modified by degree shifts! Note that the identification of $(\xi,X)$ with an element of $\Gamma_{\mathcal{E}^{\ast}}(M)$ is $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear of degree zero, hence it defines a graded vector bundle isomorphism. We will henceforth use this identification.
Now, let us define $g_{\mathcal{E}}: \mathcal{E} \rightarrow \mathcal{E}^{\ast}$ as follows. For each $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$, set
\begin{equation}
g_{\mathcal{E}}(X,\xi) := (\xi,X).
\end{equation}
First, observe that $|g_{\mathcal{E}}(X,\xi)| = |(\xi,X)| = |X| = |(X,\xi)| + \ell$, and for each $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$, one has
\begin{equation}
\begin{split}
g_{\mathcal{E}}(f \triangleright (X,\xi)) = & \ g_{\mathcal{E}}( (-1)^{|f|\ell}f X, f \xi) = (f \xi, (-1)^{|f|\ell} f X) \\
= & \ (-1)^{|f|\ell} ((-1)^{-|f|\ell} f \xi, fX) = (-1)^{|f|\ell} f \triangleright (\xi, X) \\
= & \ (-1)^{|f|\ell} f \triangleright g_{\mathcal{E}}(X,\xi).
\end{split}
\end{equation}
This proves that $g_{\mathcal{E}}$ is a $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear isomorphism of degree $\ell$. Using (\ref{eq_dualgentangentident}), the formula (\ref{eq_gEformErelation}) gives
\begin{equation} \label{eq_canpairing}
\<(X,\xi), (Y,\eta) \>_{\mathcal{E}} = \xi(Y) + (-1)^{|X||Y|} \eta(X),
\end{equation}
for all $(X,\xi), (Y,\eta) \in \Gamma_{\mathcal{E}}(M)$. It is easy to see that $\<\cdot,\cdot\>_{\mathcal{E}}$ satisfies (\ref{eq_gradedsymmetricform}). We conclude that $g_{\mathcal{E}}$ defines a fiber-wise metric of degree $\ell$ on $\mathcal{E}$.
\end{example}
We can now proceed to the main definition of this section. Let $\ell \in \mathbb{Z}$ be any integer.
\begin{definice} \label{def_gCourant}
$(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ is called a \textbf{graded Courant algebroid of degree $\ell$}, if
\begin{enumerate}[(i)]
\item $\mathcal{E}$ is a graded vector bundle over $\mathcal{M}$;
\item $\rho: \mathcal{E} \rightarrow T\mathcal{M}$ is a vector bundle map of degree $\ell$ called the \textbf{anchor};
\item $g_{\mathcal{E}}$ is a fiber-wise metric on $\mathcal{E}$ of degree $\ell$;
\item $[\cdot,\cdot]_{\mathcal{E}}: \Gamma_{\mathcal{E}}(M) \times \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ is an $\mathbb{R}$-bilinear map of degree $\ell$, that is $|[\psi,\psi']_{\mathcal{E}}| = |\psi| + |\psi'| + \ell$ for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$.
\end{enumerate}
Moreover, the operations are subject to the following set of axioms. All conditions have to hold for all involved sections and functions. Let $\hat{\rho}(\psi) := (-1)^{(|\psi|+\ell)\ell} \rho(\psi)$.
\begin{enumerate}[(c1)]
\item The bracket satisfies the \textbf{graded Leibniz rule}
\begin{equation} \label{eq_CALeibniz}
[\psi, f\psi']_{\mathcal{E}} = \hat{\rho}(\psi)(f) \psi' + (-1)^{|f|(|\psi|+\ell)} f [\psi,\psi']_{E}.
\end{equation}
\item The fiber-wise metric and the bracket are compatible, that is
\begin{equation} \label{eq_CApairingcomp}
\hat{\rho}(\psi)\<\psi',\psi''\>_{\mathcal{E}} = \< [\psi,\psi']_{\mathcal{E}}, \psi''\>_{\mathcal{E}} + (-1)^{(|\psi|+\ell)(|\psi'|+\ell)} \< \psi', [\psi,\psi'']_{\mathcal{E}} \>_{\mathcal{E}}.
\end{equation}
\item The bracket is graded skew-symmetric up to an extra term, that is
\begin{equation} \label{eq_CAskewsym}
[\psi,\psi']_{\mathcal{E}} + (-1)^{(|\psi|+\ell)(|\psi'|+\ell)} [\psi',\psi]_{\mathcal{E}} = (-1)^{|\psi|+|\psi'|} \mathrm{D} \<\psi,\psi'\>_{\mathcal{E}},
\end{equation}
where $\mathrm{D}: \mathcal{C}^{\infty}_{\mathcal{M}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ is a degree zero $\mathbb{R}$-linear map defined as $\mathrm{D} := g_{\mathcal{E}}^{-1} \circ \rho^{T} \circ \mathrm{d}$.
\item The bracket satisfies the \textbf{graded Jacobi identity}
\begin{equation} \label{eq_CAgJI}
[\psi,[\psi',\psi'']_{\mathcal{E}}]_{\mathcal{E}} = [[\psi,\psi']_{\mathcal{E}}, \psi'']_{\mathcal{E}} + (-1)^{(|\psi|+\ell)(|\psi'|+\ell)} [\psi', [\psi, \psi'']_{\mathcal{E}}]_{\mathcal{E}}.
\end{equation}
\end{enumerate}
\end{definice}
\begin{rem}
Our main intention was to have a bracket operation $[\cdot,\cdot]_{\mathcal{E}}$ of degree $\ell$. The graded Leibniz rule (\ref{eq_CALeibniz}) then requires the anchor $\rho$ to be a degree $\ell$ map. Axioms (\ref{eq_CApairingcomp}) and (\ref{eq_CAskewsym}) could be in principle formulated for a fiber-wise metric of an arbitrary (independent of $\ell$) degree. However, this freedom can be eliminated by considering a degree shifted vector bundle $\mathcal{E}[k]$ where $k \in \mathbb{Z}$ can be always chosen so that $g_{\mathcal{E}}$ has the same degree as $[\cdot,\cdot]_{\mathcal{E}}$.
Note that after assuming that the bracket and $g_{\mathcal{E}}$ have the same degree, no further degree shifting is allowed. In other words, the degree $\ell$ is an ``intrinsic'' property of graded Courant algebroids. Observe that $[\cdot,\cdot]_{\mathcal{E}}$ defines a so called Loday bracket on $\Gamma_{\mathcal{E}}(M)$, see \S 2.1 of \cite{Kosmann1996}.
\end{rem}
Similarly to ordinary Courant algebroids, one can find some immediate consequences of the axioms. Let us name a few.
\begin{tvrz} \label{tvrz_CAconsequences}
The anchor $\rho$ is a bracket homomorphism, that is
\begin{equation} \label{eq_rhoishom}
\rho( [\psi,\psi']_{\mathcal{E}}) = [\rho(\psi), \rho(\psi')],
\end{equation}
for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. One has $\rho \circ \mathrm{D} = 0$ and thus $\rho \circ \rho^{\ast} = 0$, where $\rho^{\ast} := g_{\mathcal{E}}^{-1} \circ \rho^{T}$. Consequently, there is a sequence of graded vector bundles over $\mathcal{M}$ and degree zero vector bundle maps over $\mathbbm{1}_{\mathcal{M}}$
\begin{equation}
\begin{tikzcd} \label{eq_gCAsequence}
0 \arrow{r} & T^{\ast}\mathcal{M} \arrow{r}{\rho^{\ast}} & \arrow{r}{\rho} \mathcal{E} & T[\ell]\mathcal{M} \arrow{r} & 0
\end{tikzcd}
\end{equation}
forming a cochain complex. Finally, for any $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $\psi \in \Gamma_{\mathcal{E}}(M)$, one has
\begin{equation} \label{eq_bracketswithD}
[\psi, \mathrm{D}{f}]_{\mathcal{E}} = (-1)^{|f|+\ell} \mathrm{D} \<\psi, \mathrm{D}{f}\>_{\mathcal{E}}, \; \; [\mathrm{D}{f}, \psi]_{\mathcal{E}} = 0.
\end{equation}
\end{tvrz}
\begin{proof}
The proof is completely analogous to the one for ordinary Courant algebroids, see e.g. \cite{1999math.....10078R, Kotov:2010wr}. The equation (\ref{eq_rhoishom}) follows easily from the consistency of (\ref{eq_CAgJI}) with the graded Leibniz rule (\ref{eq_CALeibniz}). Using (\ref{eq_CALeibniz}) and (\ref{eq_CAskewsym}), one can derive the rule
\begin{equation}
[f\psi,\psi']_{\mathcal{E}} = f[\psi,\psi']_{\mathcal{E}} - (-1)^{(|f|+|\psi|+\ell)(|\psi'|+\ell)} \hat{\rho}(\psi')(f) \psi + (-1)^{\ell + |f|(|\psi|+|\psi'|+\ell + 1)} \<\psi,\psi'\>_{\mathcal{E}} \mathrm{D}{f}.
\end{equation}
Applying $\rho$ on both sides of this equation, using (\ref{eq_rhoishom}) and the fact that $g_{\mathcal{E}}$ is an isomorphism, one can prove that $\rho \circ \mathrm{D} = 0$. Since $\mathrm{D}{f} = \rho^{\ast}( \mathrm{d}{f})$ and $\mathrm{d}{f}$ locally generate $\Omega^{1}_{\mathcal{M}}$, this already implies $\rho \circ \rho^{\ast} = 0$. The claim about (\ref{eq_gCAsequence}) follows immediately. To prove the first equation in (\ref{eq_bracketswithD}), one uses the fact that $\<\phi,\mathrm{D}{f}\>_{\mathcal{E}} = (-1)^{|f|+\ell} \hat{\rho}(\phi)f$ for all $\phi \in \Gamma_{\mathcal{E}}(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. By applying $\hat{\rho}(\psi)$ on this equation and using (\ref{eq_CApairingcomp}) and (\ref{eq_rhoishom}), one finds $\<\phi, [\psi,\mathrm{D}{f}]_{\mathcal{E}} \> = (-1)^{|f|+\ell} \<\phi, \mathrm{D}{ \<\psi, \mathrm{D}{f}\>_{\mathcal{E}}} \>_{\mathcal{E}}$. The claim then follows from the fact that $g_{\mathcal{E}}$ is an isomorphism. The other of the two equations then follows from the first one together with (\ref{eq_CAskewsym}).
\end{proof}
It is obvious that the definition of graded Courant algebroid was tailored in order for the following pivotal example to fit in. Let $\ell \in \mathbb{Z}$ be fixed.
\begin{example}[\textbf{Degree $\ell$ Dorfman bracket}] \label{ex_gDorfman}
Let $\mathcal{E} = T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$ be equipped with the fiber-wise metric $g_{\mathcal{E}}$ introduced in Example \ref{ex_canpairing}. Let $H \in \Omega^{3}_{\mathcal{M}}(M)$ with $|H| = -\ell$. Set
\begin{equation} \label{eq_gDorfman}
[(X,\xi),(Y,\eta)]_{D}^{H} := ([X,Y], (-1)^{|X|\ell} \Li{X}\eta - (-1)^{|Y|(|X|+\ell)} i_{Y} \mathrm{d}{\xi} + (-1)^{\ell} H(X,Y,\cdot)),
\end{equation}
for all $(X,\xi),(Y,\eta) \in \Gamma_{\mathcal{E}}(M)$. Let $\rho(X,\xi) := X$ for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$. We claim that $(\mathcal{E}, \rho, g_{\mathcal{E}}, [\cdot,\cdot]_{D}^{H})$ becomes a graded Courant algebroid of degree $\ell$, if and only if $\mathrm{d}{H} = 0$.
First, observe that $|\rho(X,\xi)| = |X| = |(X,\xi)| + \ell$, whence $\rho$ is a map of degree $\ell$. One has
\begin{equation}
\rho(f \triangleright (X,\xi)) = \rho( (-1)^{|f|\ell} fX, f\xi) = (-1)^{|f| \ell} fX = (-1)^{|f| \ell} f \rho(X,\xi),
\end{equation}
for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$ and $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. This proves that $\rho$ is $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear of degree $\ell$.
Next, for every $\alpha \in \Omega^{1}_{\mathcal{M}}(M)$ and every $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$, one finds
\begin{equation}
[ \rho^{T}(\alpha)](X,\xi) \equiv (-1)^{|\alpha|\ell} \alpha( \rho(X,\xi)) = (-1)^{|\alpha| \ell} \alpha(X).
\end{equation}
Using (\ref{eq_dualgentangentident}), we can thus write $\rho^{T}(\alpha) = ((-1)^{\ell} \alpha, 0) \in (\Omega^{1}_{\mathcal{M}}[-\ell] \oplus \mathfrak{X}_{\mathcal{M}})(M)$. Hence
\begin{equation}
\rho^{\ast}(\alpha) = (g_{\mathcal{E}}^{-1} \circ \rho^{T})(\alpha) = g_{\mathcal{E}}^{-1}( (-1)^{\ell} \alpha, 0) = (0, (-1)^{\ell} \alpha).
\end{equation}
In particular, we find that $\mathrm{D}{f} = (0, (-1)^{\ell} \mathrm{d}{f})$ for every $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. This is indeed an $\mathbb{R}$-linear map of degree zero.
Let us turn our attention to the bracket. Recall that for any $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$, we have $|(X,\xi)| = |X| - \ell = |\xi|$. We must verify that $[(X,\xi),(Y,\eta)]_{D}^{H}$ is a section of $\mathcal{E}$ of degree $|(X,\xi)| + |(Y,\eta)| + \ell$. In particular, $[X,Y]$ must be a vector field of degree $(|(X,\xi)| + |(Y,\eta)| + \ell) + \ell$. But this is true as
\begin{equation}
|[X,Y]| = |X| + |Y| = (|(X,\xi)| + \ell) + (|(Y,\eta)| + \ell).
\end{equation}
Next, all $1$-forms on the right-hand side of (\ref{eq_gDorfman}) must have a degree $|(X,\xi)| + |(Y,\eta)| + \ell$. But
\begin{align}
|\Li{X}\eta| = & \ |X| + |\eta| = (|(X,\xi)| + \ell)) + |(Y,\eta)|, \\
|i_{Y}\mathrm{d}{\xi}| = & \ |Y| + |\xi| = (|(Y,\eta)| + \ell) + |(X,\xi)|, \\
|H(X,Y,\cdot)| = & \ |X| + |Y| + |H| = (|(X,\xi)| + \ell) + (|(Y,\eta)| + \ell) - \ell.
\end{align}
Note that $H(X,Y,\cdot) \in \Omega^{1}_{\mathcal{M}}(M)$ can be also written using the interior product:
\begin{equation}
H(X,Y,\cdot) = (-1)^{-(|X|-1)\ell + (|Y|-1)(|X| - \ell)} i_{Y} i_{X}H.
\end{equation}
Let us verify the graded Leibniz rule. Again, one has to pay attention to the fact that the $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module on the sections of $T[\ell]\mathcal{M}$ is different from the original one on $\mathfrak{X}_{\mathcal{M}}(M)$. We write $\triangleright$ for the $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$ action on $\Gamma_{\mathcal{E}}(M)$ to distinguish it from the original actions on $\mathfrak{X}_{\mathcal{M}}(M)$ and $\Omega^{1}_{\mathcal{M}}(M)$. Thus
\begin{equation} \label{eq_gDorfmanleibniz}
\begin{split}
[(X,\xi), f \triangleright (Y,\eta)]_{D}^{H} = & \ [(X,\xi), ((-1)^{|f|\ell} fY, f\eta)]_{D}^{H} \\
= & \ \big( (-1)^{|f|\ell} [X,fY], (-1)^{|X|\ell} \Li{X}(f\eta) \\
& - (-1)^{(|f|+|Y|)|X|} i_{fY} \mathrm{d}{\xi} + (-1)^{(|f|+1)\ell} H(X,fY,\cdot)\big).
\end{split}
\end{equation}
At this moment, one uses the properties of graded commutator and operators on forms defined in Section \ref{sec_extmutvectoralg}. One finds the expressions
\begin{align}
[X,fY] = & \ X(f)Y + (-1)^{|X||f|} f [X,Y], \\
\Li{X}(f\eta) = & \ X(f) \eta + (-1)^{|X||f|} f \Li{X}\eta, \\
i_{fY} \mathrm{d}{\xi} = & \ f i_{Y} \mathrm{d}{\xi}, \\
H(X,fY,\cdot) = & \ (-1)^{|f|(|X|-\ell)} f H(X,Y,\cdot).
\end{align}
Plugging these expressions back into (\ref{eq_gDorfmanleibniz}) gives
\begin{equation}
\begin{split}
[(X,\xi), f \triangleright (Y,\eta)]_{D}^{H} = & \ ( (-1)^{|f|\ell} X(f)Y, (-1)^{|X|\ell} X(f) \eta) + (-1)^{|X||f|} \big( (-1)^{|f|\ell} f [X,Y], \\
& f ( (-1)^{|X|\ell} \Li{X}\eta - (-1)^{|Y|(|X|+\ell)} i_{Y} \mathrm{d}{\xi} + (-1)^{\ell} H(X,Y,\cdot))\big) \\
= & \ (-1)^{|X|\ell} X(f) \triangleright (Y,\eta) + (-1)^{|X||f|} f \triangleright [(X,\xi),(Y,\eta)]_{D}^{H}.
\end{split}
\end{equation}
But this is precisely (\ref{eq_CALeibniz}) since $|X| = |(X,\xi)| + \ell$ and $\hat{\rho}(X,\xi) = (-1)^{|X|\ell} X$. We leave the verification of (\ref{eq_CApairingcomp}) and (\ref{eq_CAskewsym}) to the reader. One employs just the Cartan relations (\ref{eq_Cartan1} - \ref{eq_Cartan5}) and definitions in the process. Finally, after a significant amount of work (especially to get the correct signs), one can show that the graded Jacobi identity (\ref{eq_CAgJI}) holds for sections $(X,\xi),(Y,\eta),(Z,\zeta) \in \Gamma_{\mathcal{E}}(M)$, if and only if the section $(0, \mathrm{d}{H}(X,Y,Z,\cdot)) \in \Gamma_{\mathcal{E}}(M)$ vanishes.
We conclude that $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{D}^{H})$ forms a graded Courant algebroid of degree $\ell$, if and only if $\mathrm{d}{H} = 0$. We call $[\cdot,\cdot]_{D}^{H}$ a \textbf{degree $\ell$ graded Dorfman bracket twisted by $H$}.
Let us conclude this example with the following observation. Let $\omega \in \Omega^{2}_{\mathcal{M}}(M)$ with $|\omega| = - \ell$. One can use it to define a $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear vector bundle map $\omega^{\flat}: T[\ell]\mathcal{M} \rightarrow T^{\ast}\mathcal{M}$ of degree zero given by $[\omega^{\flat}(X)](Y) := \omega(X,Y)$. Equivalently, $\omega^{\flat}(X) = (-1)^{(|X|-1)\ell} i_{X}\omega$. Define
\begin{equation} \label{eq_etoomega}
e^{\omega}(X,\xi) := (X, \xi + \omega^{\flat}(X)),
\end{equation}
for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$. It follows that $e^{\omega}: \mathcal{E} \rightarrow \mathcal{E}$ is a degree zero vector bundle isomorphism with the inverse $e^{-\omega}$. Using definitions and the Cartan relations (\ref{eq_Cartan1}, \ref{eq_Cartan3}), one can show that
\begin{equation} \label{eq_gDorfmanomegatwist}
e^{\omega}( [(X,\xi),(Y,\eta)]_{D}^{H + \mathrm{d}{\omega}}) = [e^{\omega}(X,\xi), e^{\omega}(Y,\eta)]_{D}^{H},
\end{equation}
for all $(X,\xi),(Y,\eta) \in \Gamma_{\mathcal{E}}(M)$. Also note that $\rho \circ e^{\omega} = \rho$ and $\< e^{\omega}(X,\xi), e^{\omega}(Y,\eta)\>_{\mathcal{E}} = \<(X,\xi),(Y,\eta)\>_{\mathcal{E}}$. Note that (\ref{eq_gDorfmanomegatwist}) was one of the most useful guides towards correct signs in (\ref{eq_gDorfman}) and Definition \ref{def_gCourant}.
\end{example}
One obtains the following graded version of the classification \cite{Severa:2017oew} of exact Courant algebroids.
\begin{tvrz}[\textbf{Graded Ševera classification}]
We say that a graded Courant algebroid $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ of degree $\ell$ is \textbf{exact}, if the sequence (\ref{eq_gCAsequence}) is exact.
Then isomorphism classes of exact Courant algebroids of degree $\ell$ over $\mathcal{M}$ are in one-to-one correspondence with de Rham cohomology classes $[H] \in H^{3}_{\mathcal{M}}(M)$ of degree $-\ell$. In particular, for $\ell \neq 0$, there is a unique isomorphism class of exact graded Courant algebroids of degree $\ell$.
\end{tvrz}
\begin{proof}
First, recall that an isomorphism of two graded Courant algebroids $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ and $(\mathcal{E}', \rho', g_{\mathcal{E}'}, [\cdot,\cdot]_{\mathcal{E}'})$ is a degree zero vector bundle isomorphism $F: \mathcal{E} \rightarrow \mathcal{E}'$, such that
\begin{equation}
\rho' \circ F = \rho, \; \; \<F(\psi),F(\psi')\>_{\mathcal{E}'} = \<\psi,\psi'\>_{\mathcal{E}}, \; \; F([\psi,\psi']_{\mathcal{E}}) = [F(\psi),F(\psi')]_{\mathcal{E}'},
\end{equation}
for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. Next, for each $p \in \mathbb{Z}$, the exterior differential $\mathrm{d}: \Omega^{p}_{\mathcal{M}}(M) \rightarrow \Omega^{p+1}_{\mathcal{M}}(M)$ preserves degrees. It follows that the $p$-th de Rham cohomology $H^{p}_{\mathcal{M}}(M)$ naturally becomes a graded vector space. However, it was observed in \cite{roytenberg2002structure} that its only non-zero component is the zeroth one, that is $[\omega] \neq 0$, only if $|\omega| = 0$. This explains the last sentence of the statement.
Now, let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a given exact graded Courant algebroid of degree $\ell$. One can use partitions of unity to show that every short exact sequence (\ref{eq_gCAsequence}) splits, that is there exists a degree zero vector bundle map $s: T[\ell]\mathcal{M} \rightarrow \mathcal{E}$ such that $\rho \circ s = \mathbbm{1}_{T[\ell]\mathcal{M}}$. One can always modify it so that $\<s(X),s(Y)\>_{\mathcal{E}} = 0$ for all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$. Such $s$ is called an \textit{isotropic splitting}.
To every such splitting, one can assign a degree zero vector bundle isomorphism $\Psi_{s}: T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M} \rightarrow \mathcal{E}$ given by $\Psi_{s}(X,\xi) = s(X) + (-1)^{\ell} \rho^{\ast}(\xi)$. One utilizes $\Psi_{s}$ to induce a graded Courant algebroid structure on $T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$ by declaring it to be an isomorphism of graded Courant algebroids. It is a straightforward calculation to show that in this way, one obtains precisely the structure from Example \ref{ex_gDorfman} for closed $H_{s} \in \Omega^{3}_{\mathcal{M}}(M)$ of degree $-\ell$ obtained from the formula
\begin{equation}
[s(X),s(Y)]_{\mathcal{E}} = s([X,Y]) + \rho^{\ast}( H_{s}(X,Y,\cdot)).
\end{equation}
This shows that every exact graded Courant algebroid of degree $\ell$ is isomorphic to the one in Example \ref{ex_gDorfman} for some $H_{s} \in \Omega^{3}_{\mathcal{M}}(M)$ of degree $-\ell$. For any other isotropic splitting $s': T[\ell]\mathcal{M} \rightarrow \mathcal{E}$, there is a unique $\omega \in \Omega^{2}_{\mathcal{M}}(M)$ with $|\omega| = -\ell$, such that $s'$ and $s$ are related by
\begin{equation}
s'(X) = s(X) + (-1)^{\ell} \rho^{\ast}( \omega^{\flat}(X)),
\end{equation}
for all $X \in \mathfrak{X}_{\mathcal{M}}(M)$. Consequently, one has $\Psi_{s'} = \Psi_{s} \circ e^{\omega}$, see (\ref{eq_etoomega}). It follows from (\ref{eq_gDorfmanomegatwist}) that $H_{s'} = H_{s} + \mathrm{d}{\omega}$. Hence to every exact graded Courant algebroid $\mathcal{E}$ of degree $\ell$, we may assign a unique class in $H^{3}_{\mathcal{M}}(M)$ of degree $-\ell$. If $F: \mathcal{E} \rightarrow \mathcal{E}'$ is an isomorphism of exact graded Courant algebroids, every isotropic splitting $s: T[\ell]\mathcal{M} \rightarrow \mathcal{E}$ induces an isotropic splitting $s' := F \circ s: T[\ell]\mathcal{M} \rightarrow \mathcal{E}'$ such that $H_{s'} = H_{s}$.
Conversely, to any class $[H] \in H^{3}_{\mathcal{M}}(M)$ of degree $-\ell$, one can use its representative $H \in \Omega^{3}_{\mathcal{M}}(M)$ to construct an exact graded Courant algebroid of degree $\ell$ as in Example \ref{ex_gDorfman}. A different choice of a representative $H' = H + \mathrm{d}{\omega}$ induces their isomorphism $e^{\omega}$ due to (\ref{eq_gDorfmanomegatwist}).
\end{proof}
\section{Dirac structures}
Let us start on graded linear algebra level. Suppose $V \in \gVect$ is a finite-dimensional graded vector space together with a degree $\ell$ isomorphism $g_{V}: V \rightarrow V^{\ast}$, such that
\begin{equation}
\<v,w\>_{V} := (-1)^{(|v|+\ell)\ell} [g_{V}(v)](w)
\end{equation}
induces a graded symmetric bilinear form $\<\cdot,\cdot\>_{V}: V \times V \rightarrow \mathbb{R}$ of degree $\ell$, that is
\begin{equation}
\<v,w\>_{V} = (-1)^{(|v|+\ell)(|w|+\ell)} \<w,v\>_{V},
\end{equation}
for all $v,w \in V$. Note that $\<v,w\>_{V} \neq 0$, only if $|v|+|w|+\ell = 0$. If $\gdim(V) = (r_{j})_{j \in \mathbb{Z}}$, then $g_{V}$ is an isomorphism, only if $r_{j} = r_{-(j+\ell)}$ for all $j \in \mathbb{Z}$. We say that $(V,g_{V})$ is a \textbf{quadratic graded vector space of degree $\ell$}.
Now, let $L \subseteq V$ be a graded subspace with $\gdim(L) = (q_{j})_{j \in \mathbb{Z}}$. First, we define the \textbf{annihilator of $L$} for each $j \in \mathbb{Z}$ by
\begin{equation}
\an(L)_{j} := \an(L_{-j}) \equiv \{ \alpha \in (V^{\ast})_{j} \; | \; \alpha(v) = 0 \text{ for all } v \in L_{-j} \}.
\end{equation}
This defines a graded subspace $\an(L) \subseteq V^{\ast}$ with $\gdim(\an(L)) = (r_{-j} - q_{-j})_{j \in \mathbb{Z}}$. Let
\begin{equation}
L^{\perp} := g_{V}^{-1}(\an(L)) = \{ v \in V \; | \; \<v,w\>_{V} = 0 \text{ for all } w \in L \}.
\end{equation}
$L^{\perp}$ is called the \textbf{orthogonal complement to $L$}. One finds $\gdim(L^{\perp}) = ( r_{-(j+\ell)} - q_{-(j+\ell)})_{j \in \mathbb{Z}}$.
\begin{definice}
Let $(V,g_{V})$ be a quadratic graded vector space of degree $\ell$, and let $L \subseteq V$ be a graded subspace. We say that $L$ is \textbf{isotropic}, if $L \subseteq L^{\perp}$. An isotropic graded subspace is \textbf{maximal}, if $L$ is not properly contained in any isotropic graded subspace. We say that $L$ is a \textbf{Lagrangian graded subspace}, if $L = L^{\perp}$.
\end{definice}
The proof of the following proposition is similar to the ordinary case and can be deduced from definitions and properties of ordinary bilinear forms. We leave it to the reader.
\begin{tvrz} \label{tvrz_maxisos1}
Let $(V,g_{V})$ be a quadratic graded vector space of degree $\ell$. Let $L \subseteq V$ be a graded subspace. Let $(r_{j})_{j \in \mathbb{Z}} = \gdim(V)$ and $(q_{j})_{j \in \mathbb{Z}} = \gdim(L)$.
\begin{enumerate}[(1)]
\item For $\ell \pmod 4 \neq 0$, the following statements are equivalent:
\begin{enumerate}
\item $L$ is maximal isotropic;
\item $L$ is Lagrangian;
\item $L$ is isotropic and $r_{j} = q_{j} + q_{-(j+\ell)}$ for all $j \in \mathbb{Z}$.
\end{enumerate}
\item Let $\ell \pmod 4 = 0$, that is $\ell = 2k$ for even $k \in \mathbb{Z}$. Then $\<\cdot,\cdot\>_{V}$ restricts to a non-degenerate symmetric bilinear form $\<\cdot,\cdot\>_{V}^{0}$ on $V_{-k}$. Let $(s,t)$ be the signature of $\<\cdot,\cdot\>_{V}^{0}$. Then the following statements are equivalent:
\begin{enumerate}
\item $L$ is maximal isotropic;
\item $L$ is Lagrangian (only for $s = t$);
\item $L$ is isotropic, $q_{-k} = \min \{ s,t \}$, and $r_{-k+i} = q_{-k+i} + q_{-k-i}$ for all $i \in \mathbb{N}$.
\end{enumerate}
\end{enumerate}
\end{tvrz}
\begin{rem}
Observe that in \textit{(2)-(c)}, the condition $L \subseteq L^{\perp}$ implies that $L_{-k} \subseteq V_{-k}$ is isotropic with respect to $\<\cdot,\cdot\>_{0}$, and the condition $q_{-k} = \min\{s,t\}$ ensures that $L_{-k}$ is maximal. Note that $(1)$ includes the case where $\ell = 2k$ for odd $k \in \mathbb{Z}$. In this case $\<\cdot,\cdot\>_{0}$ is a non-degenerate skew-symmetric bilinear form on $V_{-k}$ and $L_{-k} \subseteq V_{-k}$ is maximal isotropic, iff it is Lagrangian.
\end{rem}
Now, let us turn our attention to graded vector bundles. First, we must recall some basic facts about subbundles and their orthogonal complements.
\begin{definice}
Let $\mathcal{E}$ be a graded vector bundle over $\mathcal{M}$. Let $(r_{j})_{j \in \mathbb{Z}} = \grk(\mathcal{E})$. A \textbf{subbundle} $\mathcal{L} \subseteq \mathcal{E}$ of a graded rank $(q_{j})_{j \in \mathbb{Z}}$ consists of the following data:
\begin{enumerate}[(i)]
\item A subsheaf $\Gamma_{\mathcal{L}} \subseteq \Gamma_{\mathcal{E}}$ of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-submodules. In particular, $\Gamma_{\mathcal{L}}(U) \subseteq \Gamma_{\mathcal{E}}(U)$ is a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(U)$-submodule for each $U \in \Op(M)$.
\item Let $r := \sum_{j \in \mathbb{Z}} r_{j}$ and $q := \sum_{j \in \mathbb{Z}} q_{j}$. For each $m \in M$, there exists $U \in \Op_{m}(M)$ and a local frame $\{ \Phi_{\lambda} \}_{\lambda=1}^{r}$ for $\mathcal{E}$ over $U$, such that $\{ \Phi_{\lambda} \}_{\lambda=1}^{q}$ becomes a local frame for $\mathcal{L}$ over $U$. We say that $\{ \Phi_{\lambda} \}_{\lambda=1}^{r}$ is \textbf{adapted to $\mathcal{L}$}.
\end{enumerate}
In particular, $\Gamma_{\mathcal{L}}$ is a sheaf of sections of a graded vector bundle $\mathcal{L}$ of a graded rank $(q_{j})_{j \in \mathbb{Z}}$.
\end{definice}
For any subbundle $\mathcal{L} \subseteq \mathcal{E}$, one can define its \textbf{annihilator} $\an(\mathcal{L})$. For each $U \in \Op(M)$, let
\begin{equation} \label{eq_anihillator}
\Gamma_{\an(\mathcal{L})}(U) := \an( \Gamma_{\mathcal{L}}(U)) \subseteq \Gamma_{\mathcal{E}^{\ast}}(U).
\end{equation}
It is not difficult to see that $\Gamma_{\an(\mathcal{L})} \subseteq \Gamma_{\mathcal{E}^{\ast}}$ is a subsheaf of graded $\mathcal{C}^{\infty}_{\mathcal{M}}$-submodules. Suppose $\grk(\mathcal{E}) = (r_{j})_{j \in \mathbb{Z}}$ and $\grk(\mathcal{L}) = (q_{j})_{j \in \mathbb{Z}}$. Let $\{ \Phi_{\lambda} \}_{\lambda=1}^{r}$ be the local frame for $\mathcal{E}$ adapted to $\mathcal{L}$. It follows that the dual frame $\{ \Phi^{\lambda} \}_{\lambda=1}^{r}$ for $\mathcal{E}^{\ast}$ is adapted to $\an(\mathcal{L})$, proving that $\an(\mathcal{L})$ is a subbundle of $\mathcal{E}^{\ast}$ of a graded rank $(r_{-j} - q_{-j})_{j \in \mathbb{Z}}$.
Suppose $(\mathcal{E},g_{\mathcal{E}})$ is a quadratic graded vector bundle of degree $\ell$. One can thus define the \textbf{orthogonal complement $\mathcal{L}^{\perp}$ to $\mathcal{L}$} by declaring
\begin{equation}
\Gamma_{\mathcal{L}^{\perp}}(U) := \Gamma_{\mathcal{L}}(U)^{\perp} \equiv (g_{\mathcal{E}}|_{U})^{-1}\{ \an(\Gamma_{\mathcal{L}}(U))\},
\end{equation}
where $g_{\mathcal{E}}|_{U}: \Gamma_{\mathcal{E}}(U) \rightarrow \Gamma_{\mathcal{E}^{\ast}}(U)$ is obtained by the restriction of $g_{\mathcal{E}}$, see Remark \ref{rem_VBmorphisms}. Since $\Gamma_{\mathcal{L}^{\perp}}$ is the image of $\Gamma_{\an(\mathcal{L})}$ under a sheaf isomorphism of degree $-\ell$, it defines a subbundle $\mathcal{L}^{\perp} \subseteq \mathcal{E}$ of a graded rank $(r_{-(j+\ell)} - q_{-(j+\ell)})_{j \in \mathbb{Z}}$.
To talk about maximal isotropic subbundles, one has to have some notion of a fiber of a graded vector bundle. Since we deal with graded vector bundles only through their sheaves of sections, this requires some work. For each $m \in M$, the \textbf{fiber $\mathcal{E}_{m}$ of $\mathcal{E}$ over $m$} is the graded vector space
\begin{equation}
\mathcal{E}_{m} := \mathbb{R} \otimes_{\mathcal{C}^{\infty}_{\mathcal{M},m}} \Gamma_{\mathcal{E},m},
\end{equation}
where $\Gamma_{\mathcal{E},m} \in \gVect$ is the stalk of the sheaf $\Gamma_{\mathcal{E}}$ at $m$, and $\mathcal{C}^{\infty}_{\mathcal{M},m} \in \gcAs$ is the stalk of the sheaf of functions $\mathcal{C}^{\infty}_{\mathcal{M}}$ at $m$. $\Gamma_{\mathcal{E},m}$ has a natural graded $\mathcal{C}^{\infty}_{\mathcal{M},m}$-module structure induced from $\Gamma_{\mathcal{E}}$. Indeed, let $[f]_{m} \in \mathcal{C}^{\infty}_{\mathcal{M},m}$ and $[\psi]_{m} \in \Gamma_{\mathcal{E},m}$ be two germs represented by $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $\psi \in \Gamma_{\mathcal{E}}(M)$. Then
\begin{equation}
[f]_{m} \triangleright [\psi]_{m} := [f \psi]_{m}.
\end{equation}
The graded vector space $\mathbb{R}$ has the graded $\mathcal{C}^{\infty}_{\mathcal{M},m}$-module structure given by $[f]_{m} \triangleright \lambda := f(m)\lambda$ for every $[f]_{m} \in \mathcal{C}^{\infty}_{\mathcal{M},m}$ and $\lambda \in \mathbb{R}$. Now, for every $U \in \Op_{m}(M)$ and $\psi \in \Gamma_{\mathcal{E}}(U)$, its \textbf{value $\psi|_{m} \in \mathcal{E}_{m}$ at $m$} is defined as $\psi|_{m} := 1 \otimes [\psi]_{m}$. Let us summarize basic facts about fibers.
\begin{tvrz}
The fibers of graded vector bundles have the following properties:
\begin{enumerate}[(i)]
\item Let $m \in M$ and let $\{ \Phi_{\lambda} \}_{\lambda=1}^{r}$ be a local frame for $\mathcal{E}$ over $U \in \Op_{m}(M)$. Then $\{ \Phi_{\lambda}|_{m} \}_{\lambda=1}^{r}$ forms a total basis for $\mathcal{E}_{m}$. In particular, one has $\gdim(\mathcal{E}_{m}) = \grk(\mathcal{E})$.
\item For every graded vector bundle map $F: \mathcal{E} \rightarrow \mathcal{E}'$ and every $m \in M$, there is a graded linear map $F_{m}: \mathcal{E}_{m} \rightarrow \mathcal{E}'_{m}$ determined uniquely by $F_{m}(\psi|_{m}) := F(\psi)|_{m}$ for all $\psi \in \Gamma_{\mathcal{E}}(M)$.
\item For every $m \in M$ and $\ell \in \mathbb{Z}$, one has identifications $(\mathcal{E}^{\ast})_{m} \cong (\mathcal{E}_{m})^{\ast}$ and $(\mathcal{E}[\ell])_{m} \cong \mathcal{E}_{m}[\ell]$.
\item Let $\mathcal{L} \subseteq \mathcal{E}$ be a subbundle of $\mathcal{E}$ and let $I: \mathcal{L} \rightarrow \mathcal{E}$ be the canonical inclusion. Then for each $m \in M$, $I_{m}: \mathcal{L}_{m} \rightarrow \mathcal{E}_{m}$ is injective. Consequently, we always identify $\mathcal{L}_{m}$ with the graded subspace $I_{m}(\mathcal{L}_{m}) \subseteq \mathcal{E}_{m}$.
\item Suppose that $(\mathcal{E},g_{\mathcal{E}})$ is a quadratic graded vector bundle of degree $\ell$. For each $m \in M$, there is an induced degree $\ell$ isomorphism $g_{\mathcal{E}_{m}}: \mathcal{E}_{m} \rightarrow (\mathcal{E}_{m})^{\ast}$ and consequently a graded symmetric bilinear form $\<\cdot,\cdot\>_{\mathcal{E}_{m}}: \mathcal{E}_{m} \times \mathcal{E}_{m} \rightarrow \mathbb{R}$ of degree $\ell$. In other words, $(\mathcal{E}_{m}, g_{\mathcal{E}_{m}})$ becomes a quadratic graded vector space of degree $\ell$ for every $m \in M$.
Let $\mathcal{L} \subseteq \mathcal{E}$ be a subbundle of $\mathcal{E}$. Then every fiber of its orthogonal complement is the orthogonal complement of the fiber, $(\mathcal{L}^{\perp})_{m} = (\mathcal{L}_{m})^{\perp}$ for all $m \in M$.
\end{enumerate}
\end{tvrz}
See \S 5.2 of \cite{vysoky2021global} for details and proofs.
\begin{definice}
Let $(\mathcal{E},g_{\mathcal{E}})$ be a quadratic graded vector bundle of degree $\ell$ over $\mathcal{M}$. Let $\mathcal{L} \subseteq \mathcal{E}$ be its subbundle. We say that $\mathcal{L}$ is an \textbf{isotropic subbundle}, if $\mathcal{L} \subseteq \mathcal{L}^{\perp}$. We say that an isotropic subbundle $\mathcal{L}$ is \textbf{maximal}, if for every $m \in M$, the fiber $\mathcal{L}_{m}$ is a maximal isotropic subspace of $\mathcal{E}_{m}$. We say that $\mathcal{L}$ is \textbf{Lagrangian}, if $\mathcal{L} = \mathcal{L}^{\perp}$.
\end{definice}
We find a graded vector bundle version of Proposition \ref{tvrz_maxisos1}.
\begin{tvrz} \label{tvrz_maxisos2}
Let $(\mathcal{E},g_{\mathcal{E}})$ be a quadratic vector bundle of degree $\ell$ over $\mathcal{M}$. Let $\mathcal{L} \subseteq \mathcal{E}$ be its subbundle. Let $(r_{j})_{j \in \mathbb{Z}} = \grk(\mathcal{E})$ and $(q_{j})_{j \in \mathbb{Z}} = \grk(\mathcal{L})$.
\begin{enumerate}[(1)]
\item For $\ell \pmod 4 \neq 0$, the following statements are equivalent:
\begin{enumerate}
\item $\mathcal{L}$ is maximal isotropic;
\item $\mathcal{L}$ is Lagrangian;
\item $\mathcal{L}$ is isotropic and $r_{j} = q_{j} + q_{-(j+\ell)}$ for all $j \in \mathbb{Z}$.
\end{enumerate}
\item Let $\ell \pmod 4 = 0$, that is $\ell = 2k$ for even $k \in \mathbb{Z}$. Then for each $m \in M$, $\<\cdot,\cdot\>_{\mathcal{E}_{m}}$ restricts to the non-degenerate symmetric bilinear form $\<\cdot,\cdot\>_{\mathcal{E}_{m}}^{0}$ on $(\mathcal{E}_{m})_{-k}$. Let $(s_{m},t_{m})$ be the signature of $\<\cdot,\cdot\>_{\mathcal{E}_{m}}^{0}$. Then the following statements are equivalent:
\begin{enumerate}
\item $\mathcal{L}$ is maximal isotropic;
\item $\mathcal{L}$ is Lagrangian (only valid when $s_{m} = t_{m}$ for all $m \in M$);
\item $\mathcal{L}$ is isotropic, $q_{-k} = \min \{ s_{m},t_{m} \}$ and $r_{-k+i} = q_{-k+i} + q_{-k-i}$ for all $m \in M$ and $i \in \mathbb{N}$.
\end{enumerate}
\end{enumerate}
\end{tvrz}
\begin{rem}
All statements follow directly from Proposition \ref{tvrz_maxisos1}. Note that in the definition of isotropic subbundles, we assume that $\mathcal{L} \subseteq \mathcal{L}^{\perp}$, that is $\Gamma_{\mathcal{L}} \subseteq \Gamma_{\mathcal{L}^{\perp}}$, and not just the fiber-wise inclusion $\mathcal{L}_{m} \subseteq (\mathcal{L}_{m})^{\perp}$ for all $m \in M$. This is because by ``going fiber-wise'', one usually loses a lot of data. In other words, the isotropy of $\mathcal{L}$ implies the isotropy of the graded vector space $\mathcal{L}_{m}$ for all $m \in M$, but the converse statement may fail. Finally, observe that the signature $(s_{m},t_{m})$ is always constant in $m$ along each connected component of $M$. Since our subbundles are assumed to have a constant graded rank, we can safely assume that $(s_{m},t_{m})$ is globally constant (otherwise no maximally isotropic subbundles would exist).
\end{rem}
\begin{example}
Let $\mathcal{E} = T[\ell]\mathcal{M} \oplus T^{\ast}\mathcal{M}$ and $g_{\mathcal{E}}$ be defined as in Example \ref{ex_canpairing}. If $(n_{j})_{j \in \mathbb{Z}} = \gdim(\mathcal{M})$, then $\grk(\mathcal{E}) = (n_{-(j+\ell)} + n_{j})_{j \in \mathbb{Z}}$. Recall that the \textbf{tangent space at $m \in M$} is defined as
\begin{equation}
T_{m}\mathcal{M} = \gDer(\mathcal{C}^{\infty}_{\mathcal{M}}(M), \mathbb{R}).
\end{equation}
In other words, $v \in T_{m}\mathcal{M}$ is a degree $|v|$ graded linear map $v: \mathcal{C}^{\infty}_{\mathcal{M}}(M) \rightarrow \mathbb{R}$ satisfying
\begin{equation}
v(fg) = v(f) g(m) + (-1)^{|v||f|} f(m) v(g),
\end{equation}
for all $f,g \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. For each $m \in M$, $T_{m}\mathcal{M}$ can be canonically identified with the fiber $(T\mathcal{M})_{m}$. For every vector field $X \in \mathfrak{X}_{\mathcal{M}}(M)$, we have $X|_{m}(f) := (X(f))(m)$. Whence
\begin{equation}
\mathcal{E}_{m} = (T_{m}\mathcal{M})[\ell] \oplus T^{\ast}_{m}\mathcal{M}.
\end{equation}
Let $(v,\alpha), (w,\beta) \in \mathcal{E}_{m}$. The induced bilinear form $\<(v,\alpha), (w,\beta)\>_{\mathcal{E}_{m}}$ is non-zero only if $|(v,\alpha)| + |(w,\beta)| + \ell = 0$. Recall that $|(v,\alpha)| = |v| - \ell = |\alpha|$, hence necessarily $|v| + |\beta| = 0$ and $|w| + |\alpha| = 0$, and in this case, one has
\begin{equation}
\<(v,\alpha),(w,\beta)\>_{\mathcal{E}_{m}} = \alpha(w) + (-1)^{|v||w|} \beta(v).
\end{equation}
Whenever $\ell = 2k$ for $k \in \mathbb{Z}$, we may restrict $\<\cdot,\cdot\>_{\mathcal{E}_{m}}$ to $(\mathcal{E}_{m})_{-k}$, finding an ordinary bilinear form
\begin{equation}
\<(v,\alpha),(w,\beta)\>_{\mathcal{E}_{m}}^{0} = \alpha(w) + (-1)^{k} \beta(v),
\end{equation}
for all $(v,\alpha),(w,\beta) \in (\mathcal{E}_{m})_{-k}$. For $\ell \pmod 4 = 0$, that is $k$ even, we see that $\<\cdot,\cdot\>^{0}_{\mathcal{E}_{m}}$ is indeed a symmetric bilinear form on $(\mathcal{E}_{m})_{-k}$ with a signature $(n_{-k},n_{-k})$, where $(n_{j})_{j \in \mathbb{Z}} = \gdim(\mathcal{M})$. Hence by Proposition \ref{tvrz_maxisos2}, a subbundle $\mathcal{L} \subseteq \mathcal{E}$ is maximally isotropic, iff it is Lagrangian, $\mathcal{L} = \mathcal{L}^{\perp}$. Equivalently, it has to be isotropic and its graded rank $(q_{j})_{j \in \mathbb{Z}}$ must for all $j \in \mathbb{Z}$ satisfy
\begin{equation} \label{eq_grkmaxisostandardgenvec}
n_{j} + n_{-(j+\ell)} = q_{j} + q_{-(j+\ell)}.
\end{equation}
\end{example}
\begin{definice}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a graded Courant algebroid of degree $\ell$.
We say that a subbundle $\mathcal{L} \subseteq \mathcal{E}$ is a \textbf{Dirac structure on $\mathcal{E}$}, if it is maximally isotropic and the graded submodule $\Gamma_{\mathcal{L}}(M)$ is involutive under $[\cdot,\cdot]_{\mathcal{E}}$.
\end{definice}
\begin{example} \label{ex_PoissonasDirac}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{D}^{H})$ be a graded Courant algebroid of degree $\ell$ from Example \ref{ex_gDorfman}. This example should be considered a graded version of some ideas in the famous paper \cite{Severa:2001qm}.
Let us consider a vector bundle map $\Pi^{\sharp}: T^{\ast}\mathcal{M} \rightarrow T[\ell]\mathcal{M}$ of degree zero. Equivalently, this is a vector bundle map $\Pi^{\sharp}: T^{\ast}\mathcal{M} \rightarrow T\mathcal{M}$ of degree $\ell$. Its graph $\gr(\Pi^{\sharp})$ is a subbundle of $\mathcal{E}$ defined as
\begin{equation}
\Gamma_{\gr(\Pi^{\sharp})}(U) := \{ (\Pi^{\sharp}|_{U}(\xi),\xi) \; | \; \xi \in \Omega^{1}_{\mathcal{M}}(U) \} \subseteq \Gamma_{\mathcal{E}}(U),
\end{equation}
for each $U \in \Op(M)$. Observe that $\grk( \gr(\Pi^{\sharp})) = \grk(\Omega^{1}_{\mathcal{M}}) = (n_{j})_{j \in \mathbb{Z}}$. Hence it satisfies the constraint (\ref{eq_grkmaxisostandardgenvec}). The condition $\gr(\Pi^{\sharp}) \subseteq \gr(\Pi^{\sharp})^{\perp}$ is equivalent to
\begin{equation}
\<(\Pi^{\sharp}(\xi),\xi), (\Pi^{\sharp}(\eta),\eta) \>_{\mathcal{E}} = 0,
\end{equation}
for all $\xi,\eta \in \Omega^{1}_{\mathcal{M}}(M)$. By plugging in from (\ref{eq_canpairing}), this gives the equation
\begin{equation}
\xi(\Pi^{\sharp}(\eta)) + (-1)^{(|\xi|+\ell)(|\eta|+\ell)} \eta( \Pi^{\sharp}(\xi)) = 0.
\end{equation}
Using (\ref{eq_vfactionon1form}), one can rewrite this as $[\Pi^{\sharp}(\xi)](\eta) + (-1)^{|\xi||\eta|+\ell} [\Pi^{\sharp}(\eta)](\xi) = 0$. Let $\Pi: \mathfrak{X}_{\mathcal{M}}(M) \times \mathfrak{X}_{\mathcal{M}}(M) \rightarrow \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ be a degree $\ell$ bilinear map defined by
\begin{equation} \label{eq_PiPisharp}
\Pi(\xi,\eta) := [\Pi^{\sharp}(\xi)](\eta),
\end{equation}
for all $\xi,\eta \in \Omega^{1}_{\mathcal{M}}(M)$. In other words, we have shown that $\gr(\Pi^{\sharp})$ is maximal isotropic, if and only if $\Pi^{\sharp}$ is induced using (\ref{eq_PiPisharp}) by a skew-symmetric bivector $\Pi \in \mathfrak{X}^{2}_{\mathcal{M}}(M)$ for even $\ell$ and symmetric bivector $\Pi \in \~\mathfrak{X}^{2}_{\mathcal{M}}(M)$ for odd $\ell$. Finally, for every $\xi,\eta \in \Omega^{1}_{\mathcal{M}}(M)$, one finds
\begin{equation}
\begin{split}
[(\Pi^{\sharp}(\xi),\xi), (\Pi^{\sharp}(\eta), \eta)]_{D}^{H} = & \ \big([\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta)], (-1)^{(|\xi|+\ell)\ell} \Li{\Pi^{\sharp}(\xi)}\eta \\
& - (-1)^{(|\eta|+\ell)|\xi|} i_{\Pi^{\sharp}(\eta)} \mathrm{d}{\xi} + (-1)^{\ell} H(\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta), \cdot) \big)
\end{split}
\end{equation}
It is convenient to define an $\mathbb{R}$-bilinear bracket $[\cdot,\cdot]_{\Pi}: \Omega^{1}_{\mathcal{M}}(M) \times \Omega^{1}_{\mathcal{M}}(M) \rightarrow \Omega^{1}_{\mathcal{M}}(M)$ by
\begin{equation}
[\xi,\eta]_{\Pi} := (-1)^{(|\xi|+\ell)\ell} \Li{\Pi^{\sharp}(\xi)}\eta - (-1)^{(|\eta|+\ell)|\xi|} i_{\Pi^{\sharp}(\eta)} \mathrm{d}{\xi},
\end{equation}
for all $\xi,\eta \in \Omega^{1}_{\mathcal{M}}(M)$. Note that $|[\xi,\eta]_{\Pi}| = |\xi| + |\eta| + \ell$. We see that $\gr(\Pi^{\sharp})$ is involutive, iff
\begin{equation}
[\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta)] = \Pi^{\sharp}\big( [\xi,\eta]_{\Pi} + (-1)^{\ell} H(\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta), \cdot) \big),
\end{equation}
for all $\xi,\eta \in \Omega^{1}_{\mathcal{M}}(M)$. This can be after a lot of work rewritten into the form
\begin{equation} \label{eq_PiPiSHtwisted}
\frac{1}{2}[\Pi,\Pi]_{S}(\xi,\eta,\zeta) = -(-1)^{(|\eta|+\ell)\ell} H(\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta), \Pi^{\sharp}(\zeta)),
\end{equation}
for all $\xi,\eta,\zeta \in \Omega^{1}_{\mathcal{M}}(M)$. Note that it is important that $\Pi \in \mathfrak{X}^{2}_{\mathcal{M}}(M)$ for even $\ell$ and $\Pi \in \~\mathfrak{X}^{2}_{\mathcal{M}}(M)$ for odd $\ell$. A derivation of this formula is straightforward but a bit tedious, hence we omit it here. We see that $\gr(\Pi^{\sharp}) \subseteq \mathcal{E}$ is a Dirac structure, iff $(\mathcal{M},\Pi)$ defines an \textbf{$H$-twisted graded Poisson manifold of degree $\ell$}. For $H = 0$, it is just a graded Poisson manifold of degree $\ell$, see Example \ref{ex_gPoisson}. Recall that the bracket $\{\cdot,\cdot\}$ is obtained by $\{f,g\} = [[f,\Pi]_{S},g]_{S}$. Let
\begin{equation}
\mathbf{J}(f,g,h) = \{f, \{g,h\}\} - \{\{f,g\},h\} - (-1)^{(|f|+\ell)(|g|+\ell)} \{g, \{f,h\}\}
\end{equation}
be the Jacobiator of the bracket $\{\cdot,\cdot\}$. Using Proposition \ref{tvrz_SNB} and Proposition \ref{tvrz_SNB2}, one can prove
\begin{equation} \label{eq_JacobiatorasSchouten}
\mathbf{J}(f,g,h) = \frac{1}{2} (-1)^{|f| + |g|(1+\ell) + |h|} [\Pi,\Pi]_{S}(\mathrm{d}{f}, \mathrm{d}{g}, \mathrm{d}{h}).
\end{equation}
Now, recall that for each $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ the corresponding Hamiltonian vector field $X_{f} := [f,\Pi]_{S}$ can be written as $X_{f} = -(-1)^{|f|(1+\ell)} \Pi^{\sharp}(\mathrm{d}{f})$. Combining (\ref{eq_PiPiSHtwisted}) and (\ref{eq_JacobiatorasSchouten}) then gives
\begin{equation}
\mathbf{J}(f,g,h) = (-1)^{(|f|+|g|+|h|)\ell} H(X_{f},X_{g},X_{h}).
\end{equation}
\end{example}
\section{Generalized complex structures}
In this section, we define a graded analogue of generalized complex structures, introduced in \cite{Gualtieri:2003dx}.
Let $\mathcal{E}$ be a graded vector bundle over $\mathcal{M}$. Let $(\mathcal{C}_{\mathcal{M}}^{\infty})_{\mathbb{C}}$ denote the complexification of the structure sheaf $\mathcal{C}^{\infty}_{\mathcal{M}}$ of $\mathcal{M}$. Its sections over $U \in \Op(M)$ have the form $f + ig$, where $f,g \in \mathcal{C}^{\infty}_{\mathcal{M}}(U)$ satisfy $|f| = |g|$. The multiplication is defined in the usual way to make $(\mathcal{C}^{\infty}_{\mathcal{M}})_{\mathbb{C}}$ into the sheaf of \textit{complex} graded commutative associative algebras.
Now, let $\Gamma_{\mathcal{E}_{\mathbb{C}}} := (\Gamma_{\mathcal{E}})_{\mathbb{C}}$ be the complexification of the sheaf of sections of $\mathcal{E}$. It follows that $\Gamma_{\mathcal{E}_{\mathbb{C}}}$ becomes a sheaf of \textit{complex} graded $(\mathcal{C}^{\infty}_{\mathcal{M}})_{\mathbb{C}}$-modules. We say that $\Gamma_{\mathcal{E}_{\mathbb{C}}}$ is a sheaf of sections of the \textit{complex} vector bundle $\mathcal{E}_{\mathbb{C}}$, called the \textbf{complexification of $\mathcal{E}$}. Sections in $\Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ are of the form $\psi + i \phi$, where $\psi,\phi \in \Gamma_{\mathcal{E}}(M)$ satisfy $|\psi| = |\phi|$, and for each $f + ig \in (\mathcal{C}^{\infty}_{\mathcal{M}})_{\mathbb{C}}(M)$, one has
\begin{equation}
(f + ig)(\psi + i\phi) := f \psi - g \phi + i (f \phi + g \psi).
\end{equation}
A theory of complex graded vector bundles and their subbundles is completely analogous to the real case, except that involved graded vector spaces are complex and all graded linear maps are $\mathbb{C}$-linear. Now, there is a canonical $(\mathcal{C}^{\infty}_{\mathcal{M}})_{\mathbb{C}}(M)$-antilinear map $\Sigma: \Gamma_{\mathcal{E}_{\mathbb{C}}}(M) \rightarrow \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ of degree zero, defined for all $\psi + i \phi \in \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ as
\begin{equation}
\Sigma(\psi + i \phi) := \psi - i \phi,
\end{equation}
$\Sigma$ is called the \textbf{complex conjugation on $\mathcal{E}_{\mathbb{C}}$}. Now, for any \textit{complex} subbundle $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$, let
\begin{equation}
\Gamma_{\overline{\mathcal{L}}}(U) := \Sigma|_{U}( \Gamma_{\mathcal{L}}(U)).
\end{equation}
It follows that $\Gamma_{\overline{\mathcal{L}}}$ defines a sheaf of sections of a complex subbundle $\overline{\mathcal{L}} \subseteq \mathcal{E}$ called the \textbf{complex conjugate of $\mathcal{L}$}. One has $\grk_{\mathbb{C}}(\mathcal{L}) = \grk_{\mathbb{C}}(\overline{\mathcal{L}})$.
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a graded Courant algebroid of degree $\mathcal{E}$. All structures can be naturally complexified to obtain a \textit{complex} graded Courant algebroid $(\mathcal{E}_{\mathbb{C}}, \rho_{\mathbb{C}}, g_{\mathcal{E}_{\mathbb{C}}}, [\cdot,\cdot]_{\mathcal{E}_{\mathbb{C}}})$ of degree $\ell$, where $\rho_{\mathbb{C}}: \mathcal{E}_{\mathbb{C}} \rightarrow (T\mathcal{M})_{\mathbb{C}}$ and $g_{\mathcal{E}_{\mathbb{C}}}: \mathcal{E}_{\mathbb{C}} \rightarrow (\mathcal{E}_{\mathbb{C}})^{\ast}$ are complex vector bundle morphisms and $[\cdot,\cdot]_{\mathcal{E}_{\mathbb{C}}}: \Gamma_{\mathcal{E}_{\mathbb{C}}}(M) \times \Gamma_{\mathcal{E}_{\mathbb{C}}}(M) \rightarrow \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ is a $\mathbb{C}$-bilinear bracket. $g_{\mathcal{E}_{\mathbb{C}}}$ then induces a graded symmetric $\mathbb{C}$-bilinear form $\<\cdot,\cdot\>_{\mathcal{E}_{\mathbb{C}}}$.
\begin{definice}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a graded Courant algebroid of degree $\ell$. We say that a complex subbundle $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$ is a \textbf{generalized complex structure}, if
\begin{enumerate}[(i)]
\item $\mathcal{L}$ is isotropic with respect to $\<\cdot,\cdot\>_{\mathcal{E}_{\mathbb{C}}}$ and $\Gamma_{\mathcal{L}}(M)$ is involutive with respect to $[\cdot,\cdot]_{\mathcal{E}_{\mathbb{C}}}$;
\item one can write $\mathcal{E}_{\mathbb{C}} = \mathcal{L} \oplus \overline{\mathcal{L}}$.
\end{enumerate}
\end{definice}
\begin{rem}
Observe that contrary to the ordinary case, see e.g. Proposition 4.3 in \cite{Gualtieri:2003dx}, we do not assume that $\mathcal{L}$ is a \textit{maximal} isotropic subbundle. However, let $(r_{j})_{j \in \mathbb{Z}} = \grk(\mathcal{E}) = \grk_{\mathbb{C}}(\mathcal{E}_{\mathbb{C}})$ and $(q_{j})_{j \in \mathbb{Z}} = \grk_{\mathbb{C}}(\mathcal{L})$. The condition (ii) forces $r_{j} = 2 q_{j}$ for each $j \in \mathbb{Z}$. Since $g_{\mathcal{E}}$ is an isomorphism, by Remark \ref{rem_nondegen} we have $r_{j} = r_{-(j+\ell)}$. Hence necessarily also $q_{j} = q_{-(j+\ell)}$. It follows that $\grk_{\mathbb{C}}(\mathcal{L}) = \grk_{\mathbb{C}}(\mathcal{L}^{\perp})$ and $\mathcal{L}$ is automatically Lagrangian, hence maximal isotropic.
\end{rem}
\begin{tvrz} \label{tvrz_Jgencomplex}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a graded Courant algebroid of degree $\ell$. Let $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$ be a generalized complex structure. Then it induces a degree zero vector bundle map $\mathcal{J}: \mathcal{E} \rightarrow \mathcal{E}$ having the following properties:
\begin{enumerate}[(i)]
\item $\mathcal{J}^{2} = -\mathbbm{1}_{\mathcal{E}}$ and $\mathcal{J}$ is orthogonal with respect to $\<\cdot,\cdot\>_{\mathcal{E}}$;
\item The \textbf{Nijenhuis tensor} $N_{\mathcal{J}}$ defined for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$ by
\begin{equation}
N_{\mathcal{J}}(\psi,\psi') := [\psi,\psi']_{\mathcal{E}} - [\mathcal{J}(\psi),\mathcal{J}(\psi')]_{\mathcal{E}} + \mathcal{J}([\psi,\mathcal{J}(\psi')]_{\mathcal{E}} + [\mathcal{J}(\psi),\psi']_{\mathcal{E}})
\end{equation}
vanishes identically.
\item $\Gamma_{\mathcal{L}}(U)$ coincides with the $+i$ eigenspace of $\mathcal{J}_{\mathbb{C}}|_{U}$ for each $U \in \Op(M)$. Consequently, $\Gamma_{\overline{\mathcal{L}}}(U)$ coincides with the $-i$ eigenspace of $\mathcal{J}_{\mathbb{C}}|_{U}$ for each $U \in \Op(M)$.
\end{enumerate}
\end{tvrz}
\begin{proof}
By assumption, one can decompose every $\psi \in \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ as $\psi = \phi + \Sigma(\phi')$ for unique $\phi,\phi' \in \Gamma_{\mathcal{L}}(M)$. Define $\mathcal{J}': \Gamma_{\mathcal{E}_{\mathbb{C}}}(M) \rightarrow \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ as $\mathcal{J}'(\phi + \Sigma(\phi')) := i(\phi - \Sigma(\phi'))$. In other words, $\Gamma_{\mathcal{L}}(M)$ and $\Gamma_{\overline{\mathcal{L}}}(M)$ become the $+i$ and $-i$ eigenspaces of $\mathcal{J}'$, respectively. It is easy to see that
\begin{equation}
\mathcal{J}' \circ \Sigma = \Sigma \circ \mathcal{J}',
\end{equation}
whence $\mathcal{J}' = \mathcal{J}_{\mathbb{C}}$ for a unique degree zero $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-linear map $\mathcal{J}: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$. $\mathcal{J}$ satisfies the property $(iii)$ by definition. Moreover, since clearly $\mathcal{J}'^{2} = -\mathbbm{1}_{\mathcal{E}_{\mathbb{C}}}$ and $\mathcal{J}'$ is orthogonal with respect to $\<\cdot,\cdot\>_{\mathcal{E}_{\mathbb{C}}}$, the properties $(i)$ of $\mathcal{J}$ follow immediately.
It follows from the construction of $\mathcal{J}$ that $\Gamma_{\mathcal{L}}(M)$ is involutive with respect to $[\cdot,\cdot]_{\mathcal{E}_{\mathbb{C}}}$, iff
\begin{equation}
[\psi - i\mathcal{J}(\psi), \psi' - i \mathcal{J}(\psi')]_{\mathcal{E}_{\mathbb{C}}} \in \Gamma_{\mathcal{L}}(M),
\end{equation}
for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. The projection of this expression to $\Gamma_{\overline{\mathcal{L}}}(M)$ must vanish, that is
\begin{equation}
\begin{split}
0 = & \ (\mathbbm{1}_{\mathcal{E}_{\mathbb{C}}} + i \mathcal{J}_{\mathbb{C}})( [\psi - i\mathcal{J}(\psi), \psi' - i \mathcal{J}(\psi')]_{\mathcal{E}_{\mathbb{C}}}) \\
= & \ N_{\mathcal{J}}(\psi,\psi') + i \mathcal{J}( N_{\mathcal{J}}(\psi,\psi')).
\end{split}
\end{equation}
We see that $\Gamma_{\mathcal{L}}(M)$ is involutive with respect to $[\cdot,\cdot]_{\mathcal{E}_{\mathbb{C}}}$, iff $N_{\mathcal{J}}(\psi,\psi') = 0$ for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. This proves the claim $(ii)$.
\end{proof}
\begin{rem} \label{rem_gencomplex}
For ordinary Courant algebroids, every vector bundle map $\mathcal{J}: \mathcal{E} \rightarrow \mathcal{E}$ having the above properties $(i)$ and $(ii)$ determines the unique generalized complex structure $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$ by declaring $\mathcal{L}$ to satisfy $(iii)$. For \emph{graded} Courant algebroids, it may happen that $\Gamma_{\mathcal{L}}$ defined by
\begin{equation} \label{eq_gcLintermsofJ}
\Gamma_{\mathcal{L}} = \ker(\mathcal{J}_{\mathbb{C}} - i \mathbbm{1}_{\mathcal{E}_{\mathbb{C}}})
\end{equation}
fails to define a complex subbundle $\mathcal{L}$ of $\mathcal{E}_{\mathbb{C}}$. However, if one \textit{adds} the assumption that $\Gamma_{\mathcal{L}}$ defines a complex subbundle of $\mathcal{E}_{\mathbb{C}}$, it is not difficult to argue that $\mathcal{L}$ is a generalized complex structure based on the properties $(i)$ of $(ii)$ of $\mathcal{J}$. Note that this technical issue is closely related to the one described in Remark \ref{rem_noSerreswan}.
\end{rem}
\begin{example} \label{ex_gcomplexsymplectic}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{D})$ be the graded Courant algebroid of degree $\ell$ from Example \ref{ex_gDorfman} with $H = 0$. Let $\omega \in \Omega^{2}_{\mathcal{M}}(M)$ be a $2$-form of degree $-\ell$. Let $\omega^{\flat}: T\mathcal{M} \rightarrow T^{\ast}\mathcal{M}$ be the degree $\ell$ vector bundle map induced by $\omega$, that is
\begin{equation}
[\omega^{\flat}(X)](Y) := \omega(X,Y),
\end{equation}
for all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$. $\omega^{\flat}$ can be viewed as a vector bundle map $\omega^{\flat}: T[\ell]\mathcal{M} \rightarrow T^{\ast}\mathcal{M}$ of degree zero. Suppose that $\omega$ is a \textbf{symplectic form}, that is $\omega^{\flat}$ is an isomorphism and $\mathrm{d}{\omega} = 0$.
For each $U \in \Op(M)$, let us define the following graded $(\mathcal{C}^{\infty}_{\mathcal{M}}(U))_{\mathbb{C}}$-submodule of $\Gamma_{\mathcal{E}_{\mathbb{C}}}(U)$:
\begin{equation}
\Gamma_{\mathcal{L}}(U) := \{ (X, \omega^{\flat}|_{U}(Y)) + i(Y, -\omega^{\flat}|_{U}(X)) \; | \; X+iY \in (\mathfrak{X}_{\mathcal{M}}(U))_{\mathbb{C}} \}
\end{equation}
It is straightforward to verify that $\Gamma_{\mathcal{L}}$ defines a sheaf of sections of a complex subbundle $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$. By construction, this subbundle is isomorphic to $(T\mathcal{M})_{\mathbb{C}}$. It is straightforward to verify that $\mathcal{E}_{\mathbb{C}} = \mathcal{L} \oplus \overline{\mathcal{L}}$ and $\mathcal{L} \subseteq \mathcal{L}^{\perp}$. Finally, its involutivity can be shown to be equivalent to $\mathrm{d}{\omega} = 0$. In other words, $\mathcal{L}$ defines a generalized complex structure. The corresponding vector bundle morphism $\mathcal{J}: \mathcal{E} \rightarrow \mathcal{E}$ takes the explicit form
\begin{equation} \label{eq_Jforgencomplexsymplectic}
\mathcal{J}(X,\xi) = (-(\omega^{\flat})^{-1}(\xi), \omega^{\flat}(X)),
\end{equation}
for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$. Note that for $H \neq 0$, $\mathcal{L}$ is \textit{not} involutive for any $\omega$.
\end{example}
\begin{example}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{D}^{H})$ be the graded Courant algebroid of degree $\ell$ from Example \ref{ex_gDorfman}. Let $J: T\mathcal{M} \rightarrow T\mathcal{M}$ be a degree zero vector bundle map. We say that $J$ is a \textbf{complex structure} on $\mathcal{M}$, if it has the following properties:
\begin{enumerate}[(i)]
\item $J^{2} = -\mathbbm{1}_{T\mathcal{M}}$.
\item Let $J_{\mathbb{C}}: (T\mathcal{M})_{\mathbb{C}} \rightarrow (T\mathcal{M})_{\mathbb{C}}$ be the complexification of $J$, and define
\begin{equation}
\mathfrak{X}^{1,0}_{\mathcal{M}} := \ker( J_{\mathbb{C}} - i \mathbbm{1}_{(T\mathcal{M})_{\mathbb{C}}}).
\end{equation}
Then $\mathfrak{X}^{1,0}_{\mathcal{M}}$ must define a sheaf of sections of a complex subbundle $T^{1,0}\hspace{-0.8mm}\mathcal{M}$ of $(T\mathcal{M})_{\mathbb{C}}$, such that $\mathfrak{X}^{1,0}_{\mathcal{M}}(M)$ is involutive with respect to the complexified graded commutator $[\cdot,\cdot]_{\mathbb{C}}$.
\end{enumerate}
For ordinary manifolds, $\mathfrak{X}^{1,0}_{\mathcal{M}}$ defines a subbundle for any $J$ satisfying (i). However, in the graded setting, this is no longer true. The involutivity of $\mathfrak{X}^{1,0}_{\mathcal{M}}(M)$ is equivalent to
\begin{equation}
0 = N_{J}(X,Y) = [X,Y] - [J(X),J(Y)] + J([X,J(Y)] + [J(X),Y]),
\end{equation}
for all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$. The complex conjugate to $\mathfrak{X}^{1,0}_{\mathcal{M}}$ is $\mathfrak{X}^{0,1}_{\mathcal{M}} = \ker(J_{\mathbb{C}} + i \mathbbm{1}_{(TM)_{\mathbb{C}}})$, and let
\begin{equation}
\Omega^{1,0}_{\mathcal{M}} := \an( \mathfrak{X}^{0,1}_{\mathcal{M}}), \; \; \Omega^{0,1}_{\mathcal{M}} := \an( \mathfrak{X}^{1,0}_{\mathcal{M}}).
\end{equation}
It follows easily that $(T\mathcal{M})_{\mathbb{C}} = \mathfrak{X}^{1,0}_{\mathcal{M}} \oplus \mathfrak{X}^{0,1}_{\mathcal{M}}$ and $(T^{\ast}\mathcal{M})_{\mathbb{C}} = \Omega^{1,0}_{\mathcal{M}} \oplus \Omega^{0,1}_{\mathcal{M}}$. Let
\begin{equation}
\Gamma_{\mathcal{L}} := \mathfrak{X}^{1,0}_{\mathcal{M}}[\ell] \oplus \Omega^{0,1}_{\mathcal{M}}.
\end{equation}
Now, the complexified Courant algebroid $\mathcal{E}_{\mathbb{C}}$ can be identified with $\mathcal{E}_{\mathbb{C}} = (T\mathcal{M})_{\mathbb{C}}[\ell] \oplus (T^{\ast}\mathcal{M})_{\mathbb{C}}$. The complexified fiber-wise metric and the bracket then take the same form as in Example \ref{ex_gDorfman}, except that all operations are replaced by their complexified versions. In particular, the bracket contains the complexification $H_{\mathbb{C}}$ of the $3$-form $H$.
It is then easy to see that $\Gamma_{\mathcal{L}}(M)$ is involutive, iff $\mathfrak{X}^{1,0}_{\mathcal{M}}(M)$ is involutive with respect to $[\cdot,\cdot]_{\mathbb{C}}$ (this follows from the property (ii) of complex structures), and
\begin{equation}
H_{\mathbb{C}}(X,Y,Z) = 0,
\end{equation}
for all $X,Y,Z \in \mathfrak{X}_{\mathcal{M}}^{1,0}(M)$. In terms of $H$ and $J$, this condition can be rewritten as
\begin{equation}
H(X,Y,Z) = H(J(X),J(Y),Z) + H(X,J(Y),J(Z)) + H(J(X),Y,J(Z)),
\end{equation}
for all $X,Y,Z \in \mathfrak{X}_{\mathcal{M}}(M)$. Finally, the corresponding vector bundle map $\mathcal{J}: \mathcal{E} \rightarrow \mathcal{E}$ takes the form
\begin{equation}
\mathcal{J}(X,\xi) = (J(X), -J^{T}(\xi)),
\end{equation}
for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$. Note that although we call $J$ a complex structure on $\mathcal{M}$, we do not claim that there is a graded version of the Newlander--Nirenberg theorem.
\end{example}
\section{Differential graded Courant algebroids}
A grading of a space of global sections of a graded vector bundle $\mathcal{E}$ allows one to consider maps of a non-zero degree. Suppose $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ is a graded Courant algebroid of degree $\ell \in \mathbb{Z}$.
In particular, we can consider a degree $1$ derivation of the bracket $[\cdot,\cdot]_{\mathcal{E}}$, that is an $\mathbb{R}$-linear map $\Delta: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ of degree $1$ satisfying
\begin{equation} \label{eq_Deltaderivation}
\Delta([\psi,\psi']_{\mathcal{E}}) = [\Delta(\psi),\psi']_{\mathcal{E}} + (-1)^{|\psi|+\ell} [\psi, \Delta(\psi')]_{\mathcal{E}},
\end{equation}
for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$. However, $\Gamma_{\mathcal{E}}(M)$ is not just a graded vector space, but a graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module. Moreover, the bracket $[\cdot,\cdot]_{\mathcal{E}}$ is not skew-symmetric. We thus have to make sure that the above condition is consistent with the axioms of a graded Courant algebroid. Taking this into consideration, one ends up with the following definition.
\begin{definice} \label{def_delta}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ be a graded Courant algebroid of degree $\ell$. Let $\Delta: \Gamma_{\mathcal{E}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ be a degree $1$ graded $\mathbb{R}$-linear map, such that:
\begin{enumerate}[(i)]
\item $\Delta$ squares to zero, that is $\Delta^{2} = 0$.
\item $\Delta$ is compatible with the graded $\mathcal{C}^{\infty}_{\mathcal{M}}(M)$-module structure on $\Gamma_{\mathcal{E}}(M)$, that is there exists a vector field $\ul{\Delta} \in \mathfrak{X}_{\mathcal{M}}(M)$, such that
\begin{equation} \label{eq_DeltaLeibniz}
\Delta(f\psi) = (-1)^{|f|} f \Delta(\psi) + \ul{\Delta}(f) \psi,
\end{equation}
for all $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $\psi \in \Gamma_{\mathcal{E}}(M)$.
\item $\Delta$ is compatible with the fiber-wise metric, that is for all $\psi,\psi \in \Gamma_{\mathcal{E}}(M)$, one has
\begin{equation} \label{eq_Deltametric}
\ul{\Delta} \<\psi,\psi'\>_{\mathcal{E}} = \< \Delta(\psi), \psi'\>_{\mathcal{E}} + (-1)^{|\psi|+\ell} \<\psi, \Delta(\psi')\>_{\mathcal{E}},
\end{equation}
\item $\Delta$ is compatible with the anchor, that is for all $\psi \in \Gamma_{\mathcal{E}}(M)$, one has
\begin{equation} \label{eq_Deltaanchor}
\rho( \Delta(\psi)) = (-1)^{\ell} [\ul{\Delta}, \rho(\psi)],
\end{equation}
\item $\Delta$ is a graded derivation of the bracket $[\cdot,\cdot]_{\mathcal{E}}$, that is it satisfies (\ref{eq_Deltaderivation}) for all $\psi,\psi' \in \Gamma_{\mathcal{E}}(M)$.
\end{enumerate}
We say that $\Delta$ is a \textbf{differential on $\mathcal{E}$}, and $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}},\Delta)$ is called a \textbf{differential graded Courant algebroid of degree $\ell$}.
\end{definice}
\begin{rem}
For any $\Delta$ satisfying (\ref{eq_Deltametric}), one can show that (\ref{eq_Deltaanchor}) is equivalent to
\begin{equation}
\Delta \circ \mathrm{D} + \mathrm{D} \circ \ul{\Delta} = 0.
\end{equation}
Note that the axioms above can be modified to work for $\Delta$ of any given degree $|\Delta|$. However, to keeps things simple, we only consider $|\Delta| = 1$.
\end{rem}
\begin{example} \label{ex_homologicalsection}
Let $\phi \in \Gamma_{\mathcal{E}}(M)$ be a given section of degree $|\phi| = 1 - \ell$. Then $\Delta = [\phi,\cdot]_{\mathcal{E}}$ is a graded $\mathbb{R}$-linear map of degree $1$. Let $\ul{\Delta} := \hat{\rho}(\phi) \equiv (-1)^{\ell} \rho(\phi)$. It is easy to see that (\ref{eq_Deltaderivation} - \ref{eq_Deltaanchor}) follow immediately from the axioms (c1)-(c4) in Definition \ref{def_gCourant} and the equation (\ref{eq_rhoishom}).
It only remains to verify when $\Delta^{2} = 0$. Using the graded Jacobi identity, one finds
\begin{equation} \label{eq_Deltasquareforphi}
\Delta^{2}(\psi) = \frac{1}{2} [[\phi,\phi]_{\mathcal{E}}, \psi]_{\mathcal{E}},
\end{equation}
for all $\psi \in \Gamma_{\mathcal{E}}(M)$. This does not necessarily imply that $[\phi,\phi]_{\mathcal{E}} = 0$. For example, it follows from (\ref{eq_bracketswithD}) that it suffices to ensure that $[\phi,\phi]_{\mathcal{E}} = \mathrm{D}{f}$ for some $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ with $|f| = 2 - \ell$. We say that $\phi$ is a \textbf{homological section}, if it satisfies $[[\phi,\phi]_{\mathcal{E}},\psi]_{\mathcal{E}} = 0$ for all $\psi \in \Gamma_{\mathcal{E}}(M)$.
Note that in view of the above remark, (\ref{eq_Deltasquareforphi}) is true only if $|\Delta|$ is odd.
\end{example}
\begin{example} \label{ex_Deltaforexact}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{D}^{H})$ be a graded Courant algebroid of degree $\ell$ from Example \ref{ex_gDorfman}. Suppose $\Delta$ makes it into a differential graded Courant algebroid. Let $Q := (-1)^{\ell} \ul{\Delta}$. It follows from the properties (ii)-(iv) of Definition \ref{def_delta} that $\Delta$ has to have the explicit form
\begin{equation} \label{eq_Deltaforexact}
\Delta(X,\xi) = ([Q,X], (-1)^{\ell}\{ \Li{Q}\xi + \theta^{\flat}(X) + H(Q,X,\cdot)\}),
\end{equation}
for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$. The vector bundle map $\theta^{\flat}: T\mathcal{M} \rightarrow T^{\ast}\mathcal{M}$ is induced by a $2$-form $\theta \in \Omega^{2}_{\mathcal{M}}(M)$ of degree $|\theta| = 1 - \ell$, that is $[\theta^{\flat}(X)](Y) := \theta(X,Y)$ for all $X,Y \in \mathfrak{X}_{\mathcal{M}}(M)$. The equation (\ref{eq_Deltaderivation}) is then equivalent to $\mathrm{d}{\theta} = 0$.
Now, observe that for $\ell \neq 1$, $\theta = \mathrm{d}{\vartheta}$ for some $\vartheta \in \Omega^{1}_{\mathcal{M}}(M)$ of degree $|\vartheta| = 1 - \ell$. See \S 6.5 in \cite{vysoky2021global} for details. In this case, (\ref{eq_Deltaforexact}) can be for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$ rewritten simply as
\begin{equation}
\Delta(X,\xi) = [(Q,\vartheta), (X,\xi)]_{D}^{H}.
\end{equation}
In other words, every possible $\Delta$ takes the form as in Example \ref{ex_homologicalsection} for a homological section $\phi = (Q,\vartheta)$. However, for $\ell = 1$, $\theta$ is in general not exact.
It remains to examine the condition $\Delta^{2} = 0$. First, one finds the equation $[Q,[Q,X]] = 0$ for all $X \in \mathfrak{X}_{\mathcal{M}}(M)$. This can be rewritten as $[[Q,Q],X] = 0$ for all $X \in \mathfrak{X}_{\mathcal{M}}(M)$ and, similarly to ordinary manifolds, this is equivalent $[Q,Q] = 0$. Having this in mind, the only non-trivial remaining condition becomes
\begin{equation}
\Li{Q}( \theta + i_{Q}H) = 0.
\end{equation}
Taking into account that $\theta$ and $H$ are closed, this can be further rewritten as
\begin{equation}
\mathrm{d}( i_{Q}\theta + \frac{1}{2} i_{Q}i_{Q}H) = 0.
\end{equation}
Note that if $\ell \neq 2$, this is equivalent to $i_{Q}\theta + \frac{1}{2} i_{Q}i_{Q}H = \mathrm{d}{f}$ for some $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ of degree $|f| = 2 - \ell$. We see that in any case, $\mathcal{M}$ is equipped with a \textbf{homological vector field} $Q$, thus making $(\mathcal{M},Q)$ into a differential graded manifold.
\end{example}
Having a notion of a differential $\Delta$, it makes sense to impose some its compatibility properties with additional structures on $\mathcal{E}$.
\begin{definice}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}},\Delta)$ be a differential graded Courant algebroid of degree $\ell$. Let $\mathcal{L} \subseteq \mathcal{E}$ be a Dirac structure on $\mathcal{E}$. We say that $\mathcal{L}$ is \textbf{$\Delta$-compatible}, if $\Delta( \Gamma_{\mathcal{L}}(M)) \subseteq \Gamma_{\mathcal{L}}(M)$.
\end{definice}
\begin{example}
Consider the scenario of Example \ref{ex_Deltaforexact}. In particular, $\Delta$ is given by (\ref{eq_Deltaforexact}), where $Q \in \mathfrak{X}_{\mathcal{M}}(M)$ is a degree $|Q| = 1$ homological vector field and $\theta \in \Omega^{2}_{\mathcal{M}}(M)$ is a closed $2$-form of degree $|\theta| = 1 - \ell$. Let $\mathcal{L} = \gr(\Pi^{\sharp})$ as in Example \ref{ex_PoissonasDirac}. In particular, $\Pi$ is an $H$-twisted graded Poisson structure of degree $\ell$.
The condition $\Delta( \Gamma_{\gr(\Pi^{\sharp})}(M)) \subseteq \Gamma_{\gr(\Pi^{\sharp})}(M)$ is equivalent to
\begin{equation}
(\Li{Q}\Pi)(\xi,\eta) + (-1)^{\ell(1+|\xi|)} (\theta + i_{Q}H)(\Pi^{\sharp}(\xi), \Pi^{\sharp}(\eta)) = 0.
\end{equation}
In terms of the Poisson bracket $\{f,g\} = [[f,\Pi]_{S},g]_{S}$ and Hamiltonian vector fields $X_{f} = [f,\Pi]_{S}$, this can be rewritten as
\begin{equation}
Q\{f,g\} = \{Q(f),g\} + (-1)^{|f|+\ell} \{f, Q(g)\} + (-1)^{\ell(1+|f|+|g|)} (\theta + i_{Q}H)(X_{f},X_{g}),
\end{equation}
for all $f,g \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$. We see that this example provides a more general framework for so called QP-manifolds \cite{Schwarz:1992nx, Ikeda:2011ax}, in particular for their $H$-twisted version.
\end{example}
A similar definition can be cooked for generalized complex structures.
\begin{definice}
Let $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}},\Delta)$ be a differential graded Courant algebroid of degree $\ell$. Let $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$ be a generalized complex structure on $\mathcal{E}$. We say that $\mathcal{L}$ is \textbf{$\Delta$-compatible}, if $\Delta_{\mathbb{C}}( \Gamma_{\mathcal{L}}(M)) \subseteq \Gamma_{\mathcal{L}}(M)$, where $\Delta_{\mathbb{C}}: \Gamma_{\mathcal{E}_{\mathbb{C}}}(M) \rightarrow \Gamma_{\mathcal{E}_{\mathbb{C}}}(M)$ is a complexification of $\Delta$.
If $\mathcal{J}: \mathcal{E} \rightarrow \mathcal{E}$ corresponds to $\mathcal{L}$, the above condition is equivalent to $\Delta \circ \mathcal{J} = \mathcal{J} \circ \Delta$.
\end{definice}
\begin{example}
Suppose we have a setting as in Example \ref{ex_Deltaforexact}. In particular, we have $\Delta$ given by (\ref{eq_Deltaforexact}) for a homological vector field $Q$ of degree $|Q|=1$ and closed $2$-form $\theta$ of degree $|\theta| = 1 - \ell$. Let $\mathcal{L} \subseteq \mathcal{E}_{\mathbb{C}}$ be the generalized complex structure from Example \ref{ex_gcomplexsymplectic}, that is the one determined by a symplectic form $\omega \in \Omega^{2}_{\mathcal{M}}(M)$ of degree $-\ell$. $\mathcal{J}$ is then given by (\ref{eq_Jforgencomplexsymplectic}). Note that we have to assume that $H = 0$.
The condition $\Delta \circ \mathcal{J} = \mathcal{J} \circ \Delta$ immediately forces $\theta = 0$ and $\Li{Q}(\omega) = 0$. We see that $\mathcal{L}$ is $\Delta$-compatible, if and only if $Q$ is symplectic with respect to $\omega$ and $\theta = 0$. In other word, $(\mathcal{M},Q,\omega)$ is a \textbf{differential graded symplectic manifold} (sometimes also called a QP-manifold).
\end{example}
\section{Graded Lie bialgebroids}
Standard Courant algebroids originally rose to prominence as Manin triples for Lie bialgebroids \cite{liu1997manin}. In this section, we show how this procedure can be generalized to the graded setting. First, let us introduce a notion of graded Lie algebroid.
\begin{definice}
$(\mathcal{A}, \mathsf{a}, [\cdot,\cdot]_{\mathcal{A}})$ is a called a \textbf{graded Lie algebroid of degree $\ell$}, if
\begin{enumerate}[(i)]
\item $\mathcal{A}$ is a graded vector bundle over a graded manifold $\mathcal{M}$;
\item $\mathsf{a}: \mathcal{A} \rightarrow T\mathcal{M}$ is a vector bundle map of degree $\ell$ called the anchor;
\item $[\cdot,\cdot]_{\mathcal{A}}: \Gamma_{\mathcal{A}}(M) \times \Gamma_{\mathcal{A}}(M) \rightarrow \Gamma_{\mathcal{A}}(M)$ is an $\mathbb{R}$-bilinear map of degree $\ell$, that is $|[X,Y]_{\mathcal{A}}| = |X| + |Y| + \ell$ for all $X,Y \in \Gamma_{\mathcal{A}}(M)$.
\end{enumerate}
Moreover, the operations are subject to the following set of axioms. All conditions have to hold for all involved sections and functions. Let $\hat{\mathsf{a}}(X) := (-1)^{(|X|+\ell)\ell} \mathsf{a}(X)$.
\begin{enumerate}[(c1)]
\item The bracket satisfies the \textbf{graded Leibniz rule}
\begin{equation} \label{eq_LALeibniz}
[X, fY]_{\mathcal{A}} = \hat{\mathsf{a}}(X)(f) Y + (-1)^{|f|(|X|+\ell)} f [X,Y]_{\mathcal{A}}.
\end{equation}
\item The bracket is graded skew-symmetric, that is
\begin{equation}
[X,Y]_{\mathcal{A}} = - (-1)^{(|X|+\ell)(|Y|+\ell)} [Y,X]_{\mathcal{A}}.
\end{equation}
\item The bracket satisfies the \textbf{graded Jacobi identity}
\begin{equation} \label{eq_LAgJI}
[X,[Y,Z]_{\mathcal{A}}]_{\mathcal{A}} = [[X,Y]_{\mathcal{A}}, Z]_{\mathcal{A}} + (-1)^{(|X|+\ell)(|Y|+\ell)} [Y, [X, Z]_{\mathcal{A}}]_{\mathcal{A}}.
\end{equation}
\end{enumerate}
In other words, $(\Gamma_{\mathcal{A}}(M), [\cdot,\cdot]_{\mathcal{A}})$ becomes a graded Lie algebra of degree $\ell$.
\end{definice}
Similarly to graded Courant algebroids, see Proposition \ref{tvrz_CAconsequences}, one can use (\ref{eq_LALeibniz}, \ref{eq_LAgJI}) to prove
\begin{equation}
\mathsf{a}( [X,Y]_{\mathcal{A}}) = [\mathsf{a}(X),\mathsf{a}(Y)],
\end{equation}
for all $X,Y \in \Gamma_{\mathcal{A}}(M)$.
\begin{example} \label{ex_standardgLA}
A standard example of a graded Lie algebroid of degree $\ell$ is $(T[\ell]\mathcal{M}, \mathbbm{1}_{T\mathcal{M}}, [\cdot,\cdot])$, where the identity is viewed as a vector bundle map $\mathbbm{1}_{T\mathcal{M}}: T[\ell]\mathcal{M} \rightarrow T\mathcal{M}$ of degree $\ell$, and $[\cdot,\cdot]$ is the graded commutator (\ref{eq_gcommutator}) of vector fields.
\end{example}
Now, every graded Lie algebroid induces a differential $\mathrm{d}_{\mathcal{A}}$ and two different Schouten-Nijenhuis brackets $[\cdot,\cdot]_{\mathcal{A}}$. In order to save space, we have decided to move their definitions and basic properties into the appendix \ref{sec_appendixA}. One can now use them to define a concept of a graded Lie bialgebroid. This generalizes the notion introduced in \cite{mackenzie1994} and it uses the approach taken in \cite{kosmann1995exact}.
\begin{definice}
Let $(\mathcal{A},\mathsf{a},[\cdot,\cdot]_{\mathcal{A}})$ be a graded Lie algebroid of degree $\ell$ and $(\mathcal{A}^{\ast}, \mathsf{a}', [\cdot,\cdot]_{\mathcal{A}^{\ast}})$ a graded Lie algebroid of degree $\ell'$, where $\mathcal{A}^{\ast}$ is the dual of $\mathcal{A}$.
We say that $(\mathcal{A},\mathcal{A}^{\ast})$ is a \textbf{graded Lie bialgebroid of bidegree $(\ell,\ell')$} if the induced differential $\mathrm{d}_{\mathcal{A}}$ is a graded derivation of the Schouten-Nijenhuis bracket $[\cdot,\cdot]_{\mathcal{A}^{\ast}}$. In other words:
\begin{enumerate}[(i)]
\item Suppose $\ell$ is even. For all $\xi \in \Omega^{p}_{[\mathcal{A}]}(M) \cong \mathfrak{X}^{p}_{[\mathcal{A}]}(M)$ and $\eta \in \Omega^{q}_{[\mathcal{A}]}(M) \cong \mathfrak{X}^{q}_{[\mathcal{A}]}(M)$, one has
\begin{equation} \label{eq_GLBAduality_even}
\mathrm{d}_{\mathcal{A}} [\xi,\eta]_{\mathcal{A}} = [ \mathrm{d}_{\mathcal{A}}\xi, \eta]_{\mathcal{A}^{\ast}} + (-1)^{|X|+\ell'+p-1} [\xi, \mathrm{d}_{\mathcal{A}}\eta]_{\mathcal{A}^{\ast}}.
\end{equation}
\item Suppose $\ell$ is odd. For all $\xi \in \~\Omega^{p}_{[\mathcal{A}]}(M) \cong \~\mathfrak{X}^{p}_{[\mathcal{A}]}(M)$ and $\eta \in \~\Omega^{q}_{[\mathcal{A}]}(M) \cong \~\mathfrak{X}^{q}_{[\mathcal{A}]}(M)$, one has
\begin{equation} \label{eq_GLBAduality_odd}
\mathrm{d}_{\mathcal{A}} [\xi,\eta]_{\mathcal{A}} = [ \mathrm{d}_{\mathcal{A}}\xi, \eta]_{\mathcal{A}^{\ast}} + (-1)^{|X|+\ell'} [\xi, \mathrm{d}_{\mathcal{A}}\eta]_{\mathcal{A}^{\ast}}.
\end{equation}
\end{enumerate}
\end{definice}
Since $\mathrm{d}_{\mathcal{A}}$ satisfies the graded Leibniz rule, it suffices to verify (\ref{eq_GLBAduality_even}) and (\ref{eq_GLBAduality_odd}) for $p,q \in \{0,1\}$. Let us now summarize those conditions.
\begin{tvrz}
The condition (\ref{eq_GLBAduality_even}) (for even $\ell$) and (\ref{eq_GLBAduality_odd}) (for odd $\ell$) is equivalent to the following set of conditions:
\begin{enumerate}[(i)]
\item For every $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$, one has
\begin{equation} \label{eq_GLBAduality00}
\hat{\mathsf{a}}(\mathrm{d}_{\mathcal{A}^{\ast}}f) - (-1)^{(\ell+1)(\ell'+1)} \hat{\mathsf{a}}'( \mathrm{d}_{\mathcal{A}}f) = 0.
\end{equation}
\item For every $f \in \mathcal{C}^{\infty}_{\mathcal{M}}(M)$ and $\xi \in \Omega^{1}_{[\mathcal{A}]}(M)$, one has
\begin{equation} \label{eq_GLBAduality01}
\Li{\mathrm{d}_{\mathcal{A}^{\ast}}f}^{\mathcal{A}}(\xi) - (-1)^{(\ell+1)(\ell'+1)} [\mathrm{d}_{\mathcal{A}}f,\xi]_{\mathcal{A}^{\ast}} = 0.
\end{equation}
\item For every $\xi,\eta \in \Omega^{1}_{[\mathcal{A}]}(M)$, one has
\begin{equation} \label{eq_GLBAduality11}
\mathrm{d}_{\mathcal{A}} [\xi,\eta]_{\mathcal{A}^{\ast}} = (-1)^{|\xi|+\ell'} \Li{\xi}^{\mathcal{A}^{\ast}}( \mathrm{d}_{\mathcal{A}}\eta) - (-1)^{(|\eta|+\ell')(|\xi|+\ell'+1)} \Li{\eta}^{\mathcal{A}^{\ast}}( \mathrm{d}_{\mathcal{A}}\xi).
\end{equation}
\end{enumerate}
\end{tvrz}
\begin{proof}
This follows immediately from definitions. Observe that (\ref{eq_GLBAduality11}) is the equation of objects in $\Omega^{2}_{[\mathcal{A}]}(M)$ for even $\ell$, and in $\~\Omega^{2}_{[\mathcal{A}]}(M)$ for odd $\ell$.
\end{proof}
Similarly to the ordinary case, one obtains the following important observation.
\begin{tvrz} \label{tvrz_gLBAdual}
Let $(\mathcal{A},\mathcal{A}^{\ast})$ be a graded Lie bialgebroid of bidegree $(\ell,\ell')$.
Then $(\mathcal{A}^{\ast},\mathcal{A})$ is a graded Lie bialgebroid of bidegree $(\ell',\ell)$.
\end{tvrz}
\begin{proof}
The condition (\ref{eq_GLBAduality00}) is manifestly symmetric in $(\mathcal{A},\ell) \leftrightarrow (\mathcal{A}^{\ast},\ell')$. The evaluation of (\ref{eq_GLBAduality01}) on $X \in \Gamma_{\mathcal{A}}(M)$ and using (\ref{eq_GLBAduality00}) gives the condition $(\ref{eq_GLBAduality01})$ for $(\mathcal{A}^{\ast},\mathcal{A})$ evaluated on $\xi \in \Gamma_{\mathcal{A}^{\ast}}(M)$. Finally, the evaluation of (\ref{eq_GLBAduality11}) on $X,Y \in \Gamma_{\mathcal{A}}(M)$ and using (\ref{eq_GLBAduality00}, \ref{eq_GLBAduality01}) gives the condition (\ref{eq_GLBAduality11}) for $(\mathcal{A}^{\ast},\mathcal{A})$ evaluated on $\xi,\eta \in \Gamma_{\mathcal{A}^{\ast}}(M)$. This is an analogue of Theorem 3.10 in \cite{mackenzie1994}, except a lot of signs is involved. We \textit{do not} encourage the reader to verify this statement.
\end{proof}
It turns out a graded Lie bialgebroid can be used to construct a graded Courant algebroid.
\begin{theorem} \label{thm_doubleofgLBA}
Let $(\mathcal{A},\mathcal{A}^{\ast})$ be a graded Lie bialgebroid of bidegree $(\ell,\ell')$ over a graded manifold $\mathcal{M}$. Consider the following data:
\begin{enumerate}[(i)]
\item Let $\mathcal{E} := \mathcal{A}[\ell'] \oplus \mathcal{A}^{\ast}[\ell]$.
\item Define $\rho: \mathcal{E} \rightarrow T\mathcal{M}$ as $\rho(X,\xi) := \mathsf{a}(X) + \mathsf{a}'(\xi)$ for all $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$.
\item Let $g_{\mathcal{E}}: \mathcal{E} \rightarrow \mathcal{E}^{\ast}$ be for any $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$ given by $g_{\mathcal{E}}(X,\xi) := (\xi, (-1)^{\ell \ell'} X)$, where we use the identification $\mathcal{E}^{\ast} \cong \mathcal{A}^{\ast}[-\ell'] \oplus \mathcal{A}[-\ell]$.
\item Consider the $\mathbb{R}$-bilinear bracket
\begin{equation}
\begin{split}
[(X,\xi),(Y,\eta)]_{\mathcal{E}} = \big( & [X,Y]_{\mathcal{A}} + (-1)^{(|X|+\ell)(\ell+\ell')} \Li{\xi}^{\mathcal{A}^{\ast}} Y - (-1)^{|X|+\ell'} (\mathrm{d}_{\mathcal{A}^{\ast}}X)(\eta,\cdot), \\
& [\xi,\eta]_{\mathcal{A}^{\ast}} + (-1)^{(|X|+\ell)(\ell+\ell')} \Li{X}^{\mathcal{A}} \eta - (-1)^{|X|+\ell'} (\mathrm{d}_{\mathcal{A}}\xi)(Y,\cdot) \big).
\end{split}
\end{equation}
for all $(X,\xi),(Y,\eta) \in \Gamma_{\mathcal{E}}(M)$.
\end{enumerate}
Then $(\mathcal{E},\rho,g_{\mathcal{E}},[\cdot,\cdot]_{\mathcal{E}})$ becomes a graded Courant algebroid of degree $\ell + \ell'$ called a \textbf{double of a Lie bialgebroid $(\mathcal{A},\mathcal{A}^{\ast})$}. Both $\mathcal{A}[\ell']$ and $\mathcal{A}^{\ast}[\ell]$ are Dirac structures on $\mathcal{E}$.
\end{theorem}
\begin{proof}
First, observe that for any $(X,\xi) \in \Gamma_{\mathcal{E}}(M)$, one has $|(X,\xi)| = |X| - \ell' = |\xi| - \ell$, where $|X|$ and $|\xi|$ always denote the original degrees in $\Gamma_{\mathcal{A}}(M)$ and $\Gamma_{\mathcal{A}^{\ast}}(M)$, respectively. Then
\begin{equation}
|\mathsf{a}(X)| = |X| + \ell = |(X,\xi)| + \ell + \ell', \; \; |\mathsf{a}'(\xi)| = |\xi| + \ell' = |(X,\xi)| + \ell + \ell'.
\end{equation}
This shows that $\rho$ is indeed a graded vector bundle map of degree $\ell + \ell'$. Similarly, for any $(\xi,X) \in \Gamma_{\mathcal{E}^{\ast}}(M)$, one has $|(\xi,X)| = |\xi| + \ell' = |X| + \ell$, whence
\begin{equation}
|g_{\mathcal{E}}(X,\xi)| = |(\xi,(-1)^{\ell \ell'}X)| = |\xi| + \ell' = |(X,\xi)| + \ell + \ell'.
\end{equation}
This shows that $g_{\mathcal{E}}$ is a graded vector bundle isomorphism of degree $\ell + \ell'$. The corresponding fiber-wise metric $\<\cdot,\cdot\>_{\mathcal{E}}$ acquires the form
\begin{equation}
\<(X,\xi),(Y,\eta)\>_{\mathcal{E}} = (-1)^{(|X|+\ell)\ell} \xi(Y) + (-1)^{|X|(|Y|+\ell)} \eta(X),
\end{equation}
for all $(X,\xi),(Y,\eta) \in \Gamma_{\mathcal{E}}(M)$. It is not difficult to see that it is graded symmetric of degree $\ell + \ell'$. Finally, it is clear that $|[(X,\xi),(Y,\eta)]_{\mathcal{E}}| = |(X,\xi)| + |(Y,\eta)| + \ell + \ell'$. We conclude that all operations have the correct degree $\ell + \ell'$. The induced map $\mathrm{D}: \mathcal{C}^{\infty}_{\mathcal{M}}(M) \rightarrow \Gamma_{\mathcal{E}}(M)$ reads
\begin{equation} \label{eq_dDdoubleLiebialgexpl}
\mathrm{D}{f} = ( (-1)^{\ell} \mathsf{a}'^{T}(\mathrm{d}{f}), (-1)^{(\ell+1)\ell'} \mathsf{a}^{T}(\mathrm{d}{f})) = (-1)^{\ell + \ell'} ( (-1)^{|f|\ell'} \mathrm{d}_{\mathcal{A}^{\ast}}f, (-1)^{(|f|+\ell')\ell} \mathrm{d}_{\mathcal{A}}f).
\end{equation}
It remains to prove the graded Courant algebroid axioms (see Definition \ref{def_gCourant}). Let us only sketch the proof and leave the details to the reader.
\begin{enumerate}[(i)]
\item The graded Leibniz rule (\ref{eq_CALeibniz}) follows from the graded Leibniz rules for Lie algebroids $\mathcal{A}$ and $\mathcal{A}^{\ast}$ and consequent properties of the induced operations.
\item The compatibility (\ref{eq_CApairingcomp}) of $[\cdot,\cdot]_{\mathcal{E}}$ with $\<\cdot,\cdot\>_{\mathcal{E}}$ reduces to the definition of Lie derivatives $\Li{}^{\mathcal{A}}$ and $\Li{}^{\mathcal{A}^{\ast}}$. One also uses the fact that $(\mathrm{d}_{\mathcal{A}}\xi)(X,Y) + (-1)^{\ell + |X||Y|} (\mathrm{d}_{\mathcal{A}}\xi)(X,Y) = 0$ and its dual counterpart using $\mathrm{d}_{\mathcal{A}^{\ast}}$.
\item The proof of (almost) graded skew-symmetry (\ref{eq_CAskewsym}) uses (\ref{eq_dDdoubleLiebialgexpl}) together with Cartan relations (\ref{eq_LACartan1}, \ref{eq_LACartan1b}) and their dual counterparts.
\item The most complicated bit is naturally the graded Jacobi identity (\ref{eq_CAgJI}). Its validity is equivalent to the graded Jacobi identities for both $\mathcal{A}$ and $\mathcal{A}^{\ast}$, together with the duality conditions (\ref{eq_GLBAduality01}, \ref{eq_GLBAduality11}) and their dual counterparts (they hold thanks to Proposition \ref{tvrz_gLBAdual}).
\end{enumerate}
The claim about $\mathcal{A}[\ell']$ and $\mathcal{A}^{\ast}[\ell]$ being Dirac structures is obvious.
\end{proof}
\begin{rem}
One can see that $\mathcal{E}$, the anchor $\rho$, and the bracket $[\cdot,\cdot]_{\mathcal{E}}$ are symmetric with respect to $(\mathcal{A},\ell) \leftrightarrow (\mathcal{A}^{\ast},\ell')$. However, for the pairing, one can write
\begin{equation}
\<(X,\xi),(Y,\eta)\>_{\mathcal{E}} = (-1)^{\ell \ell'} \{ (-1)^{(|\xi|+\ell')\ell'} X(\eta) + (-1)^{|\xi|(|\eta|+\ell')} Y(\xi) \}.
\end{equation}
This shows that the double of the dual Lie bialgebroid $(\mathcal{A}^{\ast},\mathcal{A})$ is $(\mathcal{E}, \rho, (-1)^{\ell \ell'} g_{\mathcal{E}}, [\cdot,\cdot]_{\mathcal{E}})$.
\end{rem}
\begin{tvrz}
Let $(\mathcal{A},\mathcal{A}^{\ast})$ be a graded Lie bialgebroid of bidegree $(\ell,\ell')$. Then for any $q \in \mathbb{Z}$, $(\mathcal{A}[q], \mathcal{A}[q]^{\ast})$ is a graded Lie bialgebroid of bidegree $(\ell+q, \ell'-q)$.
\end{tvrz}
\begin{proof}
It is easy to see that $(\mathcal{A}[q],\mathsf{a},[\cdot,\cdot]_{\mathcal{A}})$ forms a graded Lie algebroid of degree $\ell + q$. Since $\mathcal{A}[q]^{\ast} \cong \mathcal{A}^{\ast}[-q]$, we may view $\mathcal{A}[q]^{\ast}$ as a graded Lie algebroid of degree $\ell' - q$. Since
\begin{equation}
\mathcal{E} = \mathcal{A}[\ell'] \oplus \mathcal{A}^{\ast}[\ell] \cong (\mathcal{A}[q])[\ell'-q] \oplus (\mathcal{A}^{\ast}[-q])[\ell+q],
\end{equation}
we may compare the structures induced by both $(\mathcal{A},\mathcal{A}^{\ast})$ and $(\mathcal{A}[q],\mathcal{A}[q]^{\ast})$ on $\mathcal{E}$ as in Theorem \ref{thm_doubleofgLBA} $(i)$-$(iv)$. It turns out that that the anchors $\rho$ and the brackets $[\cdot,\cdot]_{\mathcal{E}}$ are completely the same, whereas the fiber-wise metrics $\<\cdot,\cdot\>_{\mathcal{E}}$ differ only by an overall sign. In particular, the graded Jacobi identity for $[\cdot,\cdot]_{\mathcal{E}}$ implies the duality condition (\ref{eq_GLBAduality_even}) or (\ref{eq_GLBAduality_odd}) for $(\mathcal{A}[q], \mathcal{A}[q]^{\ast})$.
\end{proof}
Let us now turn our attention to examples.
\begin{example}
Consider the graded Lie algebroid $\mathcal{A} = T\mathcal{M}$ of degree $\ell = 0$, see Example \ref{ex_standardgLA}.
One can view the dual $\mathcal{A}^{\ast} = T^{\ast}\mathcal{M}$ as a graded Lie algebroid of degree $\ell'$ with the trivial anchor of degree $\ell'$ and the trivial bracket of degree $\ell'$. This may seem strange, but since graded vector spaces are (in our viewpoint) just sequences of vector spaces, they have a zero vector in each degree. Consequently, we can have trivial graded linear maps of any given degree.
Clearly $(\mathcal{A},\mathcal{A}^{\ast})$ is a graded Lie bialgebroid of bidegree $(0,\ell')$. Its double $\mathcal{E} = T[\ell']\mathcal{M} \oplus T^{\ast}\mathcal{M}$ is precisely the degree $\ell'$ graded Dorfman bracket from Example \ref{ex_gDorfman} (for $H = 0$).
\end{example}
\begin{example}
Let $(\mathcal{M},\Pi)$ be a graded Poisson manifold of degree $\ell'$, see Example \ref{ex_gPoisson} and Example \ref{ex_PoissonasDirac} for details and notation. We assume $H = 0$. Observe that $(T^{\ast}\mathcal{M}, \Pi^{\sharp}, [\cdot,\cdot]_{\Pi})$ then becomes a graded Lie algebroid of degree $\ell'$.
There is thus the corresponding Lie algebroid differential $\mathrm{d}_{\Pi}$ acting on multivector fields. It turns out that it can be written simply as
\begin{equation} \label{eq_dPiformula}
\mathrm{d}_{\Pi} X = (-1)^{\ell'} [\Pi,X]_{S},
\end{equation}
for all $X \in \mathfrak{X}^{p}_{\mathcal{M}}(M)$ (for even $\ell')$ or for all $X \in \~\mathfrak{X}^{p}_{\mathcal{M}}(M)$ (for odd $\ell')$. Since both $\mathrm{d}_{\Pi}$ and $(-1)^{\ell'}[\Pi,\cdot]_{S}$ satisfy the graded Leibniz rule (\ref{eq_dAgLeibniz}) or (\ref{eq_dAgLeibniz2}), it suffices to consider $p \in \{0,1\}$. The rest is a straightforward verification using the properties of $[\cdot,\cdot]_{S}$ and definitions.
Now, $[\cdot,\cdot]_{S}$ is precisely the Schouten-Nijenhuis bracket associated to the graded Lie algebroid $(T\mathcal{M}, \mathbbm{1}_{T\mathcal{M}}, [\cdot,\cdot])$ of degree $0$. It follows from (\ref{eq_dPiformula}) and the graded Jacobi identity (\ref{eq_LASNB_gJacobi}) or (\ref{eq_LASNB_gJacobi2}) that $\mathrm{d}_{\Pi}$ is the graded derivation of $[\cdot,\cdot]_{S}$. In particular, this proves that $(T\mathcal{M}, \mathbbm{1}_{T\mathcal{M}}, [\cdot,\cdot])$ and $(T^{\ast}\mathcal{M}, \Pi^{\sharp}, [\cdot,\cdot]_{\Pi})$ form a graded Lie bialgebroid $(T\mathcal{M},T^{\ast}\mathcal{M})$ of bidegree $(0,\ell')$.
\end{example}
\begin{example}
Suppose $\mathcal{M} = \{ \ast \}$ is a singleton graded manifold. Then every graded Lie algebroid of degree $\ell$ reduces to a graded Lie algebra $(\mathfrak{g}, [\cdot,\cdot]_{\mathfrak{g}})$ of degree $\ell$. A graded Lie bialgebroid is in this case called a \textbf{graded Lie bialgebra}. Equivalently, it can be reformulated as follows.
For any graded vector space $V$, the space of all graded linear endomorphisms $\Lin(V)$ forms a graded Lie algebra of degree zero. A graded linear map $\varrho: \mathfrak{g} \rightarrow \Lin(V)$ of degree $|\varrho| = \ell$ is called a \textbf{representation of $\mathfrak{g}$ on $V$}, if $\varrho([x,y]_{\mathfrak{g}}) = [\varrho(x),\varrho(y)]$ for all $x,y \in \mathfrak{g}$. Define
\begin{equation}
(\ad^{\ast}_{x} \xi)(y) := - (-1)^{(|x|+\ell)|\xi|} \xi( [x,y]_{\mathfrak{g}}),
\end{equation}
for all $\xi \in \mathfrak{g}^{\ast}$ and $x,y \in \mathfrak{g}$. This defines the \textbf{coadjoint representation} of $\mathfrak{g}$ on $\mathfrak{g}^{\ast}$. For each $p \in \mathbb{N}_{0}$, let $\mathfrak{m}_{p}(\mathfrak{g}^{\ast})$ denote the space of all graded $p$-linear maps from $\mathfrak{g}^{\ast}$ to $\mathbb{R}$. There is an \textbf{adjoint representation } of $\mathfrak{g}$ on $\mathfrak{m}_{p}(\mathfrak{g}^{\ast})$ given for each $\omega \in \mathfrak{m}_{p}(\mathfrak{g}^{\ast})$ and $x \in \mathfrak{g}$ by
\begin{equation}
[\ad_{x}^{(p)} \omega](\xi_{1},\dots,\xi_{p}) = -\sum_{r=1}^{p} (-1)^{(|x|+\ell)(|\omega|+ |\xi_{1}| + \dots + |\xi_{r-1}|)} \omega( \xi_{1}, \dots, \ad^{\ast}_{x} \xi_{r}, \dots, \xi_{p}),
\end{equation}
for all $\xi_{1},\dots,\xi_{p} \in \mathfrak{g}^{\ast}$. For any graded vector space $V$ and $p \in \mathbb{N}_{0}$, there is a a space $\mathfrak{c}^{p}(\mathfrak{g},V)$ of all graded $p$-linear maps $\omega: \mathfrak{g} \times \dots \times \mathfrak{g} \rightarrow V$ satisfying for each $i \in \{1,\dots,p-1\}$ the conditions
\begin{equation} \label{eq_cpgVdefined}
\omega(\dots,x_{i},x_{i+1},\dots) = -(-1)^{|x_{i}||x_{i+1}|} \omega(\dots, x_{i+1},x_{i}, \dots)
\end{equation}
and $|\omega(x_{1},\dots,x_{p})| = |\omega| + |x_{1}| + \dots + |x_{p}|$ for all $x_{1},\dots,x_{p} \in \mathfrak{g}$. We declare $\mathfrak{c}^{0}(\mathfrak{g},V) = V$. Similarly, one defines a graded vector space $\~\mathfrak{c}^{p}(\mathfrak{g},V)$ of symmetric $p$-linear maps, where the sign on the right-hand side of (\ref{eq_cpgVdefined}) is opposite. Finally, for each $p \in \mathbb{N}_{0}$, there is a degree $\ell$ graded linear map $\Delta_{\mathfrak{g}}: \mathfrak{c}^{p}(\mathfrak{g},V) \rightarrow \mathfrak{c}^{p+1}(\mathfrak{g},V)$ (for even $\ell$) and $\Delta_{\mathfrak{g}}: \~\mathfrak{c}^{p}(\mathfrak{g},V) \rightarrow \~\mathfrak{c}^{p+1}(\mathfrak{g},V)$ (for odd $\ell$). It is defined using the formulas similar to (\ref{eq_dAeven}) of (\ref{eq_dAodd}), with $\mathsf{a}$ replaced by $\varrho$ and $[\cdot,\cdot]_{\mathcal{A}}$ by $[\cdot,\cdot]_{\mathfrak{g}}$. It follows that $\Delta_{\mathfrak{g}}^{2} = 0$. In this way, we obtain a cochain complex $(\mathfrak{c}^{\bullet}(\mathfrak{g},V), \Delta_{\mathfrak{g}})$ (for even $\ell$) or $(\~\mathfrak{c}^{\bullet}(\mathfrak{g},V), \Delta_{\mathfrak{g}})$ (for odd $\ell$) of graded vector spaces.
Now, suppose that we have a graded Lie algebra structure $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}})$ of degree $\ell'$, and consider a degree $\ell'$ graded linear map $\delta: \mathfrak{g} \rightarrow \mathfrak{m}_{2}(\mathfrak{g}^{\ast})$ defined as
\begin{equation}
[\delta(x)](\xi,\eta) := (-1)^{|x|(\ell+\ell'+1)} (\mathrm{d}_{\mathfrak{g}^{\ast}} x)(\xi,\eta) \equiv -(-1)^{|x|(\ell+\ell') + |\xi|\ell} x([\xi,\eta]_{\mathfrak{g}^{\ast}}).
\end{equation}
We can interpret $\delta$ as a $1$-cochain of degree $|\delta| = \ell'$ in $\mathfrak{c}^{1}(\mathfrak{g}, \mathfrak{m}_{2}(\mathfrak{g}^{\ast}))$ or $\~\mathfrak{c}^{1}(\mathfrak{g}, \mathfrak{m}_{2}(\mathfrak{g}^{\ast}))$, depending on the parity of $\ell$. One can with some effort (which we leave to the reader) arrive to the following statement: $(\mathfrak{g},\mathfrak{g}^{\ast})$ is a graded Lie bialgebra of bidegree $(\ell,\ell')$, iff $\Delta_{\mathfrak{g}}\delta = 0$.
By Theorem \ref{thm_doubleofgLBA}, the graded vector space $\mathfrak{d} := \mathfrak{g}[\ell'] \oplus \mathfrak{g}^{\ast}[\ell]$ can be then equipped with a graded Lie algebra bracket $[\cdot,\cdot]_{\mathfrak{d}}$ of degree $\ell + \ell'$. Explicitly, it is given by
\begin{equation} \label{eq_bialgebradouble}
\begin{split}
[(x,\xi),(y,\eta)]_{\mathfrak{d}} = \big( & [x,y]_{\mathfrak{g}} + (-1)^{(|x|+\ell)(\ell+\ell')} \ad^{\ast}_{\xi}y - (-1)^{(|x|+\ell')(|y|+\ell)} \ad^{\ast}_{\eta}x, \\
& [\xi,\eta]_{\mathfrak{g}^{\ast}} + (-1)^{(|x|+\ell)(\ell+\ell')} \ad^{\ast}_{x}\eta - (-1)^{(|x|+\ell')(|y|+\ell)} \ad^{\ast}_{y}\xi \big),
\end{split}
\end{equation}
for all $(x,\xi),(y,\eta) \in \mathfrak{d}$. It follows that the graded Jacobi identities for $[\cdot,\cdot]_{\mathfrak{d}}$ are equivalent to the graded Jacobi identities for both $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$, together with the duality condition $\Delta_{\mathfrak{g}}\delta = 0$.
Let us finish with an actual example of a graded Lie bialgebra. Suppose $\mathfrak{g}$ for some $q \in \mathbb{Z} - \{0\}$ satisfies $\dim(\mathfrak{g}_{q}) = \dim(\mathfrak{g}_{0}) = 1$ and $\dim(\mathfrak{g}_{j}) = 0$ for $j \notin \{0,q\}$. Let $(t_{1},t_{2})$ be a total basis of $\mathfrak{g}$ with $|t_{1}| = q$ and $|t_{2}| = 0$. Then
\begin{equation}
[t_{1},t_{2}]_{\mathfrak{g}} = t_{1}, \; \; [t_{1},t_{1}]_{\mathfrak{g}} = 0, \; \; [t_{2},t_{2}]_{\mathfrak{g}} = 0
\end{equation}
makes $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ into a graded Lie algebra of degree $\ell = 0$. If $(t^{1},t^{2})$ is the dual total basis of $\mathfrak{g}^{\ast}$, we have $|t^{1}| = -q$ and $|t^{2}| = 0$. It follows that by declaring
\begin{equation}
[t^{1},t^{2}]_{\mathfrak{g}^{\ast}} = t^{2}, \; \; [t^{1},t^{1}]_{\mathfrak{g}^{\ast}} = 0, \; \; [t^{2},t^{2}]_{\mathfrak{g}^{\ast}} = 0,
\end{equation}
we make $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}})$ into a graded Lie algebra of degree $\ell' = q$. The ``mixed'' brackets od $\mathfrak{d}$ can be calculated from (\ref{eq_bialgebradouble}), giving the expressions
\begin{align}
[(t_{1},0),(0,t^{1})]_{\mathfrak{d}} = & \ (0,-t^{2}), \\
[(t_{1},0),(0,t^{2})]_{\mathfrak{d}} = & \ (0,0), \\
[(t_{2},0),(0,t^{2})]_{\mathfrak{d}} = & \ (-t_{1},0), \\
[(t_{2},0),(0,t^{1})]_{\mathfrak{d}} = & \ (t_{2},t^{1}).
\end{align}
It is straightforward to verify that $(\mathfrak{d},[\cdot,\cdot]_{\mathfrak{d}})$ constitutes a graded Lie algebra of degree $q$, whence $(\mathfrak{g},\mathfrak{g}^{\ast})$ forms a graded Lie bialgebra of bidegree $(0,q)$.
\end{example}
\section*{Acknowledgments}
First and foremost, I would like to express gratitude to my family for their patience and support.
Next, I would like to thank Branislav Jurčo, Alexei Kotov and Vít Tuček for helpful discussions.
The author is grateful for a financial support from MŠMT under grant no. RVO 14000.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,906
|
#ifndef LAMBDACOMMON_RESOURCES_H
#define LAMBDACOMMON_RESOURCES_H
#include "system/fs.h"
#include "lstring.h"
#include <utility>
#ifdef LAMBDA_WINDOWS
# pragma warning(push)
# pragma warning(disable:4251)
#endif
namespace lambdacommon
{
using namespace std::rel_ops;
/*!
* Identifier
*
* Represents a resource identifier.
*/
class LAMBDACOMMON_API Identifier : public Object
{
private:
std::string _namespace;
std::string _name;
public:
Identifier(const std::string& name);
Identifier(std::string domain, std::string name) noexcept;
Identifier(const Identifier& other);
Identifier(Identifier&& other) noexcept;
/*!
* Gets the namespace of the resource.
* @return The namespace of the resource.
*/
const std::string& get_domain() const;
/*!
* Gets the name of the resource.
* @return The name of the resource.
*/
const std::string& get_name() const;
/*!
* Creates a new {@code Identifier} from this resource name.
* @param name The path to append.
* @return The new {@code Identifier} with the appended path.
*/
Identifier sub(const std::string& name) const;
/*!
* Gets the resource name as a string.
* @return The resource name as a string.
*/
std::string to_string() const override;
/*!
* Creates a new {@code Identifier} from this resource name.
* @param path The path to append.
* @return The new {@code Identifier} with the appended path.
*/
inline Identifier operator/(const std::string& path) const {
return sub(path);
}
Identifier& operator=(const Identifier& other);
Identifier& operator=(Identifier&& other) noexcept;
bool operator==(const Identifier& other) const;
bool operator<(const Identifier& other) const;
template<std::size_t N>
decltype(auto) get() const {
if constexpr (N == 0) return this->_namespace;
else if constexpr (N == 1) return this->_name;
}
};
/*!
* ResourcesManager
*
* Represents a resources manager.
*/
class LAMBDACOMMON_API ResourcesManager
{
protected:
u32 _id;
public:
ResourcesManager();
ResourcesManager(const ResourcesManager& other);
ResourcesManager(ResourcesManager&& other) noexcept;
/*!
* Gets the identifier of the resources manager.
* @return The identifier.
*/
u32 get_id() const;
/*! @brief Checks whether the resources exists or not.
*
* @param resource The resource to check.
* @return True of the resource exists, else false.
*/
[[nodiscard]] virtual bool has_resource(const Identifier& resource) const = 0;
/*! @brief Checks whether the resource exists or not.
*
* @param resource The resource to check.
* @param extension The extension of the resource file.
* @return True if the resource exists, else false.
*/
[[nodiscard]] virtual bool has_resource(const Identifier& resource, const std::string& extension) const = 0;
/*! @brief Loads the resource content into a string value.
*
* @param resource The resource to load.
* @param extension The extension of the resource file.
* @return The resource content if successfully loaded,X else an empty string.
*/
[[nodiscard]] virtual std::string load_resource(const Identifier& resource, const std::string& extension) const = 0;
bool operator==(const ResourcesManager& other) const;
/*!
* Loads the resource content into a string value.
* @param resource The resource to load.
* @return The resource content if successfully loaded, else an empty string.
*/
[[nodiscard]] virtual std::string load_resource(const Identifier& resource) const = 0;
};
class LAMBDACOMMON_API FileResourcesManager : public ResourcesManager
{
private:
lambdacommon::fs::path _working_directory;
public:
explicit FileResourcesManager(lambdacommon::fs::path working_directory = std::move(fs::current_path()));
FileResourcesManager(const FileResourcesManager& other);
FileResourcesManager(FileResourcesManager&& other) noexcept;
/*! @brief Gets the working directory of the resource manager.
*
* @return The working directory.
*/
const lambdacommon::fs::path& get_working_directory() const;
bool has_resource(const Identifier& resource) const override;
bool has_resource(const Identifier& resource, const std::string& extension) const override;
lambdacommon::fs::path
get_resource_path(const Identifier& resource, const std::string& extension) const;
std::string load_resource(const Identifier& resource) const override;
std::string load_resource(const Identifier& resource, const std::string& extension) const override;
FileResourcesManager& operator=(const FileResourcesManager& other);
FileResourcesManager& operator=(FileResourcesManager&& other) noexcept;
bool operator==(const FileResourcesManager& other) const;
};
}
// Structured bindings for lambdacommon::Identifier.
namespace std
{
template<>
struct tuple_size<lambdacommon::Identifier> : std::integral_constant<std::size_t, 2>
{
};
template<std::size_t N>
struct tuple_element<N, lambdacommon::Identifier>
{
using type = decltype(std::declval<lambdacommon::Identifier>().get<N>());
};
}
#ifdef LAMBDA_WINDOWS
# pragma warning(pop)
#endif
#endif //LAMBDACOMMON_RESOURCES_H
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,864
|
\section{Introduction}
LOFAR (the LOw Frequency ARray) is a new kind of radio telescope for astronomical observations in the frequency range of $\approx (10-240)$ MHz with high sensitivity and high spatial resolution (http://www.lofar.org). It uses a large number of simple dipole antennas instead of the traditional big parabolic dishes. It consists of $40$ stations in the Netherlands, $5$ in Germany and one each in Great Britain, France and Sweden covering a total area of more than $1000$ km in diameter.
Though primarily design as an astronomical telescope, LOFAR can also be used for the detection of very high-energy cosmic rays (CRs) in the interesting energy region above $10^{16}$ eV where the transition of galactic to extra-galactic CRs is expected (H\"{o}randel et al. 2009, Horneffer et al. 2010). This will be done by looking at extensive air showers which are essentially cascades of energetic secondary particles produce by the interaction of CR primaries with the nuclei present in the atmosphere. A large fraction of these secondaries are electrons and positrons which produce radio synchrotron emission in the presence of the Earth's magnetic field (Falcke et al. 2005). Due to coherence effects, this emission can give strong signals on the ground in the frequency range of $\approx (10-80)$ MHz detectable by the LOFAR low band antennas.
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=8.7cm]{lofar_CR.eps}
\end{center}
\caption{Detectable CR energy ranges for LOFAR. LOFAR VHECR refers to the detection using air showers and UHEP to the detection using the Moon. Also shown are the energy ranges for the LOPES and the Auger radio experiments.}
\end{figure}
In addition, LOFAR can also look for the highest energy CRs or neutrinos above around $10^{21}$ eV (Scholten 2010). This will be carried out by detecting coherent radio Cherenkov emission from particle cascades at the Moon induced by those CRs or neutrinos. At frequencies of $\approx (100-200)$ MHz which is within the range of the LOFAR high band antennas, the angular spread of this emission becomes wider leading to more detectable signals at the Earth. The different energy ranges detectable by LOFAR are shown in Fig. 1 where the VHECR (Very High Energy CRs) refers to the detection using air showers and the UHEP (Ultra High Energy Particles) refers to that using the Moon. Fig. 1 also shows the energy ranges for the LOPES (LOFAR Prototype Station) experiment which is located at the KASCADE-Grande experimental site (Apel et al. 2010) and the currently building AERA (Auger Engineering Radio Array) experiment at the Pierre Auger Observatory (van den berg et al. 2009).
One important goal of the LOFAR CR experiment is to push the radio detection technique towards an independent way of detecting very high energy CRs. By detecting the radio signals with better sensitivity and better spatial resolution in a wider frequency range, we strongly believe that LOFAR will provide better understanding of the measured signals, their emission mechanisms and their relations with the air shower parameters, thereby leading to better estimates of the properties of the primary particle. Compared to the LOPES (30 antennas) and the CODALEMA (24 antennas) experiments (Ardrouin et al. 2005), LOFAR has $18$ stations in its core (an area of $\approx 2\times 3$ km$^2$) with each station consisting of $96$ low band antennas and $48$ high band antennas. However, at this stage, it is still quite early for radio detection experiments to do a stand-alone study on CRs. Therefore, we are also setting up a small particle detector array called LORA (LOfar Radboud Air shower array). Its main role will be to trigger and confirm the radio detection of air showers with the LOFAR antennas. It will also help in the reconstruction of several important air shower parameters like the primary energy, the shower core location, the arrival direction, the lateral density distribution etc.
\section{LORA set-up}
LORA is setting up inside the LOFAR core within an area of $\sim 300$ m diameter. It consists of 5 stations with 4 particle detectors each, placed at a separation of $\sim (50-100)$ m between them. It is expected to detect CRs with energies greater than $\sim 10^{15}$ eV at an event rate of around once every few minutes. Detailed simulations about LORA as well as combined studies including the LOFAR radio data will be carried out as soon as possible. The schematic layout of the LORA detectors arrangement is shown in Fig. 2 where the red stars represent the positions of the detectors and the circles denote the positions of the LOFAR antennas fields. Data collection in individual stations are controlled locally by station computers which are then controlled by a master computer where the overall data processing is done.
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=7.0cm]{lora_layout.eps}
\end{center}
\caption{Layout of the LOfar Radboud Air shower array (LORA) in the LOFAR core. The red stars denote the positions of the particle detectors and the circles denote the positions of the antenna fields. The detectors are contained within an area of $\sim 300$ m diameter with relative spacings of $\sim (50-100)$ m.}
\end{figure}
\subsection{Detectors}
The detectors for LORA are taken over from the KASCADE experiment (Antoni et al. 2003). Each detector is of $(98\times 125)$ cm in size and consists of two slabs of $3$ cm thick plastic scintillators. When a charged particle passes through a scintillator slab, the light which are produced are collected into a photomultiplier tube (PMT) through a wavelength shifter bar. The electrical signals from the PMTs (one PMT for each slab) are then combined and fed into the electronics where they are converted into digital signals.
\subsection{Electronics}
The electronics was developed originally for the HISPARC experiment (http://www.hisparc.nl). An electronics unit (hereafter referred to as HISPARC unit) can handle two channels (one channel corresponds to one detector) and there are two units per station (a master and a slave). The master sets the event trigger condition and also provides the time stamp with a GPS receiver along with a $200$ MHz clock counter. Data in each channel are handled by two $12$-bit ADCs (so there are 4 ADCs per HISPARC unit) which can measure voltages in the range of $(0-2)$ V. Event data (signal) in a channel are sampled with a time resolution of $2.5$ ns and are stored in a total time window of $10$ $\mu$s. Fig. 3 shows a typical signal of a charged particle passing through one of our detectors. Data from the electronics are sent to the station computer through USB.
Each HISPARC unit has a FPGA circuit built into it. This gives an observer control over several parameters required by the electronics as well as the detectors. For instance, one can set the trigger thresholds, high voltages, trigger condition, coincidence time window etc. at any time during the observation through a user software running at the station computer.
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=8.7cm]{trace_paper.eps}
\end{center}
\caption{Example of a signal trace when a charged particle passes through one of our detectors.}
\end{figure}
\subsection{Data acquisition}
All the PMTs are gain calibrated by adjusting their voltages to match the response for single muons. When the signals in the two HISPARC units in a station satisfy some minimum trigger condition, the signals are sent to the station computer which then sends them to the master computer. The master computer combines the signals from all stations and checks for good air shower events. When an air shower candidate is found, a first-level online analysis is performed to calculate shower parameters like the arrival direction, shower core position, event size etc., and a trigger information will be sent to the LOFAR antenna to dump the corresponding radio data for the air shower. The master computer takes less than $10$ ms to process an event. This is necessary because the antennas have to dump their radio data corresponding to a particular event before they are overwritten in a memory ring-buffer, known as Transient Buffer Board (TBB).
The DAQ software for LORA is developed in C/C++ language for Linux based operating systems. The software (particularly the online monitoring tool) uses several features of the ROOT package (http://root.cern.ch). On the monitoring panel, important information is displayed which is useful to monitor the performance of the system electronics and of the detectors during observations.
The final data are stored in ROOT format and they consist of four kinds of data. The first are the event data which are generated whenever an air shower event is detected. The data contain the event time stamp for each station and the signal trace in ADC counts for each detector. The second kind of data stores the so-called one second messages from the HISPARC unit. This data is generated every second by the master device and it contains information about the number of times the analog signal went over the threshold in the last second for each of the four channels. It also has important timing information which can be used for calculating an event time stamp with nanosecond accuracy. The third data kind comprises several control parameters for the observation run. This data is stored every interval of time fixed by the observer at the start of the run. The fourth kind of data contains information about the noise level in each channel averaged over some fixed interval of time.
The DAQ software and a preliminary data analysis software have been tested successfully on a LORA prototype installed at Radboud University (RU) Nijmegen. The results from the prototype are presented in the following section.
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=10.5cm]{core_paper.eps}
\end{center}
\caption{Reconstructed air shower core distribution for 5463 events collected with the LORA prototype.}
\end{figure}
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=9.2cm]{theta_phi_paper.eps}
\end{center}
\caption{Arrival direction distribution $(\theta,\phi)$ for the 5463 events collected with the LORA prototype.}
\end{figure}
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\includegraphics[width=8.7cm]{theta_paper.eps}\\
\includegraphics[width=8.7cm]{phi_paper.eps}
\end{center}
\caption{Zenith $\theta$ (top) and azimuthal angles $\phi$ (bottom) distribution for the measured showers. The $\theta$ distribution is fitted with a function given by Eq. (1). The $\phi$ distribution is fitted with a constant.}
\end{figure}
\section{Results from the LORA prototype}
The LORA prototype that we set up to test the electronics and the DAQ software consists of 4 detectors in a $(20\times 80)$ m arrangement. With that, we collected more than $5000$ air shower events in a total observation time of $\sim 236$ hrs ($\sim 10$ days) during April $2010$. This corresponds to an event rate of around once every $3$ minutes. We have performed a preliminary analysis of the data to reconstruct the shower core position on the ground, the arrival direction and the energy of the primary. The primary energy is calculated from the total number of charge particles on the ground using the parameterization given in H\"{o}randel 2007. This simple estimate gives the energy threshold of the prototype to be less than $10^{15}$ eV. The shower core position is calculated using the center of gravity method and the primary arrival direction is calculated using the relative pulse timing informations in the detectors assuming that the shower front arrives in a plane on the ground. The normal to the shower plane gives the arrival direction and it is characterized by two angles: the zenith angle $\theta$ measured from the vertical direction and the azimuthal angle $\phi$ measured in the horizontal plane. Though detailed simulation is still yet to be performed, the energy resolution of LORA is expected to be $\lesssim 30\%$ and the angular resolution to be $\lesssim 1^\circ$.
In Figs. 4 and 5, we show the distributions of the reconstructed core positions and the arrival directions $(\theta,\phi)$ respectively. In Fig. 6, we plot the $\theta$ (top) and the $\phi$ (bottom) distributions separately. The $\theta$ distribution is fitted using the following function,
\begin{equation}
\frac{dN}{d\theta}=a\;\mathrm{sin}\theta\;\mathrm{cos}^{b}\theta.
\end{equation}
The fit parameters are found to be $a=636\pm14.9$ and $b=6.35\pm 0.14$. For the $\phi$ distribution, we fit it to a constant as
\begin{equation}
\frac{dN}{d\phi}=c.
\end{equation}
Eq. (2) is expected if the arrival direction of the CRs are isotropic. However, in our case we see a sinusoidal distribution with two dips one at $\sim 90^\circ$ and the other at $\sim 270^\circ$ which are due to the narrow arrangement of our detector set-up.
It should be mentioned that at this stage, we do not aim to perform a detailed analysis and derive a final set of fit parameters which determine important properties of the CRs. As already mentioned, our primary aim was to test our electronics and the DAQ software. The preliminary results from the prototype are in general consistent with what we expect from CR observations with air shower arrays.
\section{Current status and future plans}
The electronics and the DAQ software for LORA have been tested successfully. The installation of all the LORA stations in the field is already completed and the set-up is currently under testing. On the other hand, the LOFAR CR pipeline software is under development and we are expecting to start the simultaneous observation of CRs with the radio antennas before summer 2011.\\
\\
\textbf{Acknowledgements}: The authors are deeply thankful to the KASCADE collaboration for kindly providing us some of the scintillators previously used for their experiment. The authors are also grateful to both the anonymous referees for constructive comments.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,575
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Francis, Corodon S. was born 23 November 1948, is male, registered as Republican Party of Florida, residing at 7206 Alafia Ridge Rd, Riverview, Florida 33569. Florida voter ID number 110388349. His telephone number is 1-813-677-6748. This is the most recent information, from the Florida voter list as of 31 March 2019.
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31 December 2018 voter list: Cynthia M. Francis, 701 N Ocean St, 512, Jacksonville, FL 32202 Florida Democratic Party.
30 June 2015 voter list: Cynthia M. Francis, 701 Ocean St N, 512, Jacksonville, FL 32202 Florida Democratic Party.
FRANCIS, CYNTHIA MARIA was born 20 August 1966, is female, registered as Florida Democratic Party, residing at 3450 Westford Dr, Apopka, Florida 32712. Florida voter ID number 112847986. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Cynthia Marie born 30 April 1959, Florida voter ID number 108522233 See Chandler, Cynthia Marie. CLICK HERE.
Francis, Cynthia Veronica was born 27 December 1937, is female, registered as Florida Democratic Party, residing at 4917 Nw 43Rd St, Lauderdale Lakes, Florida 33319. Florida voter ID number 124891856. Her telephone number is 1-954-485-6291. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, CYRIL ANTHONY was born 7 June 1944, is male, registered as Florida Democratic Party, residing at 2339 Bellarosa Cir, Royal Palm Beach, Florida 33411. Florida voter ID number 112637500. The voter lists a mailing address and probably prefers you use it: 2339 BELLAROSA CIR ROYAL PALM BEACH FL 33411-1472. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 May 2012 voter list: CYRIL ANTHONY FRANCIS, 401 MICHIGAN PL, WEST PALM BEACH, FL 33409 Florida Democratic Party.
Francis, Cyril Junior was born 16 May 1998, is male, registered as Florida Democratic Party, residing at 2601 Nw 16Th Street Rd, Apt 740, Miami, Florida 33125. Florida voter ID number 122874540. His email address is sneakerheadcj@gmail.com. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dacia Magalene was born 19 April 1981, is female, registered as Republican Party of Florida, residing at 999999 Brevard Co Cthouse, Titusville, Florida 32780. Florida voter ID number 100833423. Her telephone number is 1-913-297-0121. The voter lists a mailing address and probably prefers you use it: 4809 Hunters Xing Valdosta GA 31602-4940. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dacia Marsonia was born 25 December 1972, is female, registered as Florida Democratic Party, residing at 4340 Nw 36Th Ave, Lauderdale Lakes, Florida 33309-4132. Florida voter ID number 114409895. Her telephone number is 1-954-232-0985. Her email address is daclpn@hotmail.com. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 March 2017 voter list: Dacia M. Francis-Myrie, 4340 NW 36th Ave, Lauderdale Lakes, FL 333094132 Florida Democratic Party.
30 June 2015 voter list: Dacia M. Francis-Myrie, 4340 NW 36th Ave, Lauderdale Lakes, FL 33309 Florida Democratic Party.
Francis, Daeshawn Malik was born 20 June 1997, is male, registered as No Party Affiliation, residing at 16537 Saguaro Ln, Spring Hill, Florida 34610. Florida voter ID number 126224788. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DAHLIA C. was born 12 March 1955, is female, registered as Florida Democratic Party, residing at 2762 Sw Oakner St, Pt St Lucie, Florida 34953. Florida voter ID number 121532268. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dahlia Carmelita was born 28 May 1961, is female, registered as Florida Democratic Party, residing at 10056 Winding Lake Rd, Apt 101 E, Sunrise, Florida 33351. Florida voter ID number 101935787. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dahlia F. was born 1 June 1945, is female, registered as Florida Democratic Party, residing at 1501 Grand Isle Dr, Brandon, Florida 33511-4779. Florida voter ID number 113992697. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DAINA SHENEAL was born 26 December 1996, is female, registered as Florida Democratic Party, residing at 524 Susannah Leigh Ln, Apt 306, Orlando, Florida 32818. Florida voter ID number 122286206. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DAISY H. was born 8 November 1935, is female, registered as Florida Democratic Party, residing at 241 Lake Meryl Dr, West Palm Beach, Florida 33411. Florida voter ID number 123567782. This is the most recent information, from the Florida voter list as of 31 January 2019.
FRANCIS, DAKKAR ANDRE was born 14 October 1992, is male, registered as Florida Democratic Party, residing at 5025 Cassia Dr, Pensacola, Florida 32506. Florida voter ID number 120770514. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dakota Lee was born 1 September 1997, is male, registered as Republican Party of Florida, residing at 207 Cope Rd, Chipley, Florida 32428. Florida voter ID number 122892764. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DALE ANTHONY was born 4 September 1967, is male, registered as Republican Party of Florida, residing at 5918 Bilek Dr, Pensacola, Florida 32526. Florida voter ID number 104129208. His telephone number is 1-850-453-2945. This is the most recent information, from the Florida voter list as of 31 March 2019.
30 November 2018 voter list: DALE A. FRANCIS, 3114 SEAFARERS WAY, PENSACOLA, FL 32526 Republican Party of Florida.
Francis, Dale H. was born 18 September 1938, is male, registered as No Party Affiliation, residing at 4361 Summersun Dr, New Port Richey, Florida 34652. Florida voter ID number 106641030. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DALE HENRY was born 12 December 1979, is male, registered as Florida Democratic Party, residing at 2407 Venetian Way, Boynton Beach, Florida 33426. Florida voter ID number 119284320. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 August 2017 voter list: DALE HENRY FRANCIS, 16170 POPPY SEED CIR, UNIT-902, DELRAY BEACH, FL 33484 Florida Democratic Party.
FRANCIS, DALE M. was born 7 May 1996, is male, registered as Republican Party of Florida, residing at 3869 Becontree Pl, Oviedo, Florida 32765. Florida voter ID number 122435734. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dale Milton was born 1 December 1959, is male, registered as Republican Party of Florida, residing at 8880 Old Kings Rd S, #117, Jacksonville, Florida 32257. Florida voter ID number 103312808. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dale Person was born 4 September 1960, is female, registered as Florida Democratic Party, residing at 110 Nw 116Th St, Miami, Florida 33168. Florida voter ID number 109180743. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dale Richard was born 9 December 1954, is male, registered as No Party Affiliation, residing at 2511 Nw 73Rd Ter, Chiefland, Florida 32626. Florida voter ID number 105242244. His telephone number is 1-352-493-4850. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 May 2012 voter list: Dale Richard Francis, 10831 SW 69th St, Cedar Key, FL 32625 No Party Affiliation.
Francis, Dale Timothy was born 24 June 1958, is male, registered as No Party Affiliation, residing at 2007 Hallmark Ct, Lakeland, Florida 33803. Florida voter ID number 122090196. This is the most recent information, from the Florida voter list as of 31 March 2019.
FRANCIS, DALE VAUGHN was born 12 September 1974, is male, registered as No Party Affiliation, residing at 4146 Worlington Ter, Fort Pierce, Florida 34947. Florida voter ID number 101654965. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 July 2018 voter list: DALE V. FRANCIS, 4146 WORLINGTON TER, FORT PIERCE, FL 34947 No Party Affiliation.
Francis, Dale Vaughn was born 12 August 1974, is male, registered as Florida Democratic Party, residing at 1121 Sw 85Th Ter, Pembroke Pines, Florida 33025. Florida voter ID number 102174045. This is the most recent information, from the Florida voter list as of 31 October 2015.
Francis, Dalphinis was born 24 December 1937, is male, registered as Florida Democratic Party, residing at 326 Lincoln Ave, Dundee, Florida 33838-0000. Florida voter ID number 113543862. The voter lists a mailing address and probably prefers you use it: PO BOX 1459 Dundee FL 33838. This is the most recent information, from the Florida voter list as of 31 December 2018.
31 March 2015 voter list: DALPHINIS FRANCIS, 326 LINCOLN AVE, DUNDEE, FL 33838 Florida Democratic Party.
Francis, Dalton A. was born 7 November 1948, is male, registered as No Party Affiliation, residing at 5720 Nw 12Th St, Lauderhill, Florida 33313-6239. Florida voter ID number 101848815. His telephone number is 1-954-303-3018. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dalton B. was born 2 July 1960, is male, registered as No Party Affiliation, residing at 890 Nw 5Th Ave, Miami, Florida 33136. Florida voter ID number 116169762. This is the most recent information, from the Florida voter list as of 31 May 2012.
Francis, Dalton O. was born 21 June 1990, is male, registered as No Party Affiliation, residing at 1545 Nw 180Th Ter, Miami Gardens, Florida 33169. Florida voter ID number 115924905. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Damaris B. was born 19 May 1992, is female, registered as Florida Democratic Party, residing at 15825 E Bunche Park Dr, Miami Gardens, Florida 33054. Florida voter ID number 119285144. This is the most recent information, from the Florida voter list as of 31 March 2019.
30 June 2017 voter list: Damaris B. Francis, 15829 E Bunche Park DR, Miami Gardens, FL 33054 Florida Democratic Party.
30 November 2016 voter list: Damaris Beatrice Francis, 533 SW 110TH LN, APT 303, Pembroke Pines, FL 330256991 Florida Democratic Party.
30 June 2015 voter list: Damaris Beatrice Francis, 533 SW 110TH LN, APT 303, Pembroke Pines, FL 33025 Florida Democratic Party.
31 May 2012 voter list: Damaris Beatrice Francis, 1020 NW 149Th ST, Miami, FL 33168 Florida Democratic Party.
FRANCIS, DAMEYENOE D. was born 28 March 1984, is male, registered as Florida Democratic Party, residing at 509 W Obispo Ave, Clewiston, Florida 33440. Florida voter ID number 104324478. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Damian was born 3 January 1983, registered as Florida Democratic Party, residing at 107 Sw 18Th Ave, Ft Lauderdale, Florida 33312. Florida voter ID number 117025647. This is the most recent information, from the Florida voter list as of 30 November 2016.
31 May 2012 voter list: Damian Francis, 3076 SW 142Nd AVE, Miramar, FL 33027 Florida Democratic Party.
Francis, Damian Jude was born 31 October 1977, is male, registered as Florida Democratic Party, residing at 5733 Tuscany Way, Tamarac, Florida 33321. Florida voter ID number 117060059. His telephone number is 1-954-636-4032. This is the most recent information, from the Florida voter list as of 31 March 2019.
30 September 2018 voter list: DAMIAN JUDE FRANCIS, 5841 ASHDALE RD, LAKE WORTH, FL 33463 Florida Democratic Party.
31 May 2017 voter list: Damian Jude Francis, 5733 Tuscany Way, Tamarac, FL 333214457 Florida Democratic Party.
30 June 2015 voter list: Damian Jude Francis, 5733 Tuscany Way, Tamarac, FL 33321 Florida Democratic Party.
FRANCIS, DAMIEN JERMAINE was born 25 November 1980, is male, registered as No Party Affiliation, residing at 714 Buttonwood Ln, Boynton Beach, Florida 33436. Florida voter ID number 115321084. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Damion was born 14 November 1979, registered as No Party Affiliation, residing at 1059 Sw 147Th Ave, Pembroke Pines, Florida 33027. Florida voter ID number 124414102. This is the most recent information, from the Florida voter list as of 30 November 2016.
Francis, Damion Leonard was born 14 November 1979, is male, registered as Florida Democratic Party, residing at 7458 Nw 33Rd St, Lauderhill, Florida 33319. Florida voter ID number 120371682. His telephone number is 1-754-213-0681. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 October 2016 voter list: Damion Leonard Francis, 1059 SW 147th AVE, Pembroke Pines, FL 33027 Florida Democratic Party.
30 September 2016 voter list: Damion Leonard Francis, 420 SW 80TH AVE, North Lauderdale, FL 33068 Florida Democratic Party.
30 November 2015 voter list: Damion Leonard Francis, 7458 NW 33Rd St, Lauderhill, FL 333194946 Florida Democratic Party.
30 June 2015 voter list: Damion Leonard Francis, 7458 NW 33Rd St, Lauderhill, FL 33319 Florida Democratic Party.
Francis, Damond Alphonso was born 23 June 1969, is male, registered as Florida Democratic Party, residing at 3301 Spanish Moss Ter, Apt 404, Lauderhill, Florida 33319-5002. Florida voter ID number 120692333. His telephone number is 1-786-523-3679. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 October 2015 voter list: Damond Alphonso Francis, 4001 N University Dr, APT A 206, Sunrise, FL 33351 Florida Democratic Party.
Francis, Damontra Tyrail was born 4 August 1998, is male, registered as Florida Democratic Party, residing at 10 Deer Trail Cir, Bronson, Florida 32621. Florida voter ID number 125503381. His telephone number is 1-352-507-2468. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, Dana Lance was born 11 February 1951, is male, registered as No Party Affiliation, residing at 3912 Scarborough Ct, Clermont, Florida 34711. Florida voter ID number 119231184. This is the most recent information, from the Florida voter list as of 31 March 2019.
30 November 2015 voter list: Dana Lance Francis, 4018 Newland St, Clermont, FL 34711 No Party Affiliation.
FRANCIS, DANA NOEL was born 18 December 1964, is female, registered as Florida Democratic Party, residing at 1339 Homestead Way, Palm Harbor, Florida 34683. Florida voter ID number 107078942. This is the most recent information, from the Florida voter list as of 31 March 2019.
30 September 2015 voter list: DANA NOEL FRANCIS, 540 SPRING LAKE CIR, TARPON SPRINGS, FL 34688 Florida Democratic Party.
31 March 2014 voter list: DANA NOEL FRANCIS, 1424 SEAGULL DR, #203, PALM HARBOR, FL 34685 Florida Democratic Party.
Francis, Danavon Barrington was born 6 September 1971, is male, registered as Florida Democratic Party, residing at 1565 Cove Lake Rd, North Lauderdale, Florida 33068-4634. Florida voter ID number 118414913. His telephone number is 1-347-656-5092. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 January 2018 voter list: Danavon Barrington Francis, 6548 RACQUET CLUB DR, Lauderhill, FL 33319 Florida Democratic Party.
31 December 2015 voter list: Donavon B. Francis, 6630 Racquet Club DR, Lauderhill, FL 333195026 Florida Democratic Party.
31 May 2013 voter list: Donavon B. Francis, 4026 INVERRARY BLVD, APT 1208, Lauderhill, FL 33319 Florida Democratic Party.
31 May 2012 voter list: Donavon Francis, 4026 Inverrary Blvd, APT 1208, Lauderhill, FL 33319 Florida Democratic Party.
Francis, Dana Wesley was born 19 October 1969, is male, registered as Republican Party of Florida, residing at 4601 34Th Avenue Dr W, Bradenton, Florida 34209-6339. Florida voter ID number 105373001. This is the most recent information, from the Florida voter list as of 31 March 2019.
31 October 2017 voter list: Dana Wesley Francis, 4601 34th Avenue Dr W, Bradenton, FL 34209 Republican Party of Florida.
Francis, D'Andra Amanda was born 14 February 1991, is female, registered as Florida Democratic Party, residing at 8000 Hampton Blvd, Apt 303, North Lauderdale, Florida 33068. Florida voter ID number 124138180. Her telephone number is 1-954-415-7924. This is the most recent information, from the Florida voter list as of 31 March 2019.
Francis, D'Andra Amanda was born 14 February 1991, registered as Florida Democratic Party, residing at 8000 Hampton Blvd, Apt 303, North Lauderdale, Florida 33068-5626. Florida voter ID number 116137041. This is the most recent information, from the Florida voter list as of 31 August 2016.
30 June 2015 voter list: D'Andra Amanda Francis, 8000 Hampton Blvd, APT 303, North Lauderdale, FL 33068 Florida Democratic Party.
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{
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Recommitting to a Quaker truth.
Chris Morrissey still holds membership in South Bend (Ind.) Meeting and came to Friends in Santa Monica (Calif.) Meeting. He currently attends several meetings in the Portland, Ore., area. His first book, Christianity and American State Violence in Iraq: Priestly or Prophetic?, comes out next spring.
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}
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\subsection{Proof of Lemma~\ref{lem:multi_set_rois}}
Given a stochastic graph $\mathcal{G}$, recall that $\Omega_\mathcal{G}$ is the space of all possible sample graphs $g \sim \mathcal{G}$ and $\mathbb{Pr}[g]$ is the probability that $g$ is realized from $\mathcal{G}$. In a sample graph $g \in \Omega_\mathcal{G}$, $\eta_g(S,v) = 1$ if $v$ is reachable from $S$ in $g$. Consider the graph sample space $\Omega_\mathcal{G}$, based on a node $v \in V \backslash S$, we can divide $\Omega_\mathcal{G}$ into two partitions: 1) $\Omega^\emptyset_\mathcal{G}(v)$ contains those samples $g$ in which $v$ has \textit{no incoming live-edges}; and 2) $\bar \Omega^\emptyset_\mathcal{G}(v) = \Omega_{\mathcal{G}} \backslash \Omega^\emptyset_\mathcal{G}$. We start from the definition of influence spread as follows,
\begin{align}
\I(S) & = \sum_{v \in V} \sum_{g \in \Omega_\mathcal{G}} \eta_g(S,v) \mathbb{Pr}[g] \nonumber \\
& = \sum_{v \in V} \Big (\sum_{g \in \Omega^\emptyset_\mathcal{G}(v)} \eta_g(S,v) \mathbb{Pr}[g] + \sum_{g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \eta_g(S,v) \mathbb{Pr}[g]\Big ). \nonumber
\end{align}
In each $g \in \Omega^\emptyset_\mathcal{G}(v)$, the node $v$ does not have any incoming nodes, thus, $\eta_g(S,v) = 1$ only if $v \in S$. Thus, we have that $\sum_{v \in V} \sum_{g \in \Omega^\emptyset_\mathcal{G}(v)} \eta_g(S,v) \mathbb{Pr}[g] = \sum_{v \in S} \sum_{g \in \Omega^\emptyset_\mathcal{G}(v)} \mathbb{Pr}[g]$. Furthermore, the probability of a sample graph which has no incoming live-edge to $v$ is $ \sum_{g \in \Omega^\emptyset_\mathcal{G}(v)} \mathbb{Pr}[g] = 1-\gamma_v$. Combine with the above equiation of $\I(S)$, we obtain,
\begin{align}
\label{eq:lem6_eq1}
\I(S) = \sum_{v\in S} (1-\gamma_v) + \sum_{v \in V} \sum_{g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \eta_g(S,v) \mathbb{Pr}[g \in \Omega_\mathcal{G}].
\end{align}
Since our \textsf{IIS}{} sketching algorithm only generates samples corresponding to sample graphs from the set $\bar \Omega^\emptyset_\mathcal{G}(v)$, we define $\bar \Omega^\emptyset_\mathcal{G}(v)$ to be a graph sample space in which the sample graph $\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)$ has a probability $\mathbb{Pr}[\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)] = \frac{\mathbb{Pr}[\bar g \in \Omega_\mathcal{G}]}{\gamma_v}$ of being realized (since $\sum_{\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)}\mathbb{Pr}[\bar g \in \Omega_\mathcal{G}] = \gamma_v$ is the normalizing factor to fulfill a probability distribution of a sample space). Then, Eq.~\ref{eq:lem6_eq1} is rewritten as follows,
\begin{align}
\I(S) & = \sum_{v \in V} \sum_{g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \eta_g(S,v) \frac{\mathbb{Pr}[g \in \Omega_\mathcal{G}]}{\gamma_v} \gamma_v + \sum_{v\in S} (1-\gamma_v) \nonumber \\
& = \sum_{v \in V} \sum_{\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \eta_{\bar g}(S,v) \mathbb{Pr}[\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)] \gamma_v + \sum_{v\in S} (1-\gamma_v) \nonumber
\end{align}
Now, from the node $v$ in a sample graph $\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)$, we have a \textsf{IIS}{} sketch $R_j(\bar g, v)$ starting from $v$ and containing all the nodes that can reach $v$ in $\bar g$. Thus, $\eta_{\bar g}(S,v) = \textbf{1}_{R_j(\bar g,v) \cap S \neq \emptyset}$ where $\textbf{1}_{x}$ is an indicator function returning 1 iff $x \neq 0$. Then,
\begin{align}
& \sum_{\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \eta_{\bar g}(S,v) \mathbb{Pr}[\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)] \nonumber \\
& = \sum_{\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)} \textbf{1}_{R_j(\bar g,v) \cap S \neq \emptyset} \mathbb{Pr}[\bar g \in \bar \Omega^\emptyset_\mathcal{G}(v)] = \mathbb{Pr}[R_j(v) \cap S \neq \emptyset] \nonumber
\end{align}
\noindent where $R_j(v)$ is a random \textsf{IIS}{} sketch with $\textsf{src}(R_j(v)) = v$. Plugging this back into the computation of $\I(S)$ gives,
\begin{align}
& \I(S) = \sum_{v \in V} \mathbb{Pr}[R_j(v) \cap S \neq \emptyset] \gamma_v + \sum_{v\in S} (1-\gamma_v) \nonumber \\
& = \sum_{v \in V} \mathbb{Pr}[R_j(v) \cap S \neq \emptyset] \frac{\gamma_v}{\Gamma} \Gamma + \sum_{v\in S} (1-\gamma_v) \nonumber \\
& = \sum_{v \in V} \mathbb{Pr}[R_j(v) \cap S \neq \emptyset] \mathbb{Pr}[\textsf{src}(R_j) = v]\Gamma + \sum_{v\in S} (1-\gamma_v) \nonumber \\
& = \mathbb{Pr}[R_j \cap S \neq \emptyset] \cdot \Gamma + \sum_{v\in S} (1-\gamma_v)
\end{align}
\noindent That completes the proof.
\subsection{Proof of Lemma~\ref{lem:zvar}}
From the basic properties of variance, we have,
\begin{align}
\mathrm{Var}[Z_j(S)] & = \mathrm{Var}[\frac{X_j(S) \cdot \Gamma + \sum_{v \in S}(1-\gamma_v)}{n}] \nonumber \\
& = \frac{\Gamma^2}{n^2} \mathrm{Var}[X_j(S)] \nonumber
\end{align}
Since $X_j(S)$ is a Bernoulli random variable with its mean value $\E[X_j(S)] = \frac{\I(S)-\sum_{v\in S}(1-\gamma_v)}{\Gamma}$, the variance $\mathrm{Var}[X_j(S)]$ is computed as follows,
\begin{align}
& \mathrm{Var}[X_j(S)] \nonumber \\
& = \frac{\I(S)-\sum_{v\in S}(1-\gamma_v)}{\Gamma} (1 - \frac{\I(S)-\sum_{v\in S}(1-\gamma_v)}{\Gamma}) \nonumber \\
& = \frac{\I(S)}{\Gamma} - \frac{\I^2(S)}{\Gamma^2} - \frac{\sum_{v \in S}(1-\gamma_v)}{\Gamma^2}(\Gamma + \sum_{v \in S}(1-\gamma_v) - 2\I(S)) \nonumber
\end{align}
Put this back into the variance of $Z_j(S)$ proves the lemma.
\subsection{Proof of Lemma~\ref{lem:var_z}}
Since $Z_j(S)$ takes values of either $\frac{\sum_{v \in S}(1-\gamma_v)}{n}$ or $\frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n}$ and the mean value $\E[Z_j(S)] = \frac{\I(S)}{n}$, i.e. $\frac{\sum_{v \in S}(1-\gamma_v)}{n} \leq \frac{\I(S)}{n} \leq \frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n}$. The variance of $Z_j(S)$ is computed as follows,
\begin{align}
\mathrm{Var}&[Z_j(S)] \nonumber \\
& = \Big( \frac{\I(S)}{n} - \frac{\sum_{v \in S}(1-\gamma_v)}{n} \Big) \Big( \frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n} - \frac{\I(S)}{n} \Big)\nonumber \\
& \leq \frac{\I(S)}{n} \Big( \frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n} - \frac{\sum_{v \in S}(1-\gamma_v)}{n} \Big) \nonumber \\
& = \frac{\I(S)}{n} \frac{\Gamma}{n}
\end{align}
\subsection{Proof of Lemma~\ref{lem:bound}}
Lemma~2 in \cite{Tang15} states that:
\begin{Lemma}
Let $M_1, M_2, \dots$ be a martingale, such that $|M_1| \leq a$, $|M_j - M_{j-1}| \leq a$ for any $j \in [2, T]$, and
\begin{align}
\mathrm{Var}[M_1] + \sum_{j = 2}^{T} \mathrm{Var}[M_j | M_1, M_2, \dots, M_{j-1}] \leq b,
\end{align}
where $\mathrm{Var}[.]$ denotes the variances of a random variable. Then, for any $\lambda > 0$,
\begin{align}
\label{eq:mar_cher}
\mathbb{Pr}[M_T - \E[M_T] \geq \lambda] \leq \exp\Big( - \frac{\lambda^2}{\frac{2}{3}a\lambda + 2b} \Big)
\end{align}
\end{Lemma}
Note that uniform random variables are also a special type of martingale and the above lemma holds for random variable as well.
Let $p = \frac{\I(S)}{n}$. For \textsf{RIS}{} samples, since
\begin{itemize}
\item $|M_1| \leq 1$,
\item $|M_j - M_{j-1}| \leq 1, \forall j \in [2,T]$,
\item $\mathrm{Var}[M_1] + \sum_{j = 2}^{T}\mathrm{Var}[M_j | M_1, \dots, M_{j-1}] = \sum_{j = 1}^{i} \mathrm{Var}[Y_j(S)] = T p(1-p) \leq T p$,
\end{itemize}
applying Eq.~\ref{eq:mar_cher} for $\lambda = \epsilon Tp$ gives the following Chernoff's bounds,
\begin{align}
\label{eq:mar_cher_ris_1}
&\mathbb{Pr}\Big[\sum_{j = 1}^{T} X_j(S) - Tp \geq \epsilon Tp\Big] \leq \exp\Big( -\frac{\epsilon^2}{2+\frac{2}{3}\epsilon} Tp \Big), \\
&\text{and,} \nonumber \\
\label{eq:mar_cher_ris_2}
&\mathbb{Pr}\Big[\sum_{j = 1}^{T} X_j(S) - Tp \leq -\epsilon Tp\Big] \leq \exp\Big( -\frac{\epsilon^2}{2} Tp \Big).
\end{align}
However, for \textsf{IIS}{} samples in \textsf{SKIS}{} sketch, the corresponding random variables $Z_j(S)$ replace $Y_j(S)$ and have the following properties:
\begin{itemize}
\item $|M_1| \leq \frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n} \leq 1$,
\item $|M_j - M_{j-1}| \leq 1, \forall j \in [2,T]$,
\item The sum of variances:
\begin{align}
&\mathrm{Var}[M_1] + \sum_{j = 2}^{T}\mathrm{Var}[M_j | M_1, \dots, M_{j-1}] \nonumber \\
&= \sum_{j = 1}^{i} \mathrm{Var}[Z_j(S)] = T p \frac{\Gamma}{n}
\end{align}
\end{itemize}
Thus, applying the general bound in Eq.~\ref{eq:mar_cher} gives,
\begin{align}
&\mathbb{Pr}\Big[\sum_{j = 1}^{T} X_j(S) - Tp \geq \epsilon Tp\Big] \leq \exp\Big( -\frac{\epsilon^2}{2\boldsymbol{\frac{\Gamma}{n}}+\frac{2}{3}\epsilon} Tp \Big), \\
&\text{and,} \nonumber \\
&\mathbb{Pr}\Big[\sum_{j = 1}^{T} X_j(S) - Tp \leq -\epsilon Tp\Big] \leq \exp\Big( -\frac{\epsilon^2}{2\boldsymbol{\frac{\Gamma}{n}}} Tp \Big).
\end{align}
Note the factor $\frac{\Gamma}{n}$ is added in the denominator of the terms in the $\exp(.)$ function. Since $2\frac{\Gamma}{n}$ dominates $\frac{2}{3}\epsilon$, the concentration bounds for $Z_j(S)$ for \textsf{SKIS}{} are tighter than those of $Y_j(S)$ for \textsf{RIS}{} given in Eqs.~\ref{eq:mar_cher_ris_1} and \ref{eq:mar_cher_ris_2}.
\subsection{Proof of Lemma~\ref{lem:sub}}
Since the function $\hat \I_\R(S)$ contains two additive terms, it is sufficient to show that each of them is monotone and submodular. The second term $\sum_{v \in S}(1-\gamma_v)$ is a linear function and thus, it is monotone and submodular. For the first additive term, we see that $\frac{\Gamma}{|\R| \cdot n}$ is a constant and only need to show that $\mathrm{C}_\R(S)$ is monotone and submodular. Given the collection of \textsf{IIS}{} samples $\R$ in which $R_j \in \R$ is a list of nodes, the function $\mathrm{C}_\R(S)$ is just the count of \textsf{IIS}{} samples that intersect with the set $S$. In other words, it is equivalent to a covering function in a set system where \textsf{IIS}{} samples are elements and nodes are sets. A set covers an element if the corresponding node is contained in the corresponding \textsf{IIS}{} sample. It is well known that any covering function is monotone and submodular \cite{Vazirani01} and thus, the $\mathrm{C}_\R(S)$ has the same properties.
\subsection{Improvements of \textsf{SSA}/\textsf{DSSA}{} using \textsf{SKIS}{} sketch}
Recall that the original Stop-and-Stare strategy in \cite{Nguyen163} uses two independent sets of \textsf{RIS}{} samples, called $\R$ and $\R^c$. The greedy algorithm is applied on the first set $\R$ to find a candidate set $\hat S_k$ along with an estimate $\hat \I_\R(\hat S_k)$ and the second set $\R^c$ is used to reestimate the influence of $\hat S_k$ by $\hat \I_{\R^c}(\hat S_k)$. Now, \textsf{SSA}{} and \textsf{DSSA}{} have different ways to check the solution quality.
\textbf{\textsf{SSA}{}.} It assumes a set of fixed precision parameters $\epsilon_1, \epsilon_2, \epsilon_3$ such that $\frac{\epsilon_1 + \epsilon_2 + \epsilon_1 \epsilon_2 + \epsilon_3}{(1+\epsilon_1)(1+\epsilon_2)} (1-1/e) \leq \epsilon$. The algorithm stops when two conditions are met:
\begin{itemize}
\item[1)] $\mathrm{C}_\R(\hat S_k) \geq \Lambda_1$ where $ \Lambda_1 = O(\log\frac{\delta}{t_{max}}\epsilon_3^{-2})$ and $t_{max}$ is a precomputed number depending on the size of the input graph $\mathcal{G}$.
\item[2)] $\hat \I_\R(\hat S_k) \leq (1+\epsilon_1) \hat \I_{\R^c}(\hat S_k)$.
\end{itemize}
$\Rightarrow$ \textit{Improvements using \textsf{SKIS}{}:} Replacing \textsf{RIS}{} samples by \textsf{IIS}{} samples to build $\R$ and $\R^c$ helps in both stopping conditions:
\begin{itemize}
\item Reduce $\Lambda_1$ to $\Lambda_1 = O(\frac{\Gamma}{n}\log\frac{\delta}{t_{max}}\epsilon_3^{-2})$ using the tighter form of the Chernoff's bounds in Lemma~\ref{lem:bound}.
\item Since \textsf{IIS}{} samples have better influence estimation accuracy, $\hat \I_\R(\hat S_k)$ and $\hat \I_{\R^c}(\hat S_k)$ are closer to the true influence $\I(\hat S_k)$. Thus, the second condition is met earlier than using \textsf{RIS}{} samples.
\end{itemize}
\textbf{\textsf{DSSA}.} Instead of assuming precision parameters, \textsf{DSSA}{} dynamically compute the error bounds $\epsilon_1, \epsilon_2$ and $\epsilon_3$ as follows:
\begin{itemize}
\item $\epsilon_1 = \frac{\hat \I_{\R}(\hat S_k)}{\hat \I_{\R^c}(\hat S_k)} - 1$.
\item $\epsilon_2 = \epsilon \sqrt{\frac{n(1+\epsilon)}{2^{t-1}\hat \I_{\R^c}(\hat S_k)}}$.
\item $\epsilon_3 = \epsilon \sqrt{\frac{n(1+\epsilon)(1-1/e-\epsilon)}{(1+\epsilon/3)2^{t-1}\hat \I_{\R^c}(\hat S_k)}}$.
\end{itemize}
Here, $\epsilon_1$ measures the discrepancy of estimations using two different sketches $\R$ and $\R^c$ while $\epsilon_2$ and $\epsilon_3$ are the error bounds of estimating the influences of $\hat S_k$ and the optimal solution $S^*_k$ using the number of samples contained in $\R$ and $\R^c$. The algorithm stops when two conditions are met:
\begin{itemize}
\item $\mathrm{C}_\R(\hat S_k) \geq \Lambda_2$ where $\Lambda_2 = O(\log\frac{\delta}{t_{max}} \epsilon^{-2})$.
\item $(\epsilon_1 + \epsilon_2+\epsilon_1\epsilon_2)(1-1/e-\epsilon)+(1-1/e)\epsilon_3 \leq \epsilon$.
\end{itemize}
$\Rightarrow$ Improvement using \textsf{SKIS}{}: Similarly to \textsf{SSA}{}, applying \textsf{SKIS}{} helps in both stopping conditions:
\begin{itemize}
\item Reduce $\Lambda_2$ to $\Lambda_2 = O(\frac{\Gamma}{n}\log\frac{\delta}{t_{max}} \epsilon^{-2})$.
\item Reduce the value of $\epsilon_1, \epsilon_2$ and $\epsilon_3$ due to better influence estimations of $\hat \I_{\R}(\hat S_k)$ and $\hat \I_{\R^c}(\hat S_k)$ by \textsf{SKIS}{} that leads to earlier satisfaction of the second condition.
\end{itemize}
\section{Experiments}
\label{sec:exps}
We demonstrate the advantages of our \textsf{SKIS}{} sketch through a comprehensive set of experiments on the key influence estimation and maximization problems. Due to space limit, we report the results under the IC model and partial results for the LT model. However, the implementations for all models will be released on our website to produce complete results
\renewcommand{1.1}{0.9}
\subsection{Experimental Settings}
\setlength\tabcolsep{3pt}
\begin{table}[!h] \small
\caption{Datasets' Statistics}
\label{tab:data_sum}
\centering
\begin{tabular}{ l r r r}\toprule
\textbf{Dataset} & \bf \#Nodes& \bf \#Edges & \bf Avg. Degree\\\midrule
NetPHY & $37\cdot10^3$ & $181\cdot10^3$ & 9.8\\
Epinions & $75\cdot10^3$ & $841\cdot10^3$ & 22.4\\
DBLP & $655\cdot10^3$ & $2\cdot10^6$ & 6.1\\
Orkut & $3\cdot10^6$ & $234\cdot10^6$ & 78.0\\
Twitter \cite{Kwak10} & $41.7\cdot10^6$ & $1.5\cdot10^9$ & 70.5\\
Friendster & $65.6\cdot10^6$ & $3.6\cdot10^9$ & 109.6\\
\bottomrule
\hline
\end{tabular}
\end{table}
\renewcommand{1.1}{1}
\textbf{Datasets.}
We use 6 real-world datasets from \cite{snap,Kwak10} with size ranging from tens of thousands to as large as 65.6 million nodes and 3.6 billion edges. Table \ref{tab:data_sum} gives a summary.
\textbf{Algorithms compared.} On influence estimation, we compare our \textsf{SKIS} sketch with:
\begin{itemize}
\item \textsf{RIS}{} \cite{Borgs14}: The well-known \textsf{RIS}{} sketch.
\item \textsf{SKIM}{} \cite{Cohen14}: Combined reachability sketch. We run \textsf{SKIM}{} with default parameters in \cite{Cohen14} ($k = l = 64$). \textsf{SKIM}{} is modified to read graph from files instead of internally computing the edge weights.
\end{itemize}
Following \cite{Ohsaka16}, we generate samples into \textsf{SKIS}{} and \textsf{RIS}{} until the total size of all the samples reaches $h \cdot n \log n$ where $h$ is a constant. Here, $h$ is chosen in the set $\{5, 10\}$.
On influence maximization, we compare:
\begin{itemize}
\item \textsf{PMC} \cite{Ohsaka14}: A Monte-Carlo simulation pruned method with no guarantees. It only works on the IC model.
\item \textsf{IMM}{} \cite{Tang15}: \textsf{RIS}{}-based algorithm with quality guarantees.
\item \textsf{DSSA}{} \cite{Nguyen163}: The current fastest \textsf{RIS}{}-based algorithm with approximation guarantee.
\item \textsf{DSSA}{}+\textsf{SKIS}: A modified version of \textsf{DSSA}{} where \textsf{SKIS}{} sketch is adopted to replace \textsf{RIS}{}.
\end{itemize}
We set $\epsilon = 0.5, \delta = 1/n$ for the last three algorithms. For \textsf{PMC}, we use the default parameter of 200 DAGs.
\textbf{Metrics.}
We compare the algorithms in terms of the solution quality, running time and memory usage. To compute the solution quality of a seed set, we adopt the relative difference which is defined as $\frac{|\hat \I(S) - \I(S)|}{\max\{\I(S), \hat \I(S)\}} \cdot 100\%$, where $\hat \I(S)$ is an estimate, and $\I(S)$ is the ``ground-truth" influence of $S$.
\textbf{Ground-truth Influence.} Unlike previous studies \cite{Tang15,Cohen14,Ohsaka14} using a constant number of cascade simulations, i.e. 10000, to measure the ground-truth influence with \textit{unknown accuracy}, we adopt the Monte-Carlo Stopping-Rule algorithm \cite{Dagum00} that guarantees an estimation error less than $\epsilon$ with probability at least $1-\delta$ where $\epsilon = 0.005, \delta = 1/n$. Specifically, let $W_j$ be the size of a random influence cascade and $Z_j = \frac{W_j}{n}$ with $\E[Z_j] = \I(S)/n$ and $0 \leq Z_j \leq 1$. The Monte-Carlo method generates sample $Z_j$ until
$\sum_{j = 1}^{T} Z_j \geq 4(e-2)\ln(\frac{2}{\delta})\frac{1}{\epsilon^2}$
and returns $\hat \I(S) = \frac{\sum_{j = 1}^{T} Z_j}{T}n$ as the ground-truth influence.
For Twitter and Friendster dataset, we set $\epsilon = 0.05$, and $\delta = 1/n$ to compute ground-truth due to the huge computational cost in these networks. For the other networks, we keep the default setting of $\epsilon$ and $\delta$ as specified above.
\textbf{Weight Settings.}
We consider two widely-used models:
\begin{itemize}
\item \textit{Weighted Cascade} (\textsf{WC}) \cite{Tang14,Cohen14,Tang15,Nguyen163}: The weight of edge $(u,v)$ is inversely proportional to the in-degree of node $v$, $d_{in}(v)$, i.e. $w(u,v) = \frac{1}{d_{in}(v)}$.
\item \textit{Trivalency} (\textsf{TRI}) \cite{Cohen14, Chen10, Jung12}: The weight $w(u,v)$ is selected randomly from the set $\{0.1, 0.01, 0.001\}$.
\end{itemize}
\textbf{Environment.}
We implemented our algorithms in C++ and obtained the implementations of others from the corresponding authors. We conducted all experiments on a CentOS machine with Intel Xeon E5-2650 v3 2.30GHz CPUs and 256GB RAM. We compute the ground-truth for our experiments in a period of 2 months on a cluster of 16 CentOS machines, each with 64 Intel Xeon CPUs X5650 2.67GHz and 256GB RAM.
\renewcommand{1.1}{1}
\setlength\tabcolsep{0.5pt}
\begin{table}[ht] \small
\vspace{0.05in}
\centering
\caption{Average relative differences (\textsf{dnf}: ``did not finish" within 24h). \textsf{SKIS}{} almost always returns the lowest errors.}
\begin{tabular}{llrrrrrrrr|r r rrrrr}
\toprule
& & \multicolumn{7}{c}{\textbf{\textsf{WC} Model}} && \multicolumn{7}{c}{\textbf{\textsf{TRI} Model}} \\
\cmidrule{3-9} \cmidrule{11-17}
& & \multicolumn{2}{c}{\textsf{SKIS}{}} && \multicolumn{2}{c}{\textsf{RIS}{}} && \textsf{SKIM}{} && \multicolumn{2}{c}{\textsf{SKIS}{}} && \multicolumn{2}{c}{\textsf{RIS}{}} && \textsf{SKIM}{} \\
\cmidrule{3-4} \cmidrule{6-7} \cmidrule{9-9} \cmidrule{11-12} \cmidrule{14-15} \cmidrule{17-17}
\multirow{1}{*}{{$|S|$}} & \textsf{Nets} & $h(5)$ & $h(10)$ && $h(5)$ & $h(10)$ && $k(64)$ && $h(5)$ & $h(10)$ && $h(5)$ & $h(10)$ && $k(64)$ \\
\midrule
\multirow{6}{*}{l}& PHY & \cca{6.2} & \cca{3.7} && \cca{14.0} & \cca{7.8} && \cca{7.5} && \cca{1.7} & \cca{1.3} && \cca{11.8} & \cca{8.2} && \cca{4.5} \\
& Epin. & \cca{4.7} & \cca{3.0} && \cca{15.7} & \cca{11.8} && \cca{19.6} && \cca{16.6} & \cca{14.2} && \cca{55.3} & \cca{47.4} && \cca{27.7} \\
& DBLP & \cca{3.8} & \cca{4.1} && \cca{13.7} & \cca{11.6} && \cca{5.0} && \cca{0.9} & \cca{0.7} && \cca{9.4} & \cca{6.4} && \cca{3.5} \\
& Orkut & \cca{10.3} & \cca{9.2} && \cca{13.5} & \cca{8.8} && \cca{77.6} && \cca{9.3} & \cca{9.9} && \cca{14.5} & \cca{10.8} && \cellcolor{red} \textsf{dnf} \\
& Twit. & \cca{10.9} & \cca{10.5} && \cca{21.4} & \cca{16.0} && \cca{29.1} && \cca{81.4} & \cca{81.9} && \cca{80.8} & \cca{81.5} && \cellcolor{red} \textsf{dnf} \\
& Frien. & \cca{15.9} & \cca{10.2} && \cca{22.2} & \cca{13.3} && \cellcolor{red} \textsf{dnf} && \cca{29.8} & \cca{21.3} && \cca{28.5} & \cca{23.6} && \cellcolor{red} \textsf{dnf} \\
\midrule
\multirow{6}{*}{$10^2$}& PHY & \cca{0.9} & \cca{0.6} && \cca{1.0} & \cca{0.7} && \cca{2.1} && \cca{0.3} & \cca{0.2} && \cca{1.1} & \cca{0.9} && \cca{1.8} \\
& Epin. & \cca{1.0} & \cca{0.7} && \cca{1.0} & \cca{1.0} && \cca{7.6} && \cca{0.2} & \cca{1.5} && \cca{4.4} & \cca{1.8} && \cca{2.8} \\
& DBLP & \cca{0.9} & \cca{0.6} && \cca{1.9} & \cca{1.4} && \cca{5.0} && \cca{0.8} & \cca{0.7} && \cca{5.5} & \cca{5.3} && \cca{5.5} \\
& Orkut & \cca{0.9} & \cca{0.6} && \cca{1.1} & \cca{0.7} && \cca{56.5} && \cca{0.1} & \cca{0.2} && \cca{4.2} & \cca{0.9} && \cellcolor{red} \textsf{dnf} \\
& Twit. & \cca{1.1} & \cca{1.2} && \cca{1.3} & \cca{1.1} && \cca{60.2} && \cca{4.3} & \cca{3.1} && \cca{6.4} & \cca{5.5} && \cellcolor{red} \textsf{dnf} \\
& Frien. & \cca{0.9} & \cca{0.7} && \cca{0.9} & \cca{0.7} && \cellcolor{red} \textsf{dnf} && \cca{1.9} & \cca{1.9} && \cca{0.6} & \cca{2.0} && \cellcolor{red} \textsf{dnf} \\
\midrule
\multirow{6}{*}{$10^3$}{} & PHY & \cca{0.6} & \cca{0.8} && \cca{0.9} & \cca{1.0} && \cca{0.6} && \cca{0.3} & \cca{0.4} && \cca{1.2} & \cca{1.3} && \cca{1.1} \\
& Epin. & \cca{0.6} & \cca{0.6} && \cca{0.6} & \cca{0.7} && \cca{2.3} && \cca{2.3} & \cca{0.3} && \cca{1.9} & \cca{4.6} && \cca{1.5} \\
& DBLP & \cca{0.2} & \cca{0.3} && \cca{0.2} & \cca{0.2} && \cca{1.7} && \cca{0.1} & \cca{0.0} && \cca{0.3} & \cca{0.2} && \cca{0.3} \\
& Orkut & \cca{0.3} & \cca{0.3} && \cca{0.3} & \cca{0.3} && \cca{50.7} && \cca{2.5} & \cca{1.1} && \cca{6.8} & \cca{2.1} && \cellcolor{red} \textsf{dnf} \\
& Twit. & \cca{0.9} & \cca{0.9} && \cca{1.0} & \cca{0.9} && \cca{36.3} && \cca{0.9} & \cca{2.4} && \cca{4.1} & \cca{2.8} && \cellcolor{red} \textsf{dnf} \\
& Frien. & \cca{0.3} & \cca{0.3} && \cca{0.3} & \cca{0.2} && \cellcolor{red} \textsf{dnf} && \cca{1.9} & \cca{1.9} && \cca{0.6} & \cca{2.0} && \cellcolor{red} \textsf{dnf} \\
\bottomrule
\end{tabular}
\label{tbl:diff2_wc_all}
\end{table}
\renewcommand{1.1}{1}
\setlength\tabcolsep{0.4pt}
\begin{table}[ht] \small
\centering
\caption{Sketch construction time and index memory of algorithms on different edge models. \textsf{SKIS}{} and \textsf{RIS}{} uses roughly the same time and memory and less than that of \textsf{SKIM}{}.}
\begin{tabular}{cl rrrrrrrr | r rrrrrr}
\toprule
& &\multicolumn{7}{c}{\textbf{Index Time}} && \multicolumn{7}{c}{\textbf{Index Memory}} \\
& &\multicolumn{7}{c}{[second (or \textsf{h} for hour)]} & & \multicolumn{7}{c}{[MB (or \textsf{G} for GB)]} \\
\cmidrule{3-9}\cmidrule{11-17}
& & \multicolumn{2}{c}{\textsf{SKIS}{}} && \multicolumn{2}{c}{\textsf{RIS}{}} && \textsf{SKIM}{} && \multicolumn{2}{c}{\textsf{SKIS}{}} && \multicolumn{2}{c}{\textsf{RIS}{}} && \textsf{SKIM}{} \\
\cmidrule{3-4} \cmidrule{6-7} \cmidrule{9-9} \cmidrule{11-12} \cmidrule{14-15} \cmidrule{17-17}
\textbf{M} & \textbf{Nets} & $h(5)$ & $h(10)$ && $h(5)$ & $h(10)$ && $k(64)$ && $h(5)$ & $h(10)$ && $h(5)$ & $h(10)$ && $k(64)$ \\
\midrule
\multirow{6}{*}{\textsf{WC}}
& PHY & 0 & 1 && 1 & 1 && 2 && 41 & 83 && 52 & 105 && 105 \\
& Epin. & 1 & 1 && 1 & 1 && 10 && 63 & 126 && 81 & 162 && 220 \\
& DBLP & 10 & 18 && 7 & 14 && 37 && 702 & 1G && 848 & 2G && 2G \\
& Orkut & 92 & 157 && 69 & 148 && 0.6h && 2G & 5G && 3G & 5G && 9G \\
& Twit. & 0.6h & 0.9h && 0.4h & 1.0h && 5.2h && 38G & 76G && 42G & 84G && 44G \\
& Frien. & 0.8h & 1.8h && 0.8h & 1.9h && \textsf{dnf} && 59G & 117G && 61G & 117G && \textsf{dnf} \\
\midrule
\multirow{6}{*}{\textsf{TRI}}
& PHY & 0 & 1 && 1 & 2 && 1 && 46 & 90 && 97 & 194 && 99 \\
& Epin. & 1 & 1 && 1 & 1 && 29 && 41 & 82 && 41 & 84 && 230 \\
& DBLP & 11 & 34 && 18 & 36 && 22 && 1G & 2G && 2G & 5G && 2G \\
& Orkut & 88 & 206 && 89 & 197 && \textsf{dnf} && 2G & 4G && 2G & 4G && \textsf{dnf} \\
& Twit. & 0.6h & 1.2h && 0.5h & 1.3h && \textsf{dnf} && 36G & 69G && 36G & 69G && \textsf{dnf} \\
& Frien. & 0.9h & 2.3h && 1.0h & 2.4h && \textsf{dnf} && 54G & 108G && 54G & 108G && \textsf{dnf} \\
\bottomrule
\end{tabular}
\label{tbl:sk_cstr_all}
\vspace{-0.2in}
\end{table}
\begin{figure*}[!ht]
\centering
\subfloat[\textsf{TRI} model]{
\includegraphics[width=0.27\linewidth]{figures/ie/epinions_tri_n.pdf}
}
\subfloat[\textsf{SKIS}{}-\textsf{WC} model]{
\includegraphics[width=0.22\linewidth]{figures/ie/epinions_skinfest_e1_percent.pdf}
}
\subfloat[\textsf{RIS}{}-\textsf{WC} model]{
\includegraphics[width=0.22\linewidth]{figures/ie/epinions_skris_e1_percent.pdf}
}
\subfloat[\textsf{SKIM}{}-\textsf{WC} model]{
\includegraphics[width=0.22\linewidth]{figures/ie/epinions_skim_e1_percent.pdf}
}
\caption{a) Relative difference on Epinions under \textsf{TRI} model and b), c), d) error distributions under \textsf{WC} model with $|S| = 1$. \textsf{SKIS}{} has the lowest relative errors which highly concentrates around 0 while \textsf{RIS}{}'s and \textsf{SKIM}{}'s errors widely spread out.}
\label{fig:epi_err_wc}
\vspace{-0.3in}
\end{figure*}
\begin{figure*}[!ht]
\centering
\subfloat[Epinions]{
\includegraphics[width=0.24\linewidth]{figures/im/e1/epinions_inf_2.pdf}
}
\subfloat[NetPHY]{
\includegraphics[width=0.24\linewidth]{figures/im/e1/phy_inf_2.pdf}
}
\subfloat[DBLP]{
\includegraphics[width=0.24\linewidth]{figures/im/e1/dblp_inf_2.pdf}
}
\subfloat[Twitter]{
\includegraphics[width=0.24\linewidth]{figures/im/e1/twitter_inf_2.pdf}
}
\caption{Efficiency of \textsf{SKIS}{} and \textsf{RIS}{} sketches in finding the maximum seed sets. \textsf{SKIS}{} sketch is up to 80\% more efficient.}
\label{fig:im_inf_wc}
\vspace{-0.25in}
\end{figure*}
\input{body/inf_est}
\setlength\tabcolsep{3pt}
\begin{table*}[!h] \small
\centering
\caption{Performance of \textsf{IM}{} algorithms with $k=100$ (\textsf{dnf}: ``did not finish" within 6h, \textsf{mem}: ``out of memory"). }
\begin{tabular}{cl rrrr r rrrr r rrrr r rrr}
\toprule
& & \multicolumn{4}{c}{\textbf{Running Time [s (or h)]}} && \multicolumn{4}{c}{\textbf{Total Memory [M (or G)]}} && \multicolumn{4}{c}{\textbf{Expected Influence (\%)}} && \multicolumn{3}{c}{\textbf{\#Samples [$\times 10^3$] }} \\
\cmidrule{3-6} \cmidrule{8-11} \cmidrule{13-16} \cmidrule{18-20}
\multirow{2}{0.2in}{\textbf{}}& \multirow{2}{0.25in}{\textbf{Nets}} & \textsf{IMM} & \textsf{PMC}& \textsf{DSSA}{} & \textsf{DSSA}{} && \textsf{IMM} & \textsf{PMC} & \textsf{DSSA}{} & \textsf{DSSA}{} && \textsf{IMM} & \textsf{PMC}& \textsf{DSSA}{} & \textsf{DSSA}{} && \textsf{IMM} & \textsf{DSSA}{} & \textsf{DSSA}{} \\
& && & & +\textsf{SKIS} &&& & & +\textsf{SKIS} &&&& & +\textsf{SKIS}& & & & +\textsf{SKIS} \\
\midrule
\multirow{6}{*}{\textsf{WC}}
& PHY & 0.1 & 3.1 & \textbf{0.0} & \textbf{0.0} && 31 & 86 & 26 & \textbf{9} && 6.64 & 6.7& 5.33 & 5.34 && 103.3 & 8.9 & \textbf{3.8} \\
& Epin. & 0.2 & 10.5 & \textbf{0.0} & \textbf{0.0} && 39 & 130 & 34 & \textbf{17} && 19.4 & 19.8 & 17.9 & 16.6 && 39.8 & 4.48 & \textbf{0.9} \\
& DBLP & 1.1 & 137.4 & \textbf{0.1} & \textbf{0.1} && 162 & \textbf{60} & 136 & 113 && 10.8 &11.2 & 9.3 & 8.5 && 93.0 & 5.4 & \textbf{2.6}\\
& Orkut & 24.1 & 1.4h & 2.6 & \textbf{0.9} && 4G & 6G & \textbf{2G} & \textbf{2G} && 6.7 & 8.7 & 5.7 & 5.1 && 174.4 & 11.52 & \textbf{2.6} \\
& Twit. & 67.3 & \textsf{mem} & 5.5 & 6.3 && 30G & \textsf{mem} & 17G & \textbf{16G} && 25.80 & \textsf{mem} & 24.1 & 21.0 && 54.0 & 18.0 & \textbf{0.8} \\
& Frien. & \textsf{dnf} & \textsf{mem} & 78.3 & \textbf{43.6} && \textsf{dnf} & \textsf{mem} & \textbf{35G} & 36G && \textsf{dnf} & \textsf{mem} & 0.35 & 0.35 && \textsf{mem} & 215.0 & \textbf{102.4}\\
\midrule
\multirow{6}{*}{\textsf{TRI}}
& PHY & 0.2 & 1.5 & \textbf{0.0} & \textbf{0.0} && 50 & 61 & 30 & \textbf{9} && 1.77 & 1.73 & 1.4 & 1.5 && 370.1 & 35.8 & \textbf{3.8} \\
& Epin. & 13.9 & 6.9 & 2.0 & \textbf{0.6} && 483 & 40 & 72 & \textbf{33} && 5.7 & 5.9 & 5.47 & 5.46 && 123.0 & 8.9 & \textbf{0.5} \\
& DBLP & 3.2 & 20.1 & 0.3 & \textbf{0.2} && 389 & \textbf{54} & 191 & 118 && 0.32 & 0.31 & 0.28 & 0.24 && 3171.0 & 348.2 & \textbf{20.5} \\
& Orkut & \textsf{dnf} & 0.3h & 1.3h & \textbf{0.2h} && \textsf{dnf} & 16G & 28G & \textbf{11G} && \textsf{dnf} & 67.3 & 67.9 & 67.8 && \textsf{dnf} & 1.4 & \textbf{0.3} \\
& Twit. & \textsf{dnf} & \textsf{mem} & 5.2h & \textbf{0.6h} && \textsf{dnf} & \textsf{mem} & 100G & \textbf{28G} && \textsf{dnf} & \textsf{mem} & 24.2 & 24.4 && \textsf{dnf} & 3.4 & \textbf{0.4} \\
& Frien. & \textsf{dnf} & \textsf{mem} & \textsf{mem} & \textbf{3.1h} && \textsf{dnf} & \textsf{mem} & \textsf{mem} & \textbf{99G} && \textsf{dnf} & \textsf{mem} & \textsf{mem} & 40.1 && \textsf{dnf} & \textsf{mem} & \textbf{0.2} \\
\bottomrule
\end{tabular}
\label{tab:compare_im}
\vspace{-0.2in}
\end{table*}
\subsection{Influence Maximization}
This subsection illustrates the advantage of \textsf{IIS}{} sketch in finding the seed set with maximum influence. The results show that \textsf{IIS}{} samples drastically speed up the computation time. \textsf{DSSA}{}+\textsf{SKIS}{} is the first to handle billion-scale networks on the challenging \textsf{TRI} edge model. We limit the running time for algorithms to 6 hours and put ``\textsf{dnf}'' if they cannot finish.
\subsubsection{Identifiability of the Maximum Seed Sets}
We compare the ability of the new \textsf{IIS}{} with the traditional \textsf{RIS}{} sampling in terms of identifying the seed set with maximum influence. We fix the number of samples generated to be in the set $\{1000,10000, 100000\}$ and then apply the \textsf{Greedy} algorithm to find solutions. We recompute the influence of returned seed sets using Monte-Carlo method with precision parameters $\epsilon = 0.005, \delta = 1/n$. The results is presented in Figure~\ref{fig:im_inf_wc}.
From Figure~\ref{fig:im_inf_wc}, we observe a recurrent consistency that \textsf{IIS}{} samples return a better solution than \textsf{RIS}{} over all the networks, $k$ values and number of samples. Particularly, the solutions provided by \textsf{IIS}{} achieve up to $80\%$ better than those returned by \textsf{RIS}{}. When more samples are used, the gap gets smaller.
\subsubsection{Efficiency of \textsf{SKIS}{} on \textsf{IM}{} problem}
Table~\ref{tab:compare_im} presents the results of \textsf{DSSA}{}-\textsf{SKIS}, \textsf{DSSA}{}, \textsf{IMM}{} and \textsf{PMC} in terms of running time, memory consumption and samples generated.
\textbf{Running Time.} From Table~\ref{tab:compare_im}, the combination \textsf{DSSA}{}+\textsf{SKIS}{} outperforms the rest by significant margins on all datasets and edge models. \textsf{DSSA}{}-\textsf{SKIS}{} is up to 10x faster than the original \textsf{DSSA}. \textsf{DSSA}{}+\textsf{SKIS}{} is the first and only algorithm that can run on the largest network on \textsf{TRI} model.
\begin{figure}[!ht]
\vspace{-0.2in}
\centering
\subfloat[Epinions]{
\includegraphics[width=0.47\linewidth]{figures/im/tri/epinions_time.pdf}
}
\subfloat[DBLP]{
\includegraphics[width=0.47\linewidth]{figures/im/tri/dblp_time.pdf}
}
\caption{Running time of algorithms under the IC model.}
\label{fig:im_time_mod}
\vspace{-0.35in}
\end{figure}
\begin{figure}[!ht]
\centering
\subfloat[Epinions]{
\includegraphics[width=0.47\linewidth]{figures/im/tri/epinions_time_lt.pdf}
}
\subfloat[DBLP]{
\includegraphics[width=0.47\linewidth]{figures/im/tri/dblp_time_lt.pdf}
}
\caption{Running time of algorithms under the LT model.}
\label{fig:im_time_mod_lt}
\vspace{-0.35in}
\end{figure}
Figure~\ref{fig:im_time_mod} compares the running time of all \textsf{IM}{} algorithms across a wide range of budget $k = 1..20000$ under IC and \textsf{TRI} edge weight model. \textsf{DSSA}{}+\textsf{SKIS}{} always maintains significant performance gaps to the other algorithms, e.g. 10x faster than \textsf{DSSA}{} or 1000x faster than \textsf{IMM}{} and \textsf{PMC}.
\textbf{Number of Samples and Memory Usage.} On the same line with the running time, the memory usage and number of samples generated by \textsf{DSSA}{}+\textsf{SKIS}{} are much less than those required by the other algorithms. The number of samples generated by \textsf{DSSA}{}+\textsf{SKIS}{} is up to more 10x smaller than \textsf{DSSA}{} on \textsf{TRI} model, 100x less than \textsf{IMM}{}. Since the memory for storing the graph is counted into the total memory, the memory saved by \textsf{DSSA}{}+\textsf{SKIS}{} is only several times smaller than those of \textsf{DSSA}{} and \textsf{IMM}{}. \textsf{PMC} exceptionally requires huge memory and is unable to run on two large networks.
\renewcommand{1.1}{0.8}
\textbf{Experiments on the Linear Threshold (LT) model.} We carry another set of experiments on the LT model with multiple budget $k$. Since in LT, the total weights of incoming edge to every node are bounded by 1, for each node, we first normalized the weights of incoming edges and then multiply them with a random number uniformly generated in $[0,1]$.
The results are illustrated in Figure~\ref{fig:im_time_mod_lt}. Similar observations to the IC are seen in the LT model that \textsf{DSSA}+\textsf{SKIS}{} runs faster than the others by orders of magnitude.
Overall, \textsf{DSSA}{}+\textsf{SKIS}{} reveals significant improvements over the state-of-the-art algorithms on influence maximization. As a result, \textsf{DSSA}{}+\textsf{SKIS}{} is the only algorithm that can handle the largest networks under different models.
\section{Importance Sketching}
\label{sec:sketching}
This section introduces our core construction of \textit{Importance Sketching} algorithm to generate random \textit{non-singular} samples with probabilities proportional to those in the original sample space of reverse influence samples and normalized by the probability of generating a non-singular ones.
\begin{algorithm} \small
\caption{Importance Influence Sampling (\textsf{IIS}{}) Alg.}
\label{alg:eris}
\KwIn{Graph $\mathcal{G} = (V,E,w)$}
\KwOut{$R_j$ - A random \textsf{IIS}{} sample}
Pick a node $v \in V$ as the source with probability in Eq.~\ref{eq:probu};\\
Select an in-neighbor $u_i$ of $v$, $u_i \in N^{in}(v)$, with probability of selecting $u_i$ given in Eq.~\ref{eq:probn};\\
Initialize a queue $Q = \{u_i\}$ and a node set $R_j = \{v,u_i\}$;\\
\ForEach{$u_t \in N^{in}(v), t \neq i$}{
With probability $w(u_t, v)$: \\
\qquad $Q.\textsf{push}(u_t); R_j \leftarrow R_j \cup \{u_t\};$
}
\While{$Q$ is not empty}{
$v = Q.\textsf{pop}()$;\tcp{get the longest inserted node}
\ForEach{$u \in N^{in}(v)\backslash (R_j \cup Q)$}{
With probability $w(u,v)$:\\
\qquad $Q.\textsf{push}(u); R_j \leftarrow R_j \cup \{u\}$; \tcp{insert $u$}
}
}
\textbf{return} $R_j$;
\end{algorithm}
\vspace{-0.05in}
\subsection{Importance Influence Sampling (\textsf{IIS}{})}
\textbf{Sample Spaces and Desired Property.} Let $\Omega_{\textsf{RIS}}$ be the sampling space of reverse influence samples (\textsf{RIS}{}) with probability $\mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}}]$ of generating sample $R_j$. Let $\Omega_{\textsf{SKIS}}$ be a subspace of $\Omega_{\textsf{RIS}}$ and corresponds to the space of \textit{only} non-singular reverse influence samples in $\Omega_{\textsf{RIS}}$. Since $\Omega_{\textsf{SKIS}}$ is a subspace of $\Omega_{\textsf{RIS}}$, the probability $\mathbb{Pr}[R_j \in \Omega_{\textsf{SKIS}}]$ of generating a non-singular sample from $\Omega_{\textsf{SKIS}}$ is larger than that from $\Omega_{\textsf{RIS}}$. Specifically, for a node $v \in V$, let $\gamma_v$ be the probability of generating a non-singular sample if $v$ is selected as the source and $\Gamma = \sum_{v \in V} \gamma_v$. Then, since the sample sources are selected randomly, the ratio of generating a non-singular sample to generating any sample in $\Omega_{\textsf{RIS}}$ is $\frac{\Gamma}{n}$ and thus, the probability $\mathbb{Pr}[R_j \in \Omega_{\textsf{SKIS}}]$ is as follows,
\begin{align}
\mathbb{Pr}[R_j \in \Omega_{\textsf{SKIS}}] = \frac{n}{\Gamma}\mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}}].
\end{align}
Our upcoming \textsf{IIS}{} algorithm aims to achieve this desired property of sampling non-singular samples from $\Omega_{\textsf{SKIS}}$.
\textbf{Sampling Algorithm.} Our Importance Influence Sampling (\textsf{IIS}{}) scheme involves three core components:
\begin{itemize}
\item[1)] \textit{Probability of having a non-singular sample.} For a node $v \in V$, a sample with source $v$ is singular if no in-neighbor of $v$ is selected, that happens with probability $\Pi_{u \in N^{in}(v)}(1-w(u,v))$. Hence, the probability of having a non-singular sample from a node $v$ is the complement:
\begin{align}
\label{eq:gamma_v}
\gamma_v = 1-\Pi_{u \in N^{in}(v)}(1-w(u,v)).
\end{align}
\item[2)] \textit{Source Sampling Rate.} Note that the set of non-singular samples is just a subset of all possible samples and we want to generate uniformly random samples from that subset. Moreover, each node $v$ has a probability $\gamma_v$ of generating a non-singular sample from it. Thus, in order to generate a random sample, we select $v$ as the source with probability $\mathbb{Pr}[\textsf{src}(R_j) = v]$ computed as follows,
\begin{align}
\label{eq:probu}
\mathbb{Pr}[\textsf{src}(R_j) = v] = \frac{\gamma_v}{\sum_{u \in V} \gamma_u} = \frac{\gamma_v}{\Gamma},
\end{align}
where $\Gamma = \sum_{u \in V} \gamma_u$, and then generate a uniformly random non-singular sample from the specific source $v$ as described in the next component.
\item[3)] \textit{Sample a non-singular sample from a source.} From the $\textsf{src}(R_j) = v$, we generate a non-singular sample $R_j$ from $v$ uniformly at random. Let $N^{in}(v) = \{ u_1, u_2, \dots, u_l \}$ be a fixed-order set of in-neighbors of $v$. We divide the all possible non-singular samples from $v$ into $l$ buckets: bucket $B_i, 1 \leq i \leq l$ contains those samples that have the first node from $N^{in}(v)$ being $u_i$. That means all the nodes $u_1, \dots, u_{i-1}$ are not in the sample but $u_i$ is in for certain. The other nodes from $u_{i+1}$ to $u_l$ may appear and will be sampled following the normal \textsf{RIS}{} sampling. Now we select the bucket that $R_j$ belongs to with the probability of selecting $B_i$ being as follows,
\begin{align}
\label{eq:probn}
\mathbb{Pr}[\text{select }B_i]=\frac{\prod_{t = 1}^{i-1}(1-w(u_t,v)) w(u_i,v)}{\gamma_v}.
\end{align}
\noindent For $i = 1$, we have $\mathbb{Pr}[\text{select } B_1] = w(u_1,v)$. Note that $\sum_{i = 1}^l \mathbb{Pr}[\text{select } B_i] = 1$. Assume bucket $B_i$ is selected and, thus, node $u_i$ is added as the second node besides the source into $R_j$. For each other node $u_t, t \neq i$, $u_t$ is selected into $R_j$ with probability $w(u_t, v)$ following the ordinary \textsf{RIS}{} for the IC model.
\end{itemize}
These three components guarantee a non-singular sample. The detailed description of \textsf{IIS}{} sampling is in Alg.~\ref{alg:eris}. The first step selects the source of the \textsf{IIS}{} sample among $V$. Then, the first incoming node to the source $v$ is picked (Line~2) following the above description of the component 3). Each of the other incoming neighbors also tries to influence the source (Lines~4-6). The rest performs similarly as in \textsf{RIS}{} \cite{Borgs14}. That is for each newly selected node, its incoming neighbors are randomly added into the sample with probabilities equal to their edge weights. It continues until no newly selected node is observed. Note that Line~3 only adds the selected neighbors $u_i$ of $v$ into $Q$ but adds both $v$ and $u_i$ to $R_j$. The loop from Lines~7-11 mimics the BFS-like sampling procedure of \textsf{RIS}{}.
Let $\mathbb{Pr}[R_j]$ be the probability of generating a non-singular sample $R_j$ using \textsf{IIS}{} algorithm. We have
\begin{align}
\mathbb{Pr}[R_j] & = \sum_{v \in V}\mathbb{Pr}[\textsf{src}(R_j) = v] \mathbb{Pr}[\text{generate } R_j \text{ from }v] \nonumber \\
& = \sum_{v \in V}\frac{\gamma_v}{\Gamma} \frac{\mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}} \text{ and } \textsf{src}(R_j) = v]}{\gamma_v} \nonumber \\
& = \frac{n}{\Gamma} \sum_{v \in V} \frac{1}{n} \mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}} \text{ and } \textsf{src}(R_j) = v] \nonumber \\
& = \frac{n}{\Gamma} \mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}}] = \mathbb{Pr}[R_j \in \Omega_{\textsf{SKIS}}], \nonumber
\end{align}
\noindent where $\mathbb{Pr}[\text{generate } R_j \text{ from }v] = \frac{\mathbb{Pr}[R_j \in \Omega_{\textsf{RIS}} \text{ and } \textsf{src}(R_j) = v]}{\gamma_v}$ due to the selection of the bucket that $R_j$ belongs to in \textsf{IIS}{}. Thus, the output $R_j$ of \textsf{IIS}{} is an random sample from non-singular space $\Omega_{\textsf{SKIS}}$ and we obtain the following lemma.
\begin{Lemma}
Recall that $\Omega_{\textsf{SKIS}{}}$ is the sample space of non-singular reverse influence samples. \textsf{IIS}{} algorithm generates a random non-singular sample from sample space $\Omega_{\textsf{SKIS}{}}$.
\end{Lemma}
\textbf{Connection between IIS Samples and Influences.}
We establish the following key lemma that connects our \textsf{IIS}{} samples with the influence of any seed set $S$.
\begin{Lemma}
Given a random \textsf{IIS}{} sample $R_j$ generated by Alg.~\ref{alg:eris} from the graph $\mathcal{G}=(V,E,w)$, for any set $S \subseteq V$, we have,
\begin{align}
\label{eq:lem6}
\I(S) = \mathbb{Pr}[R_j \cap S \neq \emptyset] \cdot \Gamma + \sum_{v \in S} (1-\gamma_v),
\end{align}
\noindent where $\gamma_v$ and $\Gamma$ are defined in Eqs.~\ref{eq:gamma_v} and \ref{eq:probu}.
\label{lem:multi_set_rois}
\end{Lemma}
The proof is presented in our extended version \cite{extended}. The influence $\I(S)$ of any set $S$ comprises of two parts: 1) $\mathbb{Pr}[R_j \cap S \neq \emptyset] \cdot \Gamma$ depends on the randomness of $R_j$ and 2) the fixed amount $\sum_{v \in S} (1-\gamma_v)$ that is inherent to set $S$ and accounts for the contribution of singular samples in $\Omega_{\textsf{RIS}}$ to the influence $\I(S)$. Lemma~\ref{lem:multi_set_rois} states that instead of computing or estimating the influence $\I(S)$ directly, we can equivalently compute or estimate $\mathbb{Pr}[R_j \cap S \neq \emptyset] \cdot \Gamma + \sum_{v \in S} (1-\gamma_v)$ using \textsf{IIS}{} samples.
\textbf{Remark:} Notice that we can further generate samples of larger sizes and reduce the variance as shown later, however, the computation would increase significantly.
\section{Influence Oracle via IIS Sketch (SKIS)}
\label{sec:oracle}
We use \textsf{IIS}{} sampling to generate a sketch for answering influence estimation queries of different node sets. We show that the random variables associated with our samples have much smaller variances than that of \textsf{RIS}{}, and hence, lead to better concentration or faster estimation with much fewer samples required to achieve the same or better quality.
\textbf{\textsf{SKIS}{}-based Influence Oracle.}
An \textsf{SKIS}{} sketch $\R$ is a collection of \textsf{IIS}{} samples generated by Alg.~\ref{alg:eris}, i.e. $\R = \{R_1, \dots, R_T\}$. As shown in Lemma~\ref{lem:multi_set_rois}, the influence $\I(S)$ can be estimated through estimating the probability $\mathbb{Pr}[R_j \cap S \neq \emptyset]$.
Thus, from a \textsf{SKIS}{} sketch $\R = \{R_1, \dots, R_T\}$, we can obtain an estimate $\hat \I(S)$ of $\I(S)$ for any set $S$ by,
\begin{align}
\label{eq:eq21}
\hat \I_\R(S) = \frac{\mathrm{C}_\R(S)}{|\R|}\cdot \Gamma + \sum_{v \in S} (1-\gamma_v),
\end{align}
where $\mathrm{C}_\R(S)$ is coverage of $S$ on $\R$, i.e.,
\begin{align}
\mathrm{C}_\R(S) = |\{R_j \in \R | R_j \cap S \neq \emptyset\}|.
\end{align}
\begin{algorithm} \small
\caption{\textsf{SKIS}{}-based Influence Oracle}
\label{alg:oracle}
\KwIn{Graph $\mathcal{G} = (V,E,w)$}
Preprocessing: Generate a \textsf{SKIS} sketch $\R = \{R_1, \dots, R_T\}$ of \textsf{IIS}{} samples using Alg.~\ref{alg:eris}.\\
For any influence query for any set $S$: return $\hat \I_\R(S)$ (Eq.~\ref{eq:eq21}).
\end{algorithm}
We build an \textsf{SKIS}{}-based oracle for influence queries by generating a set $\R$ of $T$ \textsf{IIS}{} samples in a preprocessing step and then answer influence estimation query $\hat \I_\R(S)$ for any requested set $S$ (Alg.~\ref{alg:oracle}). In the following, we show the better estimation quality of our sketch through analyzing the variances and estimating concentration properties.
\textbf{\textsf{SKIS}{} Random Variables for Estimations.}
For a random \textsf{IIS}{} sample $R_j$ and a set $S$, we define random variables:
\newcommand{\twopartdef}[3]
{
\left\{
\begin{array}{ll}
#1 & \mbox{if } #2 \\
#3 & \mbox{otherwise.}
\end{array}
\right.
}
\begin{align}
\label{eq:varx}
X_j(S) = \twopartdef {1} {R_j \cap S \neq \emptyset} {0}, \text{and}\\
\label{eq:varzs}
Z_j(S) = \frac{X_j(S) \cdot \Gamma + \sum_{v \in S}(1-\gamma_v)}{n}.
\end{align}
Then, the means of $X_j(S)$ and $Z_j(S)$ are as follows,
\begin{align}
\label{eq:xvar}
&\E[X_j(S)] = \mathbb{Pr}[R_j \cap S \neq \emptyset] = \frac{\I(S)-\sum_{v\in S}(1-\gamma_v)}{\Gamma} \\
\label{eq:zvar}
&\E[Z_j(S)] = \E[X_j(S)] \cdot \frac{\Gamma}{n} + \frac{\sum_{v \in S} (1-\gamma_v)}{n} = \frac{\I(S)}{n}.
\end{align}
Hence, we can construct a corresponding set of random variables $Z_1(S), Z_2(S), \dots, Z_T(S)$ by Eqs.~\ref{eq:varx} and~\ref{eq:varzs}. Then, $\hat \I_\R(S)=\frac{n}{T}\sum_{j=1}^{T}Z_j(S)$ is an empirical estimate of $\I(S)$ based on the \textsf{SKIS}{} sketch $\R$.
For comparison purposes, let $Y_j(S)$ be the random variable associated with \textsf{RIS}{} sample $Q_j$ in a \textsf{RIS}{} sketch $\Q$,
\begin{align}
Y_j(S) = \twopartdef {1} {Q_j \cap S \neq \emptyset} {0}
\end{align}
From Lemma~\ref{lam:borgs14}, the mean value of $Y_j(S)$ is then,
\begin{align}
\E[Y_j(S)] = \frac{\I(S)}{n}.
\end{align}
\textbf{Variance Reduction Analysis. } We show that the variance of $Z_j(S)$ for \textsf{SKIS}{} is much smaller than that of $Y_j(S)$ for \textsf{RIS}{}. The variance of $Z_j(S)$ is stated in the following.
\begin{Lemma}
\label{lem:zvar}
The random variable $Z_j(S)$ (Eq.~\ref{eq:varzs}) has
\begin{align}
\label{eq:eq20}
\mathrm{Var}&[Z_j(S)] = \frac{\I(S)}{n}\frac{\Gamma}{n} - \frac{\I^2(S)}{n^2} \nonumber \\
& - \frac{\sum_{v \in S}(1-\gamma_v)}{n^2} (\Gamma + \sum_{v \in S}(1-\gamma_v) - 2 \I(S)).
\end{align}
\end{Lemma}
Since the random variables $Y_j(S)$ for \textsf{RIS}{} samples are Bernoulli and $\E[Y_j(S)] = \frac{\I(S)}{n}$, we have $\mathrm{Var}[Y_j(S)] = \frac{\I(S)}{n}(1-\frac{\I(S)}{n})$. Compared with $\mathrm{Var}[Z_j(S)]$, we observe that since $\frac{\Gamma}{n} \leq 1$, $\frac{\I(S)}{n}\frac{\Gamma}{n} - \frac{\I^2(S)}{n^2} \leq \frac{\I(S)}{n} - \frac{\I^2(S)}{n^2} = \mathrm{Var}[Y_j(S)]$,
\begin{align}
\mathrm{Var}[Z_j(S)]& \leq \mathrm{Var}[Y_j(S)] \nonumber \\
&- \frac{\sum_{v \in S}(1-\gamma_v)}{n^2} (\Gamma + \sum_{v \in S}(1-\gamma_v) - 2 \I(S)). \nonumber
\end{align}
In practice, most of seed sets have small influences, i.e. $\I(S) \ll \frac{\Gamma}{2}$, thus, $\Gamma + \sum_{v \in S}(1-\gamma_v) - 2 \I(S) \gg 0$. Hence, $\mathrm{Var}[Z_j(S)] < \mathrm{Var}[Y_j(S)]$ holds for most seed sets $S$.
\textbf{Better Concentrations of \textsf{SKIS}{} Random Variables.} Observe that $Z_j(S) \in \Big[\frac{\sum_{v \in S}(1-\gamma_v)}{n}, \frac{\Gamma + \sum_{v \in S}(1-\gamma_v)}{n} \Big]$, we obtain another result on the variance of $Z_j(S)$ as follows.
\begin{Lemma}
\label{lem:var_z}
The variance of random variable $Z_j(S)$ satisfies
\begin{align}
\mathrm{Var}[Z_j(S)] \leq \frac{\I(S)}{n} \frac{\Gamma}{n}.
\end{align}
\end{Lemma}
Using the above result with the general form of Chernoff's bound in Lemma~2 in \cite{Tang15}, we derive the following concentration inequalities for random variables $Z_j(S)$ of \textsf{SKIS}{}.
\begin{Lemma}
\label{lem:bound}
Given a \textsf{SKIS}{} sketch $\R = \{ R_1, \dots, R_T \}$ with random variables $Z_1(S), \dots, Z_T(S)$, we have,
\begin{align}
&\mathbb{Pr}\Big[\frac{\sum_{j = 1}^{T} Z_j(S)}{T}n - \I(S) \geq \epsilon \I(S)\Big] \leq \exp\Big( \frac{-\epsilon^2 T}{2\boldsymbol{\frac{\Gamma}{n}}+\frac{2}{3}\epsilon} \frac{\I(S)}{n} \Big) \nonumber \\
&\mathbb{Pr}\Big[\frac{\sum_{j = 1}^{T} Z_j(S)}{T}n - \I(S) \leq -\epsilon \I(S)\Big] \leq \exp\Big( \frac{-\epsilon^2 T}{2\boldsymbol{\frac{\Gamma}{n}}} \frac{\I(S)}{n}\Big). \nonumber
\end{align}
\end{Lemma}
Compared with the bounds for \textsf{RIS}{} sketch in Corollaries~1 and 2 in \cite{Tang15}, the above concentration bounds for \textsf{SKIS}{} sketch (Lemma~\ref{lem:bound}) are stronger, i.e. tighter. Specifically, we have the factor $\frac{\Gamma}{n}$ in the denominator of the $\exp(.)$ function while for \textsf{RIS}{} random variables, it is simply $1$.
\textbf{Sufficient Size of \textsf{SKIS}{} Sketch for High-quality Estimations.}
There are multiple strategies to determine the number of \textsf{IIS}{} samples generated in the preprocessing step. For example, \cite{Ohsaka16} generates samples until total size of all samples reaches $O(\frac{1}{\epsilon^3}(n+m)\log(n))$. Generating \textsf{IIS}{} samples to reach such a specified threshold is vastly faster than using \textsf{RIS}{} due to the bigger size of \textsf{IIS}{} samples. This method provides an \textit{additive} estimation error guarantee within $\epsilon$. Alternatively, by Lemma~\ref{lem:bound}, we derive the sufficient number of \textsf{IIS}{} samples to provide the more preferable $(\epsilon,\delta)$-estimation of $\I(S)$.
\begin{Lemma}
\label{lem:monte}
Given a set $S$, $\epsilon, \delta \geq 0$, if the \textsf{SKIS}{} sketch $\R$ has at least $(2\frac{\Gamma}{n}+\frac{2}{3}\epsilon) \ln(\frac{2}{\delta}) \frac{n}{\I(S)}\epsilon^{-2}$ \textsf{IIS}{} samples, $\hat \I_\R(S)$ is an $(\epsilon,\delta)$-estimate of $\I(S)$, i.e.,
\begin{align}
\mathbb{Pr}[(1-\epsilon)\I(S) \leq \hat \I_\R(S) \leq (1+\epsilon)\I(S)] \geq 1-\delta.
\end{align}
\end{Lemma}
In practice, $\I(S)$ is unknown in advance and a lower-bound of $\I(S)$, e.g. $|S|$, can be used to compute the necessary number of samples to provide the same guarantee. Compared to \textsf{RIS}{} with weaker concentration bounds, we save a factor of $O(\frac{\Gamma}{n})$.
\subsection{Properties}
\subsection{Outward Influence Properties}
\textbf{Better Influence Discrimination.} As pointed out in the introduction, outward influence is better than influence spread when comparing the relative influence among the nodes.
For example, nodes $a$ and $b$ in Fig.~\ref{fig:example} have influence spreads of $1+p+p^2$ and $1+2\cdot p$, respectively. Thus one can mistakenly think $a$ is as influential as $b$. In fact, $b$ is intuitively much more influential than $a$ since it directly connects to both $a$ and $c$. The outward influences of the two nodes accurately reflect that $b$ is almost twice as influential as $a$, since $\frac{\I_E(b)}{\I_E(a)} \approx 2$ when $p$ approaches $0$.
Thus outward influence can be adopted as a better nodes' importance measure.
\textbf{Non-monotonicity.} Outward influence as a function of seed set $S$ is not monotone. This is different from the influence spread due to the exclusion of self-influence.
Let revisit the example in Figure~\ref{fig:example}. There are 3 nodes and 2 edges with the same weights of $p=0.1$ in the graph. Consider the set having only the node $a$, $\{a\}$ with $\I_E(\{a\}) = 0.11$, if we add $b$ into it to get $\{a,b\}$, the outward influence decreases to $0.1$. However, adding node $c$ instead of $b$, the outward influence increases to $0.19$. Therefore, adding nodes may increase or decrease the outward influence and thus, the function is non-monotone.
\textbf{Submodularity.} Submodularity is a natural property that has been observed and widely applied in many real-world problems, e.g., viral marketing \cite{Kempe03}, sensor placements \cite{Krause08}, diseases outbreak detection \cite{Leskovec07}. Submodularity expresses the diminishing returns behavior of set functions, that adding nodes earlier is always more beneficial than doing that later.
Similar to the traditional influence function, the outward influence is also submodular.
\begin{Lemma}
\label{lem:submod}
Given a network $G = (V,E,w)$, the outward influence function $\I_E(S), S \in 2^{|V|},$ is submodular. For any pair of seed sets $S, T$ such that $S \subseteq T \subseteq V$ and $v \in V\backslash T$,
\begin{align}
\I_E(S \cup \{v\}) - \I_E(S) \geq \I_E(T \cup \{v\}) - \I_E(T).
\end{align}
\end{Lemma}
\begin{proof}
Recall that on a sampled graph $g \sim \mathcal{G}$, for a set $S \subseteq V$, we denote $\eta_g(S)$ to be the set of nodes, excluding the ones in $S$, that are reachable from $S$ through live edges in $g$. Alternatively, $\eta_g(S)$ is called a sampled outward influence of $S$ on sample $g$ and, consequently, we have,
\begin{align}
\I_E(S) = \sum_{g \sim \mathcal{G}} |\eta_g(S)| \mathbb{Pr}[g \sim \mathcal{G}].
\label{eq:compute_ie}
\end{align}
Let consider a sampled graph $g \sim \mathcal{G}$, two sets $S, T$ such that $S \subseteq T \subseteq V$ and a node $v \in V \backslash T$. We have three possible cases:
\begin{itemize}
\item \textbf{Case $v \in \eta_g(S)$}: then $v \in \eta_g(T)$ since $S \subseteq T$ and $v \notin T$. Thus, we have the following,
\begin{align}
\eta_g(S \cup \{v\}) - \eta_g(S) = \eta_g(T \cup \{v\}) - \eta_g(T) = -1. \nonumber
\end{align}
\item \textbf{Case $v \notin \eta_g(S)$ but $v \in \eta_g(T)$}: We have that,
\begin{align}
\eta_g(S \cup \{v\}) - \eta_g(S) = |\eta_g(\{v\}) \backslash (\eta_g(S) \cup S)| \geq 0
\end{align}
while $\eta_g(T \cup \{v\}) - \eta_g(T) = -1$. Thus,
\begin{align}
\eta_g(S \cup \{v\}) - \eta_g(S) > \eta_g(T \cup \{v\}) - \eta_g(T).
\end{align}
\item \textbf{Case $v \notin \eta_g(T)$}: Since $\forall u \in \eta_g(S) \cup S$, either $u \in \eta_g(T)$ or $u \in T$ or $\eta_g(S) \cup S \subseteq \eta_g(T) \cup T$, we have,
\begin{align}
\eta_g(S \cup \{v\}) - \eta_g(S) &= |\eta_g(\{v\}) \backslash (\eta_g(S)\cup S)| \nonumber \\
& \geq |\eta_g(\{v\}) \backslash (\eta_g(T) \cup T)| \nonumber \\
& = \eta_g(T \cup \{v\}) - \eta_g(T).
\end{align}
\end{itemize}
In all three cases, we have,
\begin{align}
\label{eq:sub_g}
\eta_g(S \cup \{v\}) - \eta_g(S) \geq \eta_g(T \cup \{v\}) - \eta_g(T).
\end{align}
Applying Eq.~\ref{eq:sub_g} on all possible $g \sim \mathcal{G}$ and taking the sum over all of these inequalities give
\begin{align}
\sum_{g \sim \mathcal{G}}(\eta_g&(S \cup \{v\}) - \eta_g(S)) \mathbb{Pr}[g \sim \mathcal{G}] \nonumber \\
&\geq \sum_{g \sim \mathcal{G}}(\eta_g(T \cup \{v\}) - \eta_g(T)) \mathbb{Pr}[g \sim \mathcal{G}],
\end{align}
or,
\begin{align}
\I_E(S \cup \{v\}) - \I_E(S) \geq \I_E(T \cup \{v\}) - \I_E(T).
\end{align}
That completes the proof.
\end{proof}
\subsection{Hardness of Computation}
Our first hardness result is that computing the outward influence of any set of nodes is \#P-hard. This is directly due to the difference by a constant $|S|$ between outward influence and the influence spread of the set $S$. \#P-hardness property of computing influence was previously proved in the work of Chen et al. \cite{Chen10} and if there exists an algorithm for computing outward influence in polynomial time, we can just add $|S|$ into the outward influence and obtain a polynomial time influence computation algorithm.
However, differ from influence spread being at least $|S| \geq 1$, the outward influence of any set $S$ can be arbitrarily small. Take an example in Figure~\ref{fig:example}, node $a$ has influence of $\I(\{a\}) = 1+p+p^2 \geq 1$ for any value of $p$. However, $a$'s outward influence $\I_E(\{a\}) = p + p^2$ can be exponentially small as $p$ approaches $0$. This creates a huge challenge in estimating the measure with high accuracy.
In particular, we are desirably interested in having an $(\epsilon,\delta)$-approximation in the form,
\begin{align}
\mathbb{Pr}[ (1 - \epsilon) \I_E(S) \leq \hat \I_E(S) \leq (1 + \epsilon) \I_E(S)] \geq 1-\delta
\end{align}
where $\hat \I_E(S)$ is an estimate of $\I_E(S)$. The smaller the values of $\epsilon,\delta$, the better $\hat \I_E(S)$.
To estimate $\I(S)$, one can use Monte-Carlo method to simulate many influence propagations and take the average results of the outward influences as an estimation. Since the influence of a seed set $S$ is at least $|S|\geq 1$, we only need a polynomial times number of samples to have an $(\epsilon, \delta)$ estimation of $\I(S)$ \cite{Dagum00}. In contrast, the number of samples necessary to obtain an ($\epsilon$, $\delta$)-approximation for $\I_E(S)$ is at least $c\frac{1}{\epsilon^{2}}\log{\frac{1}{\delta}}\frac{n}{\I_E(S)}$, where $c$ is some constant. As $\I_E(S)$ can be exponentially small, the number of samples for estimating $\I_E(S)$ can be exponentially large. Note that we cannot obtain an $(\epsilon,\delta)$-approximation of $\I_E(S)$ through $\I(S)$ since the additive $|S|$ difference does not preserve the multiplicative $1\pm\epsilon$. Even worse, the relationship between their approximation guarantees is unknown.
For the same reason, the recent advances in influence estimation in \cite{Borgs14, Lucier15} cannot be adapted to obtain polynomial-time $(\epsilon, \delta)$ estimation for outward influence. We shall address this challenging task in the next section.
\subsection{Influence Estimation}
We show that \textsf{SKIS}{} sketch consumes much less time and memory space while consistently obtaining better solution quality, i.e. very small errors, than both \textsf{RIS}{} and \textsf{SKIM}{}.
\subsubsection{Solution Quality}
Table~\ref{tbl:diff2_wc_all} and Figure~\ref{fig:epi_err_wc} present the relative estimation errors of all three sketches.
The solution quality of \textsf{SKIS}{} is consistently better than \textsf{RIS}{} and \textsf{SKIM}{} across all the networks and edge models. As shown in Table~\ref{tbl:diff2_wc_all}, the errors of \textsf{SKIS}{} are 110\% and 400\% smaller than those of \textsf{RIS}{} with $k = 1$ while being as good as or better than \textsf{RIS}{} for $k = 100, 1000$. On the other hand, \textsf{SKIM}{} shows the largest estimation errors in most of the cases. Particularly, \textsf{SKIM}{}'s error is more than 60 times higher than \textsf{SKIS}{} and \textsf{RIS}{} on Twitter when $|S| = 100$. Similar results are observed under \textsf{TRI} model. Exceptionally, on Twitter and Friendster, the relative difference of \textsf{RIS}{} is slightly smaller than \textsf{SKIS}{} with $h=5$ but larger on $h=10$. In \textsf{TRI} model, estimating a random seed on large network as Twitter produces higher errors since we have insufficient number of samples.
Figures~\ref{fig:epi_err_wc}b, c, and d draw the error distributions of sketches for estimating the influences of random seeds. Here, we generate 1000 uniformly random nodes and consider each node to be a seed set. We observe that \textsf{SKIS}{}'s errors are highly concentrated around 0\% even when the influences are small while errors of \textsf{RIS}{} and \textsf{SKIM}{} spread out widely. \textsf{RIS}{} reveals extremely high errors for small influence estimation, e.g. up to $80\%$. The error distribution of \textsf{SKIM}{} is the most widely behaved, i.e. having high errors at every influence level. Under \textsf{TRI} model (Figure~\ref{fig:epi_err_wc}a), \textsf{SKIS}{} also consistently provides significantly smaller estimation errors than \textsf{RIS}{} and \textsf{SKIM}{}.
\subsubsection{Performance}
We report indexing time and memory of different sketches in Table~\ref{tbl:sk_cstr_all}.
\textbf{Indexing Time.} \textsf{SKIS}{} and \textsf{RIS}{} use roughly the same amount of time for build the sketches while \textsf{SKIM}{} is much slower than \textsf{SKIS}{} and \textsf{RIS}{} and failed to process large networks in both edge models. On larger networks, \textsf{SKIS}{} is slightly faster than \textsf{RIS}{}. \textsf{SKIM}{} markedly spends up to 5 hours to build sketch for Twitter on \textsf{WC} model while \textsf{SKIS}{}, or \textsf{RIS}{} spends only 1 hour or less on this network.
\textbf{Index Memory.} In terms of memory, the same observations are seen as with indexing time essentially because larger sketches require more time to construct. In all the experiments, \textsf{SKIS}{} consumes the same or less amount of memory with \textsf{RIS}{}. \textsf{SKIM}{} generally uses more memory than \textsf{SKIS}{} and \textsf{RIS}{}.
In summary, \textsf{SKIS}{} consistently achieves better solution quality than both \textsf{RIS}{} and \textsf{SKIM}{} on all the networks, edge models and seed set sizes while consuming the same or less time/memory. The errors of \textsf{SKIS}{} is highly concentrated around 0. In contrast, \textsf{RIS}{} is only good for estimating high influence while incurring significant errors for small ranges.
\section{SKIS-based IM Algorithms}
\label{sec:max}
With the help of \textsf{SKIS}{} sketch that is better in estimating the influences compared to the well-known successful \textsf{RIS}{}, we can largely improve the efficiency of \textsf{IM}{} algorithms in the broad class of \textsf{RIS}{}-based methods, i.e. RIS \cite{Borgs14}, TIM/TIM+ \cite{Tang14}, IMM \cite{Tang15}, BCT \cite{Nguyen162,Nguyen172}, SSA/DSSA \cite{Nguyen163}. This improvement is possible since these methods heavily rely on the concentration of influence estimations provided by \textsf{RIS}{} samples.
\textbf{\textsf{SKIS}{}-based framework.}
Let $\R = \{R_1, R_2, \dots \}$ be a \textsf{SKIS}{} sketch of \textsf{IIS}{} samples. $\R$ gives an influence estimate
\begin{align}
\label{eq:compute_inf}
\hat \I_\R(S) = \hat \E_\R[Z_j(S)] \cdot n = \frac{\mathrm{C}_\R(S)}{|\R|} \cdot \Gamma + \sum_{v \in S}(1-\gamma_v),
\end{align}
\noindent for any set $S$. Thus, instead of optimizing over the exact influence, we can intuitively find the set $S$ to maximize the estimate function $\hat \I(S)$. Then, the framework of using \textsf{SKIS}{} sketch to solve \textsf{IM}{} problem contains two main steps:
\begin{itemize}
\item[1)] Generate a \textsf{SKIS}{} sketch $\R$ of \textsf{IIS}{} samples,
\item[2)] Find the set $S_k$ that maximizes the function $\hat \I_R(S)$ and returning $S_k$ as the solution for the \textsf{IM}{} instance.
\end{itemize}
There are two essential questions related to the above \textsf{SKIS}{}-based framework
: 1) Given a \textsf{SKIS}{} sketch $\R$ of \textsf{IIS}{} samples, how to find $S_k$ of $k$ nodes that maximizes $\hat \I_\R(S_k)$ (in Step 2)? 2) How many \textsf{IIS}{} samples in the \textsf{SKIS}{} sketch $\R$ (in Step 1) are sufficient to guarantee a high-quality
solution for \textsf{IM}?
We give the answers for the above questions in the following sections. Firstly, we adapt the gold-standard greedy algorithm to obtain an $(1-(1-1/k)^k)$-approximate solution over a \textsf{SKIS}{} sketch. Secondly, we adopt recent techniques on \textsf{RIS}{} with strong solution guarantees to \textsf{SKIS}{} sketch.
\begin{algorithm} \small
\caption{\textsf{Greedy} Algorithm on \textsf{SKIS}{} sketch}
\label{alg:cov}
\KwIn{\textsf{SKIS}{} sketch $\R$ and $k$}
\KwOut{An $(1-(1-1/k)^k)$-approximate seed set $\hat S_k$}
$\hat S_k = \emptyset$\\
\For{$i=1:k$}{
$\hat v \leftarrow \arg \max_{v\in V\backslash \hat S_k}\big (\frac{\Delta_\R(v, \hat S_k)}{|\R|}\Gamma + (1-\gamma_v)\big )$\\
Add $\hat v$ to $\hat S_k$\\
}
\textbf{return} $\hat S_k$
\end{algorithm}
\subsection{Greedy Algorithm on \textsf{SKIS}{} Sketches}
Let consider the optimization problem of finding a set $S_k$ of at most $k$ nodes to maximize the function $\hat \I_\R(S)$ on a \textsf{SKIS}{} sketch $\R$ of \textsf{IIS}{} samples under the cardinality constraint $|S| \leq k$. The function $\hat \I_\R(S)$ is monotone and submodular since it is the weighted sum of a set coverage function $\mathrm{C}_\R(S)$ and a linear term $\sum_{v \in S} (1-\gamma_v)$. Thus, we obtain the following lemma with the detailed proof in our extended version \cite{extended}.
\begin{Lemma}
Given a set of \textsf{IIS}{} samples $\R$, the set function $\hat \I_\R(S)$ defined in Eq.~\ref{eq:compute_inf} is monotone and submodular.
\label{lem:sub}
\end{Lemma}
Thus, a standard greedy scheme \cite{Nemhauser81}, which iteratively selects a node with highest marginal gain, gives an $(1-(1-\frac{1}{k})^k)$, that converges to $(1-1/e)$ asymptotically, approximate solution $\hat S_k$. The marginal gain of a node $v$ with respect to a set $S$ on \textsf{SKIS}{} sketch $\R$ is defined as follows,
\begin{align}
\label{eq:margin}
\textsf{gain}_\R(v,S) = \frac{\Delta_\R(v,\hat S_k)}{|\R|}\Gamma + (1-\gamma_v),
\end{align}
where $\Delta_\R(v,S) = \mathrm{C}_\R(S \cup \{v\}) - \mathrm{C}_\R(S)$ is called the marginal coverage gain of $v$ w.r.t. $S$ on \textsf{SKIS}{} sketch $\R$.
Given a collection of \textsf{IIS}{} samples $\R$ and a budget $k$, the Greedy algorithm is presented in Alg.~\ref{alg:cov} with a main loop (Lines~2-4) of $k$ iterations. Each iteration picks a node $\hat v$ having largest marginal gain (Eq.~\ref{eq:margin}) with respect to the current partial solution $\hat S_k$ and adds it to $\hat S_k$.
The approximation guarantee of the \textsf{Greedy} algorithm (Alg.~\ref{alg:cov}) is stated below.
\begin{Lemma}
\label{lem:greedy}
The \textsf{Greedy} algorithm (Alg.~\ref{alg:cov}) returns an $(1-(1-\frac{1}{k})^k)$-approximate solution $\hat S_k$,
\begin{align}
\hat \I_\R(\hat S_k) \geq (1-(1-\frac{1}{k})^k) \hat \I_\R(S^*_\R),
\end{align}
where $S_\R^*$ is the optimal cover set of size $k$ on sketch $\R$.
\end{Lemma}
The lemma is derived directly from the $1-(1-\frac{1}{k})^k$ approximation factor of the ordinary greedy algorithm \cite{Nemhauser81}.
\subsection{Sufficient Size of \textsf{SKIS}{} Sketch for \textsf{IM}{}}
Since the \textsf{SKIS}{} sketch offers a similar greedy algorithm with approximation ratio $(1-(1-1/k)^k)$ to the traditional \textsf{RIS}{}, we can combine \textsf{SKIS}{} sketch with any \textsf{RIS}{}-based algorithm, e.g. RIS\cite{Borgs14}, TIM/TIM+\cite{Tang14}, IMM\cite{Tang15}, BCT\cite{Nguyen172}, SSA/DSSA\cite{Nguyen163}. We discuss the adoptions of two most recent and scalable algorithms, i.e. IMM\cite{Tang15} and SSA/DSSA\cite{Nguyen163}.
\textbf{\textsf{IMM}+\textsf{SKIS}{}.} Tang et al. \cite{Tang15} provide a theoretical threshold
\begin{align}
\theta_{\textsf{RIS}} = O\Big((\log{n \choose k} + \log \delta^{-1})\frac{n}{\textsf{OPT}_k}\epsilon^{-2}\Big)
\end{align}
on the number of \textsf{RIS}{} samples to guarantee an $(1-1/e-\epsilon)$-approximate solution for \textsf{IM}{} problem with probability $1-\delta$.
Replacing \textsf{RIS}{} with \textsf{IIS}{} samples to build a \textsf{SKIS}{} sketch enables us to use the better bounds in Lemma~\ref{lem:bound}. By the approach of \textsf{IMM}{} in \cite{Tang15} with Lemma~\ref{lem:bound}, we reduce the threshold of samples to provide the same quality to,
\begin{align}
\theta_{\textsf{SKIS}{}} = O\Big(\frac{\Gamma+k}{n} \theta_{\textsf{RIS}}\Big).
\end{align}
\textbf{\textsf{SSA}/\textsf{DSSA}{}+\textsf{SKIS}{}.} More recently, Nguyen et al. \cite{Nguyen163} propose \textsf{SSA}{} and \textsf{DSSA}{} algorithms which implement the Stop-and-Stare strategy of alternating between finding candidate solutions and checking the quality of those candidates at exponential points, i.e. $2^{t}, t \geq 1$, to detect a satisfactory solution at the earliest time.
Combining \textsf{SKIS}{} with \textsf{SSA}{} or \textsf{DSSA}{} brings about multiple benefits in the checking step of \textsf{SSA}{}/\textsf{DSSA}{}. The benefits stem from the better concentration bounds which lead to better error estimations and smaller thresholds to terminate the algorithms. The details are in our extended version \cite{extended}.
\section{Extensions to other diffusion models}
\label{sec:ext}
The key step in extending our techniques for other diffusion models is devising an importance sketching procedure for each model. Fortunately, following the same designing principle as \textsf{IIS}{}, we can devise importance sketching procedures for many other diffusion models. We demonstrate this through two other equally important and widely adopted diffusion models, i.e. Linear Threshold \cite{Kempe03} and Continuous-time model \cite{Du13}.
\textbf{Linear Threshold model \cite{Kempe03}.} This model imposes a constraint that the total weights of incoming edges into any node $v \in V$ is at most 1, i.e. $\sum_{u \in N^{in}(v)} w(u,v) \leq 1$. Every node has a random activation threshold $\lambda_v \in [0,1]$ and gets activated if the total edge weights from active in-neighbors exceeds $\lambda_v$, i.e. $\sum_{u \in N^{in}(v), u \text{ is active}} w(u,v) \geq \lambda_v$. A \textsf{RIS}{} sampling for LT model \cite{Nguyen172} selects a random node as the source and iteratively picks at most one in-neighbor of the last activated node with probability being the edge weights, $w(u,v)$.
The importance sketching algorithm for the LT model has the following components:
\begin{itemize}
\item \textit{Probability of having a non-singular sample}:
\begin{align}
\gamma_v = \sum_{u \in N^{in}(v)} w(u,v)
\end{align}
\item \textit{Source Sampling Rate}:
\begin{align}
\mathbb{Pr}[\textsf{src}(R_j) = v] = \frac{\gamma_v}{\sum_{v \in V} \gamma_v}
\end{align}
\item \textit{Sample a non-singular sample from a source.}: select exactly one in-neighbor $u$ of $\textsf{src}(R_j)=v$ with probability $\frac{w(u,v)}{\gamma_v}$. The rest follows \textsf{RIS}{} sampling \cite{Nguyen162}.
\end{itemize}
\textbf{Continuous-time model \cite{Du13}.}
Here we have a deadline parameter $T$ of the latest activation time and each edge $(u,v)$ is associated with a length distribution, represented by a density function $\mathcal{L}_{(u,v)}(t)$, of how long it takes $u$ to influence $v$. A node $u$ is influenced if the length of the shortest path from any active node at time $0$ is at most $T$. The \textsf{RIS}{} sampling for the Continuous-time model \cite{Tang15} picks a random node as the source and invokes the Dijkstra's algorithm to select nodes into $\textsf{src}(R_j)$. When the edge $(u,v)$ is first visited, the activation time is sampled following its length distribution $\mathcal{L}_{(u,v)}(t)$.
From the length distribution, we can compute the probability $p(u,v,T)$ of an edge $(u,v)$ having activation time at most $T$
\begin{align}
p(u,v,T) = \int_{t=0}^{T} \mathcal{L}_{(u,v)}(t) dt
\end{align}
The importance sketching procedure for the Continuous-time model has the following components:
\begin{itemize}
\item \textit{Probability of having a non-singular sample}:
\begin{align}
\gamma_v = 1 - \prod_{u \in N^{in}(v)} (1-p(u,v,T))
\end{align}
\item \textit{Source Sampling Rate}:
\begin{align}
\mathbb{Pr}[\textsf{src}(R_j) = v] = \frac{\gamma_v}{\sum_{v \in V} \gamma_v}
\end{align}
\item \textit{Sample a non-singular sample from a source.}: Use a bucket system on $p(u,v,T)$ similarly to \textsf{IIS}{} to select the first in-neighbor $u$. The activation time of $u$ follows the normalized density function $\frac{\mathcal{L}_{(u,v)}(t)}{\gamma_v}$. Subsequently, it continues by following \textsf{RIS}{} sampling \cite{Tang15}.
\end{itemize}
\section{Introduction}
\label{sec:intro}
Online social networks (OSNs) such as Facebook and Twiter have connected billions of users, providing gigantic communication platforms for exchanging and disseminating information. For example, Facebook now has nearly 2 billions monthly active users and more than 2.5 billion pieces of content exchanged daily. Through OSNs, companies and participants have actively capitalized on the ``word-of-mouth'' effect to trigger viral spread of various kinds of information, including marketing messages, propaganda, and even fake news. In the past decade, a great amount of research has focused on analyzing how information and users' influence propagate within the network, e.g., evaluating influence of a group of individuals, aka \emph{influence estimation} \cite{Lucier15,Cohen14,Ohsaka16}, and finding a small group of influential individuals, aka \emph{influence maximization} \cite{Kempe03,Cohen14,Borgs14, Tang14,Tang15,Nguyen163}, or controlling diffusion processes via structure manipulation \cite{Tong12}.
Yet, diffusion analysis is challenging due to the sheer size of the networks. For example, state-of-the-art solutions for influence maximization, e.g., DSSA \cite{Nguyen163}, IMM \cite{Tang15}, TIM/TIM+\cite{Tang14}, cannot complete in the networks with only few million edges \cite{Arora17}. Further, our comprehensive experiments on influence estimation show that the average estimation error can be as high as 40-70\% for popular sketches such as \textsf{RIS}{} and \textsf{SKIM}{}. This happens even on a small network with only 75K nodes (Epinion). This calls for development of new approaches for analyzing influence dynamics in large-scale networks.
In this paper, we propose a new importance sketching technique, termed \textsf{SKIS}{}, that consists of \emph{non-singular} reverse influence cascades, or simply non-singular cascades. Each non-singular cascade simulates the reverse diffusion process from a source node. It is important that each non-singular cascade must include at least another node other than the source itself. Thus, our sketch, specifically, suppresses \emph{singular} cascades that die prematurely at the source. Those singular cascades, consisting of 30\%-80\% portion in the previous sketches \cite{Cohen14, Borgs14}, not only waste the memory space and processing time but also reduce estimation efficiency of the sketches. Consequently, \textsf{SKIS}{} contains samples of smaller variances providing estimations of high concentration with less memory and running time. Our new sketch also powers a new principle and scalable influence maximization class of methods, that inherits the algorithmic designs of existing algorithms on top of \textsf{SKIS}{} sketch. Particularly, \textsf{SKIS}{}-based \textsf{IM}{} methods are the only provably good and efficient enough that can scale to networks of billions of edges across different settings. We summarize of our contributions as follows:
\begin{itemize}[noitemsep,nolistsep]
\item At the central of our sketch is an importance sampling algorithm to sample non-singular cascades (Alg.~\ref{alg:eris}). For simplicity, we first present the sketch and its sampling algorithm using the popular \emph{independent cascade} model \cite{Kempe03}, and later extend them to other diffusion models.
\item We provide general frameworks to apply \textsf{SKIS}{} for existing algorithms for the influence estimation and influence maximization problems. We provide theoretical analysis to show that using \textsf{SKIS}{} leads to improved influence estimation oracle due to smaller sample variances and better concentration bounds; and that the state-of-the-art methods for influence maximization like D-SSA \cite{Nguyen163}, IMM \cite{Tang15}, and, TIM/TIM+\cite{Tang14} can also immediately benefit from our new sketch.
\item We conduct comprehensive empirical experiments to demonstrate the effectiveness of our sketch in terms of quality, memory and computational time. Using SKIS, we can design high-quality influence oracle for seed set with average estimation error up to 10x times smaller than those using \textsf{RIS}{} and 6x times those using \textsf{SKIM}{}. In addition, our influence maximization using \textsf{SKIS}{} substantially improves the quality of solutions for greedy algorithms. It achieves up to 10x times speed-up and 4x memory reduction for the fastest \textsf{RIS}-based DSSA algorithm, while maintaining the same theoretical guarantees.
\end{itemize}
\textbf{Related work.}
Sketching methods have become extremely useful for dealing with problems in massive sizes. Most notable sketches including bottom-$k$ sketches \cite{Cohen07} a summary of a set of items with nonnegative weights, count-min sketch \cite{Cormode05count} that count the frequency of different events in the stream.
Recently, Cohen et al. \cite{Cohen14} investigate the combined reachability sketch which is a bottom-$k$ min-hash sketch of the set of reachable nodes. They show small estimation errors for estimating influences. However, this method deteriorates for large influences since the size of each sketch is fixed. Similar scheme was applied for continuous-time model \cite{Du13}.
Borgs et al. \cite{Borgs14} proposed reverse influence sketch (\textsf{RIS}) which captures the influences in a reverse manner. This approach has advantage in estimating large influences and becomes very successful in finding the seed set with maximum influence, i.e. influence maximization. \cite{Ohsaka16} uses \textsf{RIS}{} sketches to estimate influences in dynamic graphs. Other related works on influences in multiplex networks and identifying the sources of influence cascades are studied in \cite{Nguyen13_l,Shen12,Nguyen164}.
\textbf{Organization.} The rest of the paper is organized as follows: In Section~\ref{sec:model}, we introduce the diffusion model and two problems of influence estimation/maximization. We propose our importance sketching scheme in Section~\ref{sec:sketching}. Applications in influence estimation/maximization is presented in Sections~\ref{sec:oracle} and \ref{sec:max}, respectively. Extensions to other diffusion models are discussed in Section~\ref{sec:ext} which is followed by experiments in Section~\ref{sec:exps} and conclusion in Section~\ref{sec:con}.
\subsection{ISSA Approximation Algorithm}
We combine the new \textsf{IIS}{} sketch and runtime approximation bound with the recent Stop-and-Stare framework \cite{Nguyen163} to propose \textsf{ISSA}{}. The combined method can provide a runtime approximation guarantee beyond $1-1/e$.
\subsubsection{Algorithm Description}
The detailed description of \textsf{ISSA}{} is presented in Alg.~\ref{alg:dpima}. In addition to the graph $\mathcal{G}$, number of selected seeds $k$, the algorithm takes in a value $\rho$ which is the desired approximation ratio that a user would like to obtain. Traditionally, $\rho = 1-1/e-\epsilon$ for some $\epsilon$, however, the runtime approximation bound gives \textsf{ISSA}{} the opportunities to provide better guarantees on a particular execution of \textsf{ISSA}{}.
Initially, it computes the value of $\epsilon$ (Line~1) which is the gap from the desired $\rho$ to $1-1/e$ and $\epsilon_0$. $\epsilon$ is used to set the maximum necessary \textsf{IIS}{} samples $N_{\max}$ (Line~2) and $\epsilon_0$ decides the initial \textsf{IIS}{} samples generated $\Lambda$ (Line~4). $t_{\max}$ is the maximum number of iterations in the main loop (Line~5-17) and computed by the logarithm based 2 of $\frac{N_{\max}}{\Lambda}$. The precomputed $\Lambda_1$ (Line~4) serves in the first stopping condition (Line~10) which provides a certain bound on the solution quality. The main body of \textsf{ISSA}{} is a loop in which two sets of \textsf{IIS}{} samples are used: $\R_t$ with $\Lambda \cdot 2^{t-1}$ \textsf{IIS}{} (Line~7) for finding the candidate seed set $\hat S_k$ by the \textsf{Greedy} with runtime approximation bound (Line~9) and $\R_t^{(c)}$ with the same number of samples for reestimating the influence of $\hat S_k$.
The error bound in iteration $t$ is calculated in Lines~11-14 and if it meets the required approximation ratio (Lines~15-16), then \textsf{ISSA}{} terminates and returns $\hat S_k$. Checking the approximation guarantee is only one of three stopping conditions. Another condition is taken from \cite{Tang15} which bounds the maximum number of samples needed to provide the bound $\rho$ in cases where $\rho \leq 1-1/e-\epsilon$ for some $0 < \epsilon < 1$. This plays as a guarding condition to stop the algorithm when $\rho > 1-1/e$ and it is not possible to achieve that desired approximation guarantee. In that case, \textsf{ISSA}{} returns a solution $\hat S_k$ with guaranteed ratio of $\max\{1-1/e-|\epsilon|,\rho_t\}$.
\subsubsection{Optimality Guarantees Analysis}
We prove that when $\rho < 1-1/e$, \textsf{ISSA}{} returns a seed set $\hat S_k$ of size $k$ that is at least $\rho$-optimal with probability at least $1-\delta$. Furthermore, in that case, \textsf{ISSA}{} requires at most the type-2 optimal number of \textsf{IIS}{} samples. The type-2 \cite{Nguyen163} is defined as the minimum samples to guarantee an $(1-1/e-\epsilon)$-optimal solution where $\epsilon = (1-1/e)-\rho$ with probability at least $1-2\delta$.
In the other cases of $\rho \geq 1-1/e$, \textsf{ISSA}{} has a chance return a solution $\hat S_k$ meeting the desired $\rho$ when the stopping condition $\rho_t \geq \rho$ is met. That depends mostly on the runtime approximation bound. In cases where $\rho_t \geq \rho$ can not be satisfied, the cap $N_{max}$ on the number of \textsf{IIS}{} samples will terminate the algorithm and provides the best-effort guarantee of $\max\{ \rho_t, 1-1/e-|\epsilon| \}$.
\textbf{Approximation Guarantee.}
For clarity, we present most of the proofs in the appendix.
Recall that \textsf{ISSA}{} stops when either 1) the number of samples exceeds the cap, i.e., $|\R_t| \geq N_{\max}$ or 2) $\rho_t \geq \rho$ for some $t \geq 1$. In the first case, $N_{\max}$ were chosen to guarantee that
$\hat S_k$ will be a $(1-1/e-\epsilon)$-approximation solution w.h.p.
\begin{Lemma}
\label{lem:cap}
Let $B^{(1)}$ be the bad event that
\[
B^{(1)} = (|\R_t| \geq N_{\max}) \cap (\I(\hat S_k) < (1-1/e-\epsilon)\emph{\textsf{OPT}}_k).
\] where $\epsilon = (1-1/e) - \rho > 0$ since we are considering $\rho < 1-1/e$. We have
\[
\mathbb{Pr}[B^{(1)}] \leq \delta/3.
\]
\end{Lemma}
This results is followed directly from \cite{Tang15} replacing $\delta$ with $\delta/3$. In the second case, the algorithm stops when $\epsilon_t \leq \epsilon$ for some
$1\leq t \leq t_{\max}$. The maximum number of iterations $t_{\max}$ is bounded by $O(\log n)$ as stated below.
\begin{Lemma}
\label{lem:tmax}
The number of iterations in \textsf{ISSA}{} is at most $t_{\max} = O(\log n)$.
\end{Lemma}
For each iteration $t$, we will bound the probabilities of the bad events that lead to inaccurate estimations of $\I(\hat S_k)$ through $\R^{c}_t$, and $\I(S^*_k)$ through $\R_t$(Lines~12 and~13).
\begin{Lemma}
\label{lem:bad2}
For each $1\leq t \leq t_{\max}$, let
\[
\hat \epsilon_t \text{ be the unique root of } f(x)=\frac{\delta}{3t_{\max}},
\]
\noindent where $f(x)=\exp{\left(-\frac{N_t \frac{\I(\hat S_k)}{n} x^2 }{2+2/3x}\right)}$, and
\[
\epsilon_t^*=
\epsilon_0 \sqrt{\frac{n}{(1+\epsilon_0/3)2^{t-1} \emph{\textsf{OPT}}_k}}.
\]
\noindent Consider the following bad events
\begin{align*}
B_t^{(2)} &= \left(
\hat \I_t^{(c)}(\hat S_k) \geq (1+\hat \epsilon_t)\I(\hat S_k) \right),
\\
B_t^{(3)} &= \left(
\hat \I_t(S^*_k) \leq (1-\epsilon_t^*) \emph{\textsf{OPT}}_k]
\right).
\end{align*}
\noindent We have
\[
\mathbb{Pr}[B_t^{(2)}], \mathbb{Pr}[B_t^{(3)}] \leq \frac{\delta}{3t_{\max}}.
\]
\end{Lemma}
\begin{Lemma}
\label{lem:e2e}
Assume that none of the bad events $B^{(1)}$, $B^{(2)}_t$, $B^{(3)}_t$ ($t =1..t_{\max}$) happen and \textsf{ISSA}{} stops with some $\epsilon_t \leq \epsilon_0$. We have
\begin{align}
&\hat \epsilon_t < \epsilon \text{ and consequently }\\
&{\hat \I}^{(c)}_t(\hat S_k) \leq (1+\epsilon_0)\I(\hat S_k)
\end{align}
\end{Lemma}
We now obtain the approximation guarantee of \textsf{ISSA}{}.
\begin{theorem}
\label{theo:approx}
Given a graph $G$, $0 < \delta < 1$, and $0 < \rho < \smash{1-1/e}$, \textsf{ISSA}{} returns an $\rho$-approximate solution with probability at least $(1-\delta)$.
\end{theorem}
\subsection{Anytime Approx. Algorithm | Beyond $(1-1/e)$}
In many cases of the input graph $\mathcal{G}$, $k$, \textsf{ISSA}{} can provide a better approximation guarantee $\rho \geq 1-1/e$ due to the runtime approximation bound. This happens when the first stopping condition $\rho_t \geq \rho$ is met. Conversely, in other cases of $\rho \geq 1-1/e$ and within $N_{\\max}$ \textsf{IIS}{} samples, \textsf{ISSA}{} cannot provide the guarantee, the second stopping condition takes effect as a guarding terminator. In those cases, the returned solution $\hat S_k$ is at least $\max\{(1-1/e-|\epsilon|), \rho_t \}$-optimal.
Moreover, due to the dynamic guarantee $\rho_t$ at every iteration, \textsf{ISSA}{} provides an \textit{anytime} approximation guarantee. \textsf{ISSA}{} can be stopped at anytime (any iteration $t$) in the middle and return the current candidate solution $\hat S_k$ with the provided guarantee of $\rho_t$. In another way, we can set the time limit for \textsf{ISSA}{} to run and it will return the best-effort solution $\hat S_k$ with a approximation guarantee of $\rho_t$ when the time limit has been used up at iteration $t$.
\section{Preliminaries}
\label{sec:model}
Consider a social network abstracted as a graph $\mathcal{G} = (V, E, w)$. Each edge $(u,v) \in E$ is associated with a real number $w(u,v) \in [0, 1]$ specifying the probability that node $u$ will influence $v$ once $u$ is influenced.
To model the influence dynamic in the network, we first focus on the popular \textit{Independent Cascade} (IC) model \cite{Kempe03} and then, discuss the extensions of our techniques to other models, e.g. Linear Threshold (LT) or Continuous-time model, later in Section~\ref{sec:ext}.
\subsection{Independent Cascade Model}
For a subset of nodes $S \subseteq V$, called seed set, the influence propagation from $S$ happens in discrete rounds $t=0, 1,...$ At round $0$, only nodes in $S$ are active (aka influenced) and the others are inactive. Each newly activated node $u$ at round $t$ will have a single chance to activate each neighbor $v$ of $u$ with probability $w(u, v)$. An activated node remains active till the end of the diffusion propagation. The process stops when no more nodes get activated.
{\bf Sample Graphs.} Once a node $u$ gets activated, it will activate each of its neighbor $v$ with probability $w(u,v)$. This can be thought of as flipping a biased coin that gives head with probability $w(u, v)$ to determine whether the edge $(u, v)$ exists. If the coin lands head for the edge $(u,v)$, the activation occurs and we call $(u,v)$ a \textit{live-edge}. Since all the influences in the IC model are independent, it does not matter when coins are flipped to determine the states of the edges. Thus, we can flip all the coins at the beginning instead of waiting until $u$ gets activated. We call the deterministic graph $g$ that contains all the live-edges resulted from a series of coin flips over all the edges in $\mathcal{G}$ a \textit{sample graph} of $\mathcal G$.
\textbf{Probabilistic Space.} The set of all sample graphs generated from $\mathcal G$ together with their probabilities define a probabilistic space $\Omega_\mathcal{G}$. Each sample graph $g \in \Omega_\mathcal{G}$ can be generated by flipping coins on all the edges to determine whether or not the edge is live or appears in $g$. That is each edge $(u, v)$ will be present in a sample graph with probability $w(u, v)$. Therefore, a sample graph $g=(V, E' \subseteq E)$ is generated from $\mathcal{G}$ with a probability $\mathbb{Pr}[g \sim \mathcal{G}]$ calculated by,
\begin{align}
\mathbb{Pr}[g \sim \mathcal{G}] = \prod_{(u,v) \in E'} w(u,v) \prod_{(u,v) \notin E'} (1-w(u,v)).
\end{align}
{\bf Influence Spread.} Given a diffusion model, the measure \emph{Influence Spread} (or simply \emph{influence}) of a seed set $S$ is defined as the expected number of active nodes in the end of the diffusion propagation, where the expectation is taken over the probabilistic space $\Omega_\mathcal{G}$. Given a sample graph $g \sim \mathcal{G}$ and a seed set $S \subset V$, we denote $\eta_g(S)$ the set of nodes reachable from $S$ (including nodes in $S$ themselves). The influence spread of $S$ is defined as follows,
\begin{align}
\I(S) = \sum_{g \sim \mathcal{G}} |\eta_g(S)| \mathbb{Pr}[g \sim \mathcal{G}].
\end{align}
The frequently used notations are summarized in Table~\ref{tab:syms}.
\renewcommand{1.1}{1.2}
\setlength\tabcolsep{3pt}
\begin{table}[hbt]\small
\centering
\caption{Table of notations}
\begin{tabular}{p{1.5cm}|p{6.5cm}}
\addlinespace
\toprule
\bf Notation & \quad \quad \quad \bf Description \\
\midrule
$n, m$ & \#nodes, \#edges of graph $\mathcal{G}=(V, E, w)$.\\
\hline
$\I(S), \hat \I(S)$ & Expected Influence of $S \subseteq V$ and an estimate.\\
\hline
$N^{in}(S)$ & Set of in-neighbor nodes of $S$.\\
\hline
$\gamma_v,\Gamma$ & $\gamma_v = 1-\Pi_{u \in N^{in}(v)}(1-w(u,v)); \Gamma = \sum_{v \in V} \gamma_v$.\\
\hline
$\gamma_0$ & $\gamma_0 = \sum_{v \in V}\gamma_v/n$.\\
\hline
$R_j, \R$ & A random \textsf{IIS}{} sample and a \textsf{SKIS}{} sketch.\\
\hline
$\mathrm{C}_\R(S)$ & $\mathrm{C}_{\R}(S) = |R_j \in \R | \R_j \cap S \neq \emptyset|$.\\
\bottomrule
\end{tabular}%
\label{tab:syms}%
\vspace{-0.1in}
\end{table}%
\subsection{Influence Estimation/Maximization Problems}
We describe the tasks of Influence Estimation and Maximization which are used to evaluate sketches' efficiency.
\begin{Definition}[Influence Estimation (\textsf{IE}{})] Given a probabilistic graph $\mathcal{G}$ and a seed set of nodes $S \subseteq V$, the \textsf{IE}{} problem asks for an estimation $\hat \I(S)$ of the set influence $\I(S)$.
\end{Definition}
\begin{Definition}[Influence Maximization (\textsf{IM}{}) \cite{Kempe03}] Given a probabilistic graph $\mathcal{G}$, a budget $k$, the \textsf{IM}{} problem asks for a set $S_k$ of size at most $k$ having the maximum influence among all other sets of size at most $k$,
\begin{equation}
S_k = \argmax_{S \subseteq V, |S| \leq k} \I(S).
\end{equation}
\end{Definition}
\subsection{Sketch-based Methods for IE/IM}
\subsubsection{Reverse Influence Sketch (RIS)}
Essentially, a random \textsf{RIS}{} sample, denoted by $R_j$, contains a \textit{random} set of nodes, following a diffusion model, that can influence a \textit{randomly selected} source node, denoted by $\textsf{src}(R_j)$. A \textsf{RIS}{} sample is generated in three steps:
\begin{itemize}[noitemsep,nolistsep]
\item[1)] Select a random node $v \in V$ which serves as $\textsf{src}(R_j)$.
\item[2)] Generate a sample graph $g \sim \mathcal{G}$.
\item[3)] Return the set $R_j$ of nodes that can reach $v$ in $g$.
\end{itemize}
Thus, the probability of generating a particular \textsf{RIS}{} sample $R_j$ can be computed based on the source selection and the sample graphs that has $R_j$ as the set of nodes that reach $\textsf{src}(R_j)$ in $g$. Let denote such set of nodes that can reach to a node $v$ in sample graph $g$ by $\eta^{-}_g(v)$. We have,
\begin{align}
\label{eq:ris_prob}
\mathbb{Pr}[R_j] = \frac{1}{n} \sum_{g, \eta^-_g(\textsf{src}(R_j)) = R_j} \mathbb{Pr}[g].
\end{align}
The key property of \textsf{RIS}{} samples for influence estimation/maximization is stated in the following lemma.
\begin{Lemma}[\cite{Borgs14}]
\label{lam:borgs14}
Given a random \textsf{RIS}{} sample $R_j$ generated from $\mathcal{G} = (V,E,w)$, for a set $S \subseteq V$ of nodes, we have,
\begin{align}
\I(S) = n \cdot \mathbb{Pr}[R_j \cap S \neq \emptyset].
\end{align}
\end{Lemma}
Thus, estimating/maximizing $\I(S)$ is equivalent to estimating/maximizing the probability $\mathbb{Pr}[R_j \cap S \neq \emptyset]$.
\textbf{Using \textsf{RIS}{} samples for \textsf{IE}{}/\textsf{IM}{}.} Thanks to Lemma~\ref{lam:borgs14}, a general strategy for \textsf{IE}{}/\textsf{IM}{} is generating a set of \textsf{RIS}{} samples, then returning an empirical estimate of $\mathbb{Pr}[R_j \cap S \neq \emptyset]$ on generated samples for \textsf{IE}{} or the set $\hat S_k$ that intersects with most samples for \textsf{IM}{}. The strong advantage of \textsf{RIS}{} is the reuse of samples to estimate influence of any seed set $S \subseteq V$. Ohsaka et al. \cite{Ohsaka16} build a query system to answer influence queries. \cite{Borgs14,Tang14,Tang15,Nguyen163,Nguyen173} recently use \textsf{RIS}{} samples in solving Influence Maximization problem with great successes, i.e. handling large networks with tens of millions of nodes and billions of edges.
\subsubsection{Combined Reachability Sketch (SKIM)}
Cohen et al. \cite{Cohen14} proposed the combined reachability sketch which can be used to estimate influences of multiple seed sets. Each node $u$ in the network is assigned a combined reachability sketch which is a bottom-$k$ min-hash sketch of the set of nodes reachable from $u$ in $l$ sample graphs. \cite{Cohen14} generates $l$ sample graphs $g$ of $\mathcal{G}$, i.e. $l = 64$ by default, and build a combined reachability sketch of size $k$ for each node.
The influence estimate of a seed set $S$ is computed by taking the bottom-$k$ sketch of the union over all the sketches of nodes in $S$ and applying the cardinality estimator \cite{Cohen14}. Using the sketches, the solution for \textsf{IM}{} is found by following the greedy algorithm which repeatedly adds a node with highest marginal influence into the solution. Here, the marginal influences are similarly estimated from node sketches.
\begin{figure}[!t]
\vspace{-0.1in}
\centering
\subfloat[Weighted Cascade (\textsf{WC})]{
\includegraphics[width=0.4\linewidth]{figures/epinions_freq_e1.pdf}
}
\subfloat[Trivalency (\textsf{TRI})]{
\includegraphics[width=0.4\linewidth]{figures/epinions_freq_tri.pdf}
}
\caption{Distribution of reversed cascade sizes on Epinions network (and on other networks as well) with two different edge weight models: Weighted Cascade (\textsf{WC}) and Trivalency (\textsf{TRI}). The majority of the cascades are singular.}
\label{fig:size_dist}
\vspace{-0.25in}
\end{figure}
\textbf{Common Shortcomings.} According to recent benchmarks \cite{Arora17} and our own empirical evaluations (details in Section~\ref{sec:exps}), both \textsf{RIS}{} and \textsf{SKIM}{} yield significant influence estimation errors. For \textsf{RIS}{}, it is due to the fact that the majority of \textsf{RIS}{} samples contain only their sources as demonstrated in Figure~\ref{fig:size_dist} with up to 86\% of such \textsf{RIS}{} samples overall. These samples, termed \textit{singular}, harm the performance in two ways: 1) they do not contribute to the influence computation of other seed sets than the ones that contain the sources, however, the contribution is known in advance, i.e. number of seed nodes; 2) these samples magnify the variance of the \textsf{RIS}{}-based random variables used in estimation causing high errors.
This motivates our Importance Sketching techniques to generate only \textit{non-singular} samples that are useful for influence estimations of many seed sets (not just ones with the sources).
\section{Conclusion}
\label{sec:con}
We propose \textsf{SKIS}{} - a novel sketching tools to approximate influence dynamics in the networks. We provide both comprehensive theoretical and empirical analysis to demonstrate the superiority in size-quality trade-off of \textsf{SKIS}{} in comparisons to the existing sketches. The application of \textsf{SKIS}{} to existing algorithms on Influence Maximization leads to significant performance boost and easily scale to billion-scale networks. In future, we plan to extend \textsf{SKIS}{} to other settings including evolving networks and time-based influence dynamics.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,372
|
Mampad is a growing town in Malappuram district, Kerala, India. located about 08 km east of Nilambur city. Nearby places include Edavanna, Areacode, Manjeri, Wandoor and Pandikkad. It is under the Wandoor Assembly Constituency. Kozhikode-Nilambur-Gudalur (CNG Road) SH pass through here. The Mampad town is now developing day to day. Most people are engaged in agriculture and business activities. Hindus, Christians and Muslims co-exist in harmony adding to the diversity in faith and religion. The land is famous for football. The land where Asif Zahir Mampad Rahman played. He contributed a lot to the Malappuram district.
Schools
GUPS Kattumunda East
GMLPS Kattumunda East
A M L P S PULLODE
A M A U P S Mampad
G L P S Mampad
G L P S Meppadam
G V H S S Mampad
M E S Higher Secondary School. Mampad
Mampad High School
Pullipadam Lp School
Rahmania Higher Secondary School Meppadam
The Springs International School
AKM LP SCHOOL PONGALOOR
Peace Public School Pallikkunnu
Colleges
Dr. Gafoor Memorial MES Mampad College, Mampad
NET Distance Education, (Under the Nest Educational Trust)
Hospitals
Life Hospital
Primary health center Mampad
maternity care hospital
River
Chaliyar
Banks
service co-operative bank
Kerala Gramin Bank
south indian bank
vanitha cooperative bank
ATM
HDFC Bank ATM
Kerala Gramin Bank ATM
south indian Bank ATM
State Bank of India, SBI ATM
Places of interest
odayikkal regulator cum bridge
Pullipadam hanging bridge
oli waterfalls
Nearby Places
Edavanna
Nilambur
Chaliyar
Vadapuram
Wandoor
Areacode
Manjeri
Kattumunda
Pulikalody
Transportation
Mampad village connects to other parts of India through Nilambur town. State Highway No.28 starts from Nilambur and connects to Ooty, Mysore and Bangalore through Highways.12,29 and 181. National highway No.66 passes through Ramanattukara and the northern stretch connects to Goa and Mumbai. The southern stretch connects to Cochin and Trivandrum. State. The nearest airport is at Kozhikode. The nearest railway station is at Nilambur
Image gallery
File:MAMPAD panjayath
References
External links
Cities and towns in Malappuram district
Nilambur area
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,991
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\section{Introduction}
The Arches cluster is a massive star cluster located within the Galactic center region, at about 11$'$ to the Galactic north-east of Sagittarius~A$^\star$. The X-ray emission of the cluster is associated with a thermal component that is thought to originate from collisions of winds from massive stars \citep{yusef-zadeh2002b,wang2006,tsujimoto2007}. A diffuse and more extended non thermal emission including the neutral iron fluorescent line at 6.4 keV, has also been detected from a region directly surrounding the star cluster.
This emission could be created either by the interaction of low-energy hadronic cosmic-rays with molecular material surrounding the cluster \citep{tatischeff2012} or by a strong X-ray irradiation as in other clouds of the central molecular zone \citep[CMZ;][and references therein]{clavel2013,ponti2013}.
Both scenarios are compatible with the \textit{XMM-Newton} observations collected up to 2009 \citep{capelli2011b,tatischeff2012} and also with the latest \textit{NuSTAR} characterization of the higher-energy emission \citep{krivonos2014}. However, due to the constant Fe~K$\alpha$\ line emission and the absence of clear molecular counterpart of the X-ray emission, the low-energy cosmic-ray protons scenario has been favored \citep{tatischeff2012}.
In this paper, we add \textit{XMM-Newton} observations spanning three years more compared to the previous studies of the Arches non-thermal emission. In particular, we include data from the 2013 monitoring of $\rm Sgr \, A^{\star}$\ and a deep observation obtained in 2012 during a scan of the CMZ. We find a significant decrease of both the 6.4 keV and the continuum emissions in the 2012 data set, suggesting that a significant part of the non-thermal emission is due to reflection.
In Section~\ref{sec:dataAnalysis}, we present the data and its reduction. The results of our analysis are detailed in Section~\ref{sec:analysisResults}, and the origin of the non-thermal emission is discussed in Section~\ref{sec:discussion}.
\section{\textit{XMM-Newton} observations and data reduction}
\label{sec:dataAnalysis}
We analyzed all the \textit{XMM-Newton}/EPIC observations available since 2000 and including the Arches cluster region (reported in Tab.\,\ref{tab:obsID}). The data reduction was carried out using the \textit{XMM-Newton} Extended Source Analysis Software \citep[ESAS,][]{snowden2008} included in version 12.0.1 of the \textit{XMM-Newton} Science Analysis Software (SAS). Calibrated event lists were produced for each exposure using the SAS \textit{emchain} and \textit{epchain} scripts and ESAS mos-filter and pn-filter were used to exclude periods affected by soft proton flaring.
Background- and continuum-subtracted mosaic images have been created in the energy band 6.32--6.48~keV (Fig.\,\ref{fig:ArchesFeKa}).
To produce them, we created the quiescent particle background (QPB) images, the count images and model exposure maps for each observation and each instrument, using \textit{mos-spectra}, \textit{pn-spectra}, \textit{mos\_back} and \textit{pn\_back} in the two energy bands 6.32--6.48~keV and 3--6~keV.
The combined exposure map was computed taking into account the different efficiencies of the three instruments. The contribution of the continuum to the 6.4~keV line was estimated using the extrapolation of the 3--6~keV emission and assuming an absorbed power-law spectrum with a photon index $\Gamma = 2$ and a column density $\rm N_{\rm H} = 7 \times 10^{22} \rm \, cm^{-2}$.
The \textit{Chandra} analysis tool \textit{reproject\_image\_grid} was used to correctly reproject the maps of each instrument and each observation within the same year and to mosaic them. For each year, the total background mosaic and the estimated continuum mosaic were then subtracted from the 6.4~keV count mosaic and normalized by the total exposure to obtain the final count rate mosaics.
All spectra used in the analysis were extracted with the ESAS \textit{mos-spectra} and \textit{pn-spectra} scripts.
In particular, the region defined to study the variability of the X-ray emission corresponds to the elliptical region where the brightest 6.4 keV emission is detected (largest ellipse in Fig.\,\ref{fig:ArchesFeKa}, referred as `Cloud' in Tab.\,\ref{tab:regions}). The Arches cluster (blue ellipse in Fig.\,\ref{fig:ArchesFeKa}, referred as `Cluster' in Tab.\,\ref{tab:regions}) has been excluded from the former region for the analysis. We point out that this spectral extraction region is very similar to the one used in \cite{tatischeff2012}.
The QPB was obtained using filter wheel closed event lists provided by the ESAS calibration database. For each region, background spectra were extracted for each EPIC camera at the same position in instrumental coordinates using the \textit{pn-spectra} and \textit{mos-spectra} tasks. These spectra were then normalized to the level of QPB in the observations, using \textit{pn\_back} and \textit{mos\_back}. Spectrum counts are grouped to have at least thirty counts per bin and then fitted using modified chi-square statistics. The errors are given by the confidence interval of the fits at 1\,$\sigma$.
In order to study the astrophysical background contributing to the measured flux we extracted spectra from several large regions surrounding the Arches cloud and fitted the corresponding spectra with a two-thermal-plasma model accounting for the Galactic thermal emission \citep[at 1 and 7\,keV, ][]{muno2004}. The ratio between the normalization of the soft and of the hot components has different values depending on the region we consider. This difference is to be linked to the strong anisotropy of the spatial distribution of the soft emission. Therefore, getting a precise estimate of the astrophysical background at the position of the Arches cloud is not straightforward. Nevertheless, using an elliptical region (referenced as `Bkg test' in Tab.\,\ref{tab:regions}) that has a sufficient coverage for all years, we fitted the normalization of the hot plasma (fixing the ratio between the two components at its mean value of 5.5). The weighted average of the hot plasma normalization is then $\rm I_{\rm 7keV} = 11.4\pm0.3 \times 10^{-4} \rm \, cm^{-5}$ with a maximal amplitude for the variation lower than 6.6\%.
\begin{table}
\centering
\caption{Elliptical regions used for the spectral extraction.}
\label{tab:regions}
\begin{scriptsize}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}c c c c c}
\hline \hline
Region & l [$^\circ$]& b [$^\circ$] & Axes [$''$] & Angle [$^\circ$] \\
\hline
Cloud & 0.124 & 0.018 & 58.9, 25.1 & 125.4 \\
Cluster (excl.) & 0.123 & 0.019 & 16.0, 14.0 & 58.6 \\
\hline
Bkg test & 0.134 & --0.031 & 100.8, 48.3 & 125.42\\
\hline
NuStar & 0.122 & 0.019 & 50, 50 & -- \\
\hline \hline
\end{tabular*}
\end{scriptsize}
\end{table}
To account for the emission of the Arches cloud region we used the following model,
\begin{equation}
\centering
\textsc{wabs}\times(\textsc{apec}+\textsc{powerlaw})+\textsc{gaussian}.
\label{eq:mod1}
\end{equation}
A similar model was used by \cite{tatischeff2012} but here we do not correct the iron line emission for the overall absorption\footnote{The absorption correction relies on the N$_{\rm H}$\ value fitted to the softer part of the spectrum, which also depends on the background subtraction. This would force the correlation between the line and the continuum emissions.}. According to the parameters derived for the Arches cloud from the overall \textit{XMM-Newton} data set \citep[Table 3 in][]{tatischeff2012}, we fixed the temperature of the plasma component (\textsc{apec}) to kT~=~2.2~keV, the cloud metallicity to Z~=~1.7~Z$_\odot$, the index of the \textsc{powerlaw} to $\Gamma$~=~1.6, the centroid energy and width of the \textsc{gaussian} to E$_{\rm 6.4keV}$~=~6.4~keV and $\Delta\rm E_{\rm 6.4keV}$~=~10~eV, respectively. The free parameters are therefore the overall absorption and the normalizations of the plasma, of the line and of the reflection continuum components. For each year, we fit these four parameters between 2 and 7.5~keV on all spectra simultaneously.
All fits were satisfactory, giving reduced $\chi^2 \sim 1$. We also tested a similar model including two thermal plasma at 1 and 7 keV, respectively, in order to better account for the astrophysical background present in the data. This second model gave consistent results regarding the Fe~K$\alpha$\ line emission. Interpreting the continuum component is more complex since it highly depends on the astrophysical background estimation. However, as discussed above, the steadiness of the astrophysical background is such that any variation larger than 10\% can only be attributed to a variation of the Arches cloud emission.
\begin{figure*}
\includegraphics[width=1.0\textwidth]{Clavel-Arches-Fig1.ps}
\caption{\textit{(Top Left)} Mopra N$_2$H$^+$\ map of the Arches cloud position integrated between --40 and --10\,km\,s$^{-1}$ \citep{jones2012}. A significant molecular contribution is there albeit a slight shift of the molecular emission to the south east compared to the X-ray emission of the Arches region. \textit{(Others)} Continuum and background subtracted Fe~K$\alpha$\ maps of the Arches cluster region for seven different years, from top left to bottom right: 2000, 2002, 2004, 2007, 2011, 2012 and 2013. The maps are displayed in Galactic coordinates and smoothed using a gaussian kernel of 20$''$
radius. The solid white ellipse is the cloud region, the cyan ellipse is the cluster region, and the two dashed ellipses are two cloud subregions: north and south. The overall cloud shows morphological variations from period to period with a clear decrease in the overall emission in 2012.
}
\label{fig:ArchesFeKa}
\end{figure*}
\begin{figure}
\centering
\vspace{-0.4cm}
\includegraphics[width=0.5\textwidth]{Clavel-Arches-Fig2.ps}
\caption{Fe~K$\alpha$\ line flux lightcurve of the Arches cloud. The emission is compatible with a constant emission up to 2011 with an average value of $\rm F_{\rm 6.4keV} = 8.2\times10^{-6}\rm \,ph\rm \,cm^{-2}\,\rm s^{-1}$ but the constant fit on the whole period is rejected at 4.3\,$\sigma$ due to a more than 30\% drop in 2012.
}
\label{fig:FeKaLC}
\end{figure}
\begin{figure}
\centering
\vspace{-0.4cm}
\includegraphics[width=0.5\textwidth]{Clavel-Arches-Fig3.ps}
\caption{Continuum flux lightcurve associated with the power-law component of the Arches cloud. The lightcurve is compatible with a constant emission up to 2011 with an average value of $\rm I_{\rm cont} = 19.3\times10^{-5}\rm \,ph \rm \,cm^{-2}\rm \,s^{-1}\rm\,keV^{-1}$ but the constant fit on the whole period is rejected at 5.6\,$\sigma$.
}
\label{fig:ContLC}
\end{figure}
\section{Variation of the X-ray non-thermal emission}
\label{sec:analysisResults}
Fig.\,\ref{fig:ArchesFeKa} presents the Fe~K$\alpha$\ line emission of the Arches region for seven different years between 2000 and 2013 along with the best correlation found for the molecular material in the region. The overall X-ray emission is distributed within an elongated ellipse, centered on the Galactic east of the Arches cluster. This shape is fully consistent with the morphology of the N$_2$H$^+$\ (J=1--0) emission seen by Mopra around \hbox{--25~km~s$^{-1}$} \citep{jones2012}. However, its exact position is shifted by about 18$''$ compared to the molecular data. This is larger than the 10$''$ pointing error mentioned by \cite{jones2012}. Nevertheless, the higher antenna temperature in the Galactic south of the Arches ellipse is consistent with the brightest emission observed in the Fe~K$\alpha$\ line. This is in agreement with the fluxes reported by \cite{capelli2011b}, and with the position shift of the hard X-ray emission detected by \cite{krivonos2014} using \textit{NuSTAR}. We also point out that this south molecular core is not seen in the CS line at the corresponding velocity \citep{tsuboi1999}. This is a hint that the CS emission might be self absorbed at this position, indicating a rather dense core.
From Fig.\,\ref{fig:ArchesFeKa} it is also clear that the Fe~K$\alpha$\ line emission is varying over the 2000--2013 time period with both morphological and intensity changes. In particular, both the north and south subregions of the Arches cloud (dashed ellipses) seem to be increasing and then decreasing in flux, with a peak in 2004 and 2007, respectively. In 2012, there is an overall decrease of the emission. In spite of a relatively shallow exposure in 2013, the overall morphology looks quite different from what is seen in the previous years, since the emission seems to surround the N$_2$H$^+$\ dense cores.
In order to confirm the variations seen in the images we performed spectral fits of the Arches cloud data (excluding the cluster) using the model detailed in Sec.\,\ref{sec:dataAnalysis}. \hbox{Figs.\,\ref{fig:FeKaLC} and \ref{fig:ContLC}} present the variations of the Fe~K$\alpha$\ line flux and of the power-law continuum emission, respectively.
Both components are varying significantly with a rejection of the constant fit at 4.3 and 5.6\,$\sigma$, respectively, and both have dropped by more than 30\% in 2012.
When a Pearson correlation test is applied to the power-law continuum flux as a function of the 6.4 keV flux (following the prescription of \citealt{pozzi2012} to take uncertainties into account), a correlation is detected at 3\,$\sigma$ confidence level and the linear fit results in a slope of $1.0\pm0.3$, indicating that most of the power-law continuum is indeed linked to the 6.4\,keV emission.
As expected in this case, the equivalent width of the Fe~K$\alpha$\ line is compatible with being constant over time with an average value ${\rm EW} = 0.9\pm0.1$\,keV
(rejection at less than 0.2\,$\sigma$).
The two other free parameters of the spectral fit are compatible with being constant over the thirteen-year period and the weighted mean values are $\rm N_{\rm H}=6.0\pm0.3\times10^{22}\rm\,cm^{-2}$ for the absorption and $\rm I_{\rm 2.2keV} = 3.6\pm0.7\times10^{-4}\rm\,cm^{-5}$ for the normalization of the thermal plasma. We point out that the normalization of the plasma given here does not consider the 2007 value that is significantly higher due to a contamination from the flaring cluster \citep{capelli2011a}.
To compare our values to the ones presented in \cite{tatischeff2012}, we also divided the 2004 data set in two time periods and performed a constant fit of the data up to 2009. For this restricted data set, both the Fe~K$\alpha$\ line flux and the power-law continuum normalization are compatible with being constant (rejection at less than $1.4\,\sigma$) and result in $\rm F_{\rm 6.4keV}=8.3\pm1.0\times10^{-6}\rm\,ph\,cm^{-2}\,\rm s^{-1}$ ($9.6\pm1.0$, if corrected for the absorption) and $\rm I_{\rm cont}=20.1\pm1.6\times10^{-5}\rm\,ph\,cm^{-2}\,\rm s^{-1}\,\rm keV^{-1}$, respectively. Both values are compatible with \cite{tatischeff2012} results even if the continuum component cannot be directly compared due to different background estimations in the two analyses.
The individual data points are also compatible within the error bars, except for the 2000 point for which values are about 2\,$\sigma$ apart. We can exclude that this discrepancy is due to a different position of the source within the field of view compared to the 2002--2009 observations. Indeed the systematic error associated\footnote{This systematic error is related to the effective area of the EPIC camera \citep{tatischeff2012}. We tested this effect on our data set by computing the flux of the Sgr~A East region as a function of its offset within the detector. We found a maximal amplitude $<$\,4\% for this systematic.} is negligible compared to the 1\,$\sigma$ error bars given by the fit.
The difference is most likely due to the poor quality of the corresponding data set but it has not been fully identified. In any case, we point out that excluding the 2000 data point does not change the significance of the constant fit rejection.
The \textit{XMM-Newton} observations of autumn 2012 can also be compared to the \textit{NuSTAR} observations collected in the same period \citep{krivonos2014}. To do so, we extracted spectra from a circular region of 50$''$ radius centered on the cluster (referred as `NuStar' in Tab.\,\ref{tab:regions}) and fitted them with the same model (eq.\,\ref{eq:mod1}) but fixing the overall absorption, the temperature of the plasma and the index of the power-law to the \textit{NuSTAR} values \citep[Table 5 and Model 1 in][]{krivonos2014}. The values found for the Fe~K$\alpha$\ line flux (corrected for absorption, $\rm F_{\rm 6.4keV}=1.02\pm0.06\times10^{-5}\,\rm ph\, cm^{-2}\,\rm s^{-1}$), for the normalization of the plasma ($\rm I_{\rm 1.76keV}=50\pm1\times10^{-4}\,\rm cm^{-5}$) and of the continuum ($\rm I_{\rm cont}=1.58\pm0.08\times10^{-12}\,\rm erg\,cm^{-2}\,\rm s^{-1}$ over 3--20 keV) are fully compatible with the values stated by \cite{krivonos2014} for the same region. Since the absolute cross calibration factor between \textit{XMM-Newton} and \textit{NuSTAR} is less than 3\% \citep{risaliti2013}, we infer that the apparent consistency between \cite{tatischeff2012} and \cite{krivonos2014} fluxes is to be linked to the larger region considered in the later work. The \textit{NuSTAR} value is therefore consistent with the 2012 emission drop.
\section{Discussion: origin of the non-thermal emission}
\label{sec:discussion}
The non-thermal emission measured in the region surrounding the Arches cluster could be created by either a particle bombardment or a strong hard X-ray irradiation by an external source. Since the spectrum of the Arches cloud does not provide enough information to discriminate between the two models \citep{capelli2011b,tatischeff2012,krivonos2014}, the remaining diagnostics are linked to the location of the emission and its variability.
The absence of correlation between this non-thermal X-ray emission and known molecular features at other wavelengths was supporting the particle bombardment scenario. In this work we identified for the first time a relevant molecular counterpart of the 6.4\,keV emission using the N$_2$H$^+$\ tracer. In particular, the overall morphology of the molecular structure with a two-lobe shape is in good agreement with the X-ray image.
This supports the reflection scenario. The peak to peak correlation is not fully verified because of the slight shift mentioned in Section\,\ref{sec:analysisResults}.
However, the exact distribution of the gas might be different than the one given by this molecular tracer, and the complex propagation of the X-ray signal within this structure may also account for the displacement \citep{clavel2013}.
Furthermore, the 30\% decrease within about one year, detected for both the 6.4 keV line and the continuum emissions, can only be explained by the reflection of hard X-ray irradiation. Indeed, the diffusion timescale and the possible rate of energy losses of the low-energy cosmic ray protons only allow for an emission decrease on much longer time scales (decades) and the energetic required for low-energy cosmic ray electrons is not realistic \citep{tatischeff2012}. Therefore, the significant variation of the non-thermal emission detected for the first time in the present work is the key element to conclude that a significant fraction of this emission is due to reflection.
The source at the origin of the X-ray emission from the Arches cloud is unlikely to be located within the Arches cluster \citep{capelli2011b,tatischeff2012}. Moreover, the variability observed is not isolated since variations have already been observed in nearby structures \citep{capelli2011b} and, on larger scale, within the Sgr~A region \citep{ponti2010,clavel2013}. A large fraction of the non-thermal emission of the Arches cloud is therefore likely due to the past activity of $\rm Sgr \, A^{\star}$, as is the emission of most of the regions listed above.
\section*{Acknowledgments}
The scientific results reported in this article are based on observations obtained with \textit{XMM-Newton}, an ESA science mission with instruments and contributions directly funded by
ESA Member States and NASA. The molecular map was obtained using the Mopra radio telescope, a part of the Australia Telescope National Facility which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. M.C. acknowledges the Universit\'e Paris Sud 11 for financial support. S.S. acknowledges the Centre National d'Etudes Spatiales (CNES) for financial support. G.P. acknowledges support via an EU Marie Curie Intra-European fellowship under contract no. FP-PEOPLE-2012-IEF-331095. Partial support through the COST action MP0905 Black Holes in a Violent Universe is acknowledged.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 576
|
es una región especial de la Metrópolis de Tokio, en Japón.
Es el más importante centro comercial y administrativo de la Metrópolis de Tokio. En el mismo, se encuentra su famosa estación de trenes, que es la más utilizada del mundo, (un promedio de 3 millones de personas emplean la estación diariamente), además del Tochō (都庁) o edificio del Gobierno Metropolitano de Tokio, el cual es el centro de la administración de Tokio y símbolo urbano más importante de la parte oriental de Tokio. En el área cercana de la estación de Shinjuku se encuentra una gran concentración de tiendas de electrónica, centros comerciales como Odakyu, cines, restaurantes y bares. Muchos hoteles internacionales poseen una sucursal en esta región especial, especialmente hacia el oeste de la región especial.
En el año 1960, la población estimada de esta región especial fue de 413.910, con una densidad poblacional de 22.708 personas por km², con un área total de 18,23 km².
Shinjuku posee la más alta tasa de inmigrantes registrados que cualquier otra región especial en la Metrópolis de Tokio. El día 1 de octubre de 2005, 29.353 personas con 107 nacionalidades estuvieron registradas en Shinjuku, siendo los habitantes de Corea (Corea del Norte y Corea del Sur), China y Francia los más recurrentes.
Su nombre significa "nueva posada" (新-Nuevo, 宿-Posada, alojamiento)
Geografía
Las regiones especiales en torno a Shinjuku son: Chiyoda al este; Bunkyo y Toshima al norte; Nakano al oeste; y Shibuya y Minato al sur. Además, Nerima está a pocos metros de distancia. El punto más alto de Shinjuku es el cerro Hakone, a 44,6 m, en el parque Toyama que se encuentra al este de las estaciones de Takadanobaba y Shin-Okubo. El punto más bajo está a 4,2 m en el área de Iidabashi.
Dentro de Shinjuku se encuentran áreas más específicas como:
Ichigaya, un área comercial, al este de Shinjuku. Se encuentra la Agencia de Defensa Japonesa.
Kabukichō, un distrito conocido por sus bares, restaurantes y como un zona roja, debido a las prostitutas y otros tipos de comercio sexual. Se encuentra al noreste de la estación de Shinjuku.
Nishi-shinjuku: en este distrito se encuentran la mayoría los rascacielos del barrio. Se encuentra al oeste de la estación de Shinjuku.
Okubo: es conocido por ser un distrito con abundancia de inmigrantes coreanos.
Shinanomachi: En la parte sureste se encuentra el Estadio Nacional, también conocido como el Estadio Olímpico.
Jardín Nacional Shinjuku Gyoen: es un gran parque con 58,3 hectáreas, con 3,5 km de radio, donde se mezclan el estilo japonés, inglés y francés en las decoraciones de los jardines.
Shinjuku ni-chome: es uno de los barrios gays de la Metrópolis de Tokio.
Waseda: se encuentra cercana a la Universidad de Waseda, que es una de las universidades privadas más prestigiosas de Japón.
Historia
En 1634, se construyó un edificio del estilo Edo, con un número de templos que se mudaron al área de Yotsuya, en el oeste del área de Shinjuku. En 1698, durante la época Edo, "Naitō Shinjuku" fue construida como un nuevo shuku (estación, posta, tambo) en el Kōshū Kaidō, que era una de las grandes vías de comunicación de la época. El señor Naitō era un daimyo (señor feudal, gobernador de provincia) cuya mansión representante en Edo estaba erigida en los terrenos; dicha finca es ahora un parque público, el Shinjuku-Gyoen.
Shinjuku comenzó a desarrollarse en su forma actual tras el gran Terremoto de Kanto en 1923. Debido a que es un área estable frente a los sismos, evitó su destrucción. Es por ello que en esta zona se pueden encontrar varios de los mayores rascacielos de Tokio.
Durante la Segunda Guerra Mundial y desde mayo hasta agosto de 1945, el 90% de los edificios en el área y alrededor de la estación de Shinjuku fueron destruidos. La región especial actual fue establecida el 15 de marzo de 1947, con la unión de los barrios de Yotsuya, Ushigome y Yodobashi.
En 1991, el Gobierno Metropolitano de Tokio se movió desde el distrito de Marunouchi en Chiyoda, al edificio actual en Shinjuku.
Gobierno
Al igual que las otras regiones especiales de la Metrópolis de Tokio, Shinjuku posee un estatus equivalente a una ciudad. En el 2005, el alcalde es Hiroki Nakaya, a. El kugiraki (o concejo), consiste de 38 miembros elegidos que se encuentran afilidados a Liberal Democracia, Nuevo Gobierno Limpio, Democrátas, Comunistas, entre otros partidos políticos, así también como independientes.
En la cultura popular
Anime y manga
En el año 2001 Shinjuku Oeste fue la escena de la serie de anime Digimon Tamers, producida por Toei Animation, y transmitida por la señal japonesa Fuji Television.
En la película Shin chan: La invasión, una invasión extraterrestre tiene su origen en un rascacielos de esta.
En la película Shin chan: en busca de las bolas perdidas, es donde el Clan de los Mariconchis tenía su local de alterne
En el manga "Gantz" cuando se provoca una matanza. Además aparece su estación de trenes. El capítulo comienza en "La mañana del crimen y termina en "Jurásico".
Aparece en el anime "Code Geass" cuando es devastado por completo por el ejército de Britannia dado al desacato y a la rebelión por parte de un pequeño grupo de terroristas, que está conformado por habitantes del "Área 11" buscando venganza por la invasión a Japón.
En la serie manga yaoi Okane ga Nai, los protagonistas viven en un prestigioso edificio de esta región especial.
El manga y Anime Get Backers ubica al dúo de recuperadores en Nishi-Shinjuku, además sitúa un enorme edificio llamado La Fortaleza Ilimitada en la misma ciudad.
En el anime Karas, los hechos transcurren en un Nishi-Shinjuku ficticio.
En el anime Death Note, ocurre un apuñalamiento, Light mata al sujeto, esto ocurre en el episodio 2 de death note.
El famoso manga y anime "City Hunter" de Tsukasa Hōjō transcurre en Nishi-Shinjuku.
El manga Demon City Shinjuku (año 1982) se desarrolla en las hipotéticas ruinas de esta zona de Tokio devastada por un terremoto.
En el anime Owari no Seraph aparece como uno de las regiones que los vampiros tienen como zona de control.
Películas
Aquí se encuentra el Hotel Park Hyatt Tokyo, que apareció en la película Lost in Traslation de Sofia Coppola (2003).
En la película de Culto Suicide Club, 54 Estudiantes se suicidan en la Plataforma 8 de la Estación de Shinjuku.
La película Ichi the killer del director Takashi Miike transcurre en esta región especial.
En la película Kimi no Na wa. de Makoto Shinkai se puede ver la Estación de Shinjuku en varias escenas, además de ser el lugar donde varios personajes toman un tren a distintas partes de Tokio.
La película Tokyo Godfathers de Satoshi Kon transcurre principalmente en Shinjuku.
Videojuegos
En el videojuego Midnight Club II (2003) se presenta esta ciudad como circuito, teniendo detalladamente varios paisajes, edificios, vehículos, calles, etc.
Una tópica representación de la región sirve de escenario en el videojuego de PS2 "Tekken 4"
En el videojuego Etrian Odyssey aparece como el 5.º estrato de este juego con el nombre de "Lost Shinjuku" (Shinjuku perdida)
En la franquicia Pokemon, Ciudad Azulona está inspirada en esta región, por sus casinos y tiendas. Y el gimnasio de Azulona está inspirado en el jardín de la región.
En la saga Yakuza de sega (Ryū ga gotoku en Japón) el escenario principal de los juegos es una recreación del barrio de Kabukichō, el barrio rojo de Shinjuku, que en el juego es conocido como Kamurocho.
En la saga de carreras de Gran Turismo , este lugar es lugar de algunos circutos.
En Persona 5, Shinjuku aparece como un lugar al que el protagonista puede visitar para realizar diversas actividades.
Literatura
En la novela Crónica del pájaro que da cuerda al mundo (2001, Tusquets) de Haruki Murakami varias partes del relato se desarrollan en las inmediaciones de la estación de Shinjuku.
Véase también
Estación de Shinjuku
Referencias
Enlaces externos
Página de la Administración de la Región Especial de Shinjuku (japonés)
Página de la Administración de la Región Especial de Shinjuku (inglés)
Barrios de Tokio
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 921
|
package storage
import (
"bytes"
"database/sql"
"fmt"
"strings"
"github.com/VividCortex/mysqlerr"
"github.com/go-sql-driver/mysql"
sqlite3 "github.com/mattn/go-sqlite3"
)
// DB a struct wrapping plain sql library with SQL dialect, to solve any feature
// difference between MySQL, which is used in production, and Sqlite, which is used
// for unit testing.
type DB struct {
*sql.DB
SQLDialect
}
// NewDB creates a DB
func NewDB(db *sql.DB, dialect SQLDialect) *DB {
return &DB{db, dialect}
}
// SQLDialect abstracts common sql queries which vary in different dialect.
// It is used to bridge the difference between mysql (production) and sqlite
// (test).
type SQLDialect interface {
// GroupConcat builds query to group concatenate `expr` in each row and use `separator`
// to join rows in a group.
GroupConcat(expr string, separator string) string
// Concat builds query to concatenete a list of `exprs` into a single string with
// a separator in between.
Concat(exprs []string, separator string) string
// Check whether the error is a SQL duplicate entry error or not
IsDuplicateError(err error) bool
// Modifies the SELECT clause in query to return one that locks the selected
// row for update.
SelectForUpdate(query string) string
}
// MySQLDialect implements SQLDialect with mysql dialect implementation.
type MySQLDialect struct{}
func (d MySQLDialect) GroupConcat(expr string, separator string) string {
var buffer bytes.Buffer
buffer.WriteString("GROUP_CONCAT(")
buffer.WriteString(expr)
if separator != "" {
buffer.WriteString(fmt.Sprintf(" SEPARATOR \"%s\"", separator))
}
buffer.WriteString(")")
return buffer.String()
}
func (d MySQLDialect) Concat(exprs []string, separator string) string {
separatorSQL := ","
if separator != "" {
separatorSQL = fmt.Sprintf(`,"%s",`, separator)
}
return fmt.Sprintf("CONCAT(%s)", strings.Join(exprs, separatorSQL))
}
func (d MySQLDialect) IsDuplicateError(err error) bool {
sqlError, ok := err.(*mysql.MySQLError)
return ok && sqlError.Number == mysqlerr.ER_DUP_ENTRY
}
// SQLiteDialect implements SQLDialect with sqlite dialect implementation.
type SQLiteDialect struct{}
func (d SQLiteDialect) GroupConcat(expr string, separator string) string {
var buffer bytes.Buffer
buffer.WriteString("GROUP_CONCAT(")
buffer.WriteString(expr)
if separator != "" {
buffer.WriteString(fmt.Sprintf(", \"%s\"", separator))
}
buffer.WriteString(")")
return buffer.String()
}
func (d SQLiteDialect) Concat(exprs []string, separator string) string {
separatorSQL := "||"
if separator != "" {
separatorSQL = fmt.Sprintf(`||"%s"||`, separator)
}
return strings.Join(exprs, separatorSQL)
}
func (d MySQLDialect) SelectForUpdate(query string) string {
return query + " FOR UPDATE"
}
func (d SQLiteDialect) SelectForUpdate(query string) string {
return query
}
func (d SQLiteDialect) IsDuplicateError(err error) bool {
sqlError, ok := err.(sqlite3.Error)
return ok && sqlError.Code == sqlite3.ErrConstraint
}
func NewMySQLDialect() MySQLDialect {
return MySQLDialect{}
}
func NewSQLiteDialect() SQLiteDialect {
return SQLiteDialect{}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,942
|
\section{Introduction}
In 1970 Esaki and Tsui published a seminal paper where they proposed a solid-state artificial structure, the superlattice (SL)~\cite{SL}. This material is engineered as a periodic modulation of the composition of an alloy, with a period much larger than the host material lattice parameter (see Fig.~\ref{schematic} (a)). This new class of man-made semiconductor structures allowed a profusion of new technological applications to appear~\cite{SL3}. Nowadays SLs are not restricted to the realm of semiconductors but are ubiquitous throughout solid state physics. Different techniques are used in its synthesis, e.g., molecular beam epitaxy (MBE), metalorganic vapor phase epitaxy deposition (MOCVD). Furthermore, SLs are fabricated from a wide range of materials, such as magnetic and non-magnetic metals, insulators and superconductors~\cite{magnetic,superconductor}.
\begin{figure}
\includegraphics[scale=0.45,angle=-90]{bands.eps}
\caption{\label{schematic} (color online) Schematic band structure for different heterostructures. (a) The regular semiconductor heterostructure. (b) Topological/normal insulator heterostructure (TI/NI) for NI thickness (L$_{NI}$) smaller than a critical value (L$_{C}$). The topologically protected boundary states (TPBS) are strongly hybridizing originating the $\Delta_{hyb}$ gap. (c) TI/NI heterostructure where L$_{NI}$ $>$ L$_{C}$ and the TPBS are decoupled. }
\end{figure}
More recently, topological insulators (TI) ~\cite{TI1,TI2,TI3} were added to the previous list. TIs form a class of materials that present an insulating bulk with robust conducting states on its boundary. A question that arises naturally is how the TI properties are affected by interfacing with trivial materials. It is well known that on a TI surface the metallic topological states are Dirac-like with a spin texture that gives rise to a spin current protected from backscattering by time-reversal symmetry. Nevertheless, TIs interfaced with a regular or normal insulator (NI) are poorly understood. Some recent studies show unequivocally the appearance of Dirac cones at the TI/NI interface~\cite{interface1,interface2,interface3,interface4,Seixas2015,miwa,rwu,bi2se3-aln}. Existing calculations~\cite{proximity1,proximity2}, and recent experiments~\cite{proximity3} predict that states at the interface, located predominantly within the normal material, could acquire spin texture due to their proximity to the topological interface states. Now, let us consider a system composed of layers of TI and NI stacked in an alternating fashion, forming a one-dimensional superlattice. Such a system has been realized recently by stacking layers of pristine and In-doped Bi$_{2}$Se$_{3}$~\cite{hasan}, and ARPES measurements on such structure seem to suggest the emergence of a one-dimensional topological phase, drawn schematically in Fig.~\ref{schematic} (c).
Motivated by these experimental and theoretical results, we have performed a systematic study of the electronic and topological properties of heterostructures composed of topological and normal insulators.
Our findings show a direct relationship between the normal insulator thickness (L$_{NI}$) and the appearance of topologically protected boundary states (TPBS). For a L$_{NI}$ smaller then a critical thickness (L$_{C}$) the heterostructure band structure will have the features of Fig.~\ref{schematic} (b), with an hybridization gap ($\Delta_{hyb}$) that is the result of TPBS interaction. For a TI/NI heterostructure, the TPBS is always present at the surface, independent of L$_{NI}$. Finally, our calculations point to an inverse relation between the NIs gap and its critical thickness.
The paper is organized as follows: in Sec.~\ref{methodology} we give a brief description of the methodology used to construct the tight-binding Hamiltonians (TB) and how the spin-orbit coupling is included; we also confront these results to fully relativistic DFT calculations for validation. The computational details are described in sec.~\ref{comp-det}. In sec.~\ref{result-diss} the TB results are discussed and compared to DFT calculations.
\begin{figure}
\includegraphics[scale=0.3,angle=-90]{structure-bands.eps}
\caption{\label{dftXtb-bulk} (color online) (a) Bi$_{2}$Se$_{3}$ bulk hexagonal crystal structure and its correspondent band structure (b) ($K-\Gamma-M$) for a fully relativistic DFT calculation (solid black line) compared to the TB+SOC (open red circles). The blue dashed line represents the Fermi level.}
\end{figure}
\section{Methodology}
\label{methodology}
\begin{figure}
\includegraphics[scale=0.43,angle=-90]{bandstrcture-het.eps}
\caption{\label{dftXtb} (color online) Comparison between the DFT-SOC and TB+SOC band structure. Bi$_{2}$Se$_{3}$ 4Qls surface band structure (a) DFT-SOC and (b) TB+SOC.
Finite heterostructure with 2Qls of Bi$_{2}$Se$_{3}$ without SOC, normal insulator (NI), between two 4Qls of Bi$_{2}$Se$_{3}$ with SOC, topological insulator (TI), (c) DFT-SOC and (d) TB+SOC. The inset of panel (c) shows the schematic representation of the TI/NI/TI heterostructure. The red (blue) regions represent the TI (NI) layers.}
\end{figure}
This section briefly describes the methodology used for the tight-binding Hamiltonians construction and the inclusion of the spin-orbit coupling (SOC) effect. The central idea is to map the plane waves (PWs) Hilbert space, in general composed of several thousand of PWs, onto the compact sub-space spanned by the pseudo-atomic orbitals $|\phi_{\mu}\rangle$ (PAOs) used to construct the atomic pseudopotentials. The Hamiltonians are constructed from the Kohn-Sham (KS) Bloch states ($|\Psi^{KS}\rangle$) projection onto the PAO basis set localized at the atomic sites. Considering the PAO basis set incompleteness some KS states are not accurately described. This manifest itself as unphysical eigenvalues and hybridizations. The solution is to apply a filtering procedure to shift these states outside the energy window of interest. The filtering procedure is based on the so called projectability number ($p_{n}$), which is defined as $p_{n}=\langle\Psi_{n}|\hat{P}|\Psi_{n}\rangle$ and the operator $\hat{P}$ projects the KS states onto PAO basis set. The interpretation of $p_{n}$ is straightforward, KS states with $p_{n}$ $\approx$ 1 ($\approx$ 0) represents states which are well (poor) described by the given PAO basis set. In this work the $p_{n}$ threshold is set to 0.95. This methodology has been successful successfully applied to different materials and properties for a full description see references~\cite{PAO1,PAO2,PAO3,PAO4,PAO5}.
Spin-orbit coupling effect is essential to describe the topological quantum states in TI. In our calculations the SOC is introduced in two different ways: standard fully-relativistic self-consistent DFT calculations~\cite{DFT-SOC} and via an effective approximation in the TB Hamiltonian~\cite{soc,PAO5}. The effective SOC included in the TB Hamiltonian can be written as
\begin{equation}
H_{SOC} = \lambda {\bf L} . {\bf S}
\end{equation}
where $\lambda$ is the, orbital and element dependent, spin-orbit coupling strength, {\bf L} and {\bf S} are the orbital and spin angular momentum operators. Figure~\ref{dftXtb-bulk} shows a comparison between DFT with SOC (solid black line) and the TB+SOC (open red circles) in Bi$_{2}$Se$_{3}$ hexagonal bulk band structure. The Bi and Se $p$-orbitals SOC parameters used are $\lambda_{Bi}=1.475$ eV and $\lambda_{Se}=0.265$ eV. These values are in good agreement with the literature~\cite{bi2se3} and were obtained via a fitting procedure of the Bi and Se BCC DFT-SOC band structure. The agreement between the DFT and TB is excellent. Same degree of agreement is observed for a Bi$_{2}$Se$_{3}$ surface with 4 quintuple-layers (Qls), see Fig.~\ref{dftXtb} (a) and (b); and in TI/NI heterostructures: to this end we considered a heterostructure composed of 2 NI Qls between 4 TI Qls, see Fig.~\ref{dftXtb} (c) inset, using Bi$_{2}$Se$_{3}$ as the material of choice. In the TB Hamiltonian such heterostructure is constructed by setting Bi and Se SOC parameters to zero in the NI region, {\it i.e.} $\lambda_{Bi}=\lambda_{Se}=0$. The DFT calculation was performed with a Bi and Se non-relativistic pseudo potential in the NI region to mimic the SOC absence. Figure~\ref{dftXtb} (c) and (d) the DFT with SOC and TB+SOC heterostructure band-structure are showed, respectively. As in the bulk and surface cases the agreement is outstanding.
\begin{figure}
\includegraphics[scale=0.32,angle=-90]{infinity.eps}
\caption{\label{infinity} (color online) Bi$_{2}$Se$_{3}$ bulk band structure (a) without and (b) with SOC. (c) The 4/8 (TI/NI) heterostructure band structure, a 42 meV hybridization gap ($\Delta_{hyb}$) is obtained. (d) The $\Delta_{hyb}$ evolution as the NI thickness is increased. (e) The relation between the NI gap ($\Delta_{Nl}$) and the critical thickness (L$_{c}$) is shown.}
\end{figure}
\section{Computational Details}
\label{comp-det}
We performed Density Functional Theory (DFT)~\cite{DFT1,DFT2} calculations using the plane wave package Quantum Espresso (QE)~\cite{QE-2009,QE-2017}, and VASP code \cite{vasp1,vasp2}. The generalized gradient approximation (GGA)~\cite{pbe} was employed to treat the correlation among electrons with the Perdew-Burke-Ernzerhof (PBE) parametrization~\cite{pbe}. The Bi and Se Ionic potentials are described using PAW pseudo potentials~\cite{PAW} with $s$, $p$ and $d$ as valence. The Bi$_{2}$Se$_{3}$ is rhombohedral crystal with space group $D^{5}_{3d}$ ($R\overline{3}m$) and a five atoms unit cell. To facilitate the interface construction and results interpretation, a Bi$_{2}$Se$_{3}$ hexagonal cell composed of 15 atoms was used, see fig.~\ref{dftXtb-bulk} (a). This cell contains 3 Qls where the intra (inter) Ql chemical bonding is of ionic/covalent (van der Waals) character. The experimental lattice parameter is adopted with a=4.134 \AA~and c=28.63 \AA. The reciprocal space sampling uses the Monkhost-pack scheme with a k-mesh of 19$\times$19$\times$3 and 19$\times$19$\times$1 for bulk and surface calculations, respectively. We also performed a fully relativistic calculation for a heterostructure of Bi$_{2}$Se$_{3}$ and a real trivial system Sb$_{2}$Se$_{3}$. Since both systems present similar lattice parameters, we use the average lattice parameter for the hexagonal plane (a=4.107~{\AA}) and the vertical c=30.90~{\AA} for Sb$_{2}$Se$_{3}$. At the interface, the average of both structures was used c=29.765~{\AA}. The calculated topological invariant using these parameters gives $Z_2=1$ for Bi$_{2}$Se$_{3}$ and $Z_2=0$ for Sb$_{2}$Se$_{3}$.
\section{Results and discussion}
\label{result-diss}
\begin{figure}
\includegraphics[scale=0.35,angle=-90]{ti-ni-ti.eps}
\caption{\label{ti-ni-ti} (color online) Spatial projected band structure for a 4/8/4 (TI/NI/TI) finite heterostructure. Projection onto the TI outmost 2 Qls at the surface (a) $\Delta_{NI}$=0.16 eV (bands aligned), (b) $\Delta_{NI}$=0.50 eV (bands aligned) and (a) $\Delta_{NI}$=0.50 eV (bands not aligned) . In (b), (e) and (g) the projection is calculated at TI/NI interface for the respective heterostructure. The shaded area of panels (a) and (b) inset shows the projected regions. In (c), (f) and (i) a schematic representation of the band alignment for the respective heterostructure.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.4,angle=-90]{ni-ti-ni.eps}
\caption{\label{ni-ti-ni} (color online) Band structure for a 2/4/2 (NI/TI/NI) finite heterostructure. (a) $\lambda_{Bi}=0.0$ eV and (b) $\lambda_{Bi}=1.10$ eV. Eigenvalues spacial projection (c) $\lambda_{Bi}=0.0$ eV and (d) $\lambda_{Bi}=1.10$ eV. Dark red (dark blue) represents high (low) spectral weight at the projected regions. The inset of panel (c), shaded region, highlights the projected region.}
\end{figure}
\subsection{$\cdots$\{/TI/NI/\}$\cdots$ in TB}
In this section, we will establish the conditions for the appearance of the TPBS in infinite, perfectly periodic TI/NI heterostructures. In our calculations, the only difference between the NI and TI is the absence of SOC within the NI region. In Fig.~\ref{infinity} the Bi$_{2}$Se$_{3}$ bulk band structure is showed without SOC (a) and with SOC (b); the SOC increases the gap from 0.16 eV (direct gap) to 0.33 eV (indirect gap), which is in agreement with previous calculations~\cite{bi2se3}. In panel (c) we show the band structure for a 4/8 (TI/NI) infinite heterostructure where a hybridization gap ($\Delta_{hyb}$) of 42 meV appears at the $\overline{\Gamma}$ point. The $\Delta_{hyb}$ gap originates from the TPBS hybridization through the NI material. Its value is reduced by increasing the NI thickness and the TPBS reemerge above a critical thickness ($L_{c}$) around 200 \AA~($\approx$ 21 Qls), see panel (d). These superlattices are chemically and structural perfect; as a consequence, there is no Rashba splitting, which would appear due to inversion symmetry breaking, as reported in Ref.~\cite{NatHet}. In a real heterostructure the NI region is formed by different materials, thus it is important to understand the relation between the NI gap ($\Delta_{NI}$) and $L_{c}$. To increase $\Delta_{NI}$ a local potential is applied to Bi and Se $p$-states. There is a inverse relation between $\Delta_{NI}$ and L$_{c}$, see Fig.~\ref{infinity} (e). Increasing $\Delta_{NI}$ to 0.5 eV the L$_{c}$ is reduced to 75\AA~($\approx$ 8 Qls). This finding is important to guide the experimental search for suitable constituent materials of heterostructures with specific properties.
\subsection{vacuum/TI/NI/TI/vacuum and vacuum/NI/TI/NI/vacuum in TB}
\begin{figure*}[ht]
\includegraphics[scale=0.65,angle=-90]{ti-ni-42.eps}
\caption{\label{ti-ni-42} (color online) Band structure for a topological/normal insulator (TI/NI) finite heterostructure. The left panel shows the heterostructure schematic representation with 4 TI Qls (red) and 2 NI Qls (blue). Where in the NI region the Bi SOC strength is varied from 50\% of its full value to 0. The eigenvalues spacial projection onto the outmost 4 Qls TI, at the surface, is showed in (a) 50\%, (b) 25\% and (c) 0\% of the full Bi SOC. In (d), (e), and (f) the projection is onto the middle region (TI and NI) of the slab. Dark red (dark blue) represents high (low) spectral weight at the projected region, see the colorbar.}
\end{figure*}
In this section we investigate heterostructures with vacuum terminations. Two different constructions will be explored: one NI between two TI regions (TI/NI/TI), and one TI between two NI (NI/TI/NI) regions as building blocks for realistic TI/NI superlattices. In Fig.~\ref{ti-ni-ti} we show the band structure for the 4/8/4 (TI/NI/TI) sandwich, projected on the interfaces. The plot shows states with high (above 60\%) spectral weight at the indicated regions, inset of panels (a) and (b). In Fig.~\ref{ti-ni-ti} (a) and (b) the NI region gap is 0.16 eV (no potential is applied, and $\lambda_{Bi}=\lambda_{Se}=0.0$) and the TI and NI bands are perfectly aligned, as indicated in panel (c). The TPBS is observed at the surface, and their dispersions are consistent with the 4Qls Bi$_{2}$Se$_{3}$ surface. However, the TPBS at the TI/NI interface is replaced by states with parabolic dispersions and a 42 meV hybridization gap. This is in perfect agreement to the situation in the 4/8 infinite superlattice, including the size of the hybridization gap. In panels (d) and (e) we investigate the effect of increasing the NI band gap on the TPBS. For a gap of 0.5 eV, no significant change is observed at the surface. At the NI interfaces, however, the TPBS is restored, with an increased dispersion and an upwards energy shift. This result strongly suggests that large NI band gaps effectively decouple the two TI/NI interfaces.
Similarly to ref.~\cite{band-alignment}, we explore the relation between band alignment and the TPBS formation. As in the previous cases, the surface TPBS is not significantly affected by the band misalignment (panel (g)) while the ones at the NI/TI interface are dramatically shifted downward in energy (panel (h)).
The next heterostructure is composed of a TI between two NI (NI/TI/NI) with equal thicknesses. With such construction, the TPBS's dispersion acquires a parabolic shape, as seen in Fig.~\ref{ni-ti-ni} (a). This larger dispersion results in the reduction of Fermi velocity when compared with the 4Qls surface linear dispersion, see Fig.~\ref{dftXtb} (b). Another important feature is the localization, in energy, of the bulk states between $\overline{\Gamma}$ and $\overline{M}$. For the 4Qls surface, this state is located 90 meV below the Dirac point (DP). Once the TI is placed in between two NI layers those states shift down to 240 meV below the DP. This effect is due to the absence of SOC at the surface since $\lambda_{Bi}=\lambda_{Se}=$0.0. As an exercise to understand this behavior the Se SOC is restored ($\lambda_{Se}=0.265$ eV) and the Bi SOC is increased to 1.10 eV, in the NI region. The bulk states are 150 meV below DP and move toward the surface value as $\lambda_{Bi}\rightarrow$ 100\%. This energy shift contributes to reduce the influence of these bulk states in the Bi$_{2}$Se$_{3}$ transport properties~\cite{bulk-states1,bulk-states2}. This has been observed for (Bi$_{x}$Sb$_{1-x}$)$_{2}$Se$_{3}$ alloys~\cite{abdalla}, where the Sb reduces the overall SOC. Moreover, panels (c) and (d) shows the spatial localization of the TPBS, a significant penetration of these states in the NI region is observed. This can be exploited in order to establish contacts for transport measurements or attain functionalization, for example.
\begin{figure}[ht]
\includegraphics[scale=0.55,angle=-90]{SL-3.eps}
\caption{\label{TI-NI-SL} (color online) (a) The TI/NI periodic superlattice schematic band structure. In (b), (c) and (d) is the spatial eigenvalue projection of a 1Ql from the TI center region, TI, and NI at the interface, respectively. }
\end{figure}
\subsection{$\cdots$~\{/TI/NI/\}$~\cdots$ in superlattice in TB}
We also investigate a superlattice composed by multilayers of topological and trivial insulators. In reference~\cite{hasan} this kind of multilayer has been presented as candidates to support a 1D chain of topological states. The modulation is performed by controlling the spin-orbit coupling, similarly to the control of topological phase by doping, as reported in ref.~\cite{hasan}. In our results, the intended 1D chain of topological states is not realized. At the TI surface the TPBS is always present. Nevertheless, at the TI/NI interface these states are absent due to a strong hybridization between subsequent interfaces. Figure~\ref{ti-ni-42}, left panel, shows the schematic representation for a 4/2 (TI/NI) heterostructure composed of 22 Qls ($\approx$ 200 \AA). In the TI region (red), full Bi and Se SOC is adopted. In the NI region (blue) the Se SOC is kept fixed at its full value and the Bi SOC is varied from 50 to 0\% of its full value. The Dirac-like states come from the TI in the surface (red in panels (a)-(c)), and no contributions come from the internal TI region (see panels (d)-(f)). We can understand that by thinking about the 4/2 (TI/NI) structure as a unit cell of a ``new'', or ``meta'' material. This meta material is indeed a topological insulator with $Z_{2}=1$, since there is only one band inversion in the first Brillouin zone at the $\Gamma$ point. In this way, for such thin films of the TI/NI metamaterial we predict surface metallic topological states and a gapped bulk, exactly as shown in Fig.~\ref{ti-ni-42}. Naturally, by increasing the NI region greater than a critical thickness, as shown above, a perturbed gapless interface TPBS will be restored. However, in this case, one can not claim uncontroversially to have obtained a 1D chain of topological states since they are effectively decoupled from each other by the intervening NI layers.
\subsection{$\cdots$~\{/Bi$_{2}$Se$_{3}$/Sb$_{2}$Se$_{3}$/\}$~\cdots$ and vacuum/Bi$_{2}$Se$_{3}$/Sb$_{2}$Se$_{3}$/Bi$_{2}$Se$_{3}$/vacuum in DFT-SOC}
In order to complement our discussion above, fully relativistic calculations of the Bi$_{2}$Se$_{3}$ (TI) interfaced with Sb$_{2}$Se$_{3}$ (NI) were performed. Since both materials are lattice matched, the heterostructure preserves the bulk topological properties. For a periodic supercell composed of 4/2 (TI/NI), a gap is formed (see Fig.~\ref{TI-NI-SL}). There are topological surface bands at the interfaces which come mostly from the TI side, however, the global gap is preserved. By cutting our TI/NI supercell to produce top and bottom surfaces, a different picture is observed, confirming our TB results above. As shown in Fig.~\ref{Slab} The TPBS re-emerge at the surface and a gap is present at the Sb$_{2}$Se$_{3}$ interface. The whole system behaves as a unique TI material, gapped inside but metallic at the surfaces. As shown in Fig.~\ref{Slab} g), the TPBS are Rashba-like sates, due to the coupling between opposite interface TPBS, as observed for surface states in thin films \cite{Shan-2010}. The asymmetric distributions of the top and bottom surface states (Fig.~\ref{Slab} (f)) arise due to a symmetry breaking induced by the NI material. It breaks the Bi$_{2}$Se$_{3}$ symmetry since the unit cell demands 3 Qls, similar to stacking faults close to the surface \cite{Abdalla-2015}.
\begin{figure}[ht]
\includegraphics[scale=0.3,angle=-90]{slab2.eps}
\caption{\label{Slab} (color online) (a) 4/2/4 (TI/NI/TI) slab structure. In (b) - (e) are spin projections on 1st, 4th, 7th and 10th Ql, respectively. Colors blue and red are projections of opposite in-plane spin texture. The Fermi level is at zero energy.}
\end{figure}
\section{Conclusions}
In conclusion, our calculations provide important insights to guide experimental realizations of TI/NI heterostructures with specific distributions of topologically protected boundary states (TPBS). Depending on the TI/NI heterostructure composition the TPBS can hybridize, through the intervening NI, leading to a bulk gap opening. We show that for NI layers thicker than a critical value (L$_{C}$) the hybridization is suppressed and the bulk TPBSs are restored. These TPBS are spatially localized at the TI/NI interface with a significant penetration in the NI region due to the proximity effect. Also, our calculations indicated an inverse relation between L$_{C}$ and the NI gap ($\Delta_{NI}$), suggesting that one should use a large gap trivial material to build the heterostructure if TPBSs in the internal TI/NII interfaces are desired. Moreover, the TI and NI band alignment will play an important role in the TPBS interface formation. With these results we expect to contribute to the understanding of the TI/NI interfacing problem which will be important in the device construction using topological insulators.
\section*{ACKNOWLEDGMENTS}
This work was supported by the Brazilian agencies FAPESP/TEM\'{ATICO} and CNPQ. We would like to acknowledge computing time provided by Laborat\'orio de Computa\c{c}\~ao Cien\'ifica Avancada (Universidade de S\~ao Paulo). We also like to thanks Prof. Yves Petroff for the fruitful discussion regarding ARPES measurements.
\bibliographystyle{apsrev4-1}
\normalbaselines
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{"url":"https:\/\/collegephysicsanswers.com\/openstax-solutions\/what-power-supplied-starter-motor-large-truck-draws-250-current-240-v-battery","text":"Question\nWhat power is supplied to the starter motor of a large truck that draws 250 A of current from a 24.0-V battery hookup?\n\n$6.0 \\textrm{kW}$\n\nSolution Video\n\n# OpenStax College Physics Solution, Chapter 20, Problem 41 (Problems & Exercises) (0:16)\n\n#### Sign up to view this solution video!\n\nView sample solution\n\n## Calculator Screenshots\n\nVideo Transcript\nThis is College Physics Answers with Shaun Dychko. The power supply to the starter motor is the current multiplied by the voltage. So it\u2019s 250 Amps we\u2019re told, multiply by 24 Volts, which is a power of 6.0 kilowatts.","date":"2019-02-18 14:17:37","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\": 0.4583416283130646, \"perplexity\": 4464.496292193983}, \"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-2019-09\/segments\/1550247486936.35\/warc\/CC-MAIN-20190218135032-20190218161032-00280.warc.gz\"}"}
| null | null |
New Victoria (2001 pop.: 1,093) is a community in Nova Scotia's Cape Breton Regional Municipality.
New Victoria is located west of New Waterford and east of Victoria Mines. It is approximately 10 kilometres north of Sydney.
It is a small, close-knit community which has a small church (St. Joseph's Church-Roman Catholic) and a lighthouse named the Low Point Lighthouse. New Victoria is also the site of Fort Petrie, a 20th-century fortification and observation post which functioned during both world wars. The fort is now a museum.
References
Communities in the Cape Breton Regional Municipality
Mining communities in Nova Scotia
General Service Areas in Nova Scotia
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UNNATURAL PRAISE FOR DAN SHAMBLE, ZOMBIE P.I. . . .
"The Dan Shamble books are great fun."
—Simon R. Green
"Sharp and funny; this zombie detective rocks!"
—Patricia Briggs
"A dead detective, a wimpy vampire, and other interesting characters from the supernatural side of the street make Death Warmed Over an unpredictable walk on the weird side. Prepare to be entertained."
—Charlaine Harris
"A darkly funny, wonderfully original detective tale."
—Kelley Armstrong
"Master storyteller Kevin J. Anderson's Death Warmed Over is wickedly funny, deviously twisted and enormously satisfying. This is a big juicy bite of zombie goodness. Two decaying thumbs up!"
—Jonathan Maberry
"Kevin J. Anderson shambles into Urban Fantasy with his usual relentless imagination, and a unique hard-boiled detective who's refreshing, if not exactly fresh. Death Warmed Over is the literary equivalent of Pop Rocks, firing up an original world with supernatural zing, bold flavor, and endlessly clever surprises."
—Vicki Pettersson
"Death Warmed Over is just plain good fun. I enjoyed every minute it took me to read it."
—Glen Cook
"Down these mean streets a man must lurch.... With his Big Sleep interrupted, Chambeaux the zombie private eye goes back to sleuthing, in Death Warmed Over, Kevin J. Anderson's wry and inventive take on the Noir paradigm. The bad guys are werewolves, the clients are already deceased, and the readers are in for a funny, action-packed adventure, following that dead man walking . . ."
—Sharyn McCrumb
"A zombie sleuth prowls the mean streets as he works a half-dozen seriously weird cases . . . Like Alexander McCall Smith's Mma Precious Ramotswe, the sleuths really do settle most of their cases, and they provide a lot of laughs along the way."
—Kirkus Reviews on Death Warmed Over
"Anderson's world-building skills shine through in his latest series, Dan Shamble, Zombie P.I. Readers looking for a mix of humor, romance, and good old-fashioned detective work will be delighted by this offering."
—RT Book Reviews (4 stars)
"Anderson's hilarious, lighthearted tale blends paranormal fiction with hardboiled detective stories to give a unique look at some of the issues that might occur should supernatural creatures populate our world. Anderson doesn't bring readers into his fictional world, he brings fictional characters into our world. The noir-y feel of this hybrid paranormal-mystery story definitely gives it the realism that makes Anderson's world-building entirely believable."
—"Elisa and Janine Dish" in RT Book Reviews
"A good detective doesn't let a little thing like getting murdered slow him down, and I got a kick out of Shamble trying to solve a series of oddball cases, including his own. He's the kind of zombie you want to root for, and his cases are good, lighthearted fun."
—Larry Correia, New York Times bestselling author of the Monster Hunter International series
"Kevin J. Anderson's Death Warmed Over and his Dan Shamble, Zombie P.I. novels are truly pure reading enjoyment—funny, intriguing—and written in a voice that charms the reader from the first page and onward. Smart, savvy—fresh, incredibly clever! I love these books."
—Heather Graham, New York Times bestselling author of the Krewe of Hunters series
. . . AND FOR KEVIN J. ANDERSON
"Kevin J. Anderson has become the literary equivalent of Quentin Tarantino in the fantasy adventure genre."
—The Daily Rotation
"Kevin J. Anderson is the hottest writer on (or off) the planet."
—Fort Worth Star-Telegram
"Kevin J. Anderson is arguably the most prolific, most successful author working in the field today."
—Algis Budrys
"I always expect more from a Kevin J. Anderson tale, and I'm yet to be disappointed."
—2 A.M. Magazine
Also by Kevin J. Anderson
Death Warmed Over
Stakeout at the Vampire Circus
Unnatural Acts
Clockwork Angels
Blood Lite anthology series (editor)
The new Dune novels (with Brian Herbert)
Hellhole series (with Brian Herbert)
The Terra Incognita trilogy
The Saga of Seven Suns series
Captain Nemo
The Martian War
Enemies and Allies
The Last Days of Krypton
Resurrection, Inc.
The Gamearth trilogy
Blindfold
Climbing Olympus
Hopscotch
Ill Wind (with Doug Beason)
Assemblers of Infinity (with Doug Beason)
The Trinity Paradox (with Doug Beason)
The Craig Kreident Mysteries (with Doug Beason)
Numerous Star Wars, X-Files, Star Trek, StarCraft novels,
movie novelizations, and collaborations
Hair Raising
Dan Shamble, Zombie P.I.
KEVIN J. ANDERSON
KENSINGTON BOOKS
www.kensingtonbooks.com
All copyrighted material within is Attributor Protected.
Table of Contents
UNNATURAL PRAISE FOR DAN SHAMBLE, ZOMBIE P.I. . . .
Also by Kevin J. Anderson
Title Page
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
CHAPTER 18
CHAPTER 19
CHAPTER 20
CHAPTER 21
CHAPTER 22
CHAPTER 23
CHAPTER 24
CHAPTER 25
CHAPTER 26
CHAPTER 27
CHAPTER 28
CHAPTER 29
CHAPTER 30
CHAPTER 31
CHAPTER 32
CHAPTER 33
CHAPTER 34
CHAPTER 35
CHAPTER 36
CHAPTER 37
CHAPTER 38
CHAPTER 39
CHAPTER 40
CHAPTER 41
CHAPTER 42
CHAPTER 43
CHAPTER 44
CHAPTER 45
CHAPTER 46
CHAPTER 47
CHAPTER 48
CHAPTER 49
CHAPTER 50
CHAPTER 51
CHAPTER 52
ACKNOWLEDGMENTS
DEATH WARMED OVER UNNATURAL ACTS STAKEOUT AT THE VAMPIRE CIRCUS
Copyright Page
Notes
This novel goes out to my fellow
Superstars Writing Seminar authors
David Farland
Brandon Sanderson
Eric Flint
Rebecca Moesta
Together, we make a great team
trying to figure out
the writing and publishing business
faster than we can teach our students.
CHAPTER 1
People do bizarre things to amuse themselves, but this illegal cockatrice-fighting ring was one of the strangest pastimes I had ever seen.
Rusty, the full-time werewolf who raised the hideous creatures and threw them together in the ring for sport, had hired me to be on the lookout for "suspicious behavior." So, there I stood in an abandoned warehouse among crowds of unnaturals who were placing bets and watching chicken-dragon-viper monstrosities tear each other apart.
What could possibly be suspicious about that?
No case was too strange for Chambeaux & Deyer Investigations, so I agreed to keep my eyes open. "You'll have a great time, Mister Shamble," Rusty said in his usual growling voice. "Tonight is family night."
"It's Chambeaux," I corrected him, though the mispronunciation may have been the result of him talking through all those teeth in his mouth, rather than not actually knowing my name.
Rusty was a gruff, barrel-chested werewolf with a full head—and I mean full head—of bristling reddish fur that stuck out in all directions. He wore bib overalls and sported large tattoos on the meat of his upper arms (although his thick fur hid most of them). He raised cockatrices in run-down coops in his backyard.
Cockatrice fighting had been denounced by many animal rights groups. (Most of the activists, however, were unfamiliar with the mythological bestiary. Despite having no idea what a cockatrice was, they were sure "cockatrice fighting" must be a bad thing from the sound of it.) I wasn't one to pass judgment; when ranked among unsavory activities in the Unnatural Quarter, this one didn't even make the junior varsity team.
Rusty insisted cockatrice fighting was big business, and he had offered me an extra ticket so Sheyenne, my ghost girlfriend, could join me. I declined on her behalf. She's not much of a sports fan.
In the cavernous warehouse, the unsettling ambient noise reflected back, making the crowd sound twice as large as it really was. Spectators cheered, growled, howled, or made whatever sound was appropriate to their particular unnatural species, getting ready for the evening's show. Several furtive humans also came to place bets and watch the violence, while hoping that violence didn't get done to them in the dark underbelly of the Quarter.
This crowd didn't come to see and be seen. I tried to blend in with the other sports fans; nobody noticed an undead guy in a bullet-riddled sport jacket. Thanks to an excellent embalming job and good hygiene habits, I'm a well-preserved zombie, and I work hard to maintain my physical condition so that I can pass for mostly human. Mostly.
Previously, the warehouse had hosted illegal raves, and I could imagine the thunderously monotonous booming beat accompanied by migraine-inducing strobe lights. After the rave craze ended, the warehouse manager had been happy to let the space be used for the next best thing.
The center of attention was a high-walled enclosure that might have been designed as a skateboard park for lawn gnomes. The barricades were high enough that snarling, venomous cockatrices could not leap over them and attack the audience—in theory at least. Although, as Rusty explained it, a few bloodthirsty attendees took out long-shot wagers that such a disaster would indeed happen; those bettors generally kept to the back rows.
While Rusty was in back wrangling the cockatrice cages to prepare the creatures for the match, his bumbling nephew Furguson went among the crowds with his notepad and tickets, taking bets. Lycanthropy doesn't run in families, but the story I heard was that Rusty had gone on a bender and collapsed half on and half off his bed. While trying to make his uncle more comfortable, Furguson had been so clumsy that he scratched and infected himself on the claws. Watching the gangly young werewolf go about his business now, I was inclined to accept that as an operating theory.
The fight attendees held tickets, scraps of paper, and printed programs listing the colorful names of the cockatrice combatants—Sour Lemonade, Hissy Fit, Snarling Shirley, and so on. The enthusiasts were a motley assortment of vampires, zombies, mummies, trolls, and a big ogre with a squeaky voice who took up three times as much space as any other audience member.
I saw werewolves of both types—full-time full-furred wolfmen (affectionately, or deprecatingly, called "Hairballs" by the other type), and the once-a-month werewolves who transformed only under a full moon but looked human most of the time (called "Monthlies" by the other side). They were all werewolves to me, but there had been friction between the two breeds for years, and it was only growing worse.
It's just human, or inhuman, nature: People will find a way to make a big deal out of their differences—the smaller, the better. It reminded me of the Montagues and the Capulets (if I wanted to think highbrow), or the Hatfields and the McCoys (if I wanted to go lowbrow) . . . or the Jets and the Sharks (if I happened to feel musical).
Rusty had asked me to pay particularly close attention to two burly Monthlies, heavily tattooed "bad biker" types named Scratch and Sniff. Even in their non-lycanthropic forms, and even among the crowd of monsters, these two were intimidating. They wore thick, dirty fur overcoats that they claimed were made of Hairball pelts—no, nothing provocative there!—coated with road dust and stained with blotches that looked like clotted blood.
Untransformed, Scratch wore big, bristly Elvis sideburns and a thick head of dark brown hair in an old-fashioned DA hairstyle; apparently, he thought this made him look tough like James Dean, but it actually succeeded only in mimicking Arthur Fonzarelli in his later shark-jumping days. His friend Sniff shaved his head for a Mr. Clean look, but he made up for it once a month when his entire body exploded in thick fur. His lower face, though, was covered with a heavy beard; he had a habit of stroking it with his fingers, then sniffing them as if to remind himself of what he had eaten last. Both had complex tattoo designs on their arms, necks, and probably other places that I did not want to imagine.
Known troublemakers, Scratch and Sniff liked to bash their victims' heads just to see what might come out. They frequently caused problems at the cockatrice fights, but since they placed large bets, Rusty tolerated them.
In recent fights, however, a lot of the money had disappeared from the betting pool, as much as 20 percent. Rusty was sure that Scratch and Sniff were somehow robbing the pot, and I was supposed to catch them. Now, these two struck me as likely perpetrators of all manner of crimes, but they didn't look to be the subtle types who would discreetly skim 20 percent of anything. My guess, they would have taken the whole pot of money and stormed away with as much ruckus as possible.
Furguson wandered among the crowd, recording the bets on his notepad, then accepting wads of bills and stuffing them into his pockets. As he collected money, he took care to write down each wager and record the ticket number. For weeks, Rusty had pored over his nephew's notations, trying to figure out why so much money went missing. He counted and recounted the bills, added and re-added the bets placed, and he simply could not find what was happening to so much of the take.
Which is why he hired me.
Suddenly, the Rocky Balboa theme blared over the old rave speakers that the warehouse owner had confiscated when the ravers stopped paying their rent. Eager fans surrounded Furguson, placing their last wagers in a flurry, shoving money at the gangly werewolf like overcaffeinated bidders on the floor of the New York Stock Exchange.
Now, I've been a private detective in the Unnatural Quarter for years, working with my legal crusader/partner Robin Deyer. We had a decent business until I was shot in the back of the head—which might have been the end of the story, for most folks. But me? I woke up as a zombie, clawed my way out of the grave, and got right back to work. Being undead is not a disadvantage in the Quarter, and the number of cases I've solved, both before and after my murder, is fairly impressive. I pride myself on being observant and persistent, and I have a good analytical mind.
Sometimes, though, I solve cases through dumb luck, which is what happened now.
While Rusty rattled the cages and gave pep talks to his violent amalgamated monsters, the Rocky theme played louder, and frantic last-minute bettors waved money at Furguson. They yelled out the names of their chosen cockatrices, snatched their tickets, and the bumbling werewolf stuffed more wads of cash into his pockets, made change, grew flustered, took more money, stuffed it into other pockets.
He was so overwhelmed that bills dropped out of his pockets onto the floor, unnoticed—by Furguson, but not by the audience members. As they pressed closer to him, they snatched up whatever random bills they could find. In fact, it was so well choreographed, the whole mess seemed like part of the evening's entertainment.
Scratch and Sniff had shouldered their way to the edge of the cockatrice ring, where they'd have the best view of the bloodshed. Despite Rusty's accusations, I could see that the big biker werewolves had nothing to do with the missing money. As the saying goes, never attribute to malice what can be explained by incompetence—and I saw the gold standard of incompetence here.
I let out a long sigh. Rusty wasn't going to like what I'd found, but at least this was something easy enough to fix. His clumsy nephew would have to either be more careful or find another line of work.
The loud fanfare fell silent, and Rusty emerged from the back in his bib overalls. His reddish fur looked mussed, as if he had gotten into a wrestling match with the vile creatures. The restless crowd pressed up against the fighting ring. Rusty shouted at the top of his lungs. "For our first match, Sour Lemonade versus Hissy Fit!"
He yanked a lever that opened a pair of trapdoors, and the two creatures squawked and flapped into the pit. Each was the size of a wild turkey, covered with scales, a head like a rooster on a bad drug trip with a serrated beak and slitted reptilian eyes. The jagged feathers looked like machetes, and sharp, angular wings gave the cockatrices the appearance of very small dragons or very large bats. Each creature had a serpentine tail with a spearpoint tip. Their hooked claws were augmented by wicked-looking razor gaffs (I didn't want to know how Rusty had managed to attach them). Forked tongues flicked out of their sawtooth beaks as they faced off.
I'd never seen anything so ugly—and these were the domesticated variety. Purebred cockatrices are even more hideous, ugly enough to turn people to stone. (It's hard to say objectively whether or not the purebreds are in fact uglier, since anyone who looked upon one became a statue and was in no position to make comparative observations. Scientific studies had been done to measure the widened eyes of petrified victims, with a standard rating scale applied to the expression of abject horror etched into the stone faces, but I wasn't convinced those were entirely reliable results.) Regardless, wild turn-you-to-stone cockatrices were outlawed, and it was highly illegal to own one. These were the kinder, gentler breed—which still looked butt-ugly.
One creature had shockingly bright lemon-yellow scales—Sour Lemonade, I presumed. The other cockatrice had more traditional snot-green scales and black dragon wings. It hissed constantly, like a punctured tire.
The two creatures flapped their angular wings, bobbed their heads, and flicked their forked tongues like wrestlers bowing to the audience. The crowd egged them on, and the cockatrices flung themselves upon each other like Tasmanian devils on a hot plate. The fury of lashing claws, pecking beaks, and spitting venom was dizzying—not exactly enjoyable, but certainly energetic. I couldn't tear my eyes away.
Sour Lemonade's barbed tail lashed out and poked a hole through Hissy Fit's left wing. The other cockatrice clamped its serrated beak on the yellow creature's scaly neck. Claws lashed and kicked, and black smoking blood spurted out from the injuries. Where it splashed the side of the pit ring, the acid blood burned and bubbled.
One large droplet splattered the face of a vampire, who yelped and backed away, swatting at his smoking skin. Scratch and Sniff howled with inappropriate laughter at the vampire's pain. The spectators cheered, shouted, and cursed. The cockatrices snarled and hissed. The sound was deafening.
Just then the warehouse door burst open, and Officer Toby McGoohan entered, wearing his full cop uniform. "This is the police!" he shouted through a bullhorn. "May I have your attention—"
The ensuing pandemonium made the cockatrice fight seem as tame as a Sunday card game by comparison.
CHAPTER 2
Shouting "Fire!" in a crowded theater is a well-known recipe for disaster. Shouting "Cops!" in the middle of an illegal cockatrice fight is ten times worse.
Officer McGoohan—McGoo to his friends (well, to me at least)—reeled in surprise at the chaos his appearance caused. His mouth dropped open as he saw dozens of unnaturals, already keyed up with adrenaline and bloodlust from the cockatrice fight, thrown into a panic.
"Cops! Everybody run!" yelled a vampire with a dramatic flourish of his cape. He turned and ran face-first into a hulking ogre. In reflex, the ogre flung the vampire against the pit ring with enough force to crack the barricade. Inside, the cockatrices still thrashed and hissed at each other.
Several zombies shambled at top speed toward the back exit. The human spectators bolted, ducking their heads to hide their identities. A bandage-wrapped mummy tripped and fell. Other fleeing monsters stepped on him, trampling his fragile antique bones and sending up puffs of old dust. After the mummy got back to his feet, the clawed foot of a lizard man caught on the bandages, and the linen strips unraveled as he ran.
At the door, McGoo waved his hands and shouted, "Wait, wait! It's not a raid!" Nobody heard him, or if they did, they refused to believe.
The skittish ogre smashed into an emergency-exit door, knocking it off its hinges. The door crashed to the ground outside, and monsters fled into the dark alleys, yelling and howling.
McGoo waved his hands and urged calm. He might as well have been asking patrons in a strip club to cover their eyes. As I headed toward him, I saw that he hadn't even brought backup.
Gangly Furguson ran about in confusion, bumped into unnaturals, and caromed off them like a pinball. Scratch and Sniff looked at each other and shared a grin. As Furguson ran within reach, they grabbed the skinny werewolf and used his own momentum to fling him into the newly cracked pit wall, which knocked down the barricade. A blizzard of haphazard currency flew out of Furguson's pockets.
The cockatrices broke free and bounded out of the pit, slashing with razor gaffs at anything that came near. They were like hyperactive whirlwinds, flailing, attacking. Sour Lemonade latched its jaws onto the shoulder of a zombie who couldn't shamble out of the way quickly enough. Hissy Fit swooped down and attacked a vampire that had been burned by acid blood; the vamp flailed his hands to get the beast away from his neatly slicked-back hair.
Taking matters into his own hands, Rusty grabbed Hissy Fit by the scaly neck, yanked it away from the vampire, stuffed the cockatrice into a burlap sack, and cinched a cord around the opening. "Furguson, get the other one! We better get out of here!"
Furguson, however, flew into a rage at Scratch and Sniff for bashing him into the wall. He bared his fangs and howled, "Shithead Monthlies!" Hurling himself on Scratch, the nearer of the two, he raked his long claws down the werewolf-pelt overcoat, ripping big gashes. With his other hand, he tore four bloody furrows in the biker werewolf's cheek.
Sniff plucked Furguson away from his friend and began punching him with a pile-driver fist. Scratch touched the blood from the wounds on his cheek, and his eyes flared. Oddly, the tangle of tattoos on his neck and face began pulsing, writhing, like a psychedelic cartoon—and the deep wounds on his face sealed together. The blood coagulated into a hard scab that flaked off within seconds, the flesh knitted itself into scar tissue, and the tattoos fell quiescent again.
I had never seen a body-imprinted healing spell before. Very cool.
Amidst the pandemonium, I reached McGoo. He looked at me in surprise. "Shamble! What are you doing here?"
"Working. What about you?"
"I'm working, too. Just answering a disturbance call, Scout's honor. Your friends sure have hair triggers! Did I catch them doing something naughty?"
Rusty tore a two-by-four from the cockatrice barricade and waded in toward Scratch and Sniff. He whacked each of the biker werewolves on the back of the head, which left them reeling while he pulled his nephew from the fray. He shoved the burlap sack into Furguson's claws. "Take this and get out of here! I'll grab the other one."
With the struggling, squirming sack in hand, the gangly werewolf bolted for the nearest door. Bills were still dropping out of his pockets as the poor klutz disappeared into the night.
The two biker werewolves shook off the daze from being battered by a two-by-four. They puffed themselves up, peeled back their lips, and faced Rusty, but the big werewolf swung the board again, cracking each man full in the face. "Want a third one? Believe me, it'll only improve your looks."
I looked at McGoo. "We may need to intervene."
"I was just thinking that." He sauntered forward, displaying the arsenal of unnatural-specific weapons that he carried on his regular beat in the Quarter.
Both biker werewolves snarled at Rusty. "Damned Hairball!" After another round of hypnotically twitching tattoos, the bloody bruises on their faces healed up.
McGoo stepped up to the troublemakers and said, "Say, know any good werewolf jokes?"
After a glance at his uniform, Scratch and Sniff snarled low in the throat, then bolted into the night as well. Outside, I heard a roar of motorcycle engines.
The less panicky, or more enterprising, spectators scurried around and grabbed fallen money from the floor, before they, too, darted out of the building. Rusty managed to seize Sour Lemonade from the zombie it had chomped down on and stuffed it into another cloth bag, which he slung over his shoulder. He loped away from the warehouse, exiting through the emergency door the ogre had shattered.
McGoo and I caught our breath, exhausted just from watching the whirlwind. He shook his head as the last of the monsters evacuated into the night. "This place looks like the aftermath of a bombing raid. Mission accomplished, I suppose."
"What mission was that?" I asked.
"One of the nearby residents called in a noise complaint. She's some kind of writer, asked me to see if they could keep the noise down, said the racket was bothering her."
"That was all?"
"That was all." McGoo shrugged, looking around the evacuated warehouse. "Should be quiet enough now."
McGoo is my best human friend, my BHF. We've known each other since college, both married women named Rhonda when we were young and stupid; later, as we got smarter, we divorced the women named Rhonda and spent a lot of guy time commiserating. I established my private-eye practice in the Unnatural Quarter; McGoo, with his politically incorrect sense of humor, managed to offend the wrong people, thus derailing a mediocre career on the outside in exchange for a less-than-mediocre career here in the Quarter. A good friend and a reliable cop, he made the best of his situation.
It took McGoo quite a while to learn how to deal with me after I was undead. He wrestled with his own prejudices against various types of monsters, and, thanks to me, he could honestly say "Some of my best friends are zombies." I didn't let him use that for any moral high ground, though.
As we surveyed the aftermath, he said, "I'd better go talk to the lady, let her know everything's under control."
"Want me to come along? I could use some calmer interaction after this." I had, after all, solved the case of the missing money, but I decided to wait for the dust to settle before I presented my results to Rusty.
I found a twenty-dollar bill on the floor and dutifully picked it up.
McGoo looked at me. "That's evidence, you know."
"Evidence of what? You were called here on a disturbing-the-peace charge." As we walked out of the warehouse, I tucked the bill into the pocket of my sport jacket. "That's our next couple of beers at the Goblin Tavern."
"Well, if it's going to a good cause, I'd say you were doing your civic duty by picking up litter."
"Works for me."
Behind the warehouse, we found a set of ramshackle apartments; I saw lights on in only two of the units, though it was full dark. A weathered sign promised "Units for Rent: Low Rates!" Low Rates was apparently the best that could be said about the place.
McGoo led me up the exterior stairs to an upper-level apartment. When he rapped on the door, a woman yanked it open, blinking furiously as she tried to see out into the night. "Stop pounding! What's with all this noise? I've already filed one complaint—I can call the police again!"
"Ma'am, I am the police," McGoo said.
The woman was a frumpy vampire, short and plump, with brown hair. She looked familiar. "Then you ought to be ashamed of yourself," she said. "The noise only got worse after I called! It was a mob scene out there."
She plucked a pair of cat's-eye glasses from a chain dangling around her neck and affixed them to her face. "How can I get any writing done with such distractions? I have to finish two more chapters before sunrise."
I knew who she was. I also knew exactly what she was writing. "Sorry for the interruption, Miss Bullwer."
McGoo looked at me. "You know this woman?"
"Who's that looming out there on my porch?" The vampire lady leaned out until she could see me, then her expression lit up as if a sunrise had just occurred on her face (which is not necessarily a good thing when speaking of vampires). "Oh, Mister Chambeaux! How wonderful. Would you like to come in and have a cup of . . . whatever it is zombies drink? I have a few more questions, details for the veracity of the literature. And you can pet my cats. They'd love to have a second lap. They can't all fit on mine, you know."
"How many cats do you have?"
"Seven," she answered quickly. "At least, I think it's seven. It's difficult to tell them apart."
I'd first met Linda Bullwer when she volunteered for the Welcome Back Wagon, a public service group that catered to the newly undead. More importantly, she had been commissioned as a ghostwriter by Howard Phillips Publishing for a series of zombie detective adventures loosely based on my own exploits, written under the pen name of Penny Dreadful. The first one was due out very soon.
Miss Bullwer gave McGoo a sweet smile, her demeanor entirely changed now. "And thank you for your assistance, Officer—I'm sure you did your best, especially with Mister Chambeaux's help." She cocked her head, lowered her voice. "Are you working on a case? Something I should write about in a future volume?"
"I doubt anyone would find it interesting," I said.
"That's what you said about the tainted Jekyll necroceuticals, and about the mummy emancipation case, and the Straight Edge hate group, and the attack on hundreds of ghosts with ectoplasmic defibrillators, and the burning of the Globe Theatre stage in the cemetery, and the golem sweatshop, and . . ."
I knew she could rattle off cases for hours, because I had spent hours telling her about them. She had taken thorough notes.
"It's nothing," I reassured her. "And we don't even know if your first book is going to sell well enough that the publisher will want to do a second one."
"They've already contracted with me for five, Mister Chambeaux. The first one is just being released—have you gotten your advance copy yet?"
"I'll check the mail when I get back to the office." I have to admit, I was uncomfortable about the whole thing. Vampires shun sunlight, and I tended to avoid limelight.
McGoo regarded me with amusement. "I believe we're done here, ma'am. Enjoy the rest of your quiet night."
"Thank you, Officer. And thank you, Mister Chambeaux, for your help." She drew a deep breath. "Ah, blessed silence—at last I can write!"
Just then, an anguished howl split the air, bestial shouts barely recognizable as words. "Help! Help me! Help!"
McGoo was already bounding down the stairs, and I did my best to keep up with him. We tracked the cries to a dark alley adjacent to the warehouse. A gangly werewolf leaned over a figure sprawled on the ground, letting out a keening howl. Beside him, two squirming cloth sacks contained the captured cockatrices; fortunately, the ties were secure.
As we ran up, Furguson swung his face toward us, his eyes wide, his tongue lolling out of his mouth. "It's Uncle Rusty!"
I recognized the bib overalls and reddish fur. The big werewolf lay motionless, sprawled muzzle-down in the alley.
"Is he dead?" McGoo asked.
Bending over, I could see that Rusty's chest still rose and fell, but he was barely conscious. The top of his head was all bloody, and it looked wrong.
Furguson let out another wail.
That's when I realized that someone must have knocked Rusty out and then, using a very sharp knife, scalped him.
CHAPTER 3
Next morning, when I got into the office, I discovered that I was suddenly a very famous zombie PI, whether I wanted to be or not.
It had been a rough night (for the undead, what else is new?). I looked more run-down than usual, or maybe "run over" is a better way to put it. While McGoo and I had combed the crime scene for evidence, an ambulance took the unconscious Rusty to the Brothers and Sisters of Mercy Hospital, the nearest medical center that could handle unnaturals. He had been stunned by an unseen assailant using a Taser and darts filled with horse tranquilizer mixed with wolfsbane. With his werewolf healing powers, Rusty would recover from the scalping—probably faster than his frantic nephew managed to calm down—although it might be a while before he regrew a proper head of hair.
As I shambled through the office door, Sheyenne was waiting for me with a big smile on her face and a sparkle in her blue eyes. Seeing her is always a good way to start a morning. I wanted to kiss her so much that I almost forgot she was insubstantial. Romance among the undead poses more than the usual share of problems.
For all her beauty and voluptuous curves, my girlfriend is a ghost, and my days of touching her whenever I wanted to were long gone. Though we had tried a few unorthodox workarounds, mostly I content myself with remembering the brief time we had together. Now her smile glowed with enough warmth to melt even the coldest ghostbuster. I knew that something was up.
"Special delivery box arrived at first light, Beaux"—that was her pet name for me—"direct from Howard Phillips Publishing. You already know what it is!" She opened the box and withdrew a copy of a book with a garish red cover. She can move objects when she puts her mind to it, but not living—or formerly living—things.
The title said, in blood-dripping letters (of course):
DEATH WARMED OVER
A Shamble & Die Mystery
I took the book from her with very little enthusiasm. "Shamble? Great, now I'll never get rid of that nickname."
"It's not so bad," Sheyenne said. "Time to move on."
I glanced up at her. "A ghost is telling me it's time to move on?"
This was the first book in the line of Penny Dreadfuls ("Now only $14.99!"). The cover art showed a sunken-eyed zombie pointing a .38 toward some unseen criminal. He wore a trench coat and a low-slung fedora that did not entirely cover the bullet hole in his forehead. Now, I had been shot in the head and I do wear a fedora (but a sport jacket, not a trench coat), but I thought I was much better-looking than this guy.
"That doesn't look at all like me." I opened the book to a random chapter and found a lurid sex scene . . . something I definitely did not remember from my real adventures. "I suppose these are going to be distributed all over the Quarter?"
"All over the country, if you can believe what Mavis Wannovich says." Sheyenne seemed delighted. "We couldn't buy advertising like this, Beaux. A private investigator who's so dedicated that even death can't keep him from his cases, partnered with a bleeding-heart human lawyer who seeks justice for all unnaturals." She picked up another copy from the box and flipped through it. "And let's not forget the detective's gorgeous girlfriend, who came back as a ghost because her love for him was so strong."
"I thought you came back to make sure I solved your murder."
"Poetic license," she said. "Don't spoil the story."
Robin emerged from her office, her nose in a copy of the novel. She looked pretty and studious. Robin Deyer—the "Die" part of the novel's fictitious "Shamble & Die"—was a young African American woman who crusaded for the rights of monsters, tackling cases that normal lawyers wouldn't consider.
"Thrilling adventures, sure, but I see it as a good platform, too. Maybe this book will call attention to the plight of unnaturals," she said. "Did you know that zombies and vampires aren't even allowed to vote? Werewolves—at least the Monthly ones—easily pass for human, unless voting happens to occur during a full moon. There was a recent case when a full-furred werewolf was turned away from the polling center based strictly on his appearance." She tapped the cover of the novel, considering. "Great works of literature can effect social change."
"I doubt this is great literature, Robin," I said.
Sheyenne was put off by my reaction. "Don't be such a curmudgeon, Beaux! Other people enjoy their fifteen minutes of fame. Look at all of those reality TV shows—Survivor: Zombie Apocalypse and Transylvania Extreme Makeover."
"That's what I'm afraid of. I'm a detective, not a celebrity." Fame and publicity could ruin my career as a private investigator. It's hard to shadow a suspect discreetly if your face is plastered all over the news.
I could see it all now. I wouldn't be able to make a move without the paparazzi shadowing me; I'd be stalked by crazy fans, get inundated with hate mail or marriage proposals. Somebody might interview me about my life story and my death story, then there'd be a scandal and an exposé—Behind the Embalming Fluid. I could become a paid commentator on news networks, offering opinions on particularly heinous cases. Fame put stars in some people's eyes, but I preferred just to solve cases and help one client at a time.
Still, I couldn't deny the excitement on Sheyenne's face as she ran through the possibilities, so I sighed and made the best of it. "All right, Spooky. With any luck, nobody will read the book, and the series will die quietly, end up on the remainder pile. Is there really an audience for this sort of book?"
Sheyenne chuckled. "You don't come out of your crypt very often, do you?"
"There's definitely an audience," Robin added. "Even I know about the big Worldwide Horror Convention coming to the Quarter. It's going to be at the Bates Hotel—they're expecting hundreds of fans, celebrity guests, media."
I had forgotten about that. "All right, I admit that I have no social life."
"We're working on that," Sheyenne said with a smile. Using her poltergeist concentration, she lifted the books out of the box and stacked them on her desk. "Fifty copies."
"What am I supposed to do with so many books?" Sheyenne probably had a scheme to give away copies as thank-you gifts to satisfied clients.
Robin picked up a pen from the desk. "These are the special limited edition. You and I both need to sign them, and they'll be sold at a charity auction to raise money for Mrs. Saldana's zombie rehab clinic."
"People are going to pay money for our autographs?" I asked.
"Good money, we hope." Robin scribbled on a sheet of scratch paper, then signed inside the first book with a flourish. "Now that I'm a board member of the Monster Legal Defense Workers, I need to show them my support in every possible way."
I grabbed a pen from a cup on Sheyenne's desk and began autographing the books as Robin finished them. Sheyenne opened the cover and held the title page out for me as I scrawled. Zombie fingers have little dexterity, but even my pre-death signature had been illegible. I thought of rock stars and famous actors swarmed by fans as they scribbled their autographs before being whisked off by chauffeurs to the next destination. People like that never had a minute's peace, and no normal life. I preferred relaxing and having a beer or two with McGoo at the Goblin Tavern.
As Sheyenne repacked the signed books in the box, I said, "I'm due for my refresher rejuvenation spell tomorrow. I'll deliver these to Mavis and Alma." The two witch sisters, former clients of ours, now worked as acquisitions editors for Howard Phillips Publishing. Sheyenne had negotiated terms with the two: In exchange for my services as consultant on their Shamble & Die series, I would receive regular magical touch-ups to keep my body in adequate shape—much better than most zombies. ("Pristine" was out of the question, so "adequate" was the best I could do.)
Early onset of decay is a serious problem for zombies. Diet doesn't matter, and exercise can do only so much to keep the muscles limber. In my line of work, I get battered more often than I want to admit. (In fact, my arm was torn off once when I fought with a gigantic monster, but the arm had been successfully reattached.) Each time Mavis Wannovich worked her magic, I felt fresh as the day I was buried, so I didn't begrudge them a little inside information for the Penny Dreadful series. A deal was a deal. As a bonus, I would return the signed books to them in person.
For now, though, I had to get ready for a new client conference. The cases don't solve themselves.
CHAPTER 4
After the mayhem last night, it was a relief to get down to business and meet with the medical examiner. And this guy wasn't just the coroner—he was a Chambeaux & Deyer client as well.
Archibald Victor (emphasis on the bald) was an intense man with pallid skin and a cadaverous complexion. He was a diminutive, even goblinesque, human of the mad-scientist variety, with eyes as large and round as ping-pong balls and thin skittering fingers that might have made him a good pianist, though he devoted his dexterity to dissection instead.
He was completely hairless, yet for some reason—I was embarrassed to ask him why—Dr. Victor wore a dark toupee on top of his scalp, meticulously combed and moussed. It lay on his head like roadkill, obvious to the point of incomprehensibility.
Robin and I sat at the conference table. "How can we help you, Dr. Victor?" she asked, ready to take notes on her yellow legal pad.
The coroner set two pickle jars on the table in front of Robin and me. One jar contained floating blobby tissue, a discolored organ that I could not identify; biology had never been my best subject in school. The larger jar held a rippled grayish mass that was obviously a human brain. He looked intently at us, narrowing his too-large eyes, keeping one hand on the jars, as if to protect them. "Before I engage your services, I must know that I can count on your absolute discretion. This is a private matter, ahem, and I wish it kept private."
"That's why I call myself a private investigator, Dr. Victor."
"And I'm bound by attorney-client privilege," Robin said. "You can speak freely."
"Good, good." He fluttered his fingers. "I'm not ashamed of my hobby—lots of people do it. Nothing abnormal about eccentric behavior."
The little coroner had my attention now. I owed him a favor anyway: During the autopsy of Snazz the gremlin, this man had proved that even though I'd been found at the scene of the murder, I was not responsible for killing the furry pawnbroker.
"We'll keep this professional, Dr. Victor." Robin clicked her pen. "How may we help you?"
He seemed embarrassed. "My wife says I'm obsessed, but in an endearing sort of way. You see, ahem, bodybuilding is a passion of mine."
I looked at his scrawny, pallid form—he was the last person I'd expect to find in a gym. And here I thought Archibald was going to say he liked to dress up in women's clothing. I wondered what strange hobby he was talking about.
McGoo had told me of a case where a psychotic dentist became obsessed with collecting human teeth; he'd only been caught because the insurance company complained that too many patients were filing claims for full sets of dentures after coming in for a routine teeth cleaning. When an insurance claims adjuster investigated, he was incapacitated with nitrous oxide and woke up giggling in the dentist's chair with bloody gums and no teeth. The dentist had tried to flee, but was caught at the airport because all the fillings in the jars of stolen teeth had set off the metal detector....
Archibald Victor tapped his spidery fingers on the conference table. "How to explain this to someone who won't understand? You know how some people like to put together models of sailing ships?"
With a wistful smile, I said, "I built a few of those plastic molded kits. You snap them together or use model cement." I let out a sigh. Ah, to be a kid again! "We used to float them in the duck pond and blow them up with firecrackers."
Archibald blinked his large eyes, touched his toupee as if to make sure it was still there, then said, "Ahem, I was talking about something more involved than that. Truly dedicated hobbyists build the ships piece by piece, sand and paint every wooden part, tie the rigging rope with tweezers, cut and string the sails. A meticulous craftsman can take up to a year to build a single model."
The hairless coroner grew even more animated. "Now, think of a hobbyist who builds a ship in a bottle, using tweezers for every step—it requires the patience of a Zen master. And when he's done, ah, the sense of pride and accomplishment . . ." He drew a deep breath, let it out slowly. "The challenge of what I do for fun is ten times harder."
I raised my eyebrows. "Bodybuilding?"
"Yes, ahem, I've become quite adept with 'build your own person' kits. You assemble a human being from the skeleton up, organ by organ, stitch by stitch."
Robin set her pencil down, concerned. "I thought that was illegal. Weren't there raids on a few mad scientist laboratories?"
"Not exactly illegal—it's still a gray area of the law," the coroner said. "I'm familiar with the cases you mentioned, but those were extraordinary circumstances, and the mad scientists in question are not members in good standing of the guild. They should have known not to use non-voluntary body parts and organs. They give us all a bad name."
"Non-voluntary body parts?" I asked. "Who'd give voluntary body parts?"
"Oh, there are sources," Archibald said. "Usually mail-order catalogs or websites."
In a short break between answering the incessant phone calls—the press release about Death Warmed Over had done its work—Sheyenne entered the conference room carrying coffee for me and the coroner, green tea for Robin. Sheyenne looked at the pickle jars on the table, glanced at the preserved brain, then focused on the lumpy organ.
"Oh, a spleen," she said. "But an odd-looking one. Is it damaged?"
Archibald brightened. "Why, yes! That's exactly what I mean—it's defective. I did not get what I ordered. And though I've complained to the company, I received no satisfaction."
"You ordered a spleen from a catalog?" Sheyenne asked.
"Online. The warehouse is local, but I prefer to remain anonymous, get my components in unmarked packages delivered right to my home. It's Tony Cralo's Spare Parts Emporium, budget prices for collectors, researchers, hospitals. It's quite a booming market, ahem."
"You learn something new every day," I said.
"At least every other day," Robin added.
"I'm building two models right now at home, getting ready for a big body-building competition coming up. I received an honorable mention last year. But when the wrong piece shows up, it throws everything out of whack." He tapped the jar lid. "This spleen is in such bad condition, even your assistant spotted it at first glance."
"I did take two years of med school," Sheyenne said.
Archibald looked at Robin. "So, I want to enlist your services, ahem. Can you get my money back and stop the company from hurting other people? Look at this!" When he lifted the larger container, the formaldehyde sloshed from side to side, and the disembodied brain bobbed as if it were nodding. "This is shit for brains! Not mint condition by any means. Look at the medulla oblongata! And can you see the damage there to the cerebellum?" He made a sound of disgust. "Looks like somebody removed the brain with a meat cleaver and a set of bacon tongs."
Sheyenne nodded. "He's right. The damage is quite apparent."
Robin said, "We'll look into the matter, Dr. Victor. We handle consumer complaints as well, and we have several options. We can threaten to bring bad publicity for the Cralo Spare Parts Emporium, maybe arrange a boycott among mad scientists."
"Oh, no, no, no!" Archibald wrung his hands. "This has to be discreet. I'd be laughed at in my job, and I couldn't take the shame. I want people to think of me as a perfectly normal person, ahem." He bowed his head, and the ridiculous toupee nearly fell off. "If you come to my home, I'll show you my kits. You'd understand better."
"I could drop by, if I'm out in that area," I said.
Robin tapped her pen on the yellow legal pad. "We'll contact the company first, visit the manager, put them on notice. They need to improve their quality control, or they could lose a large part of their customer base. Don't worry, Dr. Victor—there's usually some quick way to avoid a misunderstanding. We file a lawsuit only as a last resort."
I added, "I'll see that you get a replacement brain and spleen, Dr. Victor. Maybe even convince them to throw in a free set of lungs while we're at it."
"Oh, that would be nice."
"We can't work miracles," Robin cautioned, "but we'll do what we can."
The phone rang again, and Sheyenne flitted out to answer it. She rushed back into the conference room through the wall, ignoring the door, and from her expression I could tell that the call wasn't just another book reviewer hoping for a free copy of Death Warmed Over.
"Officer McGoohan's looking for the medical examiner. There's been a murder—and even he thinks it's unusual."
CHAPTER 5
Although death isn't necessarily permanent since the Big Uneasy, it still isn't pretty. And if this murder scene was gruesome enough to disturb even McGoo, I knew it wasn't going to be a picnic.
I gave the coroner a ride in Robin's rusty Ford Maverick, affectionately known as the Pro Bono Mobile. I invited Robin to join us, but she's squeamish about seeing murdered bodies in situ. For someone who takes pride in being called a bleeding heart, she has no stomach for real blood.
But I didn't press the issue—I'd put her through enough already. Even after all these months, I still felt guilty that she'd been the one called down to the morgue to identify my body after I was shot in an alley. You'd think that having a bullet through your brain would be the worst part, but Robin and Sheyenne felt the hurt of my dying much more acutely than I did.
Personally, I don't remember any pain whatsoever, it was so fast, although it was damned uncomfortable to wake up underground in a dark and stuffy coffin, then have to work my way out. It took forever! I couldn't breathe (not that I needed to), and by the time I poked my head back out of the fresh sod, I thought that newly turned earth had never smelled so sweet....
With Archibald Victor seat-belted in, I drove us over to the Motel Six Feet Under, known for its slogan "Dirt Cheap for a Dirt Nap: We'll leave the lights off for you." The coroner worried far too much about being seen arriving with me. "How are we going to explain to Officer McGoohan that I was at your place of business? You promised you'd be discreet. You can't tell him that I hired you!"
"Don't worry about it," I said. "There are a million reasons why a coroner could be at Chambeaux and Deyer Investigations."
Archibald wove his pale fingers together, then extricated them, then tangled them up in a different way. "But what are we going to say? What will you tell him?"
"The correct answer," I said as I pulled into the parking lot of the Motel Six Feet Under. "That it's none of his damn business."
Two police units were in the lot, lightbars flashing; a crowd had begun to form. An ambulance was already there, but the two EMTs lounged against the side, smoking cigarettes. They obviously had nothing to do, always a bad sign.
As Dr. Victor and I got out of the car, an approaching coroner's wagon rocked and swayed as it took the curves too fast, roaring down the block toward the motel, scattering curious pedestrians. The wagon carried the coroner's assistants, three ghouls who enjoyed their job far too much.
"We'd better hurry inside and have a look at the crime scene," Archibald said. "My boys tend to get so excited by a particularly gruesome setup that they scramble evidence." The body wagon pulled up with a screech of brakes, slewed sideways, and took up two spots.
The Motel Six Feet Under was a typical seedy place where rooms could be rented by the hour, week, or phase of the moon. This motel had a special relationship with the police department, due to the sheer frequency of crimes committed there. Four spots were reserved in the parking lot for handicapped parking, one for a police unit, and one for the coroner's wagon.
With the commotion clustered around the bright red door of Room #3, we assumed that was the place to go. Archibald and I stepped into the room and immediately smelled blood, large quantities of it, as if someone had purchased a double family pack of the red stuff from a price warehouse club. The coroner's concerns about being seen with me vanished with the whiff of murder and violence. McGoo wasn't concerned about somebody's private life when he had a more-than-disemboweled vampire corpse to occupy his attention.
Archibald had left his pickle jars of organs in Robin's car and now carried his standard medical bag and tool kit. He entered the crime scene and spoke sharply, maybe to distract McGoo. "What's the situation, Officer?"
"Dead vampire found on the bed." McGoo looked relieved to see me—not for the help I could provide, but just to have a friendly face around.
"Don't see many dead vampires," Archibald said.
McGoo gestured, and we saw the corpse of a fish-belly-white vampire male manacled to the bed with a set of silver handcuffs. He'd been cut open, his insides scooped out like a Halloween pumpkin.
"His heart's gone, along with everything else," McGoo said. "I guess they wanted to be sure they took all the right organs."
Archibald bent over the body. "The technical term is that he was disorganized."
The incision went from the navel to the neck. The vampire was flaccid, like a deflated balloon, because nothing remained inside—no heart, no lungs, no stomach or digestive tract, no liver, no kidneys, not even a spleen (defective or otherwise). The top of his skull had been hacked off, and his brain was gone, as if removed with a large ice-cream scoop.
"Not the way you normally see a vampire killed," I said.
"Not the most efficient way, certainly," the coroner muttered. "Since vampires heal rapidly, this must have been a lot of work. But they held him in place with the silver handcuffs, so they could take their time removing every organ. And there are limits to what even a vampire can survive."
I gave my well-considered assessment. "Eww."
"Somebody sure as hell had a grudge, Doc," McGoo said. "Do I really need a complete autopsy so you can state the cause of death?"
"I can make my preliminary assessment on the spot," Archibald said.
I had heard of people harvesting a kidney from an unsuspecting victim picked up at a singles bar. I also knew that some unscrupulous vampires would intoxicate a mark, drain a couple pints of fresh blood, then let them wake up in a room without even orange juice and cookies to help them recover.
This, though, was something entirely different.
The three ghoul assistants from the coroner's wagon shouldered their way through the door, bright-eyed and jabbering. "Nice room!"
"I wonder if they have free breakfast," said another.
The third looked at the vampire corpse on the bed. "Looks like lunch is ready."
Archibald scolded them. "Boys, be professional!"
"We're ready to take the body," said one of the ghouls. "We even hosed out the back of the wagon."
All three hurried to the bed, frowning down at the inconvenient silver handcuffs that held the body in place. "Nothing's ever easy. Should we go get a saw?"
The motel manager had been hovering outside the door, wringing his hands, trying to cope with the disaster. He, the policemen, and the coroner were all on a first-name basis. "If you cut up my headboard, you're going to pay for the damages!"
"We wouldn't cut the nice headboard. We were going to go straight through the wrists."
McGoo got out his own handcuff keys. "Standard issue. These should work." The cuffs popped open, and the dead vampire's limp wrists flopped to the bed.
"You are going to need new sheets, though," Archibald said to the manager. "And a mattress."
"No, that'll wash out. I've got a really good detergent that I buy in bulk. And the mattress has a liner—do you think I'm nuts?"
"The sheet is evidence. We keep it," McGoo said. "You know the drill by now. We try not to disrupt your business much."
"That's all right." The manager looked around, appraising the crime scene. "Murder rooms go for a premium. I can charge an extra ten bucks a night now."
"Everybody's happy, then—except for this guy," McGoo said.
The three ghouls wrapped up the eviscerated vampire in the bloodstained sheets (saving the expense of a fresh body bag) and carried the body out of the room. They knocked a lamp off the nightstand, and the manager rushed forward, yelling for them to be careful.
Seeing a vampire victim harvested of organs was certainly novel. I glanced at Archibald, remembered why he had come to Chambeaux & Deyer in the first place. I needed to have a look at that spare parts emporium as soon as possible.
CHAPTER 6
A gruesome murder scene doesn't usually whet my appetite, but it was lunchtime when I left the Motel Six Feet Under, and I decided it would be best to visit Cralo's body-parts warehouse with a full stomach and plenty of caution. After seeing the state of the vampire's corpse, it occurred to me that Tony Cralo might be involved in something far more sinister than shipping substandard organs to collectors like Archibald Victor.
Also on the day's to-do list, wrapping up a case: I had to tell Rusty what I'd learned about his missing gambling money during the cockatrice fights. Sheyenne let me know that he had already been released from the hospital and was recovering at home; werewolves are tough creatures, and Rusty, a gang leader of the Hairballs, was tougher than most. Given the assault he had suffered last night, I doubted a few missing dollars were his highest priority.
I stopped at the Ghoul's Diner for a cup of miserably bad coffee and some semblance of lunch. The sign in the window said SORRY, WE'RE OPEN. I also noticed a flyer taped to the glass door that said Welcome Worldwide Horror Convention! Yes, We Serve Humans!
The diner bustled and burbled with the lunch trade, and I settled for a solo stool at the counter, not wanting to hog one of the booths, which would have drawn the wrath of Esther, the harpy waitress. (Just being a customer was usually enough to annoy Esther.)
I cast a cursory glance at the menu and the chalkboard special without much interest—it would all taste the same to me. Albert Gould, the owner, had posted a sign on the coffeemaker, THESE GROUNDS ARE CURSED!
I pulled out a piece of scrap paper and began jotting random thoughts while I waited for somebody to take my order. In the back, Albert—a well-ripened ghoul—cooked, sweated, and oozed tendrils of runny slime into the customers' orders.
I heard the chatter of one-sided conversation as a delivery-truck driver brought in the day's load of canned goods straight from the rendering plant. The driver was a zombie in fairly good condition; he wore a green uniform and green trucker's cap. When he dropped a load of cardboard boxes on the counter, rats and cockroaches scurried out of the way. Albert turned with amazing swiftness for such a sluggish guy, using his forearm in a quick matter-of-fact gesture to sweep the bugs and rodents into a large stewpot. He stirred, then covered it with a big metal lid.
Esther came behind the counter and glared at me. Before I could say a word to her, the harpy waitress grabbed a ceramic mug and dropped it in front of me with a clatter. She splashed it half-full with the foul-smelling coffee, while also pouring a puddle across the flecked countertop.
Black and oily-green feathers stuck out from her arms like machete blades. Her hands terminated in obsidian talons, which she could use to hook several coffee mugs at a time, or snag the collars of customers who tried to depart without leaving a sufficient tip. Esther had a pointed, angular face with sharp teeth and an even sharper temper. When she skewered you with her bird-of-prey eyes, you wondered why you had come to the diner in the first place. Even so, her pale blue waitress uniform and white apron hid voluptuous and intriguing feminine curves. Unwittingly, she made you think about sex, and then made you shudder because you thought of it.
In a voice that sounded like a screech, she said, "I bet you think you're special."
In the kitchen, Albert perked up. "Special!" he slurred, and plopped a plate on the counter. "Last one."
"Just mix up some other slop," Esther called back, then she sneered at me. "Mister Famous Private Eye probably wants something that smells French."
I felt a cold dread in my stomach. "What makes you think I'm famous?"
Esther squawked again. "I saw the piece in the paper. Albert's going to want an autographed black-and-white photograph to hang on the wall." She flounced to the chalkboard listing the lunch specials and used one of her curved talons to scrape off the words with a flesh-cringing shriek of nails on slate.
I sipped my coffee while Esther bothered other customers. Maybe this wasn't such a good place to gather my thoughts after all. Investigating a suspicious body-parts emporium was sounding more and more like fun compared to lunch.
The zombie delivery-truck driver turned his cap around and came out of the back, pouring a mug of coffee for himself, demonstrating long familiarity with the diner. Esther glared daggers at him, but he paid no attention to her. Instead, he spotted me, saw the empty stool to my left, and came over. "Dan? Dan Chambeaux?"
I looked up from my piece of scrap paper that was still absent of insightful case notes. He was a blond-haired, gray-skinned blue-collar zombie. "Don't you remember me, Dan? It's Steve. We were dirt brothers. You helped me out of a tight spot—literally." When he grinned, I saw his teeth were still in relatively good condition.
Then I did remember. "Steve! Steve . . . Halsted, right?"
"Sure thing!" He clapped me on the back. "You look different out of your funeral suit, buddy. If I hadn't seen your picture on that book cover, I would never have recognized you."
Steve Halsted and I had risen from the grave on the same night. I'd emerged, dirt-encrusted and disoriented, not long before the guy in a nearby plot also crawled out. Reanimation is a tough time for everybody, and we had told each other our respective stories, exchanged contact information, and promised to stick together.
Over the months since, I had lost touch with him, though. I knew Steve left behind an ex-wife (no love lost there) and a young son, but we really didn't have much in common, so we drifted apart. I had my cases and a succession of personal emergencies, and I assumed Steve got on with his own unlife.
He said, "I was going to call you—I still have your card." I never could figure out why anyone had buried me with business cards in my pocket.
I ran my eyes over his uniform. "I see you got a job working as a delivery driver." It's always best to state the obvious.
"How did you know that?" he said, then laughed and brushed at his uniform. "Oh, I forgot, you're the detective. Yeah, that's me, delivering goods all around the Quarter. But it's about the only pleasant thing that's happened to me since coming back. I meant to give you a call to see if you or your lawyer friend could offer some advice. It's a sticky situation—and a stinky one—but I never got up the nerve to find you until I saw the piece in the newspaper talking about how you and your partner always stick up for unnaturals and never turn down a case."
"I haven't seen the article myself, but I'm pretty sure it was referring to the fictional adventures of a fictional zombie PI."
Steve chuckled. "Oh, you don't fool me, buddy!"
Albert staggered out of the back with a bowl of something that looked like a mix between magma and vomit. He opened his left hand to spill out a sprinkle of still-squirming cockroaches as garnish on top of the unappealing substance.
"New special," he said in his low, slurred voice.
Steve glanced down, took a deep sniff. "That looks good, Albert. I'll have one of those, too."
The ghoul proprietor shuffled back into the kitchen.
Talking about Steve's case seemed more appetizing than the food. "So what's the problem? Tell me about it, and I'll let you know if we can help."
Steve had a sad look as he got up and refilled both his coffee cup and mine without spilling a drop. "It's my ex, Rova. I just got this job, and I'm barely scraping by, but now she's filed a motion to garnish my wages, claiming past-due child support. I didn't even know I was supposed to keep paying, now that I'm dead. I thought that's what the life insurance money was for."
"I've never heard of postmortem wage garnishing," I said. "That's cold!" Robin might indeed have something to say about that. "I have to warn you, it'll be an uphill battle. Courts don't look kindly on deadbeat dads."
"I want to be sure that Jordan's taken care of," Steve said. "I'm not trying to get out of my responsibilities, but I don't trust Rova to use the money for our son's benefit. My insurance was supposed to cover all of his expenses, even put him through college, but Rova says that money's gone now, and she wants more from me. Worse, she's denying me visitation rights. I can't even see my own son."
I took a bite of the stewlike concoction and crunched down on something squirming and juicy.
"How does it taste?" Steve asked.
"We'd better just talk about your case."
Esther came by just as Albert served up another bowl of the new special; she swooped it from under the heat lamps and dropped it with a clatter in front of Steve. "We don't do separate checks." She tore off a ticket for both meals and set it next to my place. "Gratuity already added for parties of one or more."
I focused on Steve. "What's the problem with a zombie having visitation rights with his kid? That you might give the boy nightmares?"
"Hmm, he did scream the first time I tried to see him."
Steve seemed nearly as well-preserved as I was. "That's surprising. You don't look that scary." I meant it as a compliment, although not all monsters would have taken it as one.
"Oh, he didn't scream because I'm a zombie—it's because I'm me. I made a policy of not bad-mouthing my ex, for Jordan's sake, but Rova doesn't have the same attitude. She's poisoned my own son against me."
"And you say she spent all the insurance money?"
"Fifty thousand dollars! I thought it would be enough, but Rova took the money for herself, invested it in some beauty parlor she opened in the Quarter, said it was a surefire deal, the next best thing to printing your own money."
"She might have been a tad optimistic," I said.
Steve ate his stew, preoccupied. "I just want to make sure my kid is taken care of, you know? Could you look into it for me, buddy? See if I have any legal options?"
I glanced at my still-empty sheet of paper that should have been covered with brilliant deductions by now. "I've got appointments this afternoon, but why don't you stop by our offices later? I'll introduce you to Robin Deyer, my lawyer partner. Do you need the address?"
He pulled out the rumpled Chambeaux & Deyer Investigations card I had given him on the night we both came out of the grave. It looked as if he never took it out of his pocket, even when he laundered his uniform. He grabbed the check. "Let me get this, buddy. It's good to have a friend who understands your problems."
"We're dirt brothers, Steve. What are friends for?"
CHAPTER 7
Rusty the werewolf had been surly before, and I doubted that being scalped had improved his disposition, but I had an obligation to my client. I'd solved his case, whether or not he liked the answer. Out of consideration, though, I told Sheyenne not to send him the final bill until after he had recuperated.
Rusty lived in a dilapidated row house in a run-down part of town, not because he couldn't afford the rent elsewhere, but because the homeowners' association rules were less strict here. His front windows were boarded up since plywood was cheaper than glass, and he didn't like the view toward the street anyway. A chain-link fence surrounded his yard. A prominent warning sign on the fence said, BEWARE OF ALL MY DAMN PETS! I had been here twice before, and I knew Rusty could be more fearsome than his guard animals.
In the neighbors' front yard sat a rickety hearse on cinder blocks. Someone was burning trash in a rusty oil barrel. Dogs barked and howled incessantly because they had nothing else to do. Spiky dead weeds seemed to be the preferred form of landscaping.
When Rusty held weekend kegger parties for his rough-and-tumble Hairball gang members, he invited the neighbors, but they were afraid to come. His neighbors had their own dark secrets: coven rituals, lodge meetings, serial-killer boot camps, and quilting circles. Children didn't generally come around here selling Girl Scout cookies or Boy Scout popcorn.
I rapped on the screen door, and someone jerked it open with such force that the hinges didn't have time to squeak. Awkward-looking Furguson stood there bristling, his yellow eyes wide. He held a double-barreled shotgun that wavered from side to side. I raised my hands, not so much terrified of the gun as I was of Furguson's clumsiness. I didn't want to spend hours using tweezers to pick buckshot out of my dead skin because of a klutzy Hairball.
"Whoa, Furguson! Put that thing down, or at least point the barrel at someone who deserves to be shot!"
He seemed mortally embarrassed. "Sorry, Mister Chambeaux. Can't be too careful after what happened to my uncle." He let me into the house.
"That's who I came to see." I felt sorry for the kid, not sure how Rusty would punish his nephew, but I had been hired to solve a mystery, not to determine the consequences of the answers. "How's he doing?"
"Recovering, and just as stubborn as ever. He drank a gallon of vinegar tonic, said it was good for his constitution." Furguson was perspiring; sweaty werewolf fur smells like a wet cat under a low-powered hair dryer.
I didn't want to wake up a recovering werewolf who was in pain. "Is he resting quietly?"
"Resting?" Furguson let out a chuffing laugh. "What do you think?"
"So, out by the coops, then?"
"He spends more time with the animals than with me, I think. He says they're cuter."
I went through the small kitchen and out the back door into the fenced-in backyard. A broken fishing boat sat on a trailer, covering part of what would have been the lawn if anything had grown there. The rest of the yard was crowded with six coops covered with chicken wire and tar paper. Rusty incubated the prize cockatrice eggs, raised the chicks, and nurtured the creatures into adulthood. Then he tried to get them to kill one another.
The big werewolf stood out in the yard now, wearing his overalls, carrying a bucket of slimy entrails, which he ladled into the feed bins. "Here chick chick chick!" He slopped more intestines into the next bin. Rustling and hissing creatures swarmed forward, scales glinting in the light, beaks clacking. They squabbled as they gorged themselves. I saw a flash of lemon-yellow scales, bronze-green ones, and bright scarlet.
When Rusty turned around, even I was startled at his appearance. His scalp was a raw scab, like a skin wig turned inside out and painted with dried blood. Thanks to his lycanthropic healing abilities, the top of his head had scabbed over, though I doubted the fur would ever grow back the way it had been.
"Rusty, it's good to see you on your feet."
"Someone didn't ever want me on my feet again." He touched his head and winced. Parts of the wound that covered his cranium continued to ooze yellowish pus that crusted over. "If you're here to sell me shampoo, I won't need much of it anymore."
I laughed, because it was the polite thing to do.
Rusty growled. "Humor's a coping mechanism to keep me going until I find and kill the bastard who did this."
"Shouldn't you let the police handle it?" I asked. "Officer McGoohan was there at the scene. He won't ignore something like this."
"Hah! The same cop that tried to break up our cockatrice fight?"
"About that . . . there was a misunderstanding. McGoo just came to ask you to keep the noise down because one of the neighbors complained."
"Now you tell me. I knew I shouldn't have used those old rave speakers. It was the Rocky theme, wasn't it?"
"Among other things."
"We won't be doing the fights again for a while—I've got other priorities. Cockatrice fighting gets trumped by a blood vendetta any day. Did you find out anything about my missing money last night?"
"Afraid so," I said. "I solved the case."
Rusty growled. "Then why are you afraid?"
"Because you're not going to like the answer."
"I already don't like it—I've been losing money! Was it Scratch and Sniff, like I thought?"
"It was . . . an accident." I hesitated, and Rusty drew his own conclusions. He growled in disgust now instead of anger.
"You mean it was my nephew? He's one big walking accident."
I sighed and explained how, although Furguson had carefully recorded all the bets, he'd been sloppy stuffing the bills into his pockets.
"Damn that boy! When he tries to answer the call of the wild, all he gets is a busy signal! This is serious business. It's not as if I can take him over my knee and spank him anymore." He shook his scabby head. "My sister's son. I promised her I'd watch out for the kid, although why she'd trust her only son to a full-time werewolf makes no sense at all." He heaved a big sigh. "That kid is so clumsy, we don't dare let him use anything but these little round-ended scissors. I can't even let him trim his own facial fur."
Grumbling, he continued to ladle entrails into the cockatrice feeding troughs. Scaly, fork-tongued chicks wobbled about, flapping their not-yet-developed dragon wings and pecking at one another. Another cage held several nests built out of bent nails, which cradled leathery-shelled eggs. In three of the nests, fat warty toads sat on the eggs; in two others, vipers coiled around the clutches.
A lemon-yellow cockatrice fought with a crimson one, and they tumbled around inside the cramped cage in an orange blur, spilling offal all over the place. While they fought for the last scraps, the other cockatrices in the cage ate the ignored meal, so that when the two squabblers finished beating each other up, nothing remained.
Rusty went from cage to cage, inspecting. "I've been doing a lot of crossbreeding. Cockatrices are magnificent beasts, beautiful colors, even make nice family pets if they're raised right. I hear they're good with kids."
Two of the coops were entirely enclosed with tar paper, which blocked the view of the monsters within, but I could hear them rustling and hissing like a manifestation of intestinal distress. I figured that Rusty put some of his prized cockatrices in dark solitary confinement so they'd be hungry and angry for the next fight.
"So, I leave you to decide how to discuss the matter with your nephew," I said. "I hope you've found our services satisfactory. Sheyenne will send you a customer comment card in your final bill."
The cockatrice racket was deafening, but Rusty wasn't bothered by it. He continued to brood, flexing and unflexing his clawed hands. Finally, he threw the empty entrail bucket across the yard in a gesture of uncontrolled anger. It banged against the tar-paper side of the solitary-confinement coop, eliciting a round of bone-chilling snarls and cackles. I silently reaffirmed my desire to not see what the coops contained.
Rusty's eyes were fiery. "Oh, we're not done, Mister Shamble. I want to engage your services for a much more important matter, but I can't think what it is, off the top of my head." He tapped his raw scalpless skull, winced, then chuffed at his own joke and shook his head. "No, I don't see that using humor makes it any better." He extended his clawed hand. "I need a private detective. I want you to find out who did this to me and come up with proof."
"I'll take the case, but only if you tell me everything, especially the parts you didn't tell the doctors or the police."
"I didn't see the bastard. It was a dark alley, full night, and I was busy trying to hold my cockatrice in the bag. Those things don't like to be confined. Since I thought the fights were being raided, I was in a hurry to get out of there. Before I knew it, I got hit with two darts of full-strength animal tranquilizer laced with wolfsbane, and a Taser for good measure. Knocked me flat. Then some butcher took a knife and hacked off the top of my head."
I forced myself to look closer, telling myself it couldn't be any worse than the gutted vamp in the Motel Six Feet Under. Actually, it looked as if a neat, straight line had sliced away the scalp, causing no damage to the skull.
"I know damn well it was the Monthlies. I should have whacked Scratch and Sniff a few more times with that two-by-four. Shoulda used the end with the nails, too! But I'm a nice guy, Mister Shamble—a pacifist at heart. Look what it got me."
There was definitely bad blood between the two types of werewolves, a long-standing feud that went beyond mere rowdiness at a cockatrice fight. I guessed there would be a lot more to this case.
"You don't know for certain it was them," I cautioned.
"I know damn well for certain! I just can't prove it."
"So you're asking me to get the Monthlies," I said.
"Oh, yes—and blood will flow!"
CHAPTER 8
When you get invited to visit a mad scientist's lab—even if it's just a personal home workshop—you might think twice. But Archibald Victor wanted me to see his innocent do-it-yourself people kits.
I think he wanted to reassure me that his hobby was no more bizarre than collecting stamps or baseball cards. I'm not a judgmental guy in the first place. As a private investigator, I've represented monster brothels, a witch transformed into a sow, a bank robber ghost, an arsonist who claimed to be William Shakespeare, and a werewolf contesting her prenuptial agreement. Who was I to look askance at somebody who liked to stitch body parts together?
Still, at Chambeaux & Deyer, we try to give our clients the personal touch. I was happy to make a house call.
The coroner and his wife lived in a nice double-wide in a trailer park that had been converted from a bankrupt drive-in theater. At neighborhood get-togethers, the trailer-park management showed grainy old monster movies on the big patched screen. The tenants viewed them as comedies.
I was surprised that the Quarter's coroner could afford only a modest residence, considering the work he had to do (including all the repeat customers). But Dr. Victor and his wife were newlyweds, starting a life for themselves, and they didn't live beyond their means. I wondered how much he spent on his unusual hobby; I assumed mint-condition organs didn't come cheap.
When I rapped on the door of the mad scientist's house trailer I certainly wasn't expecting the woman who answered. I've never seen so much hair on a woman other than a werewolf. She was covered with it: long, curly locks that fell down to the middle of her back, long eyebrows that drooped to her cheeks, and a lavish beard that extended from her lower eyelids down to her ample bosom.
"You must be Mister Chambeaux! Archibald told me you'd be stopping by." She extended her hand, and her grip was warm and soft, indicating that she did not scrimp on lotions. The manicured nails were painted rose-petal pink; long and silky hair covered her forearm and the back of her hand. Everything about her had a freshly shampooed strawberry scent.
"Pleased to meet you, Mrs. Victor," I said.
"Please, call me Harriet. Come inside. My husband's in his lab." She closed the trailer door behind me and kept talking. "Yoo-hoo, Archibald! That detective is here."
"I'll be right there, love," he called back. "Just as soon as I finish installing this bronchial tube."
"Sorry the place is a mess," Harriet said. "Archibald takes over every table and countertop with his model-building hobbies and chemistry experiments. Someday, he's going to make us rich with a great discovery. He always tried to impress me when we were dating—I think he still is. He's such a sweetheart." She bustled about. "Could I get you a snack? Some crackers and cheese? I'm just getting ready for work."
"No, thank you, ma'am. I'm fine."
As I stood in the trailer's small living room, she put on a delicate necklace and bracelet, which were all but swallowed up in bodily hair. She primped her facial curls in front of a mirror. "Harriet isn't actually my real name," she said conversationally. "Just a stage name from when I worked for Oscar Kowalski's vampire circus. I was the most popular bearded lady they ever had."
"So you're not an unnatural, then?" I asked.
"No, it's just a hormone condition. I maintained a sunny disposition and made a good living, but I never liked being treated as a freak. Dear Archibald took me away from all that."
As if she had summoned him, the coroner emerged from the trailer's back room, shucking a pair of black rubber gloves and adjusting his prominent toupee to make himself presentable.
"I'm off to work, sweetie." Harriet picked up her purse and gave her husband a peck on the cheek.
He worshipfully stroked her full beard. "I love running my fingers through your hair."
Harriet giggled. "Now, don't get carried away—we have company. I'll leave you boys to your business."
After she left the trailer, Archibald stared after her like a lovesick puppy. "Don't know what I did to deserve her . . . or what she sees in me."
I had no idea how to answer that, so I didn't even try. "Let's have a look at your kits, Dr. Victor. I have another appointment later this evening." Actually it was my usual get-together with McGoo at the Goblin Tavern, but I wanted to keep this meeting on a business footing.
"Follow me to the back room. I'm just finishing a particularly challenging assembly."
Archibald's hobby area took up half of the trailer. A small bed was tucked off in the corner, and I couldn't imagine how Archibald and Harriet both fit on it, unless they snuggled tightly—which, as newlyweds, they probably did. Jars, tubes, and bottles of every conceivable type covered the countertop around the single bathroom sink, but the vanity was divided in two, as if by an imaginary line. One half was crowded with shampoos, crème rinses, mousses, hair sprays, and arcane beauty treatments—Harriet's side, I presumed. The other part contained beakers, flasks, test tubes, Bunsen burners, separation coils, differentiation cylinders, boxes of powders, jars of liquids.
"Concocting a new energy-drink recipe, Dr. Victor?" I asked, remembering his penchant for such things.
"No, no—an innovative new hair tonic. Top secret—it's going to make us rich, rich! I could rule the world!" He started to cackle, then caught himself. Embarrassed, he pulled me away from the vanity and sink. "Actually, I prefer the quiet life, never understood the appeal of ruling the world. That's not why I brought you here."
On a pair of sturdy fold-out utility tables rested two of his build-your-own-person projects in different stages of completion. One was a very tall human, nearly complete: The arms, legs, and torso were stitched together out of parts that originated from different sources; the varying shades of skin color gave the patchwork form a sort of calico appearance. The body cavity was propped open with Popsicle sticks and dowels.
I looked closer. "Why is one leg shorter than the other?"
He frowned. "Slight imperfections are to be expected. These bodies are individually handmade. It increases the collectible value."
"But won't he have trouble walking once you reanimate him?"
"Oh, these aren't for reanimation—just art objects. I'll enter them in the upcoming body-building competition, and I have a standing show over at the Night Gallery. This one still needs an acceptable spleen, as I showed you earlier. And, of course, a mint-condition brain." He tapped the kit's head, which had been sawed open, the top of the cranium set aside. The skull cavity reminded me of an empty garage waiting for a car to be parked there.
The second table held a skeleton on which some major muscle groups had been sutured with thick black threads. A set of lungs, looking like deflated balloons, had been tucked inside the rib cage, only recently attached. From the long canine fangs in the bony jaw, I assumed the head was a werewolf skull. Two eyeballs, one brown and one blue, had been installed, but the rest of the body needed a lot of work.
Archibald cracked his knuckles. "This hobby gives me an appreciation for the body's intricacies. As a coroner, I spend my days taking bodies apart, so it's gratifying to put them back together again in my spare time. Only the most skilled hobbyists start with the bare bones. And this"—Archibald tapped the long, yellowed fangs—"this is a genuine werewolf skull, very rare. Most werewolves revert to human form upon death, but this poor fellow died of a Gypsy curse—hardening of the arteries, I think—which messed up the reverse transformation. This one component cost me a fortune!"
I looked around the workshop. "If the hobby is so expensive, how do you afford it all?"
"I do consulting work to raise a little spare cash. So far, it's mostly mummies who bring their canopic jars and ask me to reinstall their organs because they feel empty inside. Once a week, I donate time to the Fresh Corpses Zombie Rehab Clinic."
I thought about the crime scene at the Motel Six Feet Under. "Do you think the murdered vampire had anything to do with hobbyists like yourself?"
The coroner gasped. "I'd be appalled to think so!"
"Why else would anyone plunder the organs?"
"You may be aware that there's quite a demand for vampire organs on the black market—they just keep going and going, no matter how detached they are from the main body." Actually, I wasn't aware of that, but it made a twisted sort of sense. "Discriminating customers pay a lot for still-functional pieces."
I walked around the hobby table, seeing all the preserved pieces just waiting to be installed in the body-building kits. "How can you be sure all these body parts are obtained legally? My partner can't make a consumer protection case on your behalf if you're buying from a shady supplier."
He shook his head so vigorously that the toupee slid askew. "Body-snatching laws have been significantly tightened since the Big Uneasy. Many zombies came back to find themselves missing vital parts. Plenty of complaints filed."
I was glad that hadn't happened to me. "I see how it would be a problem."
"That's why I buy only from reputable dealers. I'm the coroner—I wouldn't want to expose the department to a scandal! I assume Tony Cralo's Spare Parts Emporium is fully licensed."
"I'll definitely look into Mister Cralo's operation—with complete discretion," I said.
Suddenly, a cauldron-like gurgling came from the shower stall in the half bath, sounding like a drowning victim in the first stages of reanimation. Alarmed, Archibald rushed to the bathroom. "Not that drain again!" The shower drain bubbled, burbled, and splurted effluent as it backed up with a stench worse than a sewer dweller's belch. Archibald snagged a long-handled rake propped against the shower and began fishing around with the tines, uprooting long, tangled masses of hair that had refused to go down the drain. The little coroner looked like a farmer moving hay with a pitchfork.
He set the tangled wad on the floor outside the shower, then emptied a gallon jug of drain cleaner down the shower drain. Fumes began to sizzle and smoke, and the coroner flicked on the bathroom exhaust fan.
"My dear wife is beautiful, and I love her hair, but she always forgets to use the hair trap or clear the shower stall after she's finished."
"At least you don't have that problem," I quipped, before I remembered how sensitive he was about his baldness.
He blushed and tried to block my view of his chemistry experiments on the countertop. "Just wait until I find a hair tonic recipe that works."
"Good luck with that." I glanced at my watch. "Thanks for showing me your work, Dr. Victor. I'll be back in touch after I visit the Spare Parts Emporium. I'll make sure you get a replacement brain and spleen."
"And see if they'll throw in an extra set of lungs, as you promised."
"I'll do my best." I left the trailer very much looking forward to having a beer with McGoo.
CHAPTER 9
The Goblin Tavern is the Quarter's favorite watering hole, a well-known hangout for monsters of all types (and a few humans, too).
That evening, I was on my usual bar stool with a tall, cold draft in front of me and my best human friend McGoo on the adjacent stool. Francine, the regular salty-humored bartender, was back on the job after an embarrassing disagreement with the new management. Yes, all was right with the world.
"How was your day, McGoo?" I slurped the foam head off my beer.
"About the same as yesterday, Shamble."
"Lousy, you mean?"
"That's about it."
As the bartender walked by, I said, "Francine, how was your day?"
"Same as McGoo's."
"Sorry to hear it," I answered.
She shrugged. "At least I'm still alive, which makes my day better than yours."
"You've got a point."
"So, a zombie shambles into a bar," McGoo said, before I could tell him I wasn't interested in another joke. "And on his shoulder, he's got a huge red parrot with long tail feathers and a big black beak."
"Those are called macaws," I said.
McGoo gave me an impatient gesture. "He's got this big red macaw on his shoulder. The bartender takes one look and says, 'Holy cow, where did you get that thing?' "
Francine butted in, "And the macaw says, 'Out at the cemetery—there's hundreds of them!' " She let out a cackle. "Yeah, I've heard that one before." She poured gin and crushed ice into a shaker, just a drop of vermouth, and dispensed an extra-dry martini for a dapper-looking Aztec mummy who sat in the corner of the bar, reading a codex.
Francine lit a cigarette and took a puff, leaving bright pink lipstick on the filter. She was a hard-bitten woman in her late fifties who colored her hair an unnatural shade of woodchuck brown. She didn't take any disrespect from her customers, gave advice from her own personal experiences when appropriate, and cared for the Goblin Tavern patrons better than any previous bartender had—certainly better than Ilgar, the former owner.
After she'd been laid off in favor of a more unnatural employee, her regular customers protested, and Robin filed an antidiscrimination suit. Francine got her job back, after agreeing to wear a funereal black miniskirt and cobwebby fishnet stockings (honestly, not something most customers wanted to see); she accepted the terms only on the condition that she be allowed to smoke on duty. Henpecked and desperate to have the beloved bartender back, the new owner Stu admonished Francine that cigarettes were bad for her health; more importantly, he warned her to stay away from the flammable customers whenever she had a lit cigarette in hand. (She kept her distance from mummies in particular.)
Stu was a portly, good-natured human who always greeted his customers, joked with them, and pretended to be their best friend. He had cobbled together the financing to buy out the Goblin Tavern after the collapse of the Smile Syndicate. He was in over his head, but he soldiered on, sure that business would get better. McGoo and I did our part to support the local tavern by drinking there as often as possible.
Stu emerged from the back office, jaunty and cheerful. "Thought you'd need a little help behind the bar, Francine." He grinned at all of us, bobbing his head. "The tour bus is due to arrive, and it'll be full of early convention-goers. Have to make a good impression. Maybe they'll blog and post about it."
It took me a moment to remember what Robin had said. "You mean the Worldwide Horror Convention?"
"Yes, sir, Mister Chambeaux. This should be one of our busiest weekends ever—and, boy, could we use the customers. The Goblin Tavern placed a big ad in their program book, shared a page with the Full Moon Brothel. They're having a convention special, too."
"Madame Neffi wouldn't miss a chance to advertise for customers," I said.
"She wants to be plugged into the convention traffic. We've got flyers stuffed into every registration packet as well."
McGoo wasn't convinced. "Aren't the convention-goers mostly human?"
"Sure, but I'm guessing they're an adventurous lot. I made up a special sampler platter."
"Hey, Stu," McGoo said. "What's cuter than a zombie baby?"
"I don't know. . . ."
"A zombie baby with a bunny's head in its mouth."
Francine went to console a middle-aged vampire who was drowning his sorrows in a glass of AB negative. I half listened as he poured out his sob story: He'd been a successful cat burglar until he broke into the wrong house and startled the vampire who lived there; the vamp bit him on the neck and turned him. Then, once the burglar became a vampire himself, he could no longer enter a home uninvited, which immediately ended his career as a thief. Now the guy didn't know what to do with himself.
With a roar of engine noise and a puffing squeal of air brakes, a large motor coach pulled up in front of the Tavern and disgorged twenty humans with cameras and book bags, all of them wearing clip-on badges that identified them as pre-registered attendees of the Worldwide Horror Convention; three wore ribbons that said Professional Guest.
Many of the fans were dressed in elaborate monster costumes, some of which looked better than the real thing—two immaculate Bela Lugosis, a werewolf with a prosthetic muzzle and fake fur glued on her face, and two undead who seemed to think that pale foundation, liberally applied eye shadow, smears of misplaced rouge, a bit of shoe polish, and a fake scar were all they needed to turn themselves into zombies.
Stu greeted the tour group at the door, wearing a big grin. Planning for the bus's arrival, he had pushed together six tables and pulled around chairs. The rearrangement cramped the room for the dartboard and the shiny new billiards table Stu had just installed (insisting on fiberglass pool cues, so there would be no risk to vampire customers from the long, pointed wooden sticks).
Francine left the depressed cat burglar and went to take orders from the convention group. Because they were tourists and not likely to become repeat customers, Francine gave adequate but not necessarily scintillating service. For their own part, the tourists—knowing they were not going to be repeat customers—would feel no need to tip well, regardless of the service.
Usually, when a tour group left the Tavern, Francine would complain about the crazy concoctions they ordered. She blamed Stu for creating the special twelve-page "Monster Martinis" menu, and she had to keep looking up the recipes.
Hearing a rumble of motorcycle engines outside, I looked through the window to see two large choppers pull up. The burly bikers Scratch and Sniff dismounted and came inside, looming large in their matted werewolf-pelt coats. At their waists dangled meat cleavers with rusty blades and sharp edges that gleamed silver. (Apparently, the two called their trusty cleavers Little Choppers, as opposed to the big choppers they rode.) The werewolves in human form surveyed the customers, obviously not impressed. Over at the empty billiards table, they shucked off their stained fur coats and draped them on stools, which gave them a chance to show off their myriad tattoos.
"Yo, Francine!" Scratch bellowed. "Bring us the usual!"
"Yeah," said Sniff. "One of everything!"
Francine looked up from the table of tourists. "I'll be there when I'm finished taking this order."
"We already gave you our order." Scratch primped his slicked-back hair.
The intimidated tourists stared at the big men. Francine lifted her chin and turned to the two bikers. "I said, wait your turn. These people are tourists. Show them a little Unnatural Quarter hospitality."
Holding fiberglass pool cues like cudgels, the pair strode over to the table of conventioneers. Sniff sniffed at the costumes, but the fake full-furred werewolf particularly drew his ire. "Little lady, what do you want to dress like that for? It's embarrassing."
"Aww, Sniff, she looks better than most Hairball chicks. Besides, she can wash that off when she wants to. Real Hairballs are stuck with what they look like."
Francine planted herself in front of the two burly men. "I don't want any trouble here, and you're not going to give me any. Don't you know better than to piss off a bartender? Someday when you least expect it, I'll put a depilatory cream into your drink."
"Easy, Francine. We're not causing any trouble," Sniff said. "Just welcoming these people to the Quarter. Got no problems with regular humans."
Stu scuttled back to the bar, doing his best to defuse a situation that Francine had already defused. "What'll it be, boys? First order's on the house."
"Stu, don't encourage them," McGoo said. He'd been ready to intervene in case things got ugly, but I never doubted that Francine could handle it.
With exaggerated fake smiles, the two Monthly werewolves sauntered back to the billiards table and racked up a game. They ordered light beers in large mugs. I offered to take the drinks over to them, and Stu looked relieved that he didn't have to face Scratch and Sniff. Some zombies tend to lurch, but I walked over, careful not to slosh any of the beer.
The billiard balls had been painted like bloodshot eyeballs, another homey touch Stu had instituted at the Goblin Tavern. Holding pool cues and studying the balls across the table, both Monthly werewolves looked up at me.
Sniff sniffed. "Weren't you at the cockatrice fight?"
"I didn't think you'd noticed."
Scratch aimed with great care, sliding the pool stick back and forth on his posed fingers, and missed the ball entirely, whacking the felt surface of the table. His partner let out a sneering laugh. He fumbled the next shot, too, knocking the cue ball off the table. Now I understood where he had gotten the nickname of Scratch.
Scanning their myriad tattoos, I thought it would be a good icebreaker. "When you two got scratched up at the fight, I saw those tattoos come alive."
"Good healing spells, huh?" Sniff rubbed his beard, sniffed his fingers.
"They're not just tattoos." Scratch flexed his biceps to show off the pattern. "They're voodoo tattoos. I don't think they work on zombies, though."
I looked closer. "What does that mean, 'One-Percenter'?"
"Means we're one percent human," said Sniff. "I wouldn't tell you which one percent, though." They both laughed.
Always on the case, I fished for information. "Did you hear Rusty the werewolf got scalped after the fight?"
"Probably improved his looks." Sniff hooted and chuckled again.
I lowered my voice. "You guys, uh, didn't happen to have anything to do with that?"
"Scalping old Rusty?" Scratch pulled out the meat cleaver at his side. "If I was gonna do something, Little Chopper here would take his head clean off, not just his scalp."
Looking at the two, I believed them, though I doubted Rusty would.
No longer interested in me, they went back to their game. Sniff knocked an eyeball into the left corner pocket. "We should tell Miranda Jekyll to put a pool table at her werewolf sanctuary."
Scratch fumbled again and knocked a ball off the table. "When we're up there during the full moon, you wanna play billiards? That's our chance to run wild, roam the wilderness, feel the hot blood pounding."
"No, I mean during the daytime, before the moon rises," said Sniff.
"Oh, right. Sure, that's a good idea."
I left them to their game.
CHAPTER 10
Next morning, when my dirt brother Steve Halsted came to our offices, he looked forlorn and nervous, pulling off his green trucker cap and kneading it in his hands. I smiled with as much reassuring cheer as I could manage and introduced him to Robin and Sheyenne.
Steve shook Robin's hand and attempted to do the same with Sheyenne, though his hand passed directly through her ghostly form. "Sorry," he said. "Old habits die hard."
I had already told them the bare bones of Steve's situation, including a few specifics about Rova Halsted, his ex-wife. Robin liked to do everything by the book, having (sometimes unreasonable) faith that the right side would always win. I, on the other hand, was less sure of automatic happy endings. I preferred leverage.
I took Sheyenne aside to whisper in her ear, "Any interest in digging up a little dirt on the ex-wife?"
"Well, you're hard to resist when you blow in my ear. . . ." She gave me a sultry smile. "It would be my pleasure."
Robin led Steve into the conference room, and he seemed uncomfortable to be talking to an attorney, but was reassured to have me in the room. "Dan pulled me out of the grave when I was having a hard time—and now I'm having a hard time again." He drew a breath, shuffled nervously. "He said you might be able to help?"
As we sat down to discuss the case, Robin said, "If you haven't paid your original court-imposed child support, it'll be difficult to generate sympathy from the judge." She tapped her pencil on the legal pad. "Although it's highly unusual for an ex-wife to go after a dead husband."
"Ex-husband," Steve said. "Dead and divorced. I guess that makes me a double-ex-husband." He leaned forward, resting his elbows on the table. "It's not that I don't want to take care of my son. I'm a responsible parent. Jordan was the only reason I took out that life insurance policy in the first place."
Robin scribbled notes on her legal pad. "But your wife Rova received the money?"
"Ex-wife. But she's not dead, so it's only one ex. I told Rova that if anything ever happened to me, I wanted that money to go for Jordan's expenses: nice clothes, a good college."
"Was she the beneficiary?" Robin let out a barely audible sigh, but I caught it. "The status of life insurance proceeds has been muddled for years in the courts, with several lawsuits pending—zombies demanding the money for themselves, rather than their beneficiaries, and then insurance companies counter-suing to get their money back because the policyholders, while dead, are still ambulatory and are still capable of earning a living. Of course, the problem gets worse—in both cases—because the beneficiaries have usually spent all the money on funeral expenses, and new cars, long before the matter gets to court."
Once Robin got started talking about the nuances of cases, she built momentum. I loved to see her enthusiasm, but I tried to get her back on track. "Robin, so what about Steve's money for Jordan?"
She cleared her throat, embarrassed. "Sorry, Mister Halsted. In many cases, the divorce settlement voids any documents that name an ex-spouse as a beneficiary. How detailed was the paperwork delineating funds for your son's benefit?"
Steve shifted in his seat, awkward. "We just filled out the forms in one of those do-it-yourself books. Is there a standard blank for that? I wasn't really expecting to die when I did."
Now Robin's frown deepened. "You left the money to her, but there was no binding agreement that the money was to be put into a sheltered account in Jordan's name?"
"Oh, nothing formal like that. I thought I could trust her."
Robin hung her head and groaned. "Sometimes clients are their own worst enemies. In my experience, most divorced parents would rather eat glass than cooperate with each other."
"After I died, Rova took the fifty thousand dollars and went in with a partner to open a beauty salon. Now all that money's locked up in shampoos, crème rinses, and pedicure chairs. How can I make sure that it gets used for Jordan's benefit?"
Sheyenne floated into the room. "Their salon is called the Parlour (BNF)—whatever that means. Rova Halsted filed papers and all the necessary permits, founded the business with someone named Harriet Victor."
I perked up. "Harriet Victor? That's the coroner's wife—I just met her." Now that I thought about it, I wasn't surprised that the lavishly tressed bearded lady would be interested in, and have a great deal of practice with, styling hair.
"Yeah, that's the place," Steve said. "Never been in it. When I need a haircut, I go to a barber, not someone who calls herself a stylist." He shook his head. "I don't like to speak ill about Rova . . ."
Robin said, "I encourage you, Mister Halsted. Please speak ill about her to us. It could be relevant."
"To tell you the truth, she's . . . not very good at doing hair."
"That's for sure," Sheyenne said. "Rova Halsted has her stylist's license, but couldn't find a job outside the Quarter—too many complaints, even got fired from one of those haircut factories. So now she works on unnaturals."
Did I mention that Sheyenne is amazing at what she does?
"I guess monsters aren't so picky," Steve said.
"Tell that to the old-school vampires," Sheyenne said. "Some of them are as vain as high school girls on prom night."
"Rova used to cut my hair. That's why I usually wear this hat." He set his green cap on the conference table, hung his head, and admitted, "Rova's actually a dangerous hairdresser."
"How can anyone be a dangerous hairdresser?" Robin asked, jotting down the information in hopes it might be relevant. "Has she hurt people with scissors? Burned anyone with hair dye or curling irons?"
"No, I mean she's dangerous to people who see her customers. Her haircuts are so bad they've caused accidents, distracting drivers as they pass." He glanced at his watch. "I have to get to work soon. Deliveries to make, and I don't dare lose this job—especially not if I have to pay child support." Steve fished out his trucker's wallet, which was chained to his belt loop. "Here are some snapshots. Me and Jordan, good times. Rova won't let me have any other pictures. These are just the ones I was buried with."
The wallet photos were well-thumbed: a grinning boy in a Little League outfit, another one of him and his dad with fishing poles and proudly dangling very small trout they had caught. He flicked through the photos, showing another one with the boy in a rather alarming buzz cut that looked as if it had been trimmed with a lawn mower instead of hair clippers.
"Rova cuts his hair, too," Steve said.
"I gathered that," I said.
Sheyenne said, "Such a cute kid."
Steve couldn't take his eyes from the snapshots. I could tell he was about to start crying, and my heart went out to him.
Robin rose from her chair. "Let me look up some precedents. I can make the case that you've already paid fifty thousand dollars' worth of child support with the insurance, but your wife squandered it. Before we agree to pay her more, it's only fitting we have the court set up parameters, establish how exactly Rova is allowed to use the funds on your son's behalf." Seeing the determined look on her face, I began to feel that this might turn out all right after all.
"I want my boy taken care of," Steve insisted. "That's all I need to know."
"We should be able to negotiate visitation rights, if you'd like."
Steve lit up. "I'd love that, but Rova says I'm not allowed to see him anymore because I'm dead. The visitation ruling doesn't apply anymore."
"She can't have it both ways," Robin said. "If you're too dead to see your boy, then you're too dead to pay child support."
"Rova always used to confuse me with her contradictions." He forced a chuckle. "Whenever I pointed out to her that two completely opposite things couldn't be true at the same time, she just got angry."
"Lawyers are good at straightening out contradictions, Mister Halsted." Robin shook his hand again. "I'll be on this. Do you have contact information for her attorney?"
Steve shuddered. "You're not going to like him. He isn't a very likeable person."
Robin wasn't bothered. "We're professional colleagues. We don't have to like each other."
"His name is Donald Tuthery, and he works for the Addams Family Practice."
Robin started to write down the name and froze. "Oh. Well, I'll see what I can do. Sheyenne will set up a meeting with all parties as soon as possible, maybe even tomorrow." She flashed him a smile that I could tell was entirely false, but it convinced Steve. He thanked her and went off to work.
CHAPTER 11
Today was filled with reminders of the past.
Miranda Jekyll wasn't exactly an old friend, but she was a former client and a satisfied one at that. Robin had successfully broken the strictures of her prenuptial agreement with Harvey Jekyll, and we had seen to it that her evil husband was convicted of numerous horrendous crimes (which satisfied Miranda even more). The scandal had rocked Jekyll Lifestyle Products and Necroceuticals, while still leaving a fortune and a half for her.
When Miranda came for a visit, she didn't just "stop by"; rather, she arrived. Her hair was dyed a striking and potent cinnamon color, styled, swirled, massaged, and moussed into a precise sculpture that would have surpassed Rova Halsted's wildest dreams as a stylist. Miranda wore lacquered nails, layers of expensive necklaces, too much makeup, and too much pheromone-laced perfume, and the whole package was wrapped up in a tight red dress.
"Why, sweethearts!"—she pronounced it sweet-hots—"So grand to see you again! I just had to say hello, now that I'm visiting the Quarter . . . temporarily. As temporarily as I can possibly manage. You remember Hirsute, my deliciously masculine companion?"
The large hunk of male that accompanied her looked as if he had gotten carried away ripping bodices and ripped himself right off the cover painting of a romance novel. Flowing dark hair, shirt artfully torn to display his sculpted chest, square jaw, very large hands—and he didn't talk much. Though they looked human, Miranda and Hirsute had a feral glimmer in their eyes that signaled to a discriminating observer that they were both Monthly werewolves.
"Good to see you again, Mrs. Jekyll," I said.
"Oh, please, sweetheart, it's Ms. Jekyll, and I only keep Harvey's last name because of all the money involved. Besides, I'm sure it annoys the little worm no end. How is he doing by the way, after being executed and all?"
"Getting by," I said. "He just moved out of the Quarter into a small house in the suburbs." I didn't want to reveal that Robin and I had been instrumental in the matter, by filing lawsuits against locals who were trying to keep unnaturals out of their nice normal neighborhood. Even I still had a hard time believing that we had helped the loathsome man.
"I hope Harvey stays far away." She glided into the offices, holding Hirsute's muscular arm as if it were an anchor to steady herself. "Ah, I've missed the Unnatural Quarter! The seediness of this place reminds me just how wonderful it is to be somewhere else. Most of my time is spent out at the sanctuary up in Montana—that's my true calling in life. Hirsute makes it so pleasant there."
Miranda ran her long fingernails across his bare chest. He growled seductively, deep in his throat. As if to compete, Sheyenne brushed up close to me, giving me an unfortunately unfelt ectoplasmic snuggle.
"What brings you here, Ms. Jekyll?" Robin asked.
"I'm back to do my regular check-in at the factory, make sure my minions aren't ruining things." She brightened, her eyes twinkling. "And I've been invited as a celebrity guest speaker at the Worldwide Horror Convention tomorrow. They begged and begged until I finally had to agree. They're paying for our room at the Bates Hotel, even told us we could dine in the con suite, whatever that means, but it's a suite, so it must be fancy." She batted her eyes—the artificial eyelashes looked like a folded daddy longlegs. "But I see I'm not the only celebrity in town. You, Mister Chambeaux, are quite the famous detective."
We had received plenty of publicity in the wake of the JLPN scandal and our recent cases against Senator Balfour's Unnatural Acts Act, the Smile Syndicate, and solving the murder of the gremlin pawnbroker Snazz. But I feared Miranda was talking about something else.
"I saw the advertisement for your novel, sweetheart. If it's about the true adventures of a zombie detective, I wonder if I might be in there? A small cameo part perhaps? Every book like that needs a femme fatale."
"I wouldn't know, Ms. Jekyll. I haven't read the novel myself." After a brief pause, I saw an opportunity and took it—you never know where information might lead. "Since you're here, maybe you'd like to help out in another case? Recently a werewolf was attacked in the Quarter, stunned unconscious and then scalped."
"Scalped? Oh, my! But it wasn't even a full moon."
"The victim was one of the full-time werewolves," I said.
"Oh." Her alarm turned to distaste. "I have little to do with the Hairballs, if I can help it."
I pressed on. "The victim hired me to look into who attacked him. Not many leads so far, though he's suspicious of two troublemaker Monthly werewolves, Scratch and Sniff. They mentioned they've been to your sanctuary?"
"There was a time when 'troublemaker werewolves' was a redundant statement," Miranda said with a sniff. "I know those two. Rowdy, unruly boys, but they can't help it. They're just hot-blooded. At least they're real werewolves, the ones that transform under a full moon, not those other types."
"I don't understand," Robin said. "You don't consider the full-timers to be real werewolves?"
"Sweetheart, a werewolf transforms. Human during the day, majestic beast under the light of the full moon. If you're covered with hair all the time, there's no transformation, is there? Hairballs are just like trained talking dogs."
Hirsute surprised us by speaking up. "They look like Wile E. Coyote."
"I've noticed friction between the two types of werewolves," I said. "Does it go beyond rivalry? Maybe to the point of gang warfare? A blood feud? If you could tell us anything about the long-standing grievances, that might help us solve the case."
Miranda waved her colored nails in the air. "Sweethearts, do I look like I get involved in politics? Of course not. It's none of my concern. Hirsute, escort me back out onto the street. I want to walk up and down the boulevard and show you off."
"I would be most honored to do so." He took Miranda's hand and nibbled on her knuckles, much to her delight.
"I'll see you at the convention, Mister Chambeaux. Perhaps you can dine with us as my guest in the con suite?"
"I doubt I'll be going to the Worldwide Horror Convention," I said, wondering what I would do there.
But Miranda just laughed as she walked out the door. "Of course you will, Mister Chambeaux. Of course you will."
CHAPTER 12
The Unnatural Quarter is a dark, convoluted place that has bad parts of town and worse parts of town. Even after my years here, there were still plenty of backstreets and byways with which I was unfamiliar, eccentric shops to peruse, restaurants to try, nightclubs to visit, alleys to avoid entirely.
Cralo's Spare Parts Emporium had not even been on my list, and now I visited the place with a certain amount of trepidation as well as—I admit it—curiosity. Good thing Sheyenne accompanied me, not so much as a bodyguard, but as a companion. There are worse things than being haunted by a beautiful blond ghost.
The Emporium—which I now realized was just a fancy name for "warehouse"—seemed like a typical chop shop, providing used, currently unused, and unusual body parts to the entire Quarter. For ambience, the warehouse was near an abandoned skeletal railroad bridge. The location was definitely toward the latter end of the bad-to-worse part of town.
Seeing the place, Sheyenne said, "I'm not surprised Dr. Victor prefers to do his shopping online."
The corrugated Quonset hut looked like a dead caterpillar in the middle of a gravel parking lot that was dotted with oil-stained brown puddles. A billboard rode on top of the curved roof, lit by a row of bent spotlights: Tony Cralo's Spare Parts Emporium—Walk-Ins Welcome. Another sign leaned against the open front door: YES, WE SELL DIRECT TO THE PUBLIC! SHOWROOM NOW OPEN!
"It says all parts are one hundred percent organic," Sheyenne said. "That can't be all bad."
"I can think of ways it would be all bad."
We moved forward. A few cars and pickup trucks were parked haphazardly outside the Emporium. We saw two men wrestling with a lumpy rolled-up Persian rug, which they carried through a back door marked Deliveries.
Several furtive customers emerged from the front door carrying wrapped packages. An older human couple, both bespectacled, wandered inside; like a gentleman, the man held the door for his wife. They looked like a pair going antiquing on a weekend.
"I'm surprised to see so many customers," I said. "Who would have guessed there's such a demand for the spare-parts trade?"
"Body building must be a more popular hobby than we thought, Beaux. Probably mad scientists window-shopping."
I figured there must be other legitimate uses—medical research, morticians needing to fill out or fix up a client, maybe even high-end gourmet flesh-processing plants for discriminating cannibalistic customers.
The foot traffic made our job easier; we didn't have to call attention to ourselves. Sheyenne and I blended in, and entered just behind the elderly couple.
Inside, the Emporium was an amazing place, well lit, with row upon row of fully stocked shelves. A color-coded map on the wall segregated the warehouse by species (Cralo catered to humans as well as most known unnaturals).
"It's like an IKEA for body parts," I said.
"You should take me to IKEA," Sheyenne said. "We could furnish your apartment."
I picked up a small wire basket for impulse buys, so we could maintain our cover as we walked up and down the aisles. There were severed hands and claws of all types, replacement lungs and hearts. A showcase area had framed samples of flayed skin and fur "For All Uses," without specifying any of the aforementioned uses.
Aisles were labeled: Skeletal Replacement, Soft Tissue Sundries, Kidneys (Two-for-One Special); one whole alcove was devoted to eyeballs. A rack of hardware store drawers contained teeth, separated and labeled. Fingers and toes were in the bargain bins.
Sheyenne flitted ahead, fascinated. "Reminds me of my med school classes. I wish I'd known about this place when I was in school." She stopped in front of rolls labeled Bulk Muscle, Guaranteed Steroid-Free.
There were bins of bones sorted by size, as well as a grinder and a bread-making machine. A Home Fashion section advertised decorative tumors, intestines dyed various colors (sold by the foot). Feet were also sold by the foot.
Jars and cans held cranial fluid under several brand labels; spray bottles had multipurpose mucus. Self-service plasma dispensers stood next to a large aquarium in which floated glands, ducts, and small organs, kept moist and fresh. It reminded me of forlorn lobsters in a tank at a seafood restaurant.
Thinking of the eviscerated vampire corpse in the Motel Six Feet Under, I wondered if the Vampire Parts section had any new arrivals. "We should tell McGoo about this place."
"You think he wants to take up body building?" Sheyenne asked.
"I think he might be curious as to where all these body parts come from."
A floor salesman was explaining to an older necromancer, "Our specimens are of the highest quality. Flash-frozen or vacuum-sealed in bags to preserve freshness."
"But where do they all come from?" the necromancer asked, his eyes sparkling with wonder. "It's been years since I've seen a selection like this."
"Some come from murder victims, some from executed murderers, unclaimed bodies in the morgue, maybe an indigent or two. A few fell off the truck." The salesman chuckled. "Some were run over by a truck. We buy in quantity so we can keep the prices down. The titles are free and clear on everything we sell."
With a sober nod, the necromancer tapped his chin. "I'd hate to have someone come shambling by asking for their pieces back."
"That's never happened, I assure you, sir!" the salesman said, bustling off with the necromancer. "We have a complete customer satisfaction policy."
"We'll see about that," I muttered. When Archibald Victor had tried to lodge a complaint, he couldn't reach their customer service department. No one had answered the phone or responded to his written complaints.
As was typical for a large store, when I tried to find a salesperson to help us, they were entirely invisible. Finally, at the back of the warehouse Sheyenne found an office marked Floor Manager, Xandy Huff, and she guided me there.
The door was partly closed, and before I could knock, I heard shouts from the office. "We shouldn't have kept you on after the first time, Francis—you're fired! Give me your badge!"
"B-but Mister Huff—I need this job!" It was a nasal phlegmy voice, not entirely human. "How am I going to eat?"
"Eating's what got you into trouble in the first place—this isn't a restaurant! How can I justify this to Mister Cralo?"
"Please don't tell him!" The voice quavered, palpably frightened. "Just give me my paycheck, and I'll leave."
"Mister Cralo decides whether you even get your last paycheck. We'd be within our rights to deduct the lost materials from your wages. Come back tomorrow after I've discussed it with him. For now, I want you out of here!"
The door flung open, and a sick-looking, greasy-skinned ghoul tottered out, panicked and uncoordinated. He had lank, lumpy hair; his face was emaciated. His crooked and broken teeth looked like random glass chips. Sniffling and sniveling, he careened into me, then walked directly through Sheyenne's incorporeal form. "Excuse me. Sorry!"
He reeled away, and Sheyenne remarked on the oily smear the ghoul's residue had left on my sport jacket. "I suppose it needed cleaning anyway."
I pushed into the floor manager's office, taking advantage of the man's emotional state. Better to spring my problem while Huff was in a huff, when his guard might be down. "Excuse me, Mister Floor Manager . . ."
The man behind the desk had a bald pate surrounded by a fringe of dark hair, and his jowly face was punctuated by a thick black mustache. He did not look like the sort to give encouraging talks to his employees, even when he wasn't having a bad day.
"Damn ghouls, eating on the job." He stuffed an employee badge (presumably from the freshly fired ghoul) into an envelope and slid it into a metal rack of time cards on the wall behind him. "What do you want?" He ran his gaze up and down my form. "You need a job? We just had an opening. You look like you're used to handling dead things."
"I'm here for a customer."
"A customer?" Xandy Huff changed his entire demeanor; he even managed a half smile. "How can I help you? Please call me Xandy—short for Alexander, but not quite Andy."
"I'm Dan Chambeaux, private investigator, and this is my associate Sheyenne. One of our clients, a Dr. Archibald Victor, asked me to help him pursue a complaint and investigate your operations. He showed us evidence of substandard parts, damaged goods, and mislabeled items that he's ordered from your catalog."
Xandy looked uncomfortable. "I don't see how that could happen. Why didn't he contact us directly instead of hiring a detective?"
"He's called numerous times and sent several letters."
"Oh. Our complaint line has been disconnected to better serve our customers, and all complaint letters are carefully looked into, on a regular basis."
"How regular?" Sheyenne asked.
"At least every six months."
"That explains a lot," I said. "Before pursuing legal action, our client wants to invoke your complete customer satisfaction guarantee."
"Certainly!" The man seemed very solicitous. "We believe in customer satisfaction."
I took out a folded piece of paper from my jacket pocket. "This is a list of items and order numbers, primarily a brain and a spleen, both of which were damaged. Dr. Victor would like to receive appropriate replacement organs or get his money back."
"We also think he's entitled to an extra set of lungs, to make up for the inconvenience," Sheyenne added.
Annoyed, Xandy Huff took the list and skimmed it. "The spleen is no problem, got plenty of those, if you have his receipt and his original purchase order. But the brain, that's difficult. There's always a shortage of brains around here. You'll have to talk to Mister Cralo himself."
"What about one hundred percent customer satisfaction?" Sheyenne asked.
"You'll be satisfied. Mister Cralo prefers to deal with complainers face-to-face. He makes them an offer they can't refuse. So far, we've had no repeat complaints." Huff glowered at us both, as if he thought we would be intimidated, but I'm not a zombie who gives up easily.
"And where can I find Mister Cralo?" I asked. "I'll go speak with him right now."
Xandy seemed alarmed. "Really? Do you have any idea—?"
"No idea whatsoever, but I do need to get this resolved."
Sheyenne added, "Dr. Victor is our client, and at Chambeaux and Deyer we also have a customer satisfaction guarantee."
"Suit yourselves. It's your funeral."
"Already had one," I said.
"Me, too," Sheyenne said.
Xandy gave me an address. "Mister Cralo spends most of his time at the Zombie Bathhouse. You can find him there."
"Zombie . . . Bathhouse?" Sheyenne's ghostly form shuddered. "I'm afraid you're on your own for this part, Beaux."
CHAPTER 13
By the time we left the Spare Parts Emporium, a weather front had come in. A fog bank smothered the sky with miserable gray, accompanied by a cold drizzle that refilled the pothole puddles in the parking lot.
A panel truck drove off so full of body parts wrapped in white butcher paper that the back doors had to be secured with a bungee cord. The panel truck roared away, hitting every one of the potholes, splashing rooster tails of brown water into the air. Just before it left the Emporium parking lot, the bungee cord snapped, and body parts fell all over the ground. Yelling at each other, the driver and his companion got out and rushed around reloading the truck, and then hurried off again.
In the ramshackle tenements a few blocks away, a swamp creature was hanging laundry on a line to soak up the mist, freshening up lacy underwear that was not designed for any sort of body I wanted to imagine. I heard a banshee doing a gut-wrenching—and heart-stopping—rendition of "Singin' in the Rain"; windows shattered, and neighbors shouted at her to stop the racket.
Sheyenne gave me an air-kiss and flitted off to the office. "I'll let you find the Zombie Bathhouse. Need me to bring your bathing suit?"
"I'll be fine, Spooky." I couldn't convince her that it wouldn't be so bad. I couldn't convince myself of that either.
Not far from the Spare Parts Emporium, the skeletal wreck of an abandoned railroad bridge loomed out of the fog like an economy-sized gallows big enough to offer group discounts. Shadowy figures lurked under the bridge in a squalid shantytown. Homeless unnaturals gathered there, bound together by fiscal situation rather than their unnatural species.
As I shuffled out of the gravel parking lot, feeling the damp cold mist on my damp cold skin, a gruff voice called. "Shamble, is that you? Dan Shamble!"
"It's Chambeaux." I looked over at the shanties and cardboard boxes under the bridge.
Ominous figures stood around a trash fire in an oil barrel, but the voice came from a large, sagging cardboard box. A hairy werewolf crawled out of the box and stretched, putting a hand to his back as if his body ached. He motioned me over. "Don't you remember me? It's Larry." He scratched a patch of matted fur. "I used to be more intimidating."
I came over, surprised. Larry was a werewolf hit man who had served as Harvey Jekyll's personal bodyguard; now he made his home in a cardboard box from the kind of discount coffin you could buy off the shelf at one of those price warehouse clubs. He had a tattered old blanket and a pillowcase stuffed with wadded newspapers. The scruffy-looking werewolf hadn't washed or trimmed his fur in some time.
"Larry, what are you doing here?" I shook his hand, then shook my head. "What happened to you?"
"Lost my job. Jekyll let me go after he moved into his cozy house in the suburbs with their neighborhood watch and block parties—said he didn't need a bodyguard anymore." He flexed his claws. "It was a crappy job anyway. And he was a terrible boss."
"I'm not surprised. Jekyll was a terrible human being, and then he became a terrible zombie."
"I have my pride," Larry continued. "I actually quit—he didn't fire me. I quit. Never wanted to work for him in the first place. It was demeaning."
Seeing that he now lived in a cardboard box under an abandoned railroad bridge in the drizzle, I had to wonder what his definition of demeaning was.
"You couldn't find other work?" I asked. "There must be plenty of jobs for bodyguards or hit men. This is the Unnatural Quarter."
"You'd think so." Larry tried to growl, but it sounded like the phlegmy aftereffects of pneumonia. "But I'm a pariah because I worked for Harvey Jekyll."
"Well, Larry, you dig your own grave, you have to rise from it."
He brushed a paw down his chest, as if to show that he didn't care. "I'll scrape by, Shamble. I don't need much." He stretched, then gestured me to follow him. "Come here—I want you to meet a friend of mine." He moved toward one of the burning barrels.
Cold water dripped from the framework of the bridge above, and bats circled in playful mating dances. The area under the bridge itself was a trash dump piled with old cans, boxes, broken glass. Some of the debris was arranged in neat piles, even assembled in what might have been called artistic sculptures; gravity rearranged the rest.
A troll came up to us. He was a few inches shorter than me and had leprous scales, pointy ears, and a face with all the wrong angles. He wore a padded fleece coat salvaged from the garbage.
Larry introduced us. "Dan Shamble, this is my friend Tommy Underbridge. He helps the less-fortunate unnaturals here."
The troll reached out a big-knuckled claw to shake my hand. "Good to meet you, sir. Care to join us for dinner? I'm making a large community pot." A cauldron was suspended over the low flames of the barrel's garbage fire. Something bubbled, and the aroma wasn't entirely awful.
"Goat stew," said the troll. "Stringy, but delicious."
"Where do you find goats out here?"
"They come out here to eat the garbage, try to cross the bridge, and get stranded. That's when we trip 'em and trap 'em."
An ogre who wrapped himself in a stained Scooby-Doo blanket raised his big shaggy head. "Damn goats, trying to steal our garbage. Serves them right." The ogre picked a rusty can from a pile, tossed it in his large-jawed mouth, and began crunching happily.
"Goats are getting scarce around here," Larry said. "But there's always the rats—plenty of those."
To be polite, I asked, "And what's your story, Tommy? How did you become a homeless troll?"
Tommy cackled. "Homeless? This is my home, exactly where I want to be. Beautiful view, lots of friends, and a bridge to live under! What more could a troll want?" He raised his hands. "This is our community center."
Larry picked up an empty can, with which he scooped up a serving of goat stew. "Want some, Shamble?"
"No, thanks. I ate at the Ghoul's Diner yesterday. My stomach's still a little queasy."
"I understand." With a gesture of his head, Larry indicated the Spare Parts Emporium. "Don't bother shopping there. Prices are too high. If you need organs or body parts, come see me. I can set you up, cash only."
"I wasn't trying to buy anything. Just investigating a case, a customer service complaint." I narrowed my eyes, thinking of the disorganized vampire in the Motel Six Feet Under. "And where would you get body parts?"
"Donations," Larry said.
Tommy had been cheerful, but now his mood changed. "You shouldn't get into that racket, Larry. It's beneath you."
"Beneath me? I'm living in a dump, and I've got a cardboard box for a bedroom."
"And you should be happy with what you have," Tommy said. "Lots of monsters going missing these days, Mister Chambeaux—and I doubt they're all moving out to the suburbs like Harvey Jekyll did."
I said, "I was about to go meet with Tony Cralo in person. I'll mention it to him."
"Cralo? Whoa!" Larry said. The homeless monsters suddenly looked uneasy. "While you were in the Emporium, did you buy yourself a set of jumbo-sized balls? Nobody goes to see Cralo."
"I have to follow where the leads take me," I said. "The cases don't solve themselves."
"So, uh"—Larry fidgeted as he slurped his goat stew, while Tommy the troll ladled up several cans and passed them to homeless monsters—"if you have any work I could help with—on a consulting basis—just let me know. I'll be here in my box. You need the address?"
"I can find it," I said, then paused. "Come to think of it, I do have a case that involves werewolves. Rusty, the leader of the Hairballs, got assaulted and scalped the other night."
"Heard about that," Larry said. "But you probably didn't hear that it's happened before. Nobody cares when bad things happen to homeless monsters." He led me to two mangy-looking werewolves who hadn't had a bath or a grooming in quite some time.
Both homeless werewolves wore dirty stocking caps that had been knitted from colorful yarn, back at the dawn of time. The fuzzy beanies were pulled down all the way to their pointed ears. "Dan Chambeaux, this is Arnie and Ernesto, two of my werewolf pals. Guys, he's investigating what happened to Rusty." Larry nodded at the two. "Go ahead, show him. He needs to know."
Embarrassed, Ernesto and Arnie glanced at each other, then pulled off the stocking caps. Both of them had been scalped. The crowns of their heads were scabbed, rubbery-looking, angrily healed.
"They got it two nights apart, a week ago."
This made the case more complicated. So Rusty wasn't just a single revenge target. "You think it's a case of scumbags attacking the homeless? That's called trolling."
"Don't you bad-mouth trolls," said Tommy.
Larry shook his head. "It's a gang disturbance between Hairballs and Monthlies. That's where I'd put my money, if I had any money. Trouble's been brewing for a long time."
"What started it?" I asked.
"Bad blood, a personal insult, but Rusty's so embarrassed he won't even talk about it. Something to do with the Voodoo Tattoo parlor. Scratch and Sniff were behind it, Rusty never forgave them, and Hairballs and Monthlies have been at each other's throats ever since." He puffed up his chest. "I'm a proud member of the Hairballs myself, been over to Rusty's house for a couple of his kegger parties."
"Couldn't they help you out of your situation?" I asked.
Larry looked away and scratched the fur on his muzzle. "I'm not that desperate yet. I'd be ashamed for them to see me like this. But sooner or later . . ." He squared his shoulders, flexed his claws again. "If gang rivalries heat up, they'll need all the muscle they can get—and I've got muscles."
I pondered all the details I'd learned from this single visit, and for good measure—or maybe good karma—I slipped Larry a twenty-dollar bill. (I think it was the one I had found on the floor of the cockatrice-fight warehouse.) A good investment, I decided.
CHAPTER 14
Out of the fog and into the steam.
The two words Zombie Bathhouse said it all—and deterred most customers. The bathhouse was a brown brick building that covered half a block, and most of the floor space was subterranean. Sturdy arches framed doorways that looked like the mouths into a sewer. A skilled tombstone artisan had incised the address and the name of the place into the stone.
I hesitated at the doorway, then steeled myself and entered the bathhouse. As I went down the slick stone steps, a stench wafted up, overpowering, although some might have considered it inviting. The thick, humid air was pregnant with brimstone smells and mineral salts. I hoped Archibald Victor appreciated that I was willing to shamble the extra mile for his case. It was a lot of work just to file a complaint properly.
I didn't know whether this place fell on the fair-trade or rough-trade side of the equation. Some customers come into a massage parlor really and truly for a relaxing muscle massage, and I had no idea how many undead came to the Zombie Bathhouse to get a bath. Nevertheless, Tony Cralo was here. He had an intimidating reputation, but I would talk with him, man to . . . whatever, and hope I got the "happy ending" I expected.
The main area held a large swimming pool in which bobbed dozens of naked zombies in various stages of decomposition. Steam wafted up from the water. Separate whirlpool baths offered special treatments from freezing ice plunges to ultra-hot mineral soaks. One tub was a veritable cauldron, and the bubbling water foamed and swirled; only one zombie dared to venture into that tub, which looked far too much like a stewpot to me. There was even a wading pool for kiddie zombies.
An undead attendant stopped me at a turnstile gate. "Fifteen bucks for a soak." He looked as if he'd been a librarian in his previous life.
"I'm looking for a man named Tony Cralo," I said. "I was sent here."
The zombie cashier's eyes widened. "In that case, my sympathies, and I wish you the best of luck. But it'll still be fifteen bucks for a soak." He ran his gaze up and down my patched-up sport jacket. "No street clothes allowed."
"Do you have bathing suits for rent?" I asked.
"We have towels and winding sheets."
"Give me a set, then. And a locker key."
I pushed through the turnstile, and the clerk handed me a stack of fabric. "This establishment assumes no responsibility," he said. "Especially for body parts that fall off."
"I don't plan to soak that long."
"Most people don't." The cashier went back to reading his paperback book.
I took the towel and winding sheets into the changing room, claimed a locker, and disrobed. I regularly worked out at the All-Day/All-Nite Fitness Center, so I was accustomed to seeing various naked monsters horsing around in the showers. I remembered one roughhousing group of vampires cracking other patrons with gym towels, all fun and games until they inflicted permanent damage on the crumbling bones and bandages of an old mummy who had been trying to get himself back in shape.
After they disrobed, zombie customers shuffled out into the pool room. Some wrapped towels around their waists, others didn't bother. With their sagging butts and leprous blotched skin, some with ribs and scabs showing, no one spent much time looking at the customers anyway.
I was in good repair, and the series of bullet wounds across my torso had been packed and stitched up by the best sawbones and taxidermist in the Quarter, but scars and stitches don't fade from zombie skin. Maybe I would ask Mavis Wannovich if she could do anything when I saw her for my restorative appointment that afternoon.
Zombies shuffled around the pools, dropping towels and flip-flops, climbing into the relaxing waters. A bathhouse attendant walked around with a skimmer net, scooping out slime and chunks of debris that floated to the top of the pool. Some of the more limber zombies frolicked about, splashing or dunking one another. One zombie's eyeballs had fallen out of the sockets, and his friends snatched them away in an impromptu game of Marco Polo. Most of the zombies, though, just soaked away their aches and pains.
I went to the attendant as he fished out hunks of detached skin and dumped them into a reclamation bucket. "Could you tell me where to find Mister Cralo?"
He sized me up. "Private pool number two. Don't tell him I sent you."
I found private pool #2, a separate Jacuzzi. Two linebacker-sized zombies in business suits blocked my way. They were the only ones in the entire bathhouse wearing street clothes. Their suits bulged in places that shouldn't have bulged.
"Excuse me, I'm here to see Mister Cralo."
One of the goons looked over his shoulder. "You holding court, boss?"
"I'm relaxed enough," a voice drifted from the steaming whirlpool. "Who is it?"
"Dan Chambeaux, private investigator," I said.
Cralo heard me and called past his two bodyguards. "A private dick? This is a nude bathhouse, not much private dick around here." He chuckled. The bodyguards chuckled. So I chuckled, just to be part of the gang. Then Cralo said, "Check him, then send him over."
"You'll have to remove your towel, sir," said one of the bodyguards. I let the towel fall, and the two men eyed me without being impressed. "He's unarmed, boss."
They let me through, and I stood before the Jacuzzi of Tony Cralo, an imposing monstrosity of a man. He lounged back against the rim as the hot water roiled around him. "Dan Chambeaux. Climb on in and have a soak. Ease your troubles."
I'd been invited by a powerful man who seemed to intimidate everyone, so I had to agree, even though I would rather have stayed on the sidelines. But the cases don't solve themselves.
Cralo was an enormously fat zombie. His bloated skin was just on the ripe side of putrefaction, his eyes sunken into their sockets and surrounded by folds of puffy skin. He squirmed in discomfort and emitted a loud basso fart that sounded like a howitzer going off. A huge bubble broke the surface of the water, and I swear the fumes were visible like greenish-brown pipe cleaners bent into odd shapes.
"Damned outgassing," Cralo said. "Never had good digestion before I died, and now . . . with a body my size, it's only natural." He let out another much smaller fart, then sighed. "Come on in, what are you waiting for?"
I paused for as many seconds as I dared, to let the fumes clear, then lowered myself into the whirlpool next to him.
"Feels good, doesn't it? That's special hot sulfur water piped in here. Good for the skin, good for the health." He rested the back of his head on the edge of the tub.
"Yes, it does feel good, Mister Cralo," I said.
"I don't know if these aches and pains are caused by me being overweight, or being dead." He heaved another sigh. "What can I help you with? Who's trying to charge me with something now? I'll post bail and get my lawyers right on it."
"I'm not the police, Mister Cralo. Just looking into a matter for one of your Emporium customers. He's an avid body builder, and several of his orders have been defective. He can't get anyone to respond to his complaints. In particular, he needs a replacement brain and a spleen. He'd appreciate a set of lungs, too, if you can spare them."
Cralo leaned forward in the pool like a sea monster approaching for the kill. "Xandy Huff couldn't take care of you?"
"He's the one who sent me over here, said you took care of all complaints personally."
"Damned minions, can't think for themselves . . . but I guess that's not what I pay them for." He chuckled. "If that's the worst problem I have all day, then I'm going to have a good day. My Emporium does a brisk business supplying mad scientists, resurrectionists, hospitals, medical research companies, even that new zombie rehab clinic. We do good work for the community as a whole, but we try to pay attention to our smaller clients as well. One Hundred Percent Customer Satisfaction, Mister Chambeaux—that's my policy. I'll see to it your client gets a replacement brain, spleen, and whatever else he needs. Stop by and see Huff tomorrow. Tell him I said so."
I was taken aback, after all the ominous warnings and disbelief I had encountered. "That's it? You're just going to say yes?"
Cralo leaned back, closed his eyes, and concentrated on the warm brimstone water. "Nobody's ever had the nerve to lodge and follow up on a complaint before. I'll give you props for that." He was obviously finished with me.
"Thank you, sir. It's been a pleasure."
Another large, fragrant bubble shot to the surface of the whirlpool, and I climbed out of the tub as quickly as I could, retrieving my towel from the business-suited bodyguards. I went back to the locker room to clean up and get dressed.
After this, I definitely needed my restoration spell from the Wannovich sisters.
CHAPTER 15
Life had been rough for me since my death. A career as a private investigator in the Unnatural Quarter is fraught with hazards, and Robin had suggested—though not seriously—that I find a safer profession. A zombie accountant, perhaps.
I had been beaten up by demon goons, gunned down in the street, framed for the murder of a gremlin, even had my arm torn off by an ugly monster. But this is what I do, and it's not likely to get simpler. I'm not the sort of guy who could settle for a cushy desk job.
To mitigate the wear and tear on my body, I set up regular appointments at Bruno and Heinrich's Embalming Parlor to top off my fluids, touch up my makeup, and do all the cosmetic things necessary to make me presentable to the public. Unlike other well-preserved zombies, though, I had an edge—my side agreement with Mavis and Alma Wannovich. I offered the witch sisters inside details about my cases for their new Dan Shamble books, and they gave me a monthly restorative spell. Quid pro quo.
I arrived at their apartment carrying the box of autographed Death Warmed Over special editions for the charity auction to benefit the Fresh Corpses clinic. I wiped my shoes on a quaint welcome mat that said Abandon Hope, Ye Who Enter Here and rang the bell.
Mavis opened the door, wearing her stars-and-moons witch's hat, which covered an explosion of tangled black hair that looked like a smoke spell gone wrong. "Mister Chambeaux, welcome! Delighted you could make it. We're going to have a busy week—all of us." She was plump and bubbly; her black dress barely surrounded her girth.
I didn't know what she meant. "It's always a busy week." I carried the heavy box of books inside and set it on a counter. Each copy of the special edition had a small inset marble tombstone with the copy number engraved in the stone.
Mavis opened the box. "Wonderful! We'll get these over to the Worldwide Horror Convention. MLDW is running the charity auction, and these autographed first editions are sure to be hot items."
The Wannovich sisters had a small, homey place with cross-stitched samplers on the walls featuring verses from the Necronomicon, funereal lilies in a vase on the table. Odd-smelling potpourri bubbled from miniature cauldrons over black candles. A wide variety of curious roots and herbs grew in planter boxes and vases throughout the residence. A small shrine-offering sideboard stood against one wall, cluttered with unusual trinkets, amulets, vacation photographs, and even a shrunken poodle head.
Mavis's sister Alma, who had been irreversibly transformed into a sow by an attraction spell gone wrong, also worked at the Howard Phillips Publishing Company. Though the witches had offices in the main building, they took reading days and worked part time out of their home—hence the dozens of manuscripts piled on the floor.
Alma rooted through the slush pile, shoving her pig snout into the papers and nudging them around. She snuffled at the prose as if she hoped to find either truffles or some literary gem. The sow looked up at me, snorted a greeting, blinked her beady close-set eyes, and turned back to the manuscripts.
"Come on, Alma—you have to help me with Mister Chambeaux's treatment. And then we'd better clean up for our company. The ladies will be here soon."
Alma knocked over a thousand-page manuscript held together by several rubber bands. My Immortal Loves, Volume 1, by Fred Nosferatu. A more scholarly text, judging by the number of footnotes at the bottom of its manuscript pages, was Scandal in the Church, by Q. Modo.
Mavis tidied up the sitting room. She had already put a plate of cheese and crackers on the coffee table. "We're having a ladies' social gathering in about an hour, Mister Chambeaux—just some other witches for tea. It's the Pointy Hat Society."
"I won't take up too much of your time, then," I said. Alma used her snout to shove the manuscripts into a haphazard pile behind the sofa, while Mavis flipped through a well-worn spell book; particular pages were marked with sticky notes.
The restorative spell itself would not take long, although it drained Mavis by the time she was finished. "If this were easy, everyone would do it," she had once told me. "But we're very happy to help you."
Some restorative spells required nasty-smelling unguents applied to all of the body's orifices. Fortunately, my regular tune-up required none of that, just a few recited incantations in what sounded like pig Latin (and Alma joined in).
Mavis sketched designs in the air with an incense stick that roiled with lavender smoke, and I immediately sensed an improvement that turned the rigor into vigor. It felt much better than the brimstone mineral soak in the Zombie Bathhouse.
I stretched, flexed my arms and legs. "I appreciate it very much."
"And we appreciate you, Mister Chambeaux. We owe you a great deal, and not just for the cases you solved. We got our jobs because of you and Ms. Deyer, and now Howard Phillips Publishing is on the verge of its greatest success. We want you at your very best to help us with the promotion." Mavis went around the room, propping up cardboard posters that showed the garish red cover of Death Warmed Over. "We'd love it if you could do just a teensy amount of publicity for the book release. My instinct says this series will catch on like a funeral pyre."
"What . . . sort of publicity?"
Mavis continued, as if she were in a sales meeting. "We are thrilled with the response we've received so far, and the pre-order numbers are great. Have you seen the stellar reviews? And a cover quote from Charlaine Harris herself! Can you believe that? 'An unpredictable walk on the weird side. Prepare to be entertained.' Charlaine Harris!" Mavis could barely control herself.
"I'm glad Ms. Harris liked it," I said. "What sort of publicity were you expecting me to do? I'm not good at interviews."
"Oh, just an appearance—a book signing for a very specialized audience. Say . . . tomorrow? We want you to be our guest at the Worldwide Horror Convention. Howard Phillips Publishing has a large table in the dealers' room. We've taken out an ad for the novel in the program book and included a free preview copy in the registration bags." She moved around the room, setting out stacks of Death Warmed Over postcards and bookmarks, and engraved pencils shaped like wooden stakes. "Just one day at the con and one signing at our table, to meet your fans?"
"How could I have any fans? The book just came out."
"Some people are fast readers," Mavis said, "and the buzz has been phenomenal." Alma snorted in enthusiastic agreement. The witch pulled out a Special Guest badge adorned with a VIP ribbon, like the ones I had seen on the tour group at the Goblin Tavern. "We already picked up your badge."
I flexed my arms again, feeling the renewed vigor and flexibility there. After the rejuvenation spell, my whole body felt tingly and energetic, and I owed a debt of gratitude to these two witches. "Tomorrow is short notice, but I can make an appearance for a few hours. I've never been to a Worldwide Horror Convention before." I had to admit, I was interested in what Miranda Jekyll intended to talk about, since she was also a special guest.
The doorbell rang, and Mavis hurried to answer it. "Oh, the ladies are starting to arrive! I'm glad you're still here, Mister Chambeaux. Many of them want to meet you."
I looked at my watch. "You said they wouldn't be here for an hour yet."
"Some insist on showing up early."
Four other witches came in, all wearing identical pointy hats. They chattered, giggled, and complimented one another on hairstyles and scarves. With far too much pride, Mavis hurried the women over to introduce me. "These ladies are our friends, Mister Chambeaux. We're not exactly a coven, just a social group." While we were talking, several more witches arrived. It didn't take a detective to figure out that Mavis had told them to come early, just so she could show me off.
Alma waddled around the apartment, letting the guests place jackets and scarves onto her back, after which she tottered toward the spare room and shook them off onto the bed.
"We started out as a support group for Alma," Mavis continued. "They're still working to find a cure for her spell, but we love and accept Alma just the way she is."
The ladies had brought cupcakes and finger sandwiches. One particularly tall witch slipped a large silver flask out of her coat and passed it around. Mavis put a cauldron on the stove to boil.
The ladies treated me like a very special guest, asking how it felt to be famous, tittering over how dangerous and exciting my job must be. Many of them already had copies of Death Warmed Over, which they asked me to sign. Finally when the Pointy Hat Society turned their conversation to ladies' club topics that I didn't understand—primarily gossip about members who had not yet arrived for the meeting—I excused myself.
"Sorry, ladies. I have work to do. The cases don't solve themselves." I pocketed my convention guest badge and promised I would be there as agreed. All the women said goodbye in an eerie harmony.
Mavis quipped, "He's a zombie detective. He can't rest in peace until he brings criminals to justice." Then she jotted that down, thinking it might be useful as a tagline.
CHAPTER 16
Walking back to our offices in the afternoon, I soaked up the ambience, or the miasma, of the Quarter. One of the Kreepsakes chain gift shops was shuttered, its inventory of ridiculous monster souvenirs liquidated. A large UQ Tours motor coach rolled by, packed with tourists who snapped photos from behind the safety of monster-proof windows. I could hear the muffled sound of the tour guide's voice as he described noteworthy parts of town.
I did enjoy this place and felt it was my home. Years ago, I never would have imagined my career, and afterlife, would lead me here, but I felt a certain responsibility for the people, particularly the ones who came to me and Robin for help.
It was amazing, in a way, that so many monsters of various sorts—as well as a fair number of humans—managed to get along, live and let live (or maybe exist is a better term for it). There are ethnic neighborhoods generally divided by the type of unnaturals, and there are feuds, romances, lazy nights, nail-biting days, the aromas of exotic cooking (or spells), everyone finding their niche. Thankfully not everyone got along, or I would have been out of a job, but the Quarter wasn't much different from any other city, except for all the supernatural creatures.
A black-furred, full-time werewolf in a white apron cranked open the awning of his little tienda, setting out fruit baskets and marking up the prices on the spoiled fruit. Next to him, a vampire tended his newsstand in the shade, wearing gloves, oversized sunglasses, and slathers of sunscreen. A black gargoyle stopped to buy a package of peppermint gum.
The tour bus ground to a halt, and the human tourists crowded to one side, taking pictures of the gargoyle and the vampire, who looked up in annoyance. It made me think of Harriet Victor and her life as the bearded lady in a circus freak show. Given the publicity from the Dan Shamble books, I worried that UQ Tours would change the bus route so the gawkers could swing by the Chambeaux & Deyer offices.
I came upon McGoo entering the Transfusion coffee shop for his usual cinnamon latte. "Hey, Shamble. Heading back to the office?"
"In my own ponderous way."
"I'm headed that direction. Think we should pick up a coffee for Robin?"
"She'd like that." We ducked into the coffee shop, and McGoo even paid, which was one of the strangest things that had happened all day.
"So, what does a vegan zombie eat?"
"I don't know. I'm a rib-eye man myself."
He stretched out his answer. "Graaaaaiiiiiins!"
I let out a long groan, always an impressive sound, to further my image. Before he could tell another joke, I distracted McGoo by talking about work. "I've got a lead you might want to check out for that murdered vampire—Cralo's Spare Parts Emporium has a staggering variety of organs and body pieces. I don't know exactly where they get their inventory, but if you dig into their records, you might find a few organs—maybe even some vampire organs—that don't have the proper receipts and donation forms."
He sipped his coffee. "Thanks, Shamble. I'll look into that."
Emerging from Transfusion, we spotted the lanky Furguson sniffing around a fire hydrant. The owner of a used-clothing shop, a female Monthly werewolf, came out of the shop and chased him off. "Get away from that hydrant! Nothing for you there!" She made a disgusted sound. "Stupid Hairball."
Growling, Furguson slinked away. "Stupid Monthly."
Seeing him alone, I wondered if his uncle had chastised him, maybe even kicked him out of the house for losing so much money at the cockatrice fights. No doubt Rusty would have preferred to learn that the Monthlies were responsible, since he had already expended so much personal energy on the feud.
Hearing the coughing rumble of loud motorcycle engines, I looked up. "Here comes trouble," McGoo said.
"You're stereotyping again."
"I'm observant. Got no problem with bikers in general—just those two in particular."
I knew that motorcyclists could be some of the most tight-knit and sociable people you'd ever meet; they didn't judge, they lent a helping hand. There was even talk of a great rally through the Quarter, the Ghost Rider Classic.
But Scratch and Sniff were the bad apples of the bunch. They arrived on their chrome custom-built choppers, fur coats flapping behind them, rusty meat cleavers dangling at their sides. The two rode without helmets or sunglasses, catching flies in their teeth; they looked human and exuded feral energy. Gripping ape-hanger handlebars, they looked from side to side as the choppers cruised past. They let out wolf whistles at the clothing shop owner who stood at her doorway; she gave them a flirtatious wave. Turning toward the black-furred Hairball tienda owner, though, Scratch and Sniff hawked up wads of phlegm and spat like a double-barreled shotgun. The black-furred werewolf ducked out of the way in time, but the phlegm splattered the baskets of fruit.
"Sorry about that, man," Scratch yelled. "Didn't mean to hit the fruit."
Sniff chuckled and hooted. "Yeah, we just had hairballs caught in our throats."
Scowling, McGoo stepped into the street, wearing his cop uniform like armor. "Spitting in public is a hundred-dollar citation, boys. We don't want any trouble—hope you didn't bring any."
"Just going for a nice afternoon ride, Officer," Sniff said.
"We wouldn't dream of—"
Then a snarling cannonball of fur barreled toward the two bikers, all gangly arms and legs, flashing teeth. Furguson. He sprang at Scratch, the nearer of the two, but tripped at the last minute and fell on the pavement in front of the chopper. He tumbled into the front wheel, which knocked the chopper over so that Scratch collided with Sniff, and both motorcycles crashed.
Furguson sprang up in an instant, baring his teeth. "That's for what you did to my uncle Rusty!" He threw himself on Scratch and raked his claws down the biker's head and chest, ripping open a long, ugly wound.
McGoo and I rushed toward the fray, but the brawl exploded in a minute. The Hairball tienda owner bounded from his shop and dove on Sniff, who was trying to help his partner. Furguson moved in a flurry, slashing, jabbing—not a very good fighter, but certainly a frenetic one. Scratch and Sniff were bleeding profusely, but looked more annoyed than mortally wounded. The used-clothing shop owner ran to fight alongside her fellow Monthly werewolves, armed with nothing more than a coat hanger.
I pulled out my .38 and fired two shots in the air, which startled everyone long enough for McGoo to snag Furguson by his shirt collar and yank him away.
Scratch and Sniff struggled to grab their meat cleavers. The black-furred werewolf stood with muscles bulging, growling deep in his throat as he faced off with the clothes-shop owner, who menaced him with her clothes hanger.
McGoo had had enough. He reached for his belt, which held numerous defenses against both humans and unnaturals, and brandished a large canister. "Back off, all of you, or you'll get a face full of pepper spray. You think you're red-eyed werewolves now . . ."
Scratch coughed blood, pushed himself into a sitting position. He seemed more concerned about his mussed hair than the gashes. "That was assault—you all saw it!"
Furguson snarled. "They started it! They attacked my uncle Rusty!"
"Haven't proved that yet, Furguson," I said.
I watched in fascination as the kaleidoscope of tattoos on Scratch and Sniff crawled like multicolored worms, squirming, stitching skin together until the deep gouges vanished.
Sniff brushed himself off. "Gotta love those voodoo tattoos."
"Not a mark." Scratch laughed at the embarrassed Furguson, then turned toward McGoo. "We want to file charges, Officer. That's deadly assault. He tried to kill us!"
"There's not a mark on you—as you said yourself." McGoo smiled. "But I'll be happy to take you downtown and let you file a complaint. I've been meaning to have you both answer some questions anyway about this feud with the full-time werewolves—and your involvement in the very real assault on Rusty."
Scratch backed away. "We had nothing to do with that." He adjusted his hair so that it looked like the back end of a duck again. "Just like we had nothing to do with those other things you've charged us with in the past."
"Only this time, we really mean it," Sniff said.
Scratch righted his chopper. "We don't need to file paperwork, and this matter isn't going to any court. We'll handle it our own way." The two bikers swung aboard their choppers again. "This isn't over," they said and roared down the street.
Furguson looked infuriated and ashamed. His fur was still bristling.
"That's not the way to make it up to your uncle," I said.
"It was a start," he mumbled, then shook his head. "And I couldn't even do that right."
CHAPTER 17
Coming back from the dead isn't as bright and cheery as most people might think. More often than not, there are tragic consequences.
For my very first case in the Quarter, I'd been hired by a family to track down their missing uncle, who had risen from the grave and shambled off. When I did find the man, he was in such a decomposed state that the family was repulsed and never saw him again. It broke the poor guy's heart.
Next morning, when our walk-in client appeared (though it was more of a lurch-and-stagger-in client), I knew in my gut that this was going to be one of those situations.
Adriana Cruz was messed up, both physically and mentally. She had been quite vivacious—good figure, good looks, great personality, on the cheerleading squad her freshman year before moving on to pre-law and volunteering at a community legal center. Adriana had a bright future and ambitious plans, a whole life ahead of her.
Unfortunately, she also loved to send text messages to her friends while she was driving, and she wasn't as adept at multitasking as she thought she was.
"I watched the funniest kitten-and-crocodile video you ever saw, and I was just texting a message back to my friend," Adriana said. "I hit L O and was about to type the last L when I became car number five in an eight-car pileup. Or maybe I was car number six—my memory's a little fuzzy on the details."
Adriana's good looks and good figure hadn't come through the automotive trash compactor in the best condition, to put it mildly. She held up her scarred hands, her fingers splayed at all the wrong angles. Her head was cocked to one side. Although Adriana must have arrived at the funeral parlor with "closed coffin suggested" stamped on her forehead, her mortician had gone the extra mile. Unfortunately, she'd needed a lot more than a mile.
"I'm sorry for your loss," Robin said. "How can we help you?"
"Nobody plans on being mangled," she said. "We all imagine that we'll die in front of a cozy fire at age ninety. No such luck for me."
Sheyenne joined us, her expression deeply sympathetic. "Some days, I'm thankful I came back as a ghost. Given the choice . . ."
I imagined Adriana might want us to file some sort of wrongful death suit against the other drivers in the pileup, or the cell phone manufacturer, or even her friend who had sent the funny cat-and-crocodile video clip.
It turned out to be much stranger than that.
Sheyenne led the way, politely opening the door as Adriana lurched into the conference room and slumped her battered and mismatched body down into the chair. The disfigured zombie kept talking as she adjusted her position. "I'm a responsible young woman. Twenty-eight. Most people my age don't worry about death planning, but since I was in pre-law, I know what can happen if you have a sloppy will, so I left instructions for the disposal of my remains. My parents thought I was being morbid, and I told them they were being irresponsible. We, uh, didn't always see eye to eye. But I made my last wishes known. It was completely clear."
Sounding like a voice-over in a movie trailer, I said, "But something went wrong?"
"Something definitely went wrong—I mean, in addition to being killed in an eight-car pileup." Adriana fidgeted as if trying to find a comfortable position. "I requested cremation, especially after the Big Uneasy. I didn't want to come back no matter what. It's just a matter of personal preference."
"But your parents didn't follow your wishes," Robin said, pulling out a yellow legal pad and taking notes. "Sentimental reasons?"
"Complete, utter, malicious incompetence."
I began to grasp how ugly this was going to be, if Adriana wanted to sue her family for not following her final wishes.
Adriana looked from me to Robin, saw the expressions on our faces, and she said, "Oh, it's not my parents! They sent my body to Joe's Crematorium—but, as you can see, I wasn't exactly cremated."
"They'd probably take you as a walk-in," Sheyenne suggested. "Tell them what happened. I doubt they'd even charge you for the cremation."
Adriana raised herself up, horrified. "Oh, I couldn't do that now! This is . . . this is what I am."
Still trying to get all the facts, I asked, "So, you woke up before you were delivered into the furnace? That can be awkward." Reanimation rates vary widely. Some, such as Harvey Jekyll, came back to life almost immediately; in my case, it took several days.
Again, Adriana shook her head, and it wobbled from side to side like one of those bobblehead dolls that McGoo finds so amusing. "No, I woke up in a body truck at the crematorium. It was stacked high with cadavers, but I was the only one moving and moaning. It was hard for me to get up and walk because I just had a quick patch-up job. The mortician didn't bother to use any finesse, considering the condition I was in."
I had my mind focused on the mystery, though. "You were in a truck full of bodies being delivered to the crematorium?"
Adriana shook her head. "I was confused and disoriented, as you might guess. I rolled over and fell off the back of the truck—when the driver pulled out of the parking lot and drove away from Joe's Crematorium."
"Wait a minute, why would anybody take a truckload of bodies away from the crematorium?" Robin asked.
"That is the question, isn't it?" Adriana said, folding her hands on the table and getting down to the real business. I could see that she would have made a good attorney, if she'd survived law school. "I'd like to engage your services to investigate Joe's Crematorium, to make sure this doesn't happen to other zombies. Find out what's really going on."
Robin tapped her pencil on the pad. "And get your parents a full refund."
Now Adriana seemed even more agitated. "The worst part is, my family didn't even know! They received an urn that supposedly contained my ashes. They thought everything was fine."
"That's fraud, not incompetence," Robin said, her righteous anger rising, as I knew it would. "We'll find out whose ashes they really have, and we'll hold Joe's Crematorium responsible."
Adriana looked as if she wanted to cry, but found an inner strength. "I want to find out how this could have gone so wrong."
Sheyenne flitted out and returned quickly with a card for Miss Lujean Eccles's taxidermy and body-repair boutique. "You might want to give this woman a call. She's done wonders for Dan, several times."
Adriana shook her head again. "I've already tried a cosmetician, a place called the Parlour (BNF). I spent hours and a lot of money on skin wraps, lotions, various types of makeup. They even redid my hair, and it looks horrible. In my spare time, I might want to sue them as well." She looked over at Robin. "I'll keep your card."
Considering the intertwined cases, I decided it was time to look into the Parlour (BNF) before more harm was done.
CHAPTER 18
Since Harriet Victor was the co-owner, I made my visit to the Parlour (BNF) under the pretext of passing along an update on her husband's case, but I really wanted to see firsthand how Rova Halsted had spent the life insurance money. And I intended to investigate whether she was as bad a stylist as she was rumored to be.
Although Robin had contacted the opposing attorney and set up a meeting with all of us for early the next morning, Rova Halsted didn't know who I was . . . not yet. I had plausible deniability. Since her ex and I were post-death friends, she had never seen me hanging out with him. After we all met to discuss the case, however, the rules would change. This was the best time for me to gather information, when her guard might be down.
At the door of the Parlour (BNF)*, the asterisk directed me to an explanation: *(Beauty, not Funeral). There, one mystery solved.
I stepped into the Parlour, jingling the cheery bell mounted above the door. Inside, I saw several empty hair-cutting stations, a pedicure chair with a large tub for oversized feet, a manicure table complete with a display of nail polish that boasted a selection of blacks ranging from Simple Flat Black to Pitch to Oilslick. Smells assaulted my nostrils: detanglers and conditioners, bleaches and colors, perm chemicals, reptilian scale wax, freshening crèmes, fixatives, mousses, pomades, and deodorants.
The lavishly bearded Harriet Victor was tending a female demon whose entire body—and I mean entire body—was covered with pubic hair (which smelled accordingly). Harriet was giving her a full-body perm on an aesthetician's table.
A pale-skinned undead woman, who had been stitched together from various pieces, had her hair done up in a pyramidal beehive. She sat under a hair dryer, preening in front of a hand-held mirror. "I love the long white streaks. I'm going to be a bride!"
At the cash register, a female full-time werewolf was paying, and the human woman ringing up the sale handed her a receipt. "Here's a coupon for your next visit."
The werewolf flinched, as if the coupon were cursed. "No, thanks. I won't be needing that." When she turned around, I saw that her face fur was trimmed and chopped so badly that it looked to be the work of a drunken, nearsighted Edward Scissorhands. The werewolf touched her face, groaned deep in her throat, and hurried out of the Parlour (BNF).
The human woman—Rova Halsted, I assumed—closed the cash register. She could have been pretty if she hadn't practiced her questionable skills on herself. Her hair was a randomly colored mop, like a collection of sample swatches from a hair-color line. She looked up at me. "How can I help you?"
Before I had to make up a story, Harriet called out. Her lips curved upward in a broad smile that fluffed out her curly beard like catfish whiskers. "Mister Chambeaux! Are you looking for Archibald? He's at work."
I thought quickly. "He told me to be discreet, so I thought I'd ask you to pass along a message."
She clicked her tongue against her teeth. "Oh, Archibald and his silly secrets! He thinks everything is such a grand mystery. Does anybody really care? I love him all the same."
Even so, I chose my words carefully. As a general rule, I didn't want a mad scientist upset with me. "Regarding the matter of the . . . the business trouble he was having, I spoke to the owner, and I've been promised adequate replacement parts. I'll deliver them in person tomorrow."
Harriet nudged the pubic-hair demon to roll over onto her back, then continued massaging deodorant oils into the creature's shoulders and arms. The wafting stench of body odor was overpowering, even to my deadened nostrils. A forest of tangled wiry hair extended from her solar plexus all the way down to her knees. "Oooh, that feels good, don't stop," the demon said, and Harriet continued her massage.
Rova went to attend her undead bride customer, swinging aside the hair dryer so she could squeeze more bleaching chemical in a zigzag streak from the temples to the top of the beehive hairdo. "Permanent means permanent—our guarantee."
Harriet talked to me as she continued working the demon's short hairs. "You look a little pallid, Mister Chambeaux. We could offer you a tanning session for half price, a friends-and-family discount. You need to get out in the sun more."
"It's not lack of sun exposure—embalming fluid takes the color away," I said. "Do your tanning beds get much use?"
"Unfortunately, no. I . . . misjudged the market."
Sheyenne had uncovered all the business details for me already. Harriet had originally opened a tanning salon, since not all monsters hid in the shadows or avoided direct sunlight. Even so, it was a niche market and an undead tanning salon wasn't the best idea anyone had ever had—vampires couldn't use the service, werewolves didn't care, and no matter how much sun a zombie got, undead skin would retain its deathly pallor. After leaving the circus, Harriet sank her life savings into the business and nearly went bankrupt.
Then she met Rova Halsted, who was desperate for work as a beautician, unable to keep a job in the outside world. As a result of Steve's death, she had a large life insurance check, which she used to expand the tanning salon into a full-fledged parlor (beauty, not funeral).
Now Harriet spritzed freshening spray and feminine deodorant over the pubic-hair demon. She even tied a bright red ribbon in one of the wiry locks of hair that protruded from the woman's head. "I like making people beautiful, in their own particular way," she said to me in a wistful-sounding voice. "After all those years in the freak show, I had enough of people not accepting me for who I was, so I came to appreciate everyone for who they are."
After Rova replaced the hair dryer over the bride's head, she went back to look at the appointment book. "I have one more client for the day, Harriet. A full preening, but after that I have to pick Jordan up from day care by six o'clock."
"You have a son?" I asked. Disingenuous is my middle name.
She was surprised by my interest. "Yes, he's eight." She eyed me up and down as if I had tried to hit on her. "I am single, but not looking—certainly not for a zombie."
Harriet gave a scolding cluck of her tongue. "That was rude, Rova. Mister Chambeaux is a professional."
The woman gave me a brusque apology. "Sorry—you can't help that you're dead." She closed the appointment book. "I was just dreading my next appointment."
"Apology accepted. It wasn't my place to ask." I'd learned all I expected to, so I decided to take my leave. As I opened the door of the Parlour (BNF), however, I nearly ran into the next customer.
She bristled with razor-edged black-and-green pinion feathers over an unfortunately curvaceous body; she had a face that immediately set you on edge and a hard smile that would scare away the hungriest carrion bird. Esther the harpy waitress stopped me in my tracks.
"Dan Chambeaux!" she said in her piercing voice. "Stop right there—I want to have words with you."
CHAPTER 19
It's unnerving to be buttonholed by a harpy, especially an angry harpy. (Or is that redundant?)
Esther ruffled her plumage so she seemed to be more in the valkyrie category. She poked my chest with one of her talons and nudged me back into the Parlour (BNF). Now, I'm not cut out to be a larger-than-life hero, regardless of how my character might be portrayed in the Dan Shamble Penny Dreadfuls, but I saw no way to duck under her sharp feathers and escape.
Finished with the bride, Rova Halsted looked up with a decidedly forced smile. "Hello, Esther. Ready for your preening?"
"Of course I am! Why else would I come here?" Esther's sharp, raucous tone reminded me of brawling crows. "I don't have time for the usual chitchat—this famous detective and I have something to discuss. Do your work while we have a private conversation, but no listening in!"
Rova was unfazed. "Not to worry. I never listen when my customers jabber."
I asked, "What's this all about, Esther?" As far as I could remember, I had left her a tip the last time I ate at the Ghoul's Diner. She'd been known to walk the streets of the Quarter, hunting down recalcitrant customers and demanding her due.
"I'm hiring you for a case. Actually, I need that lawyer partner of yours to file a lawsuit, but she never comes into the diner, so how am I supposed to talk with her?"
I was relieved that this was a business matter instead of some personal grudge against me. "Well, you could come to our offices, like most clients do."
"What? Is she too good for us? Something wrong with the food?"
"Robin usually brings a bag lunch."
The bride examined herself in the mirror, studied her newly painted nails, touched the white-streaked hairdo. Esther rounded on her. "Stop hogging the chair. It's my appointment now. Don't you have someplace to be?"
"Yes, I do," the bride said with a sniff, then added in a syrupy voice, "I'm going to meet my husband-to-be for a romantic lunch together while we talk about our wedding." As she flounced out of the Parlour (BNF), she added over her shoulder, "You obviously need to get laid—very badly."
"How dare you!" Esther shrieked, flashing talons and fluttering her machete-like feathers. "I've been very badly laid plenty of times!"
Indignant, Esther dropped into the vacated chair while Rova tied an apron around her neck and gathered up brushes and pots of chemicals. The harpy continued her stream of bitter complaints, though I was the only one listening, and I had no interest whatsoever.
"I have a boyfriend," Esther continued, sounding defensive. "A rich and powerful one, too—my own sugar daddy! He rides in a limousine and takes me to the fanciest places. Better than any man who would've settled for that bitch."
Esther was flaunting plenty of bling that she didn't wear around the Ghoul's Diner—gold bracelets, jeweled rings on her talons, even a designer scarf around the wattles of her neck. Rova began the long process of combing the harpy's plumage, filing and sharpening her long talons, oiling the fine scales on her forehead.
"And does your . . . boyfriend have anything to do with the matter you wish to discuss?" I was looking for an excuse to leave as soon as possible.
"Of course not! Why the hell would I want to sue him? He'd stop giving me nice things if I sued him."
"So what is the lawsuit about, then?" I asked.
Esther made a disgusted noise that sounded as if she'd gotten a piece of gravel stuck in her gizzard. "I see I'm going to have to treat you like a schoolchild and start at the beginning."
"I recommend that," I said.
She made that disgusted sound again. "Zombies are known for brains, but obviously not the ones inside their skulls."
Always cheerful, Harriet Victor finished with the pubic-hair demon, who admired the red ribbon as she left the Parlour, smelling fresh as a spring morning. Rova, not much of a conversationalist herself, continued to work on the pinfeathers and then the larger feathers on the harpy's arms.
Esther jerked from side to side, trying to concentrate on me. "Somebody left me a tip—a tip that I don't like." Considering her personality, she should have been happy with any tips at all, but I wasn't stupid enough to say that out loud. "It's cursed, and that wizard left it to me on purpose. He's never been happy with my service."
Again, I refrained from pointing out that no one was happy with Esther's service.
As Rova dodged out of the way, the harpy squirmed and pulled out a small black coin purse. "Instead of leaving cash, this moron gave me a medallion." She yanked and tugged until she withdrew a gold medallion on a chain. Two simple words had been stamped on its face: For Luck.
I still didn't see what was wrong. "It's a luck charm."
"I can see it's a luck charm—but I didn't realize until later that it's a bad-luck charm! I thought he was trying to make up for being such an asshole customer, but ever since he gave this to me, I've had a run of constant bad luck, and that wizard is to blame. I've dropped countless plates and cups. My orders are screwed up. Large parties come into the diner two minutes before closing time. Customers take the wrong copies of their credit-card receipts. It's awful!"
I listened with an expression of rapt interest because that was important to Esther, but I had seen her drop dishes and cups, screw up orders, and rail at the customers long before being given any cursed jewelry items.
"Worse, he keeps coming in, acting as if nothing ever happened! He wants to see how miserable I am. I'm never accepting a tip from him again!"
Still looking for a way to slip out the door, I asked, "How exactly can Chambeaux and Deyer help you?"
"Don't be an idiot. I want to sue his ass, then I want to sue the rest of his body. Make sure you add all that pain-and-suffering crap, too."
"I'll look into the matter and give it all the attention it deserves," I said. "If you're sure it's a bad-luck charm, why don't you just get rid of it? Throw the medallion away and stop worrying about it?"
"I tried that, you moron! I threw it into a Dumpster, into a trash can, dropped it down a sewer drain—it always reappears! It's magically connected to me. As I said, the damned thing is cursed. Weren't you listening?"
"I wasn't clear on the specifics," I said.
She swiveled her head and snapped at Rova. "And you need to hurry up! The limo will be around to pick me up in half an hour, and I've got to look my best! Can't have the fat slob thinking I'm unattractive."
Though I remembered the misery that Rova Halsted was inflicting on my friend Steve, I felt sorry for her at the moment. Just a little bit. But I got over it. "We wouldn't want that," I said.
"Thank you!" the harpy said, exasperated. "At least he understands! And if a man can understand, then even an idiot stylist can!"
Rova gave her a cold smile. "Would you like your wings clipped?" She had already gotten out a large pair of hedge-trimming shears and seemed anxious to use them.
"Not today, no time. And you, Chambeaux—don't just stand there. Get on my case!" Esther thrust the medallion toward me. "Take this. You'll need it for your investigations."
I backed away as politely as I could, in case it truly was a bad-luck charm. "Not necessary. I have everything I need. Let me do some investigating."
"You better give my case the highest priority, or you can serve yourself the next time you're in the diner."
That might have been preferable, but instead of provoking Esther, I said, "I give every case the highest priority."
While she looked momentarily satisfied, I hurried out of the Parlour (BNF) into the gathering dark.
CHAPTER 20
Each client is important to us at Chambeaux & Deyer Investigations, but after that less-than-pleasant experience, diving into an abrasive harpy's problems was not the first thing I wanted to do. A gang war among werewolves seemed more important than Esther's well-deserved string of bad luck.
Twice now, I had seen the arcane tattoos come alive on Scratch and Sniff, and Larry the werewolf had mentioned that Voodoo Tattoo was partly responsible for the blood feud between Monthlies and Hairballs. I decided to stop by and see what I could find out.
Voodoo Tattoo was a little boutique business tucked away in one of the Quarter's innumerable dark alleys. More than just a place for body art, it also doubled as a collectible doll shop. The façade was run-down chic, and a pulsating neon sign flickered the words Voodoo Tattoo on and off. Signs in the windows listed the specialties: Tattoos: Temporary or Permanent (depending on species), Special Discount on Do-Overs, We Offer Piercings of All Types, and in smaller letters, Silver Available upon Request, with an even smaller disclaimer beneath that Customer Assumes All Risk, No Refunds for Fatalities.
The place was dim, lit by deep red floodlights. I could smell a thick, sour stench of seaweed-and-ganja incense mixed with cloves. Catalogs of tattoo designs sat open on a countertop, next to a heavy reprint volume of the Necronomicon. Shelves along the walls held dolls that represented specific people, although crudely fashioned (denoting either artistic license or a lack of representational skills). Tags dangling from the dolls' toes listed very high prices. According to an index card thumbtacked to a shelf, the proprietor took special commissions: Get One For Your Enemies—Better Yet, Get Two!
The Voodoo Tattooist, Antoine Stickler, was a tall, coffee-skinned Jamaican vampire with a mass of dreadlocks that Medusa might have envied. A muscle shirt showed off his biceps, which sported yin and yang symbols and peace signs. He was bent over a client sprawled on the single padded bed, and when I entered the shop, Antoine turned his dark eyes to me and flashed a smile that showed off his fangs. "Be with you in a minute." He sized me up. "Can put some bright color on that gray skin of yours."
His voice had a clear spice of Jamaican accent but with careful attention to detail so that he pronounced the words without an excessive Caribbean blur. As a vampire, he might have been working on erasing his accent for half a century or more. He bent back to his customer, humming—I swear, I'm not kidding—"Don't Worry, Be Happy."
The client was a small bookwormy vampire who had his shirt off and lay facedown on the tattooing bed. The Jamaican was finishing up a bold tattoo, covering the right shoulder blade with spiky capital letters: BITE ME! Definitely not what I'd expect to see on this little bookworm, but who could tell a person's double life? Even so-called "respectable" people sometimes sneaked off to rough bars to follow their baser desires—the definition of which was certainly broad in the Quarter. As Antoine continued to poke and swirl with his pulsating tattoo gun, the vampire squirmed and whined. "Ouch, ouch, ouch!"
"It'll be okay. Got some skin cream to take care of that—special ingredients." When he had finished the letters in BITE ME, Antoine rubbed a dark red gooey gel over the tattoo area, which hardened quickly like a large scab. "Okay, now go home, crawl into your coffin, and sleep it off. When you're ready to go party tonight, your tat'll be perfect."
The vampire sat up. "And how long will this one last?" With great care, he pulled on a formal white business shirt, buttoned it, and artfully tied his necktie. After he shrugged into a conservative gray suit, he looked like a banker, lawyer, undertaker. I suspected his coworkers would be shocked if they discovered the tattoo.
"It'll last a week, that's all," Antoine said. "Then you come back and get a new one."
"I might change it next time. Let me think about it." The vampire businessman glanced at me, looked away as if embarrassed, and scuttled out of the tattoo parlor.
"Aren't tattoos supposed to be permanent?" I asked. "Yours fade in a week?"
"It's vampires—they regenerate too much." Antoine passed a hand over his arm in a clean-slate motion. "Skin heals, ink goes away, and I have a clean dead slab all over again. Good for repeat customers." He touched his own biceps, rubbed his fingers around the yin-yang and the peace symbol. "I do myself every five days. I wish I'd gotten inked up right when I was still alive."
Antoine Stickler was a mellow guy, relaxed and patient. He seemed ready to watch the world go by, equally satisfied whether the shop was busy or empty. After introducing myself, I asked him about the self-healing tattoos I'd seen on the two biker werewolves.
"Ah, Scratch and Sniff." Antoine chuckled. "Rowdy boys, practical jokesters, but there's nothing funny about those tats. My best work, I think." He went to the illustration catalogs on the countertop, but instead picked up the Necronomicon. It was one of the deluxe illustrated and annotated editions released by Howard Phillips Publishing.
"Me, I like to expand my palette. I've done too damn many dragons and broken hearts, skulls with daggers, bulldogs, Celtic knots . . ." He made a face. "But when I saw the illustrations in the Necronomicon, I decided to diversify my menu. Scratch and Sniff wanted the new tats right away."
"So werewolf tattoos don't fade like vampire tats do?" I asked.
"No. Werewolves have tough skin, keeps the ink. Once they pay for a tat, no way does their hide let it go. Those boys are covered with healing spells, but they also have gang tattoos—proud Monthlies! Course, as soon as Monthlies started getting loyalty ink, then the Hairballs had to have theirs. I was so busy for a month, werewolves were coming one after another after another. Had to set special hours—Monthlies in the morning, Hairballs in the afternoon. Didn't want any trouble.
"I spent a lot of time listening to them chatter—I'm a good listener, and there's a lot of time for conversation when I do big, complicated designs. Heard grumbling and growling on both sides. What an earful! Messed up my attitude, day after day, but it's over now. Everybody on each side has loyalty ink, and I can get back to regular customers."
"So what was the problem?" I asked. "I heard that your tattoos started the feud somehow?"
Antoine tossed his long dreadlocks, like bullwhips. "Scratch and Sniff, they're jokesters, like I said. Really funny, too— unless you don't have a sense of humor. So, when Rusty came in to get his gang ink, they paid me for a special. I swear I misunderstood what they said . . . but I think they wanted me to." He snorted. "Making fun of my accent!"
"And how did that exacerbate the bad-blood situation between the Monthlies and the Hairballs?" I had always wanted to use the word exacerbate in a sentence.
The big Jamaican vampire seemed embarrassed. "I should have known they weren't poking fun—they were just being mean. After all, Rusty is the Hairball leader, big badass, rough and tough, and he came in here for an extra-special tat to make him important. But Scratch and Sniff, they convinced me I didn't understand him right, that Rusty really wanted a tat to make him impotent. So that's what I gave him. And werewolf tats—well, they're as permanent as permanent can be."
Antoine scratched his dreads. "Rusty wanted to rip my head off and stuff my mouth full of garlic right then and there, but he knew that would start a full-on war between werewolves and vampires. And it wasn't my fault! He knew Scratch and Sniff were behind it, so he chose that fight instead. Rusty tries to keep the impotence tat a big secret, but secrets of that sort . . . well, they're too funny to keep secret for long."
No wonder Rusty hadn't been willing to give me details. "So that's why the Monthlies and the Hairballs don't like each other."
Antoine whistled. "No, sir—not at all. Monthlies and Hairballs already couldn't stand each other! I think it's because they're so much the same." He hung his head. "Then I went and made it worse. And that Rusty, he holds a grudge!"
"You know he was attacked?" I asked. "Someone slashed his scalp clean off."
"Yeah, I heard about it. Nasty business. And three or four other Hairballs got attacked, too, every one scalped. Must be some gang thing. Doesn't make any sense. The world sure has changed since the Big Uneasy, and it needs to keep changing. Why can't we all just get along, vampires and werewolves, Monthlies and Hairballs, spiders and flies? No need for all this violence and hate."
"Some of it's just inhuman nature," I said, as if that explained things.
"But there's no need for it! We clawed our way to the top of the food chain, and we should act like it. There's alternatives, specially processed tofu human-flesh substitutes, voluntarily donated blood. I get my supply from the Talbot and Knowles Blood Bars, humanely obtained, one hundred percent organic, and free-range. Keeps me healthy." He gave me a playful comradely punch in the shoulder. "And look at you—obviously a zombie who takes care of himself. You don't put garbage in your body, do you? Who provides your embalming fluid? Is it sustainable, locally manufactured?"
"I hate to say it, but I do occasionally eat at the Ghoul's Diner."
Antoine winced and shook his head, making his dreads dance. "Aww, don't do that to yourself! You're already dead, why make it worse? You know how much cholesterol and preservatives are in that food?"
"Wouldn't preservatives just help me?" I wondered aloud.
Antoine tsked and flipped open one of the catalog binders. "Why don't you take your shirt off and let's look at some tats for you."
"No thanks. I was just here asking questions." I wasn't sure how Sheyenne would feel about me getting a tattoo. Maybe we could find some creative way to spruce up the repaired bullet holes in my torso. Make them into daisies?
Before I left, Antoine called to me. "How about a doll instead? I could make you a doll. Good likeness, too."
"I don't know what I'd do with it."
"You don't do with these dolls." Antoine grinned. "You do to these dolls. Come on, it'll be fun. Must be someone in your life who needs one of my special dolls?"
"Plenty of people, but I'd need to make a list," I said. "Thanks for your help."
CHAPTER 21
By the time I got back to the office late that night, Sheyenne had compiled research on various bad-luck charms, based on my description and a crude sketch I had made of the talisman Esther had received.
"I put more stock in your descriptive abilities than your artistic ones, Beaux," she said.
"There goes my hope for a second career as a face painter," I said.
"I found a selection of common charms. Which one looks most like what Esther had?" She handed me a stack of printouts from home-shopping channel websites and various special pendant promotions. As I sorted through the images, she said, "Each has a standard fine-print disclaimer that regulatory agencies have not independently verified the bad-luck charm claims, that such talismans are not intended to diagnose or cause any particular disease, that they have not been proven effective under all circumstances, blah, blah, blah."
Robin came out of her office. "The disclaimers are necessary. They give lawyers something to do."
I held up a sheet that matched the pendant exactly. "This one."
"Ooh, a high-end special edition," Sheyenne said. "Somebody really wanted to make sure it worked."
"Esther must have given the wizard enough incentive." Robin took the paper and read it slowly, nodding. "Though there's little doubt a vindictive wizard did indeed give our client a bad-luck charm, I'm concerned about the viability of the case. Even if we can prove the hazardous nature of the talisman, the wizard might respond to any lawsuit by saying he gave the tip 'with cause.' "
"Better not put Esther on the stand if it ever goes to trial," I said, and Robin shuddered at the suggestion. Sheyenne took the papers and the now-identified bad-luck charm and went back to her desk to continue her research, blowing me an air-kiss as she departed.
Meanwhile, after accepting Steve Halsted's case, Robin had set up an initial meeting with Don Tuthery, the attorney representing his ex-wife. It would take place first thing in the morning; Steve had asked for an early appointment because he had to do his daytime deliveries throughout the Quarter—and I had promised to make an appearance at the Worldwide Horror Convention for a few hours.
Digging into case law, Robin had found several vaguely relevant precedents, and she was sure Tuthery had done the same. But she seemed worried, which I found unsettling. She lowered her voice, as if she didn't want to hear her own doubts. "He's a pit bull, Dan—exactly the man you want representing you in a divorce or custody dispute. Unfortunately he's not on our side."
"We have an even bigger advantage," I said with a reassuring nod. "We've got Robin Deyer working for us."
She gave me that warm smile and went back to her research for the 8 A.M. meeting.
Steve arrived in his dark green overalls and trucker's cap, looking uneasy. To show that he understood the importance of this meeting, he had draped a polka-dotted necktie over his chest. I complimented him on it.
"I get nervous every time I see Rova," he said. "So much history there, so much bad blood. And her lawyer makes it worse with all his pushing and prodding."
"We'll push and prod right back, Mister Halsted," Robin said. "I promise. Dan, would you sit in with us for moral support?"
"Lucky for you, today is my day reserved for moral support." I wasn't due at the convention for another couple of hours.
Don Tuthery arrived at 8 A.M. to the minute, a tall serious human in a charcoal business suit and gold wire-rimmed glasses; his steel-gray hair was impeccably trimmed and combed, thereby proving beyond a reasonable doubt that Rova didn't cut his hair at the Parlour (BNF). His briefcase looked as if it might contain nuclear launch codes. I couldn't describe his teeth because he never showed them, never smiled, and barely moved his lips when he spoke.
Next to him, Rova Halsted wore a conservative black skirt and blazer, as if ready to attend her ex-husband's funeral all over again, rather than a "friendly discussion." Her mop of mismatched hair-color swatches added a bit of carnival to the funereal atmosphere. When she recognized me from my Parlour visit the previous day, her expression soured into a suspicious scowl.
Steve looked away and mumbled, "Hello, Rova."
When she started to respond, Don Tuthery raised a finger, stopping her like a schoolteacher rapping knuckles with a ruler. "Not one word, Ms. Halsted." He faced Robin, Steve, and me. "I will speak for my client in this interaction."
"If you're speaking for her, then could you at least say hi back?" I said. His rudeness set me on edge from the very beginning.
Tuthery frowned at me. "And who is this . . . gentleman? I object to having a second zombie attend these proceedings."
"This isn't a deposition, and you're not in court before a judge, Mister Tuthery. But feel free to object all you like, if it makes you happy," Robin said. "Mister Chambeaux is my business partner, and Mister Halsted requested his presence. No need to be concerned—humans still outnumber the zombies here, three to two."
"I'm not concerned," Tuthery said. "I'm never concerned. Normally, however, I would prefer a meeting such as this to take place on neutral ground, preferably hallowed ground, but that's difficult to find in the Quarter."
Tuthery set his briefcase on the conference room table, opened it to show that it contained not nuclear launch codes, but stacks of manila folders, legal briefs, and lovely photos of doting mother Rova and cute-as-a-button Jordan, whom I recognized from the snapshots in Steve's wallet.
Tuthery took control of the meeting, even though it was on our turf. "We can resolve this today." He pulled out papers, glanced at Robin, completely ignored Steve and me.
"I'm glad your client is willing to reconsider her allegations, Mister Tuthery," Robin said.
Steve and Rova kept looking at each other and then away, both of them uncomfortable and angry, but also . . . hopeful?
"I respect you as a professional colleague, Ms. Deyer, but have you looked at the complaint? Mister Halsted simply has no case. I expect you to withdraw your claims and objections so we waste no further time on the matter."
My already-sluggish heart sank. Oh. So that was how it was going to be.
Robin's eyes flared at his dismissive and antagonistic tone. "I was about to say the same to you. The law couldn't be more clear: A person's obligations to provide spousal and child support terminate upon death. Mister Halsted is clearly dead."
"On the contrary," Tuthery said. "The Halsteds' divorce settlement decrees that so long as Mister Halsted is gainfully employed and earning an income, he is required to provide for the care of his son."
"I want to do what's right for Jordan," Steve blurted, "but I already did that! The insurance settlement—"
Robin held up her hand to keep Steve quiet. "My client raises an important point. Mister Halsted did arrange for the care of his son from the life insurance payout, an amount totaling fifty thousand dollars, which was to remain in trust for Jordan. We can file a motion demanding Ms. Halsted's financial records and vigorously investigate any mismanagement of that trust fund. Ms. Halsted did set up the required trust fund, correct? We expect to find every penny there and available for the child's care."
Rova blanched at this. She flicked a wide-eyed glance from Steve to Mr. Tuthery and back.
"Ms. Halsted, not the child, was the designated beneficiary on the policy, and she was entitled to use the money as she saw fit," Tuthery stated.
"Not according to my client."
Steve rose partly to his feet, with difficulty. "That money was for Jordan, Rova! You know it."
Though Tuthery looked annoyed, he pressed on. "Ms. Halsted invested that money, and Jordan will benefit significantly once he comes of age."
Rova burst out, "Once the Parlour makes a million dollars, I'll be able to send Jordan to Harvard, or Stanford, or both in the same year!"
"What happens if the Parlour flops?" I asked. "You could lose everything."
Rova sniffed at me. "Not a chance. It's a beauty salon, and the Quarter needs a lot of beauty. This is a surefire bet, as guaranteed as opening a restaurant!"
Tuthery shushed his client.
"Which brings us to visitation rights," Robin said. "My client would like to see his son regularly."
"Out of the question," Tuthery said. "We shall not allow you to inflict psychological damage on an eight-year-old boy."
"You'll give him nightmares!" Rova wailed.
"But he was happy to see me the two times I showed up," Steve said.
"And that's another matter we will bring to court," Tuthery said. "Undead stalking, particularly of a minor, is a very serious matter."
"What? He's my son!"
Robin was shocked. "You have no legal grounds for that."
Glaring at Steve, Rova burst out, "You were never around to see him when you were alive! Now you want to make up for it?"
Steve spoke with surprising calm. "Yes, Rova—yes, I do." She just stared at him.
"I'm afraid visitation rights are entirely off the table," Tuthery said.
"I'm afraid that's the crux of the matter. Bottom line, Mister Tuthery: If your client wants continued child support, my client gets visitation. You will have my motion for discovery of your client's financial records by tomorrow." Robin began packing up her documents. Rova was white as . . . well, as a ghost.
"We're finished here," Tuthery replied, also packing up. He made sure we saw the beautiful, loving photos of mother and son before he put them away. "I warn you, if this goes to court, it'll be an ugly case. Think of the boy's welfare. Children shouldn't play with dead things."
Steve said, bewildered, "I just want to do what's right for my son."
Rova didn't say another word as she left with her attorney.
When we were alone in the conference room, Steve looked miserable. He tore off his polka-dot tie. "If you tell me not to worry, then I won't worry." He shook his head and adjusted his cap. "But I'm worried."
"We haven't even started to fight for you, Mister Halsted," Robin said. "Don't worry until I tell you to."
CHAPTER 22
The idea to host the Worldwide Horror Convention in the Unnatural Quarter was a stroke of genius. It must have been a particularly rambunctious meeting of the con committee when they decided to include the Bates Hotel in their convention bid. Both the convention planners and the hotel staff saw the unique relevance and attraction, even suggested making the Quarter the permanent home of the Worldwide Horror Convention, which traditionally moved from city to city.
The Chamber of Commerce hung welcoming banners on lampposts up and down the streets; the City Council proclaimed a special Monster-Loving Humans Day, and several thousand fans, writers, industry professionals, and celebrities descended upon the Bates Hotel.
According to the ribbon on my guest badge, I was in the celebrity category.
I arrived in my traditional fedora and sport jacket with crudely repaired bullet holes. I had added more mortician's putty to the bullet hole in my forehead, but I could never do as good a job as Bruno and Heinrich. I hoped the Wannovich sisters didn't expect me to wear some kind of costume.
When I entered the hotel lobby, I was surrounded by a milling mixture of normal-looking humans, humans dressed in a variety of monster costumes, real monsters wearing normal street clothes, and reporters and TV crews covering the event. It took me a few minutes just to drink it all in.
Con attendees stood at tables, filling out registration forms and getting in line to pay for their badges. The queue for the Pre-Registered badges was long and disorganized; and the crowd was growing restless, particularly when it became apparent that the line for those not pre-registered was moving faster.
A wizard sat behind the Pre-Reg table. Apparently, he had used a crystal ball to arrange the badges and registration packets in the order he predicted the guests would arrive, instead of alphabetical order. Unfortunately, his spell was flawed.
Over at the Professionals registration table, a slimy tentacle-faced creature was holding his badge and arguing with the poor human volunteer in a burbling otherworldly voice. "My name is misspelled! Yov Shuggoleth has two Gs!" He slapped the badge down, waiting for it to be reprinted. He moaned, "I'll bet it's misspelled in the program book, too."
Fortunately, Mavis and Alma had already given me my badge, so I went to the welcome table and asked for my attendee packet. The young woman handed me a plastic bag with the convention logo printed on the front. Inside, I found a flyer, one side advertising specials for the Goblin Tavern, the other side soliciting business for the Full Moon Brothel. Both had coupons. I also found a giveaway copy of Death Warmed Over. Howard Phillips Publishing was really getting behind this book.
I saw Miranda Jekyll holding court, walking in a regal procession through the lobby crowds, flanked by reporters and TV camera crews. Hirsute accompanied her like a burly bodyguard. "I tell you, sweethearts, it's so wonderful to be welcomed back to the Quarter. Since my husband's very timely demise, I have tried to rehabilitate the image of Jekyll Lifestyle Products and Necroceuticals."
At the back of the lobby, a man in an extravagant gargoyle costume, complete with a pneumatically twitching barbed tail and folding green fabric wings, tried to enter an elevator with a real scaly demon who had even larger wings. The two of them jostled back and forth, struggling to fit inside, but neither could fold their wings tightly enough. A large ogre stood waiting for the elevator, his fat lips turned downward; finally he shuffled off to take the stairs.
Hearing screams, I went instantly on the alert when a man dressed like Van Helsing with a wooden stake and mallet ran at full speed after a slender vampire woman in a flowing white gown. But they were both laughing, and I realized they were just costumed fans in a live-action role-playing game.
A particularly frayed-looking mummy reclined on a palanquin, carried by two fan volunteers from the convention. They stood at the Handicapped Services desk, requesting a special access placard. On the mezzanine level up from the lobby, signs advertised a blood drive at the convention: Sponsored by Talbot & Knowles Blood Bars. Leave a Pint of Yourself Behind in the Quarter! What Flows Here, Stays Here! Volunteers lay back on gurneys with red-filled tubes leading out of their arms while white-uniformed medical attendants took care of them. Conveniently close to the blood-drive area was a Talbot & Knowles Blood Bar refreshment stand.
Yes, this was going to be an interesting day.
I didn't know where to go or what to do first. A sign pointed me to the dealers' room, where Howard Phillips Publishing had set up their table. Maybe I could do my autographing, politely say hello to a few fans, then having fulfilled my duty for the day, I could get back to work on real cases. Robin and I already had plans to investigate Joe's Crematorium that afternoon.
Somebody called out, "Look! There's that Dan Shamble guy!"
I saw a fan raise a large camera; two other fans stopped and scrambled for their phones. Then I realized that not three feet away stood another man in a fedora, with obvious pallid makeup and dark eye shadow applied to his face. The fedora was slouched low, and a fake putty bullet hole stood out in the center of his forehead. He even had black stitches covering bullet holes in his sport jacket. He didn't look a thing like me, but he grinned in my direction. "Whoa, yours looks really realistic."
"I work hard at it," I said.
The fans clamored for the two of us to pose together for a photo. I stood patiently; we were blocking traffic, but the photographers didn't seem to care. "I, uh, I need to get to a panel," I said, and flipped open the program book.
Several tracks of programming were listed, some for professional writers, some for fans. I saw "Funniest Vampire Misconceptions," "Alternatives to Brains—New Diet for the Undead," "Most Embarrassing Zombie Moments," and "Got Wood?—Stake Design and Use." I was particularly delighted to see "Real Life in Ancient Egypt—Peeling the Bandages Off." It was one of Ramen Ho-Tep's presentations, and I knew he'd find a fascinated audience here. Then I spotted "The Hairy Truth—What Every Horror Writer Needs to Know about Werewolves."
Hmm, maybe I could get some work done while I was here.
CHAPTER 23
The self-proclaimed "World Renowned Werewolf Expert" Professor Walter Zevon was giving an hour-long presentation. Normally, academics wouldn't bother to attend frenetic conventions such as this, but since many universities refused to give tenure to non-human lecturers, unnatural scholars had been striving for respect in their various fields. Professor Zevon's lecture was in Voorhees Ballroom B, a hall with about fifty seats, two-thirds of which were filled with fans, writers, and a smattering of monsters.
I slipped in late, opening the door and distracting the crowd. I mumbled an apology and closed the door as quietly as I could. I shuffled forward and took a seat in one of the back rows next to a Wiccan who was knitting a long, pink scarf.
Professor Zevon stood at the podium, a rather frail-looking elderly gentleman, a full-time werewolf with a lavish head of gray fur, a long snout, and most of his teeth. He adjusted round spectacles and looked down at his notes, intent. His shoulders were hunched. A pair of younger furry werewolves sat in the front row; two of his academic students, I guessed.
"We provide this lecture as a service to all readers and writers, in order to promote accuracy in horror fiction. We strive to increase understanding and tolerance of lycanthropes throughout the world." Zevon's tongue lolled out, and he licked his muzzle. "We also want to save you the embarrassment of making real howlers of mistakes."
A few people politely chuckled, but not many.
"First, some definitions," the professor continued. "There's been confusion about werewolves in general. As you look around you, some of us, such as I and my students here"—he gestured to the two Hairballs in the front row—"are obvious werewolves. We wear our fur proudly and permanently. We don't hide who we are. We don't blame our condition on some Gypsy curse. We are the true werewolves, not the ones who are human most of the time and only do an occasional transformation. Those once-a-month, full-moon-only, quote-unquote werewolves are like temp workers, mere hobbyists rather than true professionals."
There was muttering in the audience, and two large figures rose out of their seats. "You'd better take that back, Professor."
Scratch and Sniff. I was surprised they had purchased badges to sit in on a lecture at the convention. They'd probably come for the sole purpose of heckling Professor Zevon.
"Boys, boys, don't pick on the poor, senile fool." I was surprised to see that Miranda Jekyll occupied a seat in the center of the room, flanked by an entourage of reporters taking notes. She didn't deign to rise to her feet. "Professor, sweetheart, I'm afraid you're toiling under the delusions of academia. Your research is flawed."
At the podium, the elderly werewolf bristled. "How so?"
"If you would study the very basics of werewolf lore, you'll note that transformation is an indispensable part of the equation. Werewolves look like humans, except during the full moon, when they transform and unleash the bestial part of their nature." Miranda flashed a broad grin, showing her bright white teeth. "Some might call it the best part." She stroked her red fingernails along the bulging arm of Hirsute, who sat next to her. "Since a Hairball never changes, you're not technically a werewolf. You're just a"—she twitched her hand as if searching for the word, but I knew Miranda had thought of her snappy comeback even before sitting down—"a large hairy dog that's been taught how to talk and dress up."
The Hairball students in the front row growled, baring their long canines and rising to their feet. One said, "Just because you can't keep your hackles up for more than a day or two at a time doesn't make you a werewolf."
Scratch and Sniff were ready to leap over the rows and charge the stage, but convention security (two golems wearing headsets connected to walkie-talkies) stepped forward. "Calm down. No trouble here. If you don't like the topic, go to another panel."
Professor Zevon regarded Miranda Jekyll as his main opponent. "And where, might I ask, did you get your degree in lycanthropy, madam?"
"I don't need a degree to know what's true, sweetheart."
The professor dismissed her and pushed a button on the podium to activate an overhead projector, showing the first slide in his presentation. "If there are no further interruptions, I would like to discuss personality types in relation to fur color."
"This is a bunch of bullshit!" Scratch said. "Let's go to the dealers' room and pick up some bootleg anime." The two big tattooed Monthlies made as much disturbance as possible as they worked their way out of their row, elbowing each other, laughing loudly before they slammed out of the room, banging the doors.
I wanted to ask about the friction between the two types of werewolves, since we'd just seen a demonstration of it, but the professor continued his lecture, going into excruciating detail about daily life as a werewolf, his preferred diet, methods of fur care, even the consistency of his bowel movements. It was far too much information, and not at all the kind I needed.
In the front row I noticed Linda Bullwer, the vampire ghostwriter, with her cat's-eye glasses in place, scribbling notes, probably for the next Dan Shamble book she would write. I hoped she would leave out the part about the bowel movements.
I decided to hold my questions until the end of the lecture, when the golems held up a sign announcing a five-minute warning, then cut off the discussion. The next panel, "A Beginner's Guide to Sewers and Catacombs," required a lot of setup. I didn't have a chance to ask anything.
Miranda Jekyll drew much of the audience with her as she flounced out of the ballroom. Too many fans crowded around the professor for me to get close, so I would have to see him later. I decided it was time to do my duty and take my stint at the Howard Phillips table, sign a couple of books, and then go back to work on my cases.
The convention dealers' room was like an Arabian bazaar of the bizarre. Used books, rare first editions, homemade jewelry, magical items, gaudy items, and gaudy magical items. A zombie was hunched over in a massage chair, moaning as the masseuse whispered soothing words into his ear and rubbed his shoulders. The Unnatural Acts adult novelty shop was doing a brisk business at their booth with chains, leather straps, whips, corsets, feathers, spikes, and truly mystifying harnesses that came with an instruction sheet sealed inside a plain brown envelope.
Another kiosk sold personal ectoplasmic defibrillators from Harvey Jekyll's spin-off company. One of the dangerous contraptions was on display, but not hooked up to any power source. An index card noted that it was "For Display Purposes Only."
The same vendor had a panoply of wooden stakes and mallets, squirt guns filled with holy water as well as refill jugs, hexes, charms, silver bullets, garlic, and wolfsbane. The garlic and wolfsbane were sealed in zipper-lock plastic bags, but several allergic vampires still sneezed, their eyes red and watery, and they complained about the stench.
I found Mavis Wannovich behind a table next to tall retractable posters for Howard Phillips Publishing and stacks of catalogs touting their new releases; Death Warmed Over was featured prominently on the catalog cover. A pyramid of the books stood in front of two empty chairs, next to a poster: Meet the Real Dan Shamble! Signing Books Here Today!
Alma rooted among the cardboard boxes behind the table, tipping them over to spill new product onto the floor, which Mavis then placed on the table. Before I could get there, Linda Bullwer scuttled through the dealers' room crowd, came around the table, and took one of the vacant seats behind the pile of books. Mavis called out, "Oh, look, everyone! Here's Penny Dreadful, the author of the Shamble and Die series!"
The vampire writer nudged her cat's-eye glasses higher on her nose, took out a pen, and sat smiling, waiting for the queue to form.
When I shuffled up, Mavis heaved a sigh of relief. "Oh, Mister Chambeaux, you're finally here. We've had so many questions. You're going to get a long line."
"If you say so."
I came around to meet the beaming gaze of Linda Bullwer, who reached out to shake my hand. "This is such an honor, Mister Chambeaux. I hope our book does well."
Mavis handed me a regular pen as well as a fountain pen and a cup of fresh blood from the Talbot & Knowles Blood Bar. "In case somebody wants you to sign with this."
To my surprise, people actually began to gather in front of the table, each one carrying a copy of Death Warmed Over. Alma used her bulk, waddling up and down the line to nudge the fans into some semblance of order.
The first fan was a young man with two copies of the book. "Could you please sign this, Mister Shamble?"
"It's Chambeaux," I said.
"Great! I've got one for a friend, too. I already read it. I loved it! Especially the part where you died!"
"No spoilers!" someone yelled from the back of the line.
"He starts out dead," the fan sneered back.
I autographed the novels, then passed them over to Linda Bullwer, who began to write a lovely inscription. The fan glanced at her. "And who are you? Why are you writing in my book?"
"She wrote it," I said.
"Oh." The fan frowned. "Well, okay then, I suppose."
The fourth person in line asked me to sign in blood, and I used the fountain pen and the cup from Talbot & Knowles. Once he started that trend, everybody wanted their book signed in blood. They asked me questions about my cases. They asked how many more Dan Shamble books I was going to write, and I deferred to Linda Bullwer, again pointing out, "This is the real Penny Dreadful. She's the author."
"I'm already working on the next volume." She smiled graciously. "The material is so rich. Provided sales go as well as we hope, we could have a long-term series."
One woman came up to me, leaned over the table, and extended a meaty forearm. "Bite me! Please, bite me! I want a souvenir of the con."
I was flustered. "No, but thank you."
"It doesn't say anywhere on the sign that you won't bite fans."
"I'm making it a new policy," I said.
One fan had a stack of twelve copies, and he insisted on signature and date only. Another one pestered me for writing advice, and I referred her to Linda Bullwer, but the fan didn't seem to understand that I hadn't actually written the novel. Two more gave me "great ideas for cases" in future books in the series.
"And Miranda Jekyll is here, too!" said a very enthusiastic older man. "Do you think she'll sign my copy, the part where she's mentioned in the book?" Knowing Miranda, I was sure she would, and told the man so.
At first I assumed that the fans were only getting in line because they'd received the book for free in their registration packets, but many of them had purchased multiple copies, and quite a few had read the whole novel already.
I had planned to be there for only an hour, but the line continued. And continued. I was truly embarrassed by all the attention.
Fortunately, or unfortunately, the line degenerated into near panic when alarms went off. At first I assumed that some overly enthusiastic gamer had pulled a fire alarm—until I heard sirens outside.
Finally one of the con security golems came in, his clay face sculpted into an expression of dismay. "It's Professor Zevon! The ambulance is taking him to the hospital right now."
"What happened?" I asked.
"Somebody assaulted him out in the parking lot. He's been scalped!"
CHAPTER 24
The ambulance rushed the freshly, and unfortunately, topless Professor Zevon to the Brothers and Sisters of Mercy Hospital. In the parking lot of the Bates Hotel, crime-scene techs stretched out yellow tape, pushing back the crowd of both real and fan unnaturals who observed the goings-on with interest. Some figured that this must be part of the convention fun; several were already reviewing the "incredibly realistic staged event."
McGoo paced around a chalk outline that marked where the professor had sprawled. The crime-scene tech had been quite meticulous, adding details of fingers, even wrinkles of clothes; for the top of Professor Zevon's formerly furred head, the tech had used dotted lines to render where the scalp had been.
McGoo shook his head. "This is crazy, Shamble. Why would anybody do this? He was just an old professor—how could he have any enemies?"
I told him about the disturbance at the werewolf lecture and what Antoine Stickler had told me about the feud between the Monthlies and the Hairballs. Rusty did seem an obvious target, but I couldn't understand why gang members would target an old lecturer (or, for that matter, the homeless werewolves who had also been scalped). In explaining violent irrational behavior, though, Just Because is often a good enough reason.
McGoo wore a queasy expression. "I'm putting Scratch and Sniff on my list of suspects, just for the hell of it."
I couldn't argue with the conclusion, but these assaults were nastier than the work of mere thugs. Personally, I thought scalping a victim also showed more imagination than one would expect from Scratch and Sniff, but a good investigator follows the clues wherever they take him.
I left the Worldwide Horror Convention, having done my duty, although I doubted my afternoon with Robin at Joe's Crematorium would be any more enjoyable.
Sheyenne briefed me on what she had learned about the proprietor, Joe Muggins.
"You're so good at uncovering background information, Spooky, I think you deserve a PI license of your own," I said.
"I do deserve one, but you can be the front man of the agency. Besides, you're the one with the fedora." She pulled up the information she had gathered. "Joe Muggins is a vampire, and his criminal rap sheet, even his juvenile record, is still available from his prior life."
"Anything interesting?"
"He's been arrested for arson several times. He liked to play with matches, build blazes, even set his sister's bedspread on fire."
"Sounds like he settled on a career that matches his skill set and interests," I said. "Not everybody's cut out to be a crematorium manager."
Armed with the pink copy of Adriana Cruz's cremation receipt, Robin and I headed off. Something smelled wrong about Joe's Crematorium.
After the Big Uneasy, cremation had become a big business. While some people hoped to come back as zombies, vampires, or other types of undead, many had a horror of joining the walking dead, so they rewrote their living wills to insist on cremation. Joe's Crematorium had capitalized on the craze, offering specials, running ads at bus stops and on public benches.
Robin and I pulled up in her Pro Bono Mobile in front of a large cinder-block building. Tall stacks curled rich black smoke into the air. A bright fabric banner was stretched across the drab walls: We Do Viking Funerals! A big signboard on the door advertised This Month Only: Two-for-One Special. Trucks had pulled up to the rear, and I saw giant piles of cordwood stacked there, as if someone were preparing for winter.
We got out, and I looked around curiously, sniffing the smoke in the air. To my surprise, I heard loud motorcycles starting up at the back of the building, and two choppers roared off with wrapped packages precariously balanced on their large saddlebags. Scratch and Sniff gunned the engines and accelerated out of the parking lot.
Robin looked after the two. "What were they doing here?"
"I'll ask next time I see them," I said. "I need to have a chat with those boys anyway."
We entered through the crematorium's homey-looking front door, which was surrounded by flower boxes and nicely trimmed hedges. The receptionist was on the phone and waved us over to a small couch. I picked up a brochure from a tabletop stand. Consider Cremation . . . For Your Family's Peace of Mind. And Your Own. Easy-listening music played over the intercom.
The receptionist put her hand over the mouthpiece and yelled through an open door to the back. "Hey Joe! Walk-in customers." She looked human, but her voice had a familiar edge that told me she was a Monthly werewolf.
A short, business-suited vampire came out—and I immediately recognized him as "Bite Me" from the Voodoo Tattoo parlor. He recognized me too, looked away in embarrassment, and reassembled his greeting-the-bereaved demeanor. "I'm so sorry to hear about your death, sir. We can take care of your every need."
"My needs have already been met," I said. "I'm here on behalf of one of our clients."
Robin opened her briefcase. "A former customer of your crematorium—a very dissatisfied customer."
Joe Muggins fidgeted. "My, my! We almost never get complaints from our customers. There should be very little left of them to complain."
"That's exactly the problem, Mister Muggins. There's a lot left of this client. Adriana Cruz is still walking around, and not very happy about it." Robin handed over the pink slip and the receipt.
"My, my!" he said again. "If you send her back, we'd be happy to cremate her at no extra charge."
"That isn't what my client wishes," Robin said in a stern voice. "And we want to make certain this does not happen again in the future."
Taking the pink slip, Joe went to the computer, moving the receptionist aside as he tapped away and called up Adriana's cremation certificate. "Ah, here it is. Yes, she was cremated five weeks ago, a no-frills ceremony, no add-ons. Her ashes were delivered to her family. They chose a streamlined budget-model urn, but still very nice."
"The family received somebody's ashes," I said, "but they weren't Ms. Cruz's."
Joe shook his head, flustered. "I don't see how this could be. We always tell prospective customers to take care of their bodies, and if they don't, we'll take care of them."
Robin's expression grew harder. "I assure you, Mister Muggins, our client is walking around as a zombie. Something went terribly wrong with your procedures. Whose ashes were delivered to the Cruz family? How did our client's cadaver not get put into the furnace, even though your records show she was? What sort of quality-control measures do you have in place?"
He swallowed hard. "Check with the Better Business Bureau and the Chamber of Commerce. I've got an A plus rating with the BBB, and I've been a member of the Chamber since I opened my business. I assure you, there have been no other complaints. I run a clean business, and from now on I'll make sure I light the fires myself."
"I'm sure you'd enjoy lighting the fires, Mister Muggins," I said in a dangerous voice. "We've looked at your prior record. And maybe we should look into your . . . extracurricular activities, too?" I knew that a mousy guy like that wouldn't get a Bite Me tattoo just to show his pet cats.
Now he seemed mortified. "There's no need to get personal, Mister Chambeaux." Then, in a very small voice, he added, "Please?"
I shrugged. "No need . . . at the moment." I had no intention of exposing his private life—any more than I worried about Archibald Victor's eccentric hobby. Unless it had some bearing on the case.
Robin continued, "We obviously have clear and substantial grounds for a lawsuit. If there've been other instances, we could file a class-action suit. And we couldn't help but notice the wood you have stacked in the back. I assume your furnaces use gas jets to ramp up and maintain the required two thousand degrees? What exactly is the cordwood for?"
Now the vampire was quite panicked. "The . . . wood? It's a premium service we offer—in addition to the Viking funerals. Some clients prefer the choice of hickory or mesquite smoked. The neighbors say it smells nice." He fidgeted, straightened his tie, handed back Adriana Cruz's pink slip. "There's no need for any unpleasantness. How can I make restitution? I've offered to cremate your client for free. We could even throw a big going-away party, champagne for everyone . . . maybe toss in a free cruise for her parents afterward? I'll certainly investigate the matter internally to ensure that the mistake is never made again."
"To begin with, you could refund the entire cost of the cremation," Robin suggested. "Plus an additional consideration for pain and suffering."
The little vampire swallowed hard, then nodded. He knew he was defeated. "Of course . . . in fact, I insist—so long as your client signs a release and an appropriate nondisclosure agreement. Bad publicity could harm our business. There are other crematoriums in town, you know." He handed us a brochure as well as discount coupons and told me to give him a call if I ever reconsidered my own cremation.
I had a growing suspicion that more was going on here than a simple administrative glitch and one accidentally uncremated body. According to her story, Adriana had woken up in a truck filled with corpses, and the truck had been driving away from the crematorium.
No, I wasn't done investigating yet.
CHAPTER 25
After leaving Joe's Crematorium, Robin drove back to the office, but I decided to walk. It had already been a long day and I wanted to stretch my legs, reduce the stiffness; this would take the place of half an hour on the treadmill at the All-Day/All-Nite Fitness Center, although it would have been better if Sheyenne had been with me, two postmortem lovers on an afternoon stroll through the dark side of town.
Besides, I needed to stop by the Spare Parts Emporium to pick up the replacement brain and spleen Tony Cralo had promised, maybe even a bonus set of lungs, and then I could wrap up Archibald Victor's case. Customer satisfaction all around.
When I arrived, I saw that another weather front of cold gray drizzle had socked in the body-parts warehouse, and brown water refilled the pothole puddles in the gravel parking lot. Like a giant's erector set, the abandoned railroad bridge loomed out of the mist, beneath which I could see huddled figures, trash fires, and the large cardboard boxes of the residential complex for homeless unnaturals.
When I entered the Spare Parts Emporium, I had to dodge a man with a hand truck piled so high with boxes that he couldn't even see over the top. Customers milled about. Passing mummies, zombies, even a gargoyle with a handwritten shopping list, I made my way toward the floor manager's office.
This time, naturally, since I didn't need any assistance, two customer service reps rushed to me and asked, "May I help you, sir?" One looked human, but I guessed he was a Monthly werewolf; the other was a stocky female zombie.
"No, thanks. I know right where I'm going," I said, but then paused. "Say, you don't happen to sell werewolf scalps, do you?"
The two looked at each other. "Why would anyone want werewolf scalps?"
I stopped myself from replying with "Why would anybody want anything in here?" deciding that would be rude. Instead, I said, "Just curious. It's a specialty item."
The female zombie suggested, "If you fill out a form at customer service, we might be able to special-order it for you. How many do you need?"
"No, thanks. I was only asking for a friend."
The two looked at each other again, raised their eyebrows, and nodded knowingly. The werewolf wiggled his bushy eyebrows and said, "Yes . . . 'for a friend.' "
Then I had another thought. "Last time I was here, there was a ghoul who helped me . . . an attractive young lad, black teeth, sunken cheeks, deathly pallor?" I knew he'd been fired, but disgruntled former employees give the very best background dirt.
"Oh, you mean Francis." Distaste was plain on the female zombie's face. "He actually helped you?"
"He did what he could."
"He doesn't work here any longer."
"Just thought I'd ask." I headed to the back of the warehouse.
When I reached the floor manager's office, Xandy Huff was on the telephone. Behind him, on the metal rack of envelopes and time cards, the slip that had belonged to Francis the ghoul was gone. I heard Huff acknowledge "receipt of the order," then he hung up and looked at me.
Before he could say anything, I quipped, "I see my friend Francis picked up his severance check."
Huff's face fell into an annoyed grimace. "That ghoul? We won't be seeing him again." I watched the slow-moving gears in his mind clank together until he remembered me. "Wait a minute, you're that zombie detective guy. Francis didn't have anything to do with you." He grew suddenly wary. "Or did he file some kind of complaint?"
"Did he have something to complain about?"
"Absolutely not." Frazzled, Huff wiped beads of perspiration from his forehead. "I have a full-time job as it is. And now the police want to triple my work. Years' worth of records and receipts! I don't see why they're on a witch hunt—if they're hunting for witches, they can just look them up in the business directory."
I didn't mention that I was personally responsible for all this extra work, having pointed McGoo toward the Emporium as a possible link.
"Mister Cralo usually hires attorneys to fight every step of the way, but this time he's taking a different approach." Huff heaved a long sigh and hung his head. "He told me to provide records of everything—everything! Hold nothing back, he said—copies of every record, every body part, every transaction, for as long as we've been in business. I had to hire an army of temp goblins in our accounting department, and four golems just to haul boxes of records."
"Think of it as doing your civic duty, Mister Huff." I knew perfectly well what Cralo was doing—trying to hide a needle of truth in a haystack of paperwork.
Huff gave me a long-suffering look. "Are you here to gripe about something else? I thought I took care of your complaint already."
"You sent me to see Mister Cralo personally."
"Yes, that usually takes care of any complaints. We never see the customer again."
"Well, I did as you suggested, I went to see your boss in the Zombie Bathhouse, and I resolved the issue. He told me to come back today and pick up a new brain and spleen, plus extras."
Huff's eyes went wide and he lowered his voice in disbelief. "You actually went to see Mister Cralo? Nobody does that!"
"He invited me into his Jacuzzi and everything." I shuddered to remember sitting in the fetid whirlpool with the fat, outgassing crime lord.
Huff swallowed hard. "You're a brave man, Mister Chambeaux." He shuffled papers on his desk.
"You should read the novel," I said. Good thing he hadn't seen me in my less-brave moments. "In addition to the replacement brain, Mister Cralo said you'd give my client a new spleen, a set of lungs, and a discount on any future orders. Then the case will require no further action on our part. My client has an important body-building competition coming up, and he needs those organs."
Huff was flustered. "I'm sure you're right. I just have to double-check that with Mister Cralo."
"One hundred percent customer satisfaction," I reminded him.
"Yes, of course. And you went through the appropriate procedures." He shook his head. "But we don't have the replacement brain ready for you yet. I can get you the spleen and lungs—ship them directly to the customer. Come back tomorrow."
"Is this how you get return customers?" I asked. "Just have them keep coming back because their order isn't ready?"
"Believe me, Mister Chambeaux, I don't want to see you any more often than necessary." He double-checked records on his computer screen, shook his head. "I really don't have a brain."
"So . . . you're saying it's a no-brainer?" He didn't laugh at my joke, but it wasn't much of a joke anyway.
The phone rang again, and he made an impatient gesture. "Please, come back tomorrow. Right now, the body parts you need are not in stock, but we'll get them. I've got about a ton and a half of paperwork to prepare."
I thought Archibald would be happy enough with that. He could get started with the spleen and lungs at least.
On my way out, I strolled up and down a few of the aisles, where my two eager customer service reps were shelving boxes in the Ghoul section. The sales reps stacked crates of various ghoul body parts, each with a bright red label that said Fresh Today!
I wondered if Francis had been not so much fired, but rather disassembled and recycled.
CHAPTER 26
When my Best Human Friend showed up at our office with a bright grin, I knew I should have been suspicious. As my second clue that something was up, McGoo started the conversation with, "I've got to call in a favor, Shamble—a big one."
I should have asked exactly what he wanted before agreeing, but what are friends for? My automatic response was, "Sure, McGoo. Anything you need."
With an embarrassed grin, he called into the hall. "Bring everything in, boys!" That was another bad sign.
Two uniformed cops and a sturdy golem wheeled in hand trucks piled with bankers' boxes. It looked as if they had come from a clearance sale at the Ark of the Covenant warehouse.
Sheyenne hovered beside me, giving me a questioning look. "What did you get us into, Beaux?"
Robin emerged from her office to see what the commotion was about. Regarding all the boxes, she said, "I haven't seen so much paper since I got my copy of the Unnatural Acts Act."
McGoo explained with a shrug. "I followed up on your hunch, Shamble, and requested records from the Spare Parts Emporium. I asked for everything they had. I didn't expect all this stuff! It's supposed to make us think the fat zombie's operation is aboveboard, that he has nothing to hide."
Sheyenne drifted over from her desk. "Anybody can fake records. We can't exactly double-check with the donors to make sure they surrendered their body parts voluntarily." Her normally beautiful smile became hard and determined. "But I'm good at digging."
"Their next of kin should have the counter-copies of the receipts," I said.
Robin looked at the mountains of documents and raised her chin. "I've seen this technique before. Sometimes the party being investigated will bury you under an avalanche of paperwork in hopes that you won't be able to sift out the one or two compelling pieces of evidence you need. Were you specific in your warrant, Officer McGoohan?"
He looked embarrassed. "I just asked for their records. This is what they gave me."
The golem thumped down his hand truck, and the uniformed policemen began unstacking the boxes in the middle of our reception area.
"What do you expect us to do with them?" I asked. "This goes far beyond a consumer complaint case." But if Tony Cralo had anything to do with the disorganized vampire corpse in the Motel Six Feet Under, we were dealing with a murder. And though Sheyenne always insisted that we concentrate on paying clients first, we might also look into the fate of Francis, the ghoul who'd been fired for eating on the job. The Spare Parts Emporium could be like a walk-in-sized closet full of skeletons.
"As I said, I need your help. Somebody's got to look through all these records to find discrepancies. Say, like a detective. The precinct doesn't have the manpower to go through it all, and you know I can't do it—I can't even balance my checkbook."
"You still have a checkbook?" I asked. "Nobody carries a checkbook anymore."
"I only use it in the express checkout line at the grocery store," McGoo said. He let out a long sigh at the sheer number of boxes. "I wasn't figuring you could do much, Shamble—but these two lovely detail-oriented ladies . . ." He gave a flirtatious smile that failed miserably.
Robin looked over at me. "Dan, did you pick up the replacement brain for our client?"
"No, I got the runaround. The floor manager told me to come back tomorrow, but he promised to send over a new spleen and lungs."
"I'll believe that when it actually happens." Robin sniffed. "We would be happy to look into this, Officer. We might identify another entire set of wronged customers so we can expand our case, if necessary. I smell class-action suit."
"I smell something," I added.
"If you can do that, Robin, then I'll owe you a coffee," McGoo said.
"You'll owe me a coffee plantation," she said. "But I'll accept the coffee for the moment. Keep me caffeinated so I can bury myself in the case."
One of the uniformed cops pulled a cardboard lid off a box. "Should be everything you need. They sent everything." He rustled among the papers. "These are receipts for junk-food purchases for vending machines in the employee break room."
The other cop held up a folder. "Monthly formaldehyde bills. And full donation receipts for charity write-offs, regular deliveries to the Fresh Corpses Zombie Rehab Clinic."
"We're going to need the whole conference room table," Robin said. "Sheyenne, do we have any intake appointments for the next day or two?"
"None scheduled."
"Good." Robin nodded at the boxes of records, and determination was carved on her face. "Let's sprawl."
"Day or two?" McGoo said. "This looks like months of work."
Robin said, "I'm a fast reader."
He looked relieved and brightened further. "What about the scalps? Does Cralo deal in werewolf scalps?"
"Not that he would say—I already checked," I answered.
McGoo shook his head, but a strange quirk of a smile hovered on the corners of his lips. "I keep looking into the murdered vampire, and I come up empty." His grin grew larger as he looked expectantly at us. "That was a joke! Get it, coming up empty—like that vampire?"
"So . . . about the case?" I asked. "Any leads?"
McGoo heaved a sigh. "You dead people have no sense of humor. We've identified the victim. Name was Ben Willard, worked part-time as a hemoglobin barista at one of the Talbot and Knowles Blood Bars. More importantly, he was active in the rough-trade clubbing scene, liked to dress up, act tough. He'd pick up anyone who was still ambulatory, whether warm-or cold-blooded, then he'd go have a good time."
"The Motel Six Feet Under is a real destination spot for that," I said. "The manager was offering 'change your own bedsheets' discounts."
"Vamps think they're invincible," McGoo said. "And whoever Ben Willard picked up that night proved too much for him."
His helpers wheeled the empty hand trucks down the hall to the stairs where they clattered, bump-bump-bump, down to street level. McGoo followed them out, calling back, "Let me know as soon as you find anything. I have to get back to my beat, but these guys will bring up the rest of the load."
I looked at the stack of boxes in our reception area. "The rest?"
"It's a big Emporium, Shamble. They wanted to be thorough."
Since I wouldn't be much help studying receipts or balances on bank statements, or calculating whether the volume of formaldehyde purchased was reasonable for the business Cralo conducted, I made myself useful by lugging the heavy boxes into the conference room.
Robin directed me like a stage manager and sighed in disgust as she tried to prioritize the cartons. "These boxes aren't even labeled! How am I going to organize all this?"
"Through talent, sheer persistence, and determination," I said.
Robin sniffed. "That goes without saying. But I still wish they were alphabetized."
Over the next couple of hours, we worked together to sort the relevant records (body-part inventories, donation forms, and receipts) from the silly records (water, sewer, and utility bills; parking-lot pothole repairs—not too many of those); and fan letters (mostly from a group of schoolchildren who had toured the Emporium on a field trip).
As expected, many organs were sold to mad scientists and body-building enthusiasts like Archibald Victor, as well as hospital transplant wards. Many scratch-and-dent or slightly damaged floor samples had been sent over to the Fresh Corpses Zombie Rehab Clinic for a (rather large) tax write-off.
I had an idea. "Maybe I should stop by the clinic, see if Mrs. Saldana can give me any background on the Emporium." When it came to tracking down the origins of those spare body parts, I was the best person to do the arm-and-leg work.
Sheyenne interrupted, "I think we should go there, Beaux. You and Robin went to the grand opening reception, but I've never seen the place. Do you think Mrs. Saldana will be working today?"
"She splits her time between the clinic and the Hope and Salvation Mission. I'd be happy to have you along, Spooky, if Robin can spare you." In fact, spending a little more time with Sheyenne would go a long way toward brightening the dreary case.
Robin was intensely focused on the folders and boxes. To me, it looked as if a document explosion had occurred in our conference room, but she had some sort of visual organization method that I didn't want to disturb. "It's all right," she said in that preoccupied voice of hers. "I'll do better by myself, if I can just stare at this and absorb it all. I can put the pieces together."
Robin had often told me how she kept up with her studies in law school days by getting into hyperfocus, immersing herself in details, and becoming One with the legal precedents. Now she was doing a similar thing, absorbing the storm of data provided by the Spare Parts Emporium and making connections that Tony Cralo never expected anyone to find. "Getting dirty in the facts," she called it.
"All right," I said. "We'll go see what we can dig up at the zombie rehab clinic."
Sheyenne slipped her incorporeal arm through mine, just for effect, and we headed out.
CHAPTER 27
Not all zombies are as well-preserved as I am. Many take their time coming out of the ground, and by then the maggots and worms have made them less than presentable. Sometimes the initial embalming job is bad, or a mortician doesn't care enough to put the pieces back together properly when it's for a closed-coffin service. Other zombies are just plain slobs who don't take care of themselves, let their bodies fester, eat a poor diet of junk food—sometimes including brains, sometimes not. It's a rotten situation all around.
Though the worst shamblers are too far gone to want any sort of help, others just need a leg up. The charity-run Fresh Corpses Zombie Rehab Clinic was now administered by the Monster Legal Defense Workers. As a member of MLDW's advisory board, Robin had done a substantial amount of pro bono legal work for the charity, and after the recent scandals with the Smile Syndicate, they were on solid humanitarian ground again.
I entered the front doors of the clinic with Sheyenne drifting by my side. The clinic was a clean, modern facility with several wings, including a private lockdown ward for recovering brain addicts. Fresh Corpses also had a day-spa section, a dental replacement room, a necro-ophthalmologist who offered a wide selection of glass eyes, and a vault filled with replacement parts, many of them—I now knew—donated by Tony Cralo.
At the front desk, a horrifically scarred and mismatched zombie girl gave us a crooked grin. Sheyenne and I both brightened. "Wendy!" I said. "I didn't know you were working here."
"Doing my part for those less fortunate," she said in a slurred voice because her teeth, palate, and tongue had not been properly reassembled. "Other zombies have it worse than I do."
"You're looking well," I said. "Good to see you out in public."
Wendy looked away and batted her eyelashes. She was shy and self-conscious about her appearance. Unlucky in love, she had thrown herself in front of a train. But suicide victims have a higher reanimation percentage than normal deaths, and her mangled pieces had come back to life. Fortunately, the taxidermist and sawbones Miss Lujean Eccles had stitched the Patchwork Princess together as best she could. Her work had gotten better with practice.
"Is Miss Eccles here today, too?" Sheyenne asked.
"She's in the Upholstery and Repair ward. Let me take you there." Wendy lurched up from her chair and tottered off. Sheyenne and I followed.
After I'd been shot down in the street—an inconvenience, even for a zombie—Miss Eccles had patched the bullet wounds in my torso, while Wendy used her seamstress abilities to repair the holes in my sport jacket. Wendy was clearly thrilled that I still chose to wear it. "You don't have to keep that jacket, you know, Mister Chambeaux. It's such a bad stitch-up job."
"Not a chance," I said, brushing the front with my free hand. "I love it."
Sheyenne drifted on the other side of us. "That jacket has character. We won't let him get rid of it." Wendy blushed.
We saw the matronly Miss Eccles inside a room with two patients. One zombie sat by himself on a table, pulling a leather glove over shredded bony fingers, testing the fit.
At the adjacent table, Miss Eccles was stitching a triangular swatch of flesh-colored leatherette onto a rotting cheek. She hummed as she tightened the tiny stitches and affixed the replacement part where it covered the zombie's exposed molars. "Now, you have to take care of this patch," she said. "Someday, you might want to upgrade to real leather."
"Can't afford it," the zombie mumbled.
She clucked at him. "Plenty of available options for afterlife insurance and postmortem-care plans. Conscientious people plan ahead."
"Didn't expect to come back as a zombie." He stretched his mouth, flexed the new skin patch, rubbed his jaw like a dental patient checking the extent of Novocain.
Miss Eccles shook her head and said in a chiding voice, " 'Didn't expect to.' I bet an insurance agent hears those words more often than anything else."
Wendy led us into the room. "Look who's here, ma'am."
Miss Eccles looked up as we entered, and her expression lit up. "Why, Mister Chambeaux! Did you get damaged again?" She eyed me up and down for unexpected leaks or tears. "You'll have to put your name on a waiting list if you need repairs."
"I wouldn't take up a spot in the free clinic," I said. "Just had my monthly restorative spell, so I feel fine."
"I'm taking care of Dan," Sheyenne said. "You use your services for these people in need. Thanks for paying back to the community."
Miss Eccles snipped off the thread next to the zombie's face, gave him a motherly peck on the cheek, and sent the repaired undead patient on his way. "The zombies appreciate it. Most of them just need a second, third, or fourth chance."
Before attending to the second patient, she patted the Patchwork Princess on her misshapen shoulder. "I don't know what I'd do without Wendy. She's been taking self-esteem classes. Isn't she positively glowing?"
"And not in a bad way," I said. "I really can see the difference."
Sheyenne lowered her voice. "We should encourage Adriana Cruz to see Miss Eccles. I think she'd benefit from it."
Wendy said in her slurred voice, "Working here makes me realize how lucky I am."
Miss Eccles adjusted the leather glove on the second zombie, tweaking the knuckles, aligning the edge on the rotted wrist. "How does that feel? Can you flex your fingers?" The zombie did so. "The shade doesn't quite match the rest of your arm, but it's difficult to get an identical putrefaction color."
"This'll do fine," the patient said. "Stylish."
As she finished attaching the leather glove to the zombie's forearm, Miss Eccles glanced at me. "So if you don't need repairs, Mister Chambeaux, did you just come to visit the clinic?"
"It's part of a case, ma'am. We're investigating one of your suppliers, Tony Cralo. We understand that a lot of your . . . raw material comes from his Emporium."
Miss Eccles was surprised. "Mister Cralo is one of our most generous donors! As a zombie himself, he suffers from aches and pains, so he understands the plight of many of our patrons. Admittedly, he provides a lot of substandard pieces and writes them off at full value for tax purposes. We get his seconds, the slightly irregular parts or the ones near expiration dates." She clucked her tongue again.
"Sounds like a factory outlet store," Sheyenne said.
With a sigh, Miss Eccles shrugged. "We take what we can get. But if you need to look at paperwork, you should speak with Mrs. Saldana—she does the administrative work for the clinic. I just do patch-ups."
Mrs. Hope Saldana was a kindly old woman who had devoted her life to helping out downtrodden monsters. She had opened the first mission in the Unnatural Quarter, feeding, clothing, and caring for indigent monsters.
Wendy led us to the clinic's admin office, but Mrs. Saldana was pulling on her coat, ready to leave. "I wish you'd come sooner, dears! I'm heading over to the mission. Jerry's been handling the work all day, but he gets overwhelmed easily."
I didn't know where she got the energy. "Before you go, could you answer just a few quick questions about your dealings with Tony Cralo?"
"Dan meant to say 'please' at the end of his sentence," Sheyenne said.
Mrs. Saldana grabbed her purse. "This clinic wouldn't last long if we had to purchase all the materials that the Spare Parts Emporium donates to us." She turned to a pair of tall file cabinets behind her desk. "We have all the paperwork right there, receipts and forms with his contributions. We're never able to decipher them. Nothing seems to match, but we accept what is freely given and do our good works with it."
"Could we have copies of those records, Mrs. Saldana?" Sheyenne asked.
I wasn't so sure. "You think Robin wants more paperwork?"
"I can compare these forms with the papers Cralo gave to the police. If there are any discrepancies, I'll spot them."
And I knew she would.
Mrs. Saldana opened the drawers and began pulling out manila folders. "You're welcome to look at them, Mister Chambeaux. As far as I'm concerned, after all the help you've given me, you don't need a warrant. I trust you. Besides, Ms. Deyer is a member of the MLDW board, so she is allowed access to any of the clinic's paperwork." The old woman lowered her voice as a concerned look crossed her face. (She was very good at concerned looks.) "Mister Cralo isn't in any trouble, is he?"
"We won't know until Sheyenne has a look at these records," I said. "He may be marketing a few body parts that the original owners were still using at the time."
It took a second for that to sink in, then Mrs. Saldana's eyes went wide. "Oh, my! Well, we keep careful records of all patients we treat here at the clinic. If you need any of the donated body parts returned . . ." She shook her head, frowning. "A lot of poor zombies are going to be so disappointed."
CHAPTER 28
By the time Sheyenne and I returned from the clinic, Robin had done a spectacular job of papering every horizontal space in the Chambeaux & Deyer offices (and used pushpins to take advantage of many vertical spaces). Piles of documents had been distributed across the conference room table, the seats of the chairs, Sheyenne's desk, my desk, even the top of the microwave and coffeemaker in the kitchenette.
"Now we're making some progress," Robin said.
All visible evidence to the contrary, she has a highly organized mind. By now she'd concocted a master plan for arranging the Emporium records and laid them out according to a certain pattern. I thought of one of those optical illusions: If you tilted it just so and held your gaze in the right way, some sort of new pattern suddenly popped up in an "Aha!" moment. (Unfortunately, I was never good at those puzzles, and I couldn't see any sort of pattern now.)
"What did you find out at the clinic?" she asked.
"Wendy says hi," I said, "and Miss Eccles and Mrs. Saldana."
Sheyenne carried the folders of receipts from the Fresh Corpses files. "And we brought more documents—donation forms we can compare with the paperwork Cralo already provided."
"Excellent!" Robin said, and she genuinely sounded excited. Go figure. She was upbeat as she bustled back and forth, carrying folders, looking through receipts and entries. She was in the zone, and we could almost hear her thoughts humming.
That was when Miranda Jekyll and Hirsute arrived, generating all the usual amount of commotion as they pushed into the offices. The two Monthly werewolves were alone, but Miranda was always followed by a large entourage of sycophants and adoring fans, in her own mind at least.
She wore a shimmering emerald dress, diamond stud earrings with stones so large and heavy they stretched out her earlobes, and enough gold necklaces to create a layer effect. Intense lipstick, intense nail polish, intense expression. She paused after walking through the door, as if waiting for applause. Heavily muscled and lavishly haired, Hirsute stood by her side like a hunk of 100 percent pure hunkness.
Miranda noticed the papers all around. "Sweethearts, you must get a better office manager."
Sheyenne took offense. "It's a work in progress, but completely under control."
"I'm working on a complex case," Robin said. "It requires a lot of my attention."
Miranda deftly dodged piles of paperwork, careful to not soil her Jimmy Choos with copy toner. "At the moment, sweethearts, I require your full attention. I understand you're working for the Monster Legal Defense Workers, Ms. Deyer? I need to engage your legal services."
Robin brightened. "Yes, I'm a member of the board. Very important work, standing up for the rights of underprivileged unnaturals."
"It's charity work, Ms. Jekyll," Sheyenne pointed out. "In your case, the normal fee structure would remain in place."
"Of course, sweetheart." Miranda sighed. "I wouldn't want word to get around that I'm a charity case."
"How can we help you, Ms. Jekyll?" I prodded, afraid she might accidentally disorganize one of Robin's careful paperwork stacks.
"It's about my werewolf sanctuary up in Montana. I originally intended it to be a place where Hirsute and I could roam and unleash our animal passions." She stroked long fingernails along the curves of his biceps, like an all-wheel-drive vehicle negotiating steep mountainous terrain. "But I hated to be so selfish. I felt obligated to open up the property, make all that wilderness acreage a refuge for my Monthly brothers and sisters during the full moon, a place where they can be free and loved, and feel safe to be who they really are. But I want to be legally protected."
Robin said, "You'd have to set up a nonprofit organization and transfer the property into the corporation's name. I'll complete and file the forms, then we can set up another meeting to review the details, go over the tax implications with you."
Miranda gave a blasé wave of her hand. "Whatever you think is best, sweetheart, but it's important. I still feel soured and unsettled by that nasty werewolf business at the Worldwide Horror Convention."
Robin gave a concerned nod. "What happened to Professor Zevon this morning was awful. I hear he's recovering in the hospital, though. He'll be all right."
Miranda grimaced, as if she had swallowed a mouse and it had gone down wrong. "Oh, I'd almost forgotten about him. I was talking about the snotty attitude and outright disrespect the Hairballs show the rest of us true werewolves."
This was as unwise as discussing politics at a family reunion, but I couldn't stop myself from reminding her, "Let's not forget that Scratch and Sniff did cause trouble on their own."
"Of course they did, sweetheart. They're troublemakers. It's what they do. But those two are anomalies—rather garish ones, if you look at their tattoos. Implying that all Monthlies are ill-tempered troublemakers is like implying that all motorcyclists are ill-behaved and destructive. Scratch and Sniff may embody the cliché, but it doesn't make it true."
Hirsute squared his shoulders. "They were unruly up in Montana last full moon, too, Miranda. We might not extend them a return invitation. I'd rather have the wilderness with you, my dear, without distractions." He lifted his square jaw and cocked his head in a well-practiced display of his profile. "When we walk among the trees under the full moon . . . and the light bathes us, and we feel the change—I love to be right there, so I can help tear your clothes off, nibble you with my fangs, taste you. Then we lope off into the underbrush. . . ."
Sheyenne rolled her eyes and flitted away. "Whoa, a little TMI there."
Miranda giggled in uncharacteristic embarrassment.
But Hirsute wasn't finished; his voice grew huskier. "I'll know that you're in heat, and I'll chase you into the meadows, where we'll make wild love. Together we will hunt prey, stalking anything that moves. And after we've killed it and torn the flesh with our fangs and claws, we'll drink the hot blood and feel energy again. We will make love a second time—and continue until the sun rises! Then we will rest for the next night of the full moon."
"Whew!" Robin fanned herself, glancing awkwardly at me.
Miranda looked as if she were about to melt, and I was afraid she'd turn into a puddle right there and ruin Robin's neat stacks of papers. She let out a long sigh. "Ah, those are my favorite days of the month." She nuzzled against Hirsute's rock-hard body. Her face was flushed, her pupils dilated, and she breathed heavily. "The full moon will be here soon, but maybe we'd better not wait to get in some practice."
She gave a quick dismissive wave to Robin. "You know what to do about the legal matters, Ms. Deyer. Call me whenever the paperwork is ready. I think . . . I think we'd better go now." She glanced at Hirsute. "Hurry."
As they left the office, Hirsute tossed his mane of hair, glanced back, and for some reason gave me a triumphant wink. He playfully grabbed Miranda's ass as they rushed into the hallway.
Robin put her hands on her hips and looked down at the piles of paperwork. "Now I've lost my train of thought."
CHAPTER 29
Walking the dark streets of the Unnatural Quarter was a habit of mine. Some might have diagnosed it as restless-corpse syndrome; I'm even a card-carrying member of AARZ—the American Association of Restless Zombies. In actuality, I just like to think while I walk. Sure, it might look like I'm wandering aimlessly—as many zombies do—but I'm really working. The cases don't solve themselves.
I walked past nightclubs, bars, gambling joints, all-night restaurants, blues clubs. I stopped before a small run-down theater, the kind that used to play second-run porno films before home video ruined the market; now the refurbished theater catered to a wider clientele. The red plastic letters on the marquee said ONE NIGHT ONLY: GOLEM MUD WRESTLING!
Directly in front of the theater, two familiar choppers were chained to a fire hydrant in a blatant Screw You gesture at parking enforcement. Raucous shouts came from inside the theater, and I knew I had to go inside (and not just to see the golem mud wrestling).
I bought a ticket from a frog creature (small body, large head: some sort of D-level demon or nondescript sewer-dweller). He put a red vinyl wristband around my arm and said in a voice that sounded like an extended belch, "You can go in and out if you want."
"I don't think I'll need the privilege."
"The night's young."
Once inside, I was bombarded by raucous cheering, loud laughter, and thick smoke—cigar smoke, burning weeds, brimstone, even a whiff of what came out of the smokestacks at Joe's Crematorium. Vampire cocktail waitresses in slinky dresses patrolled the tables carrying trays of drinks. The bartender was a fat, gray fellow with four tentacled arms, which he put to good use concocting multiple drinks at once.
A few dozen people crowded around a center stage like a boxing ring, except this one held a plastic inflatable swimming pool adorned with cartoon dolphins. The plastic pool was filled with a sloppy brown slurry that looked more like diluted manure than good clean dirt. It bubbled like a hot, cloying mud bath.
Two clay female figures faced off in the pool; they had breasts and curves sculpted in all the right places, but the figures were streamlined like giant-sized Barbie dolls; etched lines around their breasts and waists implied bikinis, but they were so scanty that even the outlines didn't make much difference. Lumpy wads of gray clay hair hung down around the golems' flat generic faces; their expressions were threatening grimaces.
The referee called out the start of the match, and the golem mud wrestlers crashed together with a wet, painful-sounding slap. The women grappled, pushing and straining until they finally drove each other into the manure-like mud. Rolling around, they splashed brown slop in all directions, spraying the disgusted audience.
The golems slammed each other back and forth with wild abandon, punching, grabbing, slapping, kicking. One golem shoved the other's head into the mud, and when she came up sputtering, her head was misshapen from the pressure. Furious, she reached up and grabbed the clay hair of her opponent, tearing off lumpy wads, which she tossed into the mud. They gouged divots out of each other's skin; one punched her opponent's left breast, caving it in. When the wrestlers finally disentangled themselves, the damaged golem took a moment to squish and reform her body, glaring at her opponent the entire time. Both dripped with brown slop.
Scratch and Sniff were crowded close to the ringside, cheering and cursing, egging on the two golems, just as they had done during the cockatrice fight. The clay opponents looked identical, as if stamped from the same mold, so I had no idea how or why the audience could cheer for one or the other.
When the crowd recoiled as mud flew, I worked my way closer to the ring. Scratch and Sniff didn't seem bothered by the muck that spattered their faces and werewolf-pelt coats.
"Hi, guys," I said. "Mind if I join you?"
The biker werewolves scowled at me and wrinkled their noses. Werewolves, even monthly ones, are sensitive to smells, but I doubted my undead scent was any less pleasant than the manure-tainted mud.
"It's that zombie guy." Scratch rubbed a palm over his slicked-back hair, as if the brown mud were a new kind of mousse.
His partner contemplatively stroked his beard, then sniffed his fingers. "Either he's trying to be a player, or he's following us."
"Maybe we just travel in the same circles," I said. "In fact, I saw you both riding away from Joe's Crematorium today. Business there?"
"Visiting a sick friend," Scratch said with a sneer. "Why do we have to tell you anything?"
Before I could answer, a raucous cheer went up. One golem slammed the other down in the mud, picked up her adversary's right leg, and bent it like a pretzel until it was tangled around the back of her neck. The other golem struggled and flopped, trying to straighten out her limbs.
"I also saw you at the Worldwide Horror Convention. You two sure have diverse interests. Renaissance men?"
"We wanted to heckle that crackpot professor with his stupid ideas," Sniff said. "Guess he got what he deserved."
I hardened my voice, no longer trying to be cheerful. "Scalping an old man is no way to respond to an intellectual disagreement."
Hearing the accusation, Scratch growled deep in his throat, even though he still looked human. "Wait a minute—Sniff and I had nothing to do with that."
I shrugged. "Do the math. You were there, and you were also at the cockatrice fight when Rusty got scalped. Quite a coincidence, don't you think? Somebody sneaked up behind him, shot him with tranquilizers, and then ran away."
Sniff cocked his head. "Mister Zombie Guy, do we look like the type to tranq someone before we cut his scalp off?"
Come to think of it, they didn't. Instead, I indicated their coats. "Didn't you skin a werewolf alive for the pelt?"
The two looked at each other, self-consciously brushed at the stained fur jackets and the long-dried red clumps. "Nah, it's just synthetic fur," Scratch said. "But it looks cool. And it's part of our . . . thing."
"The dried blood is real, though." Sniff rubbed one of the clumps, then smelled his fingers. "Anti-fur protesters dumped it on us because the coats looked so real."
I supposed that was a relief. "Don't worry, I won't ruin the secret."
The mud wrestling had grown even more furious. One golem grabbed the other's fingers and wrenched them off, holding up the severed clay digits like trophies. The ref whistled a shrill, piercing note, but the wrestling golems were too intent on their battle, pummeling, cursing each other. The ref whistled and whistled again. The crowd cheered. The bartender continued to pour drinks, shaking martinis in two of his tentacles, while pulling beer with two others.
Finally Security charged the ring with a fire hose and blasted a stream of high-pressure water. The mud-wrestling golems kept fighting until they dissolved into grayish glop that filled the mud bath.
The announcer came out to ringside. "Ladies and gentlemen, we call that match a draw, but don't worry, we'll scoop up the mud and make some more golems. We still have the animation spells!"
Scratch and Sniff obviously intended to stay for the next match. "So, what sort of work do you two do for a living?" I asked them when the tumult had died down. I tried to sound cheery and conversational, best buddies.
"We work in collections," Sniff said.
Scratch was not so forthcoming. "What's it to ya, anyway?"
"Oh, just curious, since I've been hearing a lot about you two. In fact, I was at the Voodoo Tattoo parlor—heard about the practical joke you played on Rusty."
They both laughed. "Now, that's more our style! Yes, sir, big tough Rusty sure is one impotent guy!" Sniff lowered his head, brushed some of the mud that had splattered his beard, sniffed it, then wiped his fingers on his pants.
"We're sick and tired of being accused of everything by the Hairballs," Scratch grumbled. "Full moon is in a couple of days, and then it'll be time—claw to claw, wolf mano–a–wolf mano. We'll settle this."
"Or maybe just keep the feud going," Sniff added, and the two chuckled again.
A pair of newly formed female golems came out to the ring, fists raised, to loud cheers from the audience. This time, one had been sculpted with even bigger breasts, and the second sported a decidedly larger backside. A well-muscled hunchback came out and dumped a tub of fresh, noisome brown mud into the plastic swimming pool.
Scratch and Sniff no longer paid me any attention, but I had a lot to ponder anyway. Unable to concentrate in the middle of a golem mud-wrestling match, I slipped back out into the streets. If I changed my mind and wanted to go back later and see another match, I still had the wristband.
CHAPTER 30
Next morning, I went to see Professor Zevon in the hospital. The old silverback was recovering well, thanks to his lycanthropic healing abilities, but he'd been rather frail to start with, and he had an academically low pain threshold. While Rusty had gone home within hours of being scalped, the physicians were holding the professor for further observation since they were concerned about his cholesterol levels and blood pressure.
At the Brothers and Sisters of Mercy Hospital, an old panel truck was parked in front of the emergency room. The side of the truck bore the logo and bright letters of Cralo's Spare Parts Emporium: Nobody Beats Our Deals on Transplants. I wondered if the truck was there for a pickup or a delivery.
Inside the hospital, I approached the information desk. The receptionist took one look at me and immediately tried to direct me to the morgue, but I assured her that I had already been there. A security guard and triage nurse pushing a gurney with a body bag came up and politely invited me to lie down, but I brushed them aside. "I have business to attend to." At the reception desk, I asked where I could find Professor Zevon.
When I finally got to the right room, the gray-furred werewolf was lying flat on the bed, his spectacles on the nightstand, the top of his head wrapped in gauze. Two of his Hairball students kept him company; a young furry woman was reciting from a thick book—she was pretty by werewolf standards, I supposed.
As I entered the room, I nodded toward the novel. "Reading Twilight to the professor?"
"He likes to keep up with the classics," said the student.
Zevon tried to focus on me without his spectacles. "Excuse me, do I know you?"
"I'm Dan Chambeaux, private investigator. I attended your lecture at the Worldwide Horror Convention, sir, and I've been hired by another werewolf who was also scalped. I'm sure your cases are related. Mind if I ask you some questions?"
With a clawed hand, the student folded over a corner on the page where she'd stopped reading. "We should have protected the professor."
"It was a convention full of monsters and rowdy fans. I never dreamed we wouldn't be safe." Zevon leaned forward in the bed, snuffling in the air. "There's a bit of a taint about you, Mister Chambeaux."
"I'm undead, sir, but I'm still on the job."
"Zombies can be persistent indeed." He shook his head, then winced at the pain. "I'm afraid I can't help. I was shot in the back by a tranquilizer dart out in the parking lot. I didn't see a thing."
The male student growled. "If you attended the professor's lecture, then you saw those two Monthly bullies harassing him, and that awful woman, Miranda Jekyll! They scalped the professor, just to shame him."
Professor Zevon touched the bandages on his head. "I agree with the thesis. Those . . . those part-timers publicly threatened me, and I was assaulted very soon afterward!"
The blood pressure and pulse rate on the monitor apparatus began to bleep in a shrill, annoying tone that had the opposite effect of calming the patient.
"Please don't stress, Professor," said the young werewolf student in a soothing voice. She quickly reopened the book. "Here, let me read some more. Bella's problems will cheer you up."
The professor lay back on the bed, breathing heavily.
Though I doubted I'd convince them, I said, "I've talked to both Scratch and Sniff. They enjoy mayhem, but I'm not convinced they're the culprits."
The professor turned his muzzle away. "I don't want to hear it. Who else would do such a thing?"
"I'll let you know as soon as I find out." I handed over the sentimental get-well card and stuffed teddy bear I'd bought at the hospital gift shop. The professor thought the teddy bear was adorable.
As I left the room and walked down the hall, another full-furred werewolf emerged from the elevator. Rusty flaunted his raw and lumpy scalp, which still looked like a large balding scab. His gangly nephew accompanied him, jaunty and happy. Furguson must have gotten back into his uncle's good graces; maybe his brash outdoor attack on Scratch and Sniff had earned him some street cred in the Hairball gang.
Rusty was holding a bouquet of limp lilies. "We came to make sure the professor's doing all right."
"And vow to get revenge!" Furguson added.
"Just make sure you get the right target," I said. I looked at Rusty's fur, trying to make out the tattoo designs, one of which was the mark of impotence, but I could discern only vague outlines.
Rusty narrowed his yellow eyes. "We'll give you a couple more days to solve the case, Mister Shamble—but once the full moon hits, Hairballs are taking matters into our own claws. We've got enough proof, as far as I'm concerned."
Scratch and Sniff had said something similar at the golem mud-wrestling match. I said, "You might have more of a battle than you expect. The Monthlies plan to fight back."
Rusty chuffed. "We're counting on it! Time to set things right. At least four other Hairballs were scalped. Our lost brother Larry filled us in."
Furguson growled. "Larry, grrrrr—I wish he got scalped for helping Harvey Jekyll!"
Rusty placed a clawed hand on his nephew's shoulder and squeezed like a vise. "Now isn't the time for old grudges. We have to stick together to wipe out the Monthlies once and for all."
I frowned. "Isn't that an old grudge, too?"
"Not the same thing." Rusty shook his semi-shaggy head. "Maybe the professor and his students want to join us in the fight—solidarity!—though those sissies are more likely to lecture the Monthlies than rip out a few hunks of fur."
"I'll keep investigating," I said. "Don't jump to conclusions."
"Come up with the right conclusion, and we won't have to do any jumping."
Furguson twitched his gangly legs. "I like to jump!" They went past me down the hall toward the professor's room, and I pushed the button for the elevator.
Shrieking alarms blared, loud enough to wake the dead and then send them back into their crypts for a little peace.
"Code Arterial Red," shouted a voice on the intercom. "Surgery Two!"
CHAPTER 31
Code Arterial Red was a special alarm that indicated Violent Activity with Monsters.
Hospitals had never needed to address those types of emergencies before the Big Uneasy, but with all the new crises caused by and happening to unnaturals, hospitals had to assign more colors to their alarm-category rainbow.
As hospital security responded to the blaring summons, Furguson panicked, flailing his long arms and howling along with the noise. Rusty elbowed him in the gut. "Nothing to do with us. Come on, let's deliver the flowers." Furguson yipped in embarrassment, and the two made their way to the professor's room.
The experimental cardiac surgery ward happened to be on the same floor as the werewolf recovery ward. Human security guards ran toward the surgery center, colliding with nurses and staff who fled screaming from it. Several were covered in blood; one who had been severely mauled collapsed on the floor. Two nurses and a doctor loaded her onto a gurney and ran with her toward the ER, as if it were part of a relay race.
A burly human security guard careened out of the surgery doors and slammed against the wall. His throat was torn open; his head lolled at a broken angle. The electronic doors to the experimental cardiac surgery center glided automatically shut again.
I drew my .38—normally not allowed during visiting hours in the hospital, but I did have a permit. From inside the surgery room, I heard a snarling, more screams, smashing glass, a clatter of equipment and instruments.
I'm just a detective, not a hero, not a commando, not even a cop. But I was there in the hospital, and I was armed, and there was a lot of screaming going on. I had to do something.
The elevator bell chimed, and a flustered-looking McGoo burst out, weapon drawn. He blinked at me in surprise. "Shamble, what are you doing here?"
"Visiting a sick friend."
McGoo glanced at the dead security guard and shook his head. "How'd you like to be my backup?"
"Sounds better than waiting for more cops to arrive." We ran down the hall, weapons drawn.
"I thought this was going to be a routine hospital call," McGoo muttered in disbelief. "Got complaints about a vampire photographer acting erratically, so I brought him in. Turns out he was suffering from silver contamination due to photography chemicals in his darkroom."
"Photography chemicals? Doesn't everyone use digital cameras now?"
"He's old-school. Maybe next time he'll wear gloves."
The Code Arterial Red alarms still clamored like a monkeys-with-pans rock concert, but the screams had died to low gurgles as we crept to the door of the surgery. A male nurse whose back was slashed open started to crawl out of the room on hands and knees, but collapsed on the floor, thus frustrating the electronic doors; they tried to close on his waist, then opened, then tried to close again.
McGoo extended his firearms, one in each hand: both the revolver loaded with silver bullets and his regular service piece. We flattened into a doorway.
From inside the surgery, someone or something flung body parts into the hall, like a child having a tantrum because she didn't like her toys.
Through the blood-splattered observation window in the operating theater, I could see overturned monitors and a respirator, and the surgical table leaning tilted against one wall. Other equipment lay smashed and disassembled, as well as a surgeon (also smashed and disassembled) and several nurses—it was hard to tell the exact number, since the pieces were in such disarray.
Inside the room, the alleged patient was on a rampage. An obese woman in a blood-drenched hospital gown that hung in tatters, she had low-hanging breasts and hips the size of a small Volkswagen; her hair was a mess, her wild eyes blazing orange. Her teeth had sprouted into tusk-like fangs so long that she could barely open and close her mouth. She yowled and screamed, and slashed the air with fingers that ended in long claws.
The most noticeable oddity was that her chest hung wide open, her sternum sawed in half, the ribs pried apart and propped open with a spreader. A purplish-black heart dangled from her chest cavity, roughly attached by black sutures.
"I've heard of people wearing their hearts on their sleeves," I said, "but that's ridiculous."
"It's not a human heart, Shamble," McGoo pointed out.
The monster-patient fell upon a pack of replacement blood that dangled from an IV cart. She struggled to poke her long curved fangs into it, but couldn't find the right angle; in frustration, she tore the plastic with her claws and poured the blood into her mouth. Most of it missed and splattered her cheeks.
A prissy-looking middle-aged man scuttled down the hall, shaking his head. He wore a suit and hospital ID badge. "What a mess! This is going to be difficult to explain, but I don't believe we can be held legally responsible. Thank heavens the patient signed all the release forms."
McGoo and I looked at each other, then pulled the administrator aside. McGoo said, "If we're going in to fight that thing, you better tell us what's going on."
"It was an experimental transplant procedure on a terminal case." The prissy man shrugged. "The chief surgeon didn't believe there was much chance of success, but we thought, what the heck?" His nervous smile didn't endear him to either of us. "Insurance wouldn't cover it, but the patient paid with her own life savings."
I looked at the carnage strewn around us. "What procedure are you talking about—exactly?"
"Oh, simple heart transplant—or it should have been simple. We obtained a reasonably fresh vampire heart. Compared with regular human heart transplants, this should have been a piece of cake, since it's almost impossible to make vampire hearts stop beating. So why not try it for a transplant?"
I indicated the slaughter in the operating theater. "That seems to be a good reason not to do it."
"Tissue rejection, apparently," the administrator said with a shrug. "Surgery isn't an exact science. It is called medical practice, you know. Did I mention she paid for this procedure out-of-pocket? No one is liable or on the hook."
"The families of all these victims might disagree," McGoo said.
"The hospital provides generous afterlife insurance. They'll be fine."
Looking at the mangled victims, I doubted they'd be "fine."
The monstrous transplant patient threw herself into the wall, uprooted more monitors, and crashed them to the floor. She was slavering, ravenous, inhuman. The exposed heart hung out of her open chest.
"I have silver bullets," McGoo said. "You think those will work?"
"Only if you hit the heart exactly," I said.
"Right now it's a moving target, flopping around from side to side."
"I don't like your chances, McGoo. She's wild for anything hot-blooded, and if you got killed, then who'd be my best human friend? I'll go first. I doubt embalming fluid is on the menu." He and I both knew that as an undead person, I would be little more than an unappetizing piece of walking furniture as far as a vampire patient was concerned.
The transplant patient moaned, touched fingertips to her long curved fangs as if her mouth were sore. She seemed disoriented but ravenous.
"That thing needs to be put out of her misery," McGoo said. "Let's go. I'm right behind you."
The administrator lifted his chin. "Do what you must to resolve this unfortunate situation. It's all right. As I said, she signed the proper release forms."
I was about to suggest that the administrator go in and take care of his own problem, but decided that would be unproductive.
The automatic doors opened again as I stepped over the nurse's bloody body and into the room. Shuffling forward, I was careful not to slip on the red-smeared floor. McGoo followed me, revolver drawn.
The vampire patient whirled, glared at us, and made incomprehensible sounds. As she opened and closed her mouth, the downward-curving tusks bit into the folds of her chin, leaving divots.
"Easy, girl," I said, holding out my hand. "I'm here to make it better." Flailing her clawed hands, she swayed in agony, and the dangling heart bounced around, barely attached by black sutures. McGoo tried to track the renegade organ with his silver-bullet revolver, looking for a shot. She sniffed, smelling fresh blood, turned her fiery eyes toward him.
I intervened, waving a hand to get her attention. "Gentle. It'll be all right," I continued in a soothing voice. I hated to lie to a fiendish monster, but this thing had already murdered several people and had to be stopped. "I'm sorry this happened to you." She gave a hesitant yowl.
"I wish somebody would shut off that yammering alarm," McGoo said. "It isn't making that thing any less agitated."
"Give me some room, McGoo."
The transplant patient snarled and burbled, made a moaning sound, then slid to the floor, resting on her generous buttocks.
"It'll be over soon, I promise." I inched closer. "Simple enough. Just a quick little tug."
"Watch yourself, Shamble. Just grab it—"
"I've got this."
While McGoo hung back, the creature looked at me, showing something like gratitude in her orange eyes. She leaned forward, clutching the sides of her head with clawed hands, and squeezed her eyes shut, as if she didn't want to see what I was going to do.
Taking the opportunity, I grabbed the pulsing, purplish-black vampire heart and tore it free. Even after I ripped it from the sutures, the rubbery heart kept beating and convulsing in my hand. I flung it aside to a clear area on the floor.
McGoo aimed his revolver, shooting again and again, but the heart flopped about like a fish on a dock. The silver bullets ricocheted, and he had to fire three times before hitting it. We didn't dare leave the monstrous organ alive, or the hospital administrator might decide to do more experiments.
The woman let out another pitiful moan of pain, sorrow, and confusion as she died—for real, this time. "I'm so sorry, ma'am." She collapsed into me, like a deflating inner tube.
McGoo and I emerged from the surgical theater to the administrator's delighted smile. By now, several backup officers had arrived, and they proceeded to secure and process the scene. My hands, sport jacket, and slacks were sloppy with blood. I was a mess, but in far better shape than the mangled corpses around us.
"Thank you, gentlemen," the hospital administrator said, sounding chipper. "So much for that transplant idea, though . . . back to the drawing board. Did you leave the organ intact so we can ask for a refund, if we track down where it came from?"
"Sorry," I said. "Accidentally damaged."
Eyes wide, McGoo said to the administrator, "You don't even know where the organ came from? How can you not have records?"
The man shrugged. "With such a rare transplant commodity, the chief surgeon wouldn't have been too picky. And he's dead, so we can't ask him." He shook his head. "This is going to be an administrative nightmare. You can't even begin to imagine!" He grumbled as he walked away. "You gentlemen have the easy part. I get to fill out all the paperwork."
CHAPTER 32
When Sheyenne saw me enter the office caked with dried blood, she swept forward in alarm. "Beaux! What happened? Are you all right?"
Forgetting for a moment that she was a ghost, she embraced me . . . and her arms passed right through. I gave her an air-hug back.
"I was in surgery," I said, then explained my encounter. "My guess is it was part of the vampire murdered in the Motel Six Feet Under. I wouldn't be surprised if the evidence pointed back to Tony Cralo."
"We've been finding evidence of our own, though ours is less dramatic," Robin said and turned to look around the offices. Receipts and documents were still sorted into different piles, and the project had even co-opted part of my desk, spreading like the out-of-control experiment of a mad accountant rather than a mad scientist.
Thankfully, I didn't plan to stick around the office for long. "Come up with anything useful?" I asked.
"Not much yet. But we'll get there."
I couldn't forget about the gang violence brewing among the werewolves, especially with Rusty convinced the Monthlies were behind the scalpings. If we could prove that Cralo or one of his goons was behind the scalpings, however, then the furry feud didn't have to get bloody. The fat zombie crime lord would be a bigger fish to drown. It would have been nice to wrap up the investigation, but the cases don't solve themselves, and they certainly don't solve themselves in order.
I tugged at my stained sleeve. "After I get cleaned up, I'm going to ask Adriana Cruz's parents about dealing with Joe's Crematorium. If they're willing to give me a sample of Adriana's alleged ashes, I'll call in a favor from Dr. Victor and have him run an analysis."
The Cruz house was in a quiet neighborhood outside the Quarter, a tri-level Brady Bunch model. I made myself as presentable as possible, pulled the fedora low enough to cover the bullet hole in my forehead, popped a breath mint in my mouth, and rang the doorbell.
A meek-looking middle-aged woman opened the door. Even before she spoke, I thought of a dark-haired Edith Bunker. "Hello, how can I help you?"
I introduced myself, and the woman called over her shoulder. "Sal, there's a zombie at the door. Says he's here about Adriana."
Mr. Cruz hollered from the living room in a long-suffering voice. "What's she gotten into now?" He sat in a recliner rocker with the leg rest extended. On TV, the joyful buzzers of a game show proclaimed the contestant's answer to be incorrect.
"I'm a private investigator, Mister and Mrs. Cruz. Your daughter hired us to investigate fraud committed by Joe's Crematorium."
Mrs. Cruz opened the door and invited me into the foyer. "We were quite surprised to see our daughter again, after her death and all. Do come in." Mr. Cruz muted the television but kept watching the game show with closed captions.
"After she went to college, Adriana didn't have much to do with us," Mr. Cruz said. "I've had the same job in the factory all my working career, about to retire in six years. Maria's a proud housewife, never wanted to be anything else, never changed her job title to 'domestic engineer' or crap like that."
"But Adriana was her own woman, wanted to be professional. She excelled in everything," said Mrs. Cruz, with a small sigh. "I think she was embarrassed by us, and even though we might not be go-getters like her, we were still proud of her. I wish we'd been closer . . . and when she died in that horrible car accident, we didn't know what to do. But Adriana had taken care of everything ahead of time. Such a responsible girl. All we had to do was follow the steps."
"She had a separate sheet printed up with bullet points," Mr. Cruz said. "Couldn't have asked for more. Our girl was always organized. Took her to the crematorium, just like she asked."
"It was quite a shock to us when she came back."
"Returning from the dead is unsettling for everyone concerned. I've been through it myself." Thinking they needed some reassurance, I added, "Being a zombie isn't all bad, though. Look at me—I'm well-preserved, I have a decent job. I'm a contributing member of society. Adriana can still make something of herself, once she gets used to her situation."
"She always said she wanted to volunteer for important activities, change the world," Mrs. Cruz said, putting a finger on her lips. "Do you know if the Peace Corps accepts zombie volunteers?"
"I couldn't say, ma'am."
Mr. Cruz worked the lever on the side of the recliner and dropped the footrest. He heaved himself out of the chair with sounds of obvious back pain, and then pointed toward the mantelpiece. "Have you figured out how to explain that? We got her ashes, right there, specially delivered from Joe's Crematorium. So if Adriana is still around, then who—or what—is in that urn? We couldn't bear to throw it away."
"That's why I stopped by today. Would you be willing to let me run some tests on the remains? If those ashes belong to someone else, they should go to the proper owner. You'll also be receiving a full refund from Joe's."
"Take the whole urn," grumbled Mr. Cruz. "It's pretty damned clear that isn't Adriana."
Mrs. Cruz hurried into the kitchen and came back with a brown paper grocery bag. She wrapped newspapers around the urn, set it in the bottom of the bag, and handed the bag to me. "Here you are, dear."
The bald coroner with the obvious toupee was happy to see me at the crime lab. He glanced meaningfully at the brown paper bag and whispered, "Harriet told me you'd stopped by the Parlour (BNF), but you shouldn't have brought the replacement brain here. That's a private matter."
"Your brain wasn't ready when I stopped by Cralo's," I said. "But they promised it's coming, don't worry. Did you get the new lungs and spleen? They should have been delivered to your home."
His expression became annoyed. "They arrived, and the lungs are fine, but the box with the spleen was crushed. The organ's totally useless."
I sighed. "I was going to have to go back there anyway."
"I have tomorrow off. You can bring the brain and spleen directly to my lab in the trailer park." He was filled with anticipation. "Now, what can I do for you?"
I reached into the grocery bag and pulled out Adriana's urn. "I need a quick favor, an analysis for a client."
The coroner frowned at the ashes. "It's a little too late for me to perform an autopsy once the body's been cremated."
"With all your resources I was hoping you could run some tests. These were supposed to be the ashes of a beloved daughter, but since she walked into our offices as a zombie, obviously she wasn't cremated. Could you at least tell me if these are human remains?" I kept thinking of the stacked cordwood behind the crematorium and what it might be for. "Neither the family nor the crematee is satisfied."
"I'd imagine not." Archibald Victor took the urn, popped the top, and bent close, sniffing as if he meant to inhale it for a particularly odd kind of high. "Doesn't smell like roasted human flesh." He licked his fingertip, dipped it in the ash and tasted it, like a cop at a drug bust checking for cocaine. "Doesn't taste right, either." He spat it out. "Too tangy. I can run some tests and get you a quick answer."
He took the urn to an experimental table that held bubbling flasks and a test-tube centrifuge. A heavy machine spinning larger flasks reminded me of a paint mixer in a hardware store. "The department just purchased this piece of equipment, a Necro-Centrifuge specially designed to separate human ash from wood ash."
"And what would you need that for?" I asked.
"Vampire murders. If someone stakes an old vampire, everything turns to dust in a flash. In some investigations, we need to separate the stake's ash from the vampire's. The easiest way is with this handy little gadget."
The little mad scientist dumped powder from the urn into one of the paint-can-sized containers, then connected a spiral tube to an empty canister. He sealed the tops and set the machine rattling and spinning. It made as much noise as a dryer with an uneven load of soaked bath towels.
We tried to have a conversation while we waited, although the loud machine made it difficult. Archibald said, "About that murdered vampire, Ben Willard—I finished studying his entire skin, found a couple of stains that weren't apparent at first. Ink stains."
"Did he have a pen in his pocket?" It could have gotten smashed while he was being murdered.
"It was tattoo ink," said the coroner. "Tattoos don't last long on vampire skin—they leak out and wear away as the pinpricks heal. The victim must have been sweating heavily, which diluted the ink. I suppose even a vampire perspires when he's having his organs removed one by one."
"Any way to tell what the original tattoo looked like?"
"Not a chance," Archibald said. "It was just a smeary blob of ink staining the sheets. No way to know."
"I might have a way," I said. I already knew where the vamp had gotten his tattoo. "I'll call you if I learn anything."
Archibald self-consciously adjusted his toupee and shut down the Necro-Centrifuge. When he opened the canisters, all the ash had traveled through the spiral tubing to fill the second canister, leaving nothing behind. He handed the full can to me. "Well, there's your answer, Mister Chambeaux. It's quite obvious."
Obvious wasn't the word that came to my mind. "Could you make it a little . . . more obvious?"
"The Necro-Centrifuge separated the wood ash into this canister. It's entirely wood ash, Mister Chambeaux. No human remains here at all. Definitely a crematorium scam."
"Thank you, Dr. Victor. That's what I needed to know. I owe you one."
"Just bring me my brain—and spleen—and I'll be a happy man," Archibald said. "I'm close to a major breakthrough in my hair-tonic experiments, but the body-building competition is coming up, too, and I need to have my entry ready. So much to do, so much to do."
CHAPTER 33
The following morning I met Steve Halsted at the Ghoul's Diner for a bad cup of coffee, a bad breakfast, and an update on his case (not necessarily bad). Esther the harpy was cranky, tired, and testy, as if auditioning for a PMS commercial—in other words, same as she always was. She clatter-spilled a cup of coffee for me and for Steve.
Esther butted in on our conversation. "I've already had a long shift, Chambeaux, and I'm ready to get off." Her feathers were ruffled. "Did you nail the bastard who gave me the bad-luck charm yet?"
I decided not to tell her that her case was far from our highest priority. "We've done some research. My partner believes she can prove intent on the wizard's part—that he knowingly gave the bad luck to you—although she isn't convinced that we can prove damages. The courts haven't ruled definitively on whether bad luck is transferable."
"It's sure as hell been bad luck for me!" she shrieked. "How else do you explain my mixed-up orders, all the broken dishes, all the customers being angry with me?"
Sitting in a booth, a Monthly werewolf raised his coffee cup and tried to get Esther's attention, but she lashed out at him. "You wait your turn! Can't you see I'm busy with something a lot more important than your damn cup?" She turned back to me. "Stupid customers, always demanding things. Like that giant plant we used to have in the back, 'Feed me, feed me!' Nag, nag, nag."
I interrupted her rant. "What can you tell me about the wizard who gave it to you? Do you have a name, an address? I'd like to talk with him, see if I can convince him to lift the curse."
"His name is Glenn, with two Ns. He's a wizard. What more do you need to know? You're the private detective—go track him down. If I have to do all the work, what am I paying you for?"
In the kitchen, Albert the ghoul stirred his pot, which probably hadn't been washed since the mystery stew from my previous visit, and ladled out two servings of a congealed oatmeal substance. When he placed the bowls under the heat lamp, the mounded contours looked disturbingly like brains.
Esther thumped the bowls down in front of me and Steve. "I've got to get back to my tables, so I can't stand around talking all day. Let me know as soon as you find anything." She fluttered her sharp feathers. "And pay your bill as soon as you can—I'm clocking out in a few minutes. My boyfriend's picking me up in his limo."
Even though we hadn't yet taken a bite of the food, Steve and I both fished in our wallets for the appropriate amount of cash, adding a 20 percent tip because we knew what was good for us.
When we finally got around to talking about his case, Steve seemed dejected. "So, no progress? Has Ms. Deyer found a precedent that can help me?"
"She's got an interesting idea," I said. Even while auditing the records of the Spare Parts Emporium, Robin had come up with an approach that might help resolve my friend's problem. I slurped my coffee and played with my oatmeal. "Let me try to explain. When you listed your ex-wife as beneficiary for the insurance policy, I know you intended the money to go to your underage son. Even if Rova was aware of your wishes, since it wasn't explicitly written in the will or in the insurance policy, there's no way to prove your intent. Legally, Rova could do whatever she liked with the money. Now, however, you yourself can make your postmortem instructions perfectly clear. You'll sign an affidavit that clarifies your intentions for the money."
"And then Rova has to use the money for Jordan's benefit?" he asked. "It's that easy?"
I dared to take a bite of oatmeal. "You know it's not going to be that easy."
Steve twisted his green trucker cap around. "I didn't think so."
"Even if we can clear up the disposition of the money, there isn't actually any money left. Your ex-wife invested it all in the Parlour (BNF). Robin will request an audit of the beauty salon to see if there's any way to reclaim the liquid assets, but I wouldn't hold out hope. My guess is that the Parlour isn't making much money."
"I know, I've seen Rova's haircuts," Steve said miserably. "But if all that money is gone, how am I going to take care of my son? If the Parlour goes out of business, then Rova won't be able to support him either."
I wished I had something better to tell him. Don Tuthery was going to fight tooth and nail to wring every penny out of Steve, but Robin would fight with just as many teeth and just as many nails to protect her client. Neither outcome would help the young boy, however.
Before I could answer, Esther let out a raucous shriek and rushed toward the front door. A thin, bearded wizard with a sky-blue robe and pointed hat sauntered into the diner. He held a crooked staff and looked as if he had just come from a Gandalf impersonators' convention. He froze as the harpy hurtled toward him.
"You've got a lot of nerve—how dare you come back here!" She whipped out the bad-luck charm on its chain and dangled it in front of his face. "You take this back—lift the curse now, or I'll never serve you coffee again!"
The wizard spun about and fled into the street, his robes flapping, his pointy hat flying off his head (fortunately he had tied it with a string under his bearded chin, so it simply bounced on his shoulders as he ran). Everyone in the diner watched the show.
Esther remained in the doorway, her feathers bristling, bird-bright eyes flicking back and forth in search of prey. Finally, needing to damage something, she shredded the flyer for the Worldwide Horror Convention. The diner patrons hunched down in their booths, intent on their breakfasts and avoiding her gaze. The werewolf customer, who still had an empty cup of coffee, set his mug down and drank from his water glass instead.
When Esther stormed over, I said, "That was Glenn the wizard, I take it?"
"Yes, that was him! Why did you just sit here? Why didn't you go demand satisfaction for your client?"
"You chased him away before I had the chance."
She flared up. "Now you're blaming me?"
"It might have been a good opportunity to get his name and phone number."
"Go after him!" She jabbed one of her feathered arms toward the door, but by now the wizard had fled up the street and disappeared around a corner. "Run!"
"We're called the walking dead, not the sprinting dead."
"I don't like the tone of your voice, Chambeaux."
A long black limousine pulled up outside the diner, and Esther's mood changed like a ricocheting pinball. "Oh, he's here!" She flung off her apron, ran to the time clock, and punched out. "I'm done with my shift, Albert. Handle the orders yourself."
There were no other waitresses, and I didn't think the lethargic ghoul was in any condition to take care of himself, much less a diner full of customers.
The limo driver was a small hunchbacked lab assistant who had been downsized from a research lab and now worked as a hired driver. He wore a specially fitted tux and a black cap, but he was so short he sank down in the seat and had to reach up to grab the steering wheel. He could barely see over the dashboard.
The back door of the limo opened to reveal a very large man inside, with rolls of discolored flesh, bloated cheeks, and sunken eyes. He squirmed, as if he couldn't find a comfortable position.
As Esther flounced to the diner door, she looked momentarily cheerful. I said in amazement, "Tony Cralo is your boyfriend?"
"Something wrong with that?"
"No . . . just surprised, that's all," I said.
"He's fat and disgusting, he outgasses constantly, and he treats me like shit. He's just after a piece of tail feathers. But he's rich, so I'm willing to overlook his flaws."
"I remember true love," Steve said with a sigh. "It doesn't last."
Esther turned to him. "I know that, and I intend to milk this relationship for everything I can in the meantime."
She left the diner and bounded into the backseat of the limo. Tony Cralo opened his pudgy zombie arms to embrace her. She was chirpy and cheerful for a few moments, then started to rail at him for being late as the limo door closed. The hunchbacked driver pulled away.
With Esther gone, everyone in the diner heaved a sigh of relief and began to enjoy their meals. Steve had hardly touched his. He swung off the stool. "Gotta go. A delivery to make, a shipment of eggs to some Hairball werewolf—specially fertilized under the full moon or something. For his cockatrice coops."
"Rusty?" I asked. "Mind if I ride along? He's a client of mine."
"Sure, buddy. Always happy to have someone riding shotgun."
I left my oatmeal mostly untouched as well. We went out to Steve's truck.
CHAPTER 34
As I rode in the passenger seat of Steve's delivery truck, he made conversation about how he drove a big eighteen-wheeler when he was still alive. "Used to make long hauls from Cincinnati to Livingston, Montana, and all points in between. I enjoyed the big rig—spacious and powerful. It's like sitting in a house while the rest of the world rolls by." When he grinned at me, I noticed his teeth were still in good condition. "All that driving gives you time to think, listen to music, listen to an audiobook, or just watch the scenery, although the interstate freeways aren't usually scenic routes. They're Mileage Disposal Units, designed for getting rid of miles between one place and another. Drop off a load, pick up a new one, then head back."
He drove around the outskirts of the Quarter, taking backstreet routes even I had never seen before. As a delivery truck driver, he had to know the town inside and out. On his route, many of the special deliveries went to unmarked doors and dark alleys, dungeon entertainment centers, and any number of oddball shops. Steve just looked at the clipboard and address, and he did his job.
"When you're driving across Nebraska with the cruise control on, you spend your time thinking, and it's family you think about," Steve continued. "While I was still married to Rova, I'd worry about how I needed to be a better husband, spend more time at home, create a good family environment. I always put a photo of Rova and Jordan right there on the dashboard. I remember one time, in a horrific blizzard in the middle of North Dakota, the winds were howling and I couldn't see a thing. The only way I got through that was by staring and staring at that photo while I drove." Steve kept one hand on the steering wheel while punctuating his story with the other.
"If the blizzard was so bad, shouldn't you have been looking at the road?"
"Nah." He gave a dismissive wave. "Out there, the interstate's perfectly straight."
We arrived at Rusty's house with the shipment of eggs. His neighbor across the street was having a forlorn-looking garage sale, but there weren't many customers. I spotted some children's clothing, a used chemistry set, a stuffed moose head (no home should be without one), embalming supplies and equipment, and a complete set of hazmat suits in a range of sizes to fit the entire family.
A dozen cars were haphazardly parked in front of Rusty's house, three motorcycles, even a go-kart. Full-furred werewolves milled about in the front yard, going in and out through the front door. Loud music pounded from inside the house. A beer keg sat in a large tub filled with ice; the werewolves filled red plastic cups with beer from the tap.
"Need help carrying the eggs in?" I asked Steve. "I'll be your wingman."
We each took a crate of eggs from the back of the truck and carried it toward the door. The Hairballs didn't pay much attention to us.
"Got a delivery for someone named Rusty!" Steve announced.
At the kitchen table, two werewolves strained and growled in an arm-wrestling match. In a big pot on the stove, bratwursts were boiling in beer. Steve and I set the eggs on the counter. I didn't see Rusty, but I heard cheers and wolf whistles from the living room. Steve took his delivery receipt and clipboard and followed me down the hall.
The coffee table had been pushed aside, and folding chairs were set up in a circle. In the middle of the room, a werewolf bitch swayed and danced, dressed in little more than a filmy negligee. I recognized Cinnamon, one of the ladies from the Full Moon Brothel. Werewolf men hunched on the edges of the folding chairs, howling, barking, their eyes riveted as Cinnamon did a slow striptease. Furguson sat among them, intent on the stripper, but squirming on the chair, as if he hadn't taken his werewolf Ritalin that day.
Near him, I saw scalped-but-recovering Rusty sitting in his own lounge chair. He wore a red bandanna wrapped around his damaged head and didn't seem interested in Cinnamon's performance. (More evidence of the impotence tattoo.)
Furguson's foot bumped a cup and spilled beer on the carpet, much to his embarrassment. Rusty groaned. "Furguson, watch what you're doing! Now I have to get Larry to clean that up! Larry—janitorial duties! Now!"
Larry, the former werewolf bodyguard, slinked into the room, carrying a bucket and some napkins. "Sure thing, Rusty. I'll clean it right up." He had a trash sack and had obviously been emptying wastebaskets. "Thanks for letting me come to the kegger. It's been a long time. I hope you'll invite me again and not hold my past against me."
"Depends on how useful you are," Rusty said.
Cinnamon kept dancing. When she saw me, she gave me a wink and a long, lascivious curl of her tongue around her muzzle. I had helped the brothel on another case, but she was working, and I didn't want to distract her. Steve was fascinated by the dance.
Noticing us, Rusty levered himself out of the chair and came over. "Mister Shamble, got any proof yet? Or do we proceed with the war as planned?" He sounded as if he preferred the latter alternative.
"Not just yet." I gestured back toward Cinnamon. "Don't let me interrupt you watching the striptease."
Rusty grumbled. "No interest whatsoever. Doesn't do me any damn good."
"Sorry about that. I heard about the tattoo."
When Larry finished dabbing up the spilled beer with a wad of napkins, Rusty lashed out at him. "Don't forget to take out the kitchen trash—and the toilets could use scrubbing. See if anybody's barfed in them yet."
When Larry bristled at the lowly task, I remembered how fearsome he had once been—a werewolf hit man who could have torn me limb from limb. Now he seemed like a whipped cur. Seeing his reluctance, Rusty snorted, "If you can kiss Harvey Jekyll's ass, you can clean some damn toilets."
Larry controlled himself and went off to do his work.
Rusty led us into the hall, where it was less crowded, since everyone else wanted to watch Cinnamon. He flexed his heavily furred left bicep and nudged the hairs aside so I could see the intricate webwork of his tattoo. The impressive design showed a fanged dragon curled around a fortress tower with flames all around, lightning bolts striking.
"It does look important," I said.
"Not funny, Mister Shamble. I should have killed Antoine Stickler and burned down the Voodoo Tattoo parlor."
I thought of Antoine's special dolls that catered to specific customers and/or their enemies. If they had all gone up in smoke . . . "That could have had some bad consequences."
"I knew where to place the blame," Rusty said. "Stupid Jamaican vampire . . . He's just too mellow for his own good, said he was sorry. To make up for it, he did gang tattoos for every single Hairball, no charge. Soon as the Monthlies knew we were getting gang tats, then they had to get their own, but at least Stickler charged the Monthlies for theirs. Seems he's the only one who came out ahead through all this."
Werewolves came in and out of the backyard smoking cigarettes. On the porch, someone had fired up a charcoal grill, and another werewolf took the pot of bratwursts outside to begin cooking them for lunch.
Steve waved his clipboard. "Need a signature here, sir, then we'll be on our way. I delivered the eggs you wanted."
"And they're all fertilized?" Rusty asked. "Laid by chickens under the light of a full moon?"
Steve shrugged. "That's what the manifest says. Not like I was there for the actual egg drop. I'm just the truck driver."
"You vouch for this guy, Mister Shamble?" Rusty asked me.
I also shrugged. "I don't know a thing about the eggs, but I can tell you Steve is a decent, hardworking man. He wouldn't be trying to cheat you."
"Good enough for me." Before he could sign the clipboard, though, an outcry came from the back, a squawking and shrieking racket from the cockatrice coops, then another loud commotion, followed by shouts.
Rusty was already moving toward the back door. "You stay away from them dark coops if you know what's good for you!" He bolted into the yard, letting the screen door slam behind him. Most of the werewolves headed out of the living room to see what was happening in back. Even with the diminished audience, Cinnamon continued her dance, flailing her colorful scarves, using her claws to shred them and then throwing the strands in the air.
Through the kitchen window, Steve and I watched the burly werewolves throw a blanket around a man-sized object—something so heavy it required three of them to lift. The werewolf by the grill turned the bratwursts, not at all disturbed.
Rusty came back into the house, so angry his fur bristled, and I suggested it might be best if we left quickly, but Steve was diligent. "I still need my signature—the boss insists."
I stopped Rusty before he could go into full warpath mode. "What happened? Anything I can help with?"
"We take care of our own—and we got some violence to prepare. This is our full rumble planning meeting."
"I thought it was a kegger."
"We do our best planning when we get good and drunk." Behind Rusty, through the kitchen window, I watched werewolves carry the heavy man-sized object over to the garage. "Better get out of here, Mister Shamble. Nothing you need to see."
Rusty signed Steve's clipboard, and we headed out the front door, leaving the Hairballs to their strategy session against the Monthlies.
Before we drove off, Steve stopped to peruse the items in the garage sale. A tear came to his eye when he found a baseball mitt and a softball, which he bought for five dollars. "Someday, I'm going to give these to Jordan."
CHAPTER 35
The following day I stopped by the Parlour (BNF), even though I had no appointment; I didn't intend to have services rendered anyway. Since Rova Halsted knew who I was now, I doubted I'd receive a friendly reception.
I looked presentable, for the most part. Sheyenne had gotten my jacket dry-cleaned, and most of the bloodstains from the hospital were gone. The dry cleaner wanted to repair the bullet holes, or at least replace the black thread with less-visible tan stitches, but Sheyenne insisted that I liked the jacket exactly the way it was. (And she was right.)
I was the only customer in the Parlour. Harriet sat in one of the beautician's chairs covered with a smock, her tresses spread out like a lion's mane. Foil strips had been wrapped around some of her locks. Her hair was wet and dark, smeared with something that was either conditioner or the remnants of an oil slick. A kitchen timer ticked at her side. Rova bent over her, using small scissors to trim the hair on the backs of Harriet's hands.
Seeing me, the bearded lady lifted one hand to wave. Rova glanced up and turned frosty. "What are you doing here?"
Harriet frowned. "Rova, please stop being rude! Remember what I told you about customer service."
"He's not a customer. He's working with my ex."
"Among other things," I said. "I'm also working with Harriet's husband on an unrelated matter."
"We don't take sides here," Harriet said. "Archibald doesn't need a friendly manner on the job because his customers are already dead, but ours are alive, or at least ambulatory. If we want to grow this business, we need to treat everyone well." Harriet lifted her other hand, nudging Rova to get back to work. She looked at me, batting eyelashes the size of palm fronds. "We had a slow morning, Mister Chambeaux, so that's why I'm in the chair. Rova wanted to practice her art, since customers have complained."
"They don't know what they want," Rova said.
"The customer is always right," Harriet chided again, then spoke to me. "I'm demonstrating faith in my business partner. She'll give me a facial—a facial trim—and try out new techniques on my hair. Besides, this way we can do the perming and shampooing and conditioning without clogging my drains at home. Archibald gets testy when I mess up his workshop, even though it's technically part of our bedroom and bath."
"The Emporium promised I would have the . . . materials he requested by the end of the day," I said. "I'll deliver them personally."
"Good! He's busy planning for the body-building competition, but he is obsessed with that new hair-restoration formula. He thinks I care that he's bald! Sweet, precious man. At least he doesn't have to worry about hair all the time. Ah, sometimes that would make life so much easier. But Archibald thinks his new tonic will make us all rich, and there's no talking sense into him. He's a man in a workshop, one project after another after another . . . never finishes one thing before he moves on to something else."
Rova used a ceramic flat iron on a large swatch of Harriet's hair, flattening all semblance of curls. The locks did not look so much styled as crushed.
I decided it was time to be up front. "I actually hoped I could have a word with Ms. Halsted."
After seeing how much Steve cared for his son and what an all-around good man he was, I hoped—with foolish optimism, I know—that I could crack the ice. Regardless of how hard Robin fought for her client, I did not expect Don Tuthery to back down. Lawyers in divorce and child-custody cases were not known for easing tensions or finding resolutions that satisfied all parties. With Steve and Rova at such an impasse, by the time the mess was finished, Jordan might be in college, scarred from growing up under such bitter circumstances. It didn't matter who won.
Rova's nostrils flared, and not from the fumes of the hair chemicals. "You'd better leave, Mister Chambeaux. My lawyer gave me strict instructions that I'm not to have any direct contact with Steve's attorney."
"I'm not the attorney. I'm a private detective and a friend of Steve's."
"How are you a friend of Steve's? He didn't know any private detectives."
"We came out of the grave on the same night. That sort of thing creates a special bond. He didn't ask me to talk with you. I'm here on my own initiative, to ask if we can find some way out of this without racking up more legal bills and making everyone more miserable."
"More miserable? My ex is a zombie, and he wants to play with my child! How sick is that?"
"He's just a father trying to make up for the time he should have spent with his son. Besides, he hasn't decomposed very much. Look at me—I'm a zombie, too. I'm well-preserved, I interact well with others."
Rova sounded disgusted. "It's not normal!"
"Do you think Jordan would rather have no father at all?"
"This is just Steve's way of trying to get back together. He never could accept the fact that I left him."
I shook my head. "I'm fairly sure getting back together is the last thing on his mind."
Rova sniffed. "I have no intention of letting my ex-husband see Jordan. He owes me child support. The divorce decree was perfectly clear in that."
The thing that seemed perfectly clear was that Rova didn't want to work out an equitable arrangement. Even if Steve did pay child support, she would still fight visitation rights. The problem wasn't about Steve being a zombie; it was her intention to hurt him by keeping him away from his son.
Talking with me made Rova more and more upset while she snipped off hunk after hunk of Harriet's hair, first trimming the back of her hand, then working her way up the arm. She grabbed a handful of thick locks covered with foil sheets and was about to cut them off when Harriet grabbed her hand.
"I don't need that much of a trim, Rova dear. You're letting your emotions get the best of you. I realize that not everyone's marriage can be as perfect as mine and Archibald's, but you must let the calm flow through you. The haircut is what matters. Feel the Zen of styling."
Rova breathed heavily, set the scissors down, and calmed herself.
"You'd better go, Mister Chambeaux," Harriet said in a stern maternal voice. "Rova needs to meditate and focus her energy before I let her work on my hair again."
"That is a good idea—on so many levels," I said.
Disappointed that I had accomplished nothing with my unofficial visit, I left the Parlour (BNF) and decided to focus on other cases. I wanted to make at least one client happy—even if it was only Esther the harpy.
CHAPTER 36
Without much trouble, I tracked down the identity and address of the Gandalf look-alike named Glenn. He ran a little shop next to a bakery, with a signboard that offered Wizard Services, rather vaguely defined as Palm reading, sports predictions, life coaching, general tutoring, and other services as requested.
When I entered his shop, Glenn sat at a small table playing solitaire and losing. His pointed hat sat on a shelf behind him, and his gray hair was matted into an odd shape. His long beard was thrown over his left shoulder, out of the way, so he could shuffle and deal his cards without restraint. I noticed he had several playing cards tucked up the sleeve, but he still managed to lose the game.
Assuming I was a potential customer, Glenn greeted me with great enthusiasm. He rose to his feet, grabbed his wide-brimmed hat, and placed it back on his head as part of his costume. "Welcome, friend! It's a magical world outside, but even more magical in here. Any problem you have can be solved through arcane means. I specialize in trinkets."
"Matter of fact, I'm here to inquire about a trinket." I fished out one of my Chambeaux & Deyer business cards. "I'm here on a case."
"Oh, a professional investigator?" His voice was thin and nasal, and rose a note at the end of each sentence in sudden wonderment, as if he preferred to use question marks rather than periods. His expression fell. "So, not a paying customer, then?"
"Just professional courtesy. But I do referrals."
"In that case, Mister Chambeaux . . ." Glenn the wizard produced one of his own business cards with a flourish, as if by magic (though I saw him pluck it from his sleeve). An ace of spades also fell onto the table.
After the wizard put a pot of tea on a hot plate, he sat on the other side of the table and listened to me in earnest. "You must have many exciting cases. Is this about a murder, a bank robbery? Perhaps a kidnapping or blackmail!"
"Not all of my cases are exciting." I thought of suggesting he pick up a copy of Death Warmed Over, but that would give him a distorted picture of my life as a zombie PI. Besides, it was bad form to try selling the book to someone my client wanted to sue for everything he was worth. I added, "We're both customers at the Ghoul's Diner."
The wizard perked up. "Oh, right—I've seen you there." His expression became troubled again. "Is the health department trying to shut down the place finally? Did a customer die from poisoning?"
"My client is actually Esther, the waitress. . . ."
He recoiled in horror. "Oh, I'm terribly sorry for you! Esther Pester, we call her—worst waitress in the world and the most unpleasant creature in the Unnatural Quarter. I don't understand why Albert doesn't get better help." The wizard raised his eyebrows. "Is Pester in some kind of trouble . . . I hope?"
"Actually, Glenn, you're the one who's in trouble. Esther hired us because you left her a bad-luck charm as a tip, and now her life is miserable because of it."
Glenn smiled. "Her life is miserable? How could she tell the difference?"
I tried to talk to him, man-to-man. "Look, Glenn, I don't have to tell you she can be one vindictive harpy. Once Esther gets her tail feathers in a knot, she doesn't let things go. Our research proves that the charm is indeed a bad-luck charm."
"So I left her a tip," Glenn said with a shrug. "Sure, it was a bad-luck charm, one of my most popular models. I've been working with the tattoo artist over at Voodoo Tattoo. We plan to sell his special dolls and my bad-luck charms together as a set—for when you really want to give your enemies a double whammy."
I kept my voice cold and hard. "Knowingly leaving my client a dangerous object can be considered intentional infliction of emotional distress. She'll sue you for damages, claim pain and suffering."
"And what about my pain and suffering for having to endure her service at the diner?" Glenn asked. "What about her customers? Have you ever been satisfied with her service? Could you swear to that in court?"
"Uh, no."
"Maybe I'll call you as a character witness."
"Let's not go there. If you lift the curse, maybe I could convince her to drop the suit."
Glenn looked stubborn and angry. "If Esther Pester testifies in court, don't you think any judge or jury would want to curse her themselves?"
"She isn't required to testify, Glenn." In fact, I was certain that Robin wouldn't let such an awful plaintiff speak a word in the jury's earshot.
Glenn's shoulders slumped. "All right, so I was annoyed. She messed up my order. She spilled coffee on my blue robes so I had to get them cleaned. She charged me for a dessert that she never served . . . and her voice just plain gives me a headache. Maybe I gave her a bad-luck charm, but the curse only works on people who accept the gift. She's as much at fault for being so greedy. She can't throw the charm away, it'll just reappear."
"She's already tried that. Isn't there some way you can cancel the curse?"
"No, the curse is nonrefundable, with the warranty attached to the recipient. But there is one thing. . . ." The wizard looked as if he intended to charge me for the solution—and I might have been willing to pay the price, just to be rid of Esther as a client—but Glenn relented and said, "Her bad luck will go away if she just gives it to another person. But I wouldn't expect a selfish soul like Esther to understand the concept of generosity."
"That's it? She can just hand it to someone?"
"They have to accept it."
The teakettle began to whistle on the hot plate, but I didn't plan to stay for a cup. Besides, I'm more of a coffee drinker. "Thank you, Glenn. If this all works out, I won't need to see you again."
"Don't forget about the professional referrals," he said, then shuffled his cards, ready to play another game of solitaire.
"I'll have my administrator put you on our contacts list."
As I headed out the door, Glenn called after me. "You know she deserves it, Mister Chambeaux. Esther Pester is your client, and you have to do what you have to do, but is it absolutely necessary to tell her the solution . . . right away?"
I didn't have plans to see Esther in the immediate future. "I suppose I could hold off for a while."
CHAPTER 37
Showing tremendous faith (or foolish optimism) in Robin and Sheyenne's abilities to sort swiftly through tons of loose papers, McGoo arrived that afternoon, grinning. "Found any discrepancies in the Cralo paperwork yet?"
I thought he was being a tad unrealistic. "McGoo, you brought them a dump truck filled with receipts and files!"
Now he looked sheepish. "I'm just anxious for answers. That vampire transplant heart might be only the tip of the iceberg. Vampire organs harvested from a body, werewolf scalps taken. Could be a really big case."
"Were you able to prove that the transplant heart came from Ben Willard?"
"No," McGoo said. "As obsessed as that administrator was with his paperwork, he didn't keep very good records. Apparently, that vampire heart fell off a truck."
"One of Tony Cralo's trucks?" I asked.
Sheyenne flitted in from the conference room, carrying several folders. "Of course we found something—and it's as fishy as a swamp creature's underwear."
Robin was more professional in her metaphors as she followed Sheyenne. "The paperwork does give us the big picture, and a lot of the pieces just don't fit together properly. The Emporium's admin department did such a good job of doctoring the details that this can't be explained as accidental bookkeeping errors." She looked exhausted but relieved, and didn't even seem to resent McGoo for putting her through so much work. "After collating the names of the donors, the dates of donations, and the body parts obtained, we had to break down the Emporium's inventory code system in order to determine who gave what and when."
Sheyenne held up a stack of folders in her spectral hands. "I visited the Emporium myself and did some personal shopping. I flitted up and down the aisles and kept track of the SKUs, then I backtracked them and extracted the specifics from these records."
"That sounds great!" McGoo grinned. "Now, can we do it one more time in English?"
Sheyenne opened the folders and spread out the papers to show us highlighted code numbers and her own sheet of translations. Robin pointed to names on the donation forms, then the code numbers from the warehouse shelves. "The discrepancies raise obvious questions. For instance, according to these records, the same person donated more than two arms or legs. And although it's not impossible in the Quarter for someone to have more than two eyes, this particular person gave up six, and then stopped by the following week to donate not one liver, but two."
"As a former med student," Sheyenne said, "that man is a specimen I would have liked to see in the lab."
"Definitely fishy," McGoo said. "I'd say that's enough for a warrant to do further digging."
Robin said, "I can provide a summary of the current state of body-snatching laws." She picked up several sheets of paper. "I doubt these donors exist at all—they're probably dummy names. Their body-parts inventory database manager was too lazy to create as many fake donors as they needed."
Sheyenne smiled at McGoo. "So, that means we can submit an invoice to the police department for payment as consultants, right?"
Awkwardly changing the subject, McGoo said, "Hey, here's one I just heard—what has a cat's head, a dog's tail, and brains all over its face?" He looked around, hoping someone might venture a guess, but we all refused. "A zombie in a pet store!"
After a punch line like that, I knew I had to leave. "I'm going to the Spare Parts Emporium now. I've got business of my own there."
McGoo sounded surprised. "What kind of business?"
"I need to pick up a new brain."
He scowled. "Come on, the joke wasn't that bad."
"It's not for me, it's for a friend."
"Sure it is." McGoo gave a sage nod. "They always are."
On the way to Cralo's Emporium, I passed the Voodoo Tattoo parlor. Antoine Stickler was alone in the shop, sitting at a worktable and playing with arms, legs, and heads—not life-sized ones, just small doll pieces. He was sewing clothes that might have belonged on Ken dolls, but his figures had thick tufts of hair covering their exposed skin and fingernails painted black. Another set of dolls, already finished, looked like humans; two were covered with carefully reproduced tattoos. One had a goatee, one had a thick DA hairstyle. As soon as I recognized the Scratch and Sniff action figures, I figured the others must represent Monthly werewolves as well.
Antoine looked up. "Change your mind about getting yourself a tattoo? I can liven up the pallor."
"My girlfriend and I are still deciding." Sheyenne could always get a design inked onto the lifelike inflatable doll she sometimes inhabited for me. And for her, I'd even get a Cupid tattoo (fortunately, she hadn't asked me to do that).
Antoine grinned. "So right, you've got to keep that girl of yours happy. I can always do couples tattoos. His and Hers Forever, either with permanent ink or temporary tats, whichever you prefer. A lot of guys come in here asking for Forever, but then they slip me fifty bucks to make it washable."
"The couples tattoo thing would be a problem, since my girlfriend's a ghost."
"Hmm. Let me think on it a bit," Antoine said as he continued to work on his effigy dolls. "If I figure that out, it could be a new market in the Quarter."
"Are you making an entire doll series of Monthlies and Hairballs?" I said. "A collector's set?"
"Commissioned work." Antoine tossed his dreadlocks and bent back to his project. "The Monthlies wanted voodoo dolls of every Hairball, and as soon as the Hairballs found out about it, they got angry and offered to buy 'em back, at twice the price. At the same time, they commissioned voodoo dolls of every Monthly."
"Sounds like a booming business. And you're getting work from both sides."
"Yeah, but all this negativity bothers me. I agreed to make the dolls only on the condition that they stay here for safekeeping. I'll put them on display as samples of my work. A voodoo portfolio." He looked up at me. "But those two gangs don't know I put these around all the dolls." Antoine picked up a handful of brownish-purple pods. "Balm of Gilead buds. 'Hoodoo aspirin,' white folk call it. Soothes the pain of arguments and quarrels, brings peace to the home."
"Does it work?"
"Can't hurt."
Maybe Esther the harpy would be interested in using some of that on Glenn's bad-luck charm. "I was at Rusty's house yesterday," I said. "He said that you gave all the Hairballs their gang tattoos for free."
"Least I could do after the important-impotent mix-up. I also did gang tats for the Monthlies—not the kind with magic healing, though, like Scratch and Sniff have. Too difficult—cost-prohibitive, you know. At least I got 'em all marked, just in case."
"Just in case what?"
Antoine blinked up at me. "In case the balm of Gilead doesn't work."
I said, "They're dead set on having this gang rumble during the full moon as a vendetta, even though nobody proved it was Monthlies doing the scalpings."
Antoine seemed maddeningly calm. "It'll work out."
I got down to the real reason I had stopped by. "A vampire was murdered a couple of days ago. He was handcuffed to a bed and all of his organs removed. I think he was a customer of yours."
"I get quite a few vamps—repeat customers." A troubled look crossed his face. "Not many get murdered, though."
"This one was named Ben Willard. He went out clubbing a lot."
"Oh, Mister Willard! Came in every week to get fresh tats. Always planned a big Friday night. Bad mojo, though . . . he liked it rough. Not surprised to hear he was killed."
"You saw this coming?"
"Vamps have an old saying: 'You go out in the sunlight, you're going to get burned.' A lot of vamps, especially the newly turned ones, have more testosterone than they have bloodlust. They get disappointed in sex because normal human partners are far too wimpy. So they want me to make them look edgy and tough to help them pick up some big bad creature that's just as horny as they are." Antoine went to his book, flipped to the entry for the next Friday, and drew a line through Ben Willard's standing appointment. "You change your mind about a tat of your own, I got an opening now."
"I'll think about it," I repeated.
Antoine finished his dolls, tucking a balm of Gilead bud into the folds of their clothes, adding a dab of paint to the faces. The likenesses were really quite remarkable. He smiled at his handiwork. "Turn that frown upside down, Mister Chambeaux," he chided. "Don't worry, be happy."
CHAPTER 38
Back at the Spare Parts Emporium, I made my way to Xandy Huff's office in the rear of the cavernous warehouse; I was determined to get the right parts for Dr. Victor this time.
If need be, I could spoil the floor manager's day by telling him we'd found interesting discrepancies in the mountains of Cralo paperwork. I was sure Mr. Huff had put in a great deal of effort gathering and copying all that information, boxing it up in such a reverently disorganized fashion, never expecting anyone to make sense out of the storm of numbers and forms.
When I came in, Huff was at his desk eating a foul-smelling liverwurst sandwich. I've been inside dank crypts, waded through sewers beneath the Quarter, smelled the stench of many a rotting corpse, even sat in a Jacuzzi with an outgassing fat zombie. But the smell or taste of liverwurst turns my stomach.
Huff wiped his mouth with a napkin, set the sandwich aside. "I knew you'd come. I suppose you're getting impatient for your brain."
"My client certainly is. And when my clients are impatient, I'm impatient."
Huff opened the bottom drawer of his desk and removed a large box, gift-wrapped in black paper and tied with a silver ribbon. "Mister Cralo told me to give you this with his highest regards. It's a fresh brain, Grade A, just came in this morning. I'm sure your client will be satisfied." He nodded toward the box. "Do you want to inspect it? We can have Customer Service rewrap it for you, if you like."
"I wouldn't know the difference between a good brain and a bad one," I said.
"I thought zombies were connoisseurs."
"Not me. I'm just a regular guy. Cheeseburgers and fries. Never acquired a taste for brains. But the spleen you sent over yesterday was damaged in shipping. I'll need another one of those as well."
The manager sighed, controlled an angry outburst by doing some sort of silent meditation, or just cursing under his breath, and he wrote out a slip for a complimentary spleen, which he signed. "So, that completes our business together? I don't need to see you again?"
"Not unless I come back to do some shopping," I said.
Huff looked relieved. "I'll let Mister Cralo know you're satisfied." I realized that, as floor manager, he might suffer extreme repercussions himself if customers lodged too many complaints.
Carrying the gift-wrapped brain in a box, I went to the front counter, exchanged the coupon for a new spleen, and walked out of the Emporium with my prizes in hand. I kept a close eye on the rest of the inventory, particularly noticing unmarked boxes on the high shelves.
As I walked out into the sloppy parking lot, I wondered who had donated this particular brain and why he or she had decided it was no longer a useful organ. Now that Sheyenne and Robin had uncovered so much suspicious paperwork, I found myself questioning the provenance of all these items for sale, down to the amputated little-piggy toes and vermiform appendices in the bargain bins.
But Archibald Victor didn't want me to make his case public, and since McGoo worked with the coroner, I was honor bound to keep my client's complaint off the books. A mad scientist shouldn't need to be ashamed about his laboratory activities, but Archibald had not yet come out of the closet....
I made my way over to the trailer park with spleen and brain in hand. I found the Victors' address and rapped on the door. Since it was only midafternoon, Harriet should still be at the Parlour (BNF), preparing unnaturals for a night on the town (or, more likely, fixing dissatisfied clients who had been afflicted with Rova's haircuts).
The pallid coroner yanked open the door. He looked flustered in his lab coat; goggles hung at his throat, and black neoprene gloves limited his dexterity as he fumbled to adjust his toupee. "Yes, what is it?" Then he recognized me. "Mister Chambeaux! Of course, you said you were coming."
"I have a delivery for you, Dr. Victor." I held out the gift-wrapped box and the packaged spleen. "From Tony Cralo himself, along with his full apology. His floor manager assured me this is their best brain, and he wanted you to be satisfied."
Archibald was so excited he hopped on both feet as he let me inside the small trailer. Harriet had set out two bubbling pots of potpourri in the tiny kitchen, but the rest of the trailer reeked of odd chemical mixtures. In his home lab/workshop, Archibald had a setup of bubbling chemicals, liquid-filled flasks, electrical arcs that shocked downward from metal probes.
At the far end of the trailer, on a model-building table, I saw the hulking form of one of the bodies he was assembling piece by piece. Spools of suture thread, safety pins, and duct tape were strewn haphazardly around the model, but the coroner didn't seem to be working on it at all. He was far more interested in his chemistry experiments on the bathroom countertop. I remembered Harriet talking about how he obsessed on his numerous hobbies, like any man with a workbench.
Archibald yanked off his thick black gloves and unwrapped the spleen first, letting out a sigh of satisfaction. "Finally! This organ will do just fine."
As he started to undo the silver ribbon on the gift-wrapped brain, I wandered farther into the room to look at his bubbling experiments. "What are you working on, Dr. Victor? Is it the super hair-tonic project?"
With a look of horror, he whirled. "Please don't touch anything! In fact, I'd rather you not look at all. This is a very important bit of research, and it could be worth millions."
"I don't know anything about chemistry," I said, "or about hair tonics."
"Harriet shouldn't have mentioned my little project to you at all." Out of politeness, I turned from the mysterious and colorful liquids. "I was just about to achieve a breakthrough, but I'll have to set that aside for the body-building competition. There are a lot of entries in the male model category."
He removed the top of the gift box and lifted out a transparent container filled with clear preservation fluid and a many-contoured, fresh-looking brain that reminded me of an inexpertly swirled pile of spray whipped cream.
"So, we can wrap up the case, Dr. Victor?" I asked. "I want to extricate your name from any dealings with the Spare Parts Emporium. I happen to know there's an ongoing investigation, and you don't want to be involved in that."
He looked at me, and his toupee started to slip. "I'm not surprised. Many other clients have been affected by the Emporium's poor quality control." He held the brain jar up to the light, and his already-round eyes widened further in anger. I thought he was going to dash the jar on the floor. "Unbelievable! After all my complaints, they give me this as a replacement?" He set the brain down and scurried to a small rolltop desk, which he flung open, ransacked through the papers, and came out with his receipt. "Look here. My order form plainly indicates I require a human brain!" He jabbed his fingers at the jar. "Look at that one!"
I looked, but I didn't see what he saw.
In exasperation, Archibald said, "That's not a human brain, Mister Chambeaux! It's not even compatible with the spinal column! And how am I supposed to attach that medulla oblongata?" He looked at me, furious.
"I'll take your word for it," I said.
"It's a troll brain!" Indignant, Archibald thrust the brain jar back in the box, slammed the lid back on top, and shoved it into my hands. "This is absurd. Tell Mister Cralo that I am not satisfied, and I expect to receive the merchandise I purchased!" He shook his head. "I doubt it'll arrive in time now. Maybe I should just keep working with my chemistry set." He made a loud raspberry sound that reminded me of Tony Cralo's outgassing. "Troll brains! Who in the world needs a troll brain?"
"Trolls, I suppose," I said. I felt a sudden twinge of dread, suspecting that I might, after all, have some idea where they had gotten that brain.
Leaving him with the spleen, I took the box with me and left.
CHAPTER 39
I headed back to the Spare Parts Emporium, fuming and tired of being given the runaround. What was I supposed to do with yet another defective brain?
I suspected there was more going on here than the inept shelving of bodily organs. The plot had thickened, like gravy made with too much cornstarch. I vowed to find out where this inconvenient troll brain had come from, and I knew where to start asking.
It was a long walk on my dead feet back to the Spare Parts Emporium, but it was a beautiful gloomy afternoon in the Quarter. Gray skies grew dimmer as the sun set and dusk gathered on the streets. I walked past a Talbot & Knowles Blood Bar—a place filled with polished glass, flecked countertops, and checkerboard linoleum, with a vampire working behind the counter dressed as a soda jerk from an old fifties diner. I walked past an Ucky Dogs corner hot dog stand, which was attended by a lonely reptilian vendor; the dogs looked like something even a ghoul would refuse to eat.
From his stand on the corner, a soothsayer called to anyone passing by. "I'll tell the future for five dollars!" He had a stainless-steel mixing bowl in front of him, in which he stirred various entrails, lifting them up so that they dripped and steamed. His lack of customers might have given some sense about the accuracy of his predictions, or maybe five dollars per foretelling was higher than the going rate in the Quarter. He fondled the lobes of a liver and cried, "Beware! The end of the world is coming!"
I stopped. "What's the time frame on that?"
The soothsayer held up blood- and slime-covered hands, grinning at me. "No, it's just a general statement. Nothing imminent."
"I won't worry about it, then." I tipped my fedora to him and walked on.
The soothsayer raised his voice and called out again, "The end of the world is coming! Limited-time offer to spend your five dollars and get a foretelling!" If the world was indeed coming to an end, I could think of better ways to spend five bucks.
For the Hairballs and Monthlies, though, Armageddon might indeed be imminent, since tomorrow the full moon would shine and the werewolf gangs would go to war—unless I could prove someone else was responsible for the scalpings.
Clasping the arm of Hirsute, Miranda Jekyll strolled along the boulevard, dressed to kill as always (or at least dressed to hunt). The two looked as if they were on their way to the opera. (The Phantom was holding a stage revival, casting himself in the starring role; he claimed it was for added veracity, but critics had panned the show, since the real Phantom couldn't sing.) Next to Miranda, Hirsute was protective and confident, with the constant habit of tossing his thick head of hair as if to demonstrate that his locks were more desirable than any full-time werewolf's. I wondered if he had an alibi for the scalpings. . . .
Miranda and Hirsute paused to look in the window of a jewelry store, admiring the diamond earrings, necklaces, and cursed amulets. She snuggled close to her ultra-manly companion.
A battered Chevette drove up, coughing and puttering as if competing with Robin's Pro Bono Mobile for Worst Muffler Ever. When the Chevette slowed and the passenger window rolled down, I recognized Furguson riding shotgun as another full-furred werewolf drove, hunched over a tiny-diameter ghetto steering wheel made of welded chain links. Miranda and Hirsute turned to scowl at the noisy muffler.
Furguson yelled out the window, "Hey, Monthlies! Try some of our new cologne!"
He threw a liquid-filled balloon, which struck Miranda full in the chest. When it burst, brownish-gray goop splattered her. Miranda shrieked as the stain began to fume and smoke.
The Hairballs howled with laughter, and Furguson threw three more balloons in quick succession (but he was so clumsy he missed with two of them). The third balloon grazed Hirsute's chest, striking him only because he had such massive shoulders. The noxious substance covered both of them.
With a roar of rage, Hirsute bounded after the Chevette. Furguson yelped and banged on the car door, and the driver floored the accelerator. The Chevette strained to pick up speed, puttering along. Hirsute loped after it, ready to tear the back bumper off, but the car accelerated just enough to stay out of reach of his grasping hands.
I hurried over to Miranda, who pawed at the liquid staining her dress. The smell was overpowering, and I could guess what it was: Furguson had filled the balloons with cockatrice excrement. Miranda peeled off her stylish accessory scarf and dabbed at the slime. "How dare they? I can never forgive this!"
Hirsute bounded back to rescue her. "Are you all right, Miranda?" He looked as if he wanted to rip her dress off, just to prevent the stain from the monster feces from touching her skin, but he controlled himself.
"I'm stained and smelly. And very, very angry." She looked at me. "Mister Chambeaux, you are a witness. This is an outright assault!"
"I vow revenge upon them!" Hirsute said.
"I can identify the perpetrator if you want to file charges," I said.
Furguson's actions were just plain irrational and dangerous, actively trying to incite more violence against the Monthlies. He had already attacked Scratch and Sniff on their choppers. He was a loose cannon, or at least a loose BB gun. Was he provoking this rumble, just to get back in Rusty's good graces? Could he have stunned and scalped his own uncle—and the other Hairball victims—in order to rile up the gang? That didn't make sense . . . but then, Furguson didn't make a lot of sense.
When she turned to me, Miranda's face showed such fury that I could imagine what had turned Harvey Jekyll into such an evil man. "I appreciate the suggestion, sweetheart, but Hirsute and I will pursue a more immediate alternative." She actually gnashed her teeth. "We intended to stay out of this dispute—gang rivalry has nothing to do with us. But after this . . ." In disgust she touched the cockatrice poop stains on her dress. "We are joining the full moon rumble tomorrow night."
CHAPTER 40
These days, I seemed to be spending more time at Cralo's Spare Parts Emporium than at my own cluttered desk. After I splashed through parking-lot puddles the size of lunar craters—there was no avoiding them—and entered the giant warehouse, I made my way, yet again, to the floor manager's office.
Xandy Huff's office door was closed, though. He must have known I'd received the wrong brain, so he ducked out early to avoid a vengeful zombie. Probably a good idea. Maybe he thought I'd just eat it and destroy the evidence.
Not wanting to keep the unacceptable brain longer than necessary, I marched toward the Customer Service and Returns window, where a thin ghoul sat on a stool; he looked surprised that I would bother him. Apparently few customers dared to request returns from Tony Cralo, but I wasn't intimidated by their complex bureaucracy. I'd had enough of this place.
I clunked the gift-wrapped box on the countertop. "I've returned a brain twice now for replacement, and I was promised satisfaction. My client ordered a human brain, and this is a troll brain. It took two tries to get the lungs right, three tries to get the spleen, and this is our fourth attempt to get a proper brain."
The ghoul blinked heavy-lidded eyes at me. "Do you have a receipt?"
"This was an exchange. Your floor manager, Mister Huff, gave this to me."
"Anyone could say that," said the customer service ghoul. "How do I know you didn't buy this from some other body shop?"
"It's your gift wrapping. Look at the box."
"It's similar to our gift wrapping." He still sounded skeptical. "You say someone gave this to you as an exchange?"
"Mister Huff did, as authorized by Tony Cralo himself."
The ghoul fumbled with the lid, opened the box, and withdrew the container. "This is a troll brain. Different item number."
As with so many businesses, the least-motivated and least-intelligent individuals were assigned to work the customer service desk. "I know it's a troll brain, I already told you that. I need a human brain."
"Troll brains go for a higher price. You can have this one, no extra charge." The customer service ghoul reached up to grasp the dangling string of a window shade, intending to close the window.
"I don't want a troll brain. I need to return this one and get the item my client ordered."
"Ten percent restocking fee. And you need to file a formal return request."
"Already did that," I said.
"Then you have to report to the floor manager."
"Already did that, too."
"Then I'm afraid you'll have to speak to Mister Cralo in person. He's in charge of our one hundred percent customer satisfaction guarantee."
"Already did that," I said, raising my voice.
Still trying to think of some other way to deflect me, the flustered customer service ghoul finally took the brain jar and said, "Thank you for your feedback. I'll be sure the matter gets looked into promptly. We'll send a replacement brain to the original address. Please allow up to six weeks for delivery."
Before I could argue or even demand a return receipt, the ghoul pulled down the shade and closed the customer service window. He set out a sign with a flat plastic clock, saying WILL R ETURN AT—Both of the clock hands had been snapped off.
I was ready to call in full police intervention and a company shutdown just to straighten out a simple order mix-up. On previous cases, I had faced a werewolf hit man, demon thugs, a vast genocide conspiracy against unnaturals, a Congressional act from a corrupt senator, a monster that had torn my arm off—not to mention the fact that I'd been murdered myself, as had my girlfriend. And I came through those cases all right. I was not about to be defeated by a maddening customer-satisfaction guarantee. This was a vendetta for me.
I lurched out of the Emporium and, with more determination than ever, set off toward the railroad bridge swathed in cool mist. The homeless monsters sat on the front porches of their cardboard boxes. The mangy ogre slouched on a pile of cinder blocks that had cracked under his weight; his knees were drawn up to his chest, and he stared forlornly at his hands. An exceptionally thin mummy and a tentacled demon roasted a scrawny goat carcass over a fire in a rusty barrel.
Inside the large coffin box he had claimed as his home, I found Larry lying on a deflated air mattress, his bare paws sticking out of the opening. He sat up, recognized me, and scrambled out. "Chambeaux! What are you doing here?"
"Following up on a lead," I said.
"Something to do with Rusty? Did you notice he's letting me hang out with him again? I'm going to go to the rumble tomorrow night, too! And if I do enough damage, he'll initiate me into the gang. I can even go to Voodoo Tattoo for my own Hairball tat. Things are looking up, Chambeaux."
"I'm checking on Tommy Underbridge," I said. "Have you seen him lately?"
Larry puffed his furry cheeks and blew out a long breath. "Nobody's seen that troll in two days. I think something bad happened to him." The werewolf brushed loose fur from his shirt. He appeared to be shedding. "Some of the others here think he just found a better bridge to live under, but Tommy loved it here, and he loved all of us." He gestured toward the fog-enshrouded superstructure. "What could be better than this?"
"I'm also afraid something bad might have happened to him."
"Think he was scalped, too?"
"Worse. I think he involuntarily donated his brain to commerce. But I can't prove it yet."
"It's those damned Monthlies," Larry growled. "They knew Tommy was helping us, and they must have gotten even with him. Oh, the fur is gonna fly tomorrow!"
"You know where the rumble is going to take place? I didn't get an invitation."
Larry's eyes narrowed. "They aren't handing out guest passes. It's a private matter—werewolf to werewolf."
"Rusty hired me to investigate it. I think I'd be welcome."
Larry looked calculating for a moment, then chuffed a laugh. "You know that old Smile Syndicate warehouse, the one flattened by an explosion a while back?"
"I'm familiar with it." I had been personally involved in the explosion itself, although Larry didn't need to know that.
"Warehouse district is the right place for a rumble, don't you think? Wide-open spaces, just the right part of town." He bobbed his muzzle up and down. "So better stay clear of there when the full moon rises tomorrow. Things are going to get ugly in the Quarter."
"It's always ugly in the Quarter," I said.
CHAPTER 41
The following morning Sheyenne brought me coffee at my desk and lingered. I enjoyed both her coffee and her lingering a great deal. "I miss you, Beaux. I'm starting to think you're married to your job."
"Not married to it, but it does seem to be a committed relationship." I looked up at her beautiful, semitransparent form and let out a long sigh. "I'm always glad you're here, Spooky. When these cases wrap up, I promise we'll find something exciting to do. First I have to get a breather."
"I don't have to breathe anymore," she reminded me. "And neither do you."
"Good point."
Just then, Esther stopped by to brighten our day. The harpy's plumage was ruffled, her eyes fiery—as usual—and she was dressed in her Ghoul's Diner uniform.
"I'm tired of waiting for results, especially when you're charging by the hour." When she saw the Emporium papers, folders, and notes strewn about my office she seemed mollified. "All this for my case? I didn't think it was that complicated."
"It's for another case, Esther," I said, which only annoyed her further.
"You're supposed to give priority to my problems. I thought we had an understanding."
I decided to lie. "I'm glad you're here—there's been a major break in your case. Robin, could you help me consult with a client, please?" I needed to get Esther away from the papers before she noticed we were digging into Tony Cralo's activities. (And also because I didn't want her to flap her wings in a tantrum and rearrange every scrap of paper that Robin had so painstakingly organized.)
Robin was much better at faking a professional smile than I was. Fortunately, it's easy for my undead face to remain expressionless when the situation calls for it. I explained—exaggerating the difficulty of my work—that I had tracked down Glenn the wizard and had a chat with him (I remembered just in time to use the word confront instead of chat).
"He admitted to giving you a bad-luck charm, and he's very sorry that he lost his patience with you. Even wizards have bad days. His pet canary died that morning, and I'm afraid he took out his grief on you."
I shook my head sadly. It was a good performance, worthy of fictionalizing in the next Shamble & Die mystery.
"I don't give a crap about his canary," she said. "If he brought it to the diner, we'd serve it up. Speaking of which, I have to get to work—and if I'm late punching in, I expect you to reimburse me for my lost wages."
Robin looked at the wall clock. Breakfast at the diner had been going at full steam for hours. "When were you supposed to start?"
"Ninety minutes ago, maybe more. So hurry up."
I said, "I'm surprised you keep your waitressing job. Isn't Tony Cralo taking care of you well enough?" The Ghoul's Diner, and all of Esther's customers, would be better off without her.
She snorted. "Sure, the fat zombie gives me baubles and shiny things, pampers me, takes me out to nice restaurants—but that's not enough. I'm thinking of my future. What if the relationship doesn't last?"
Sheyenne didn't like Esther (none of us did, but Robin and I covered it better). She said in an acid-sweet tone, "You must be worried he'll dump you for some other chick."
The harpy ruffled her feathers. "Dump me? You've got that backward. He's so disgusting I don't know how much more I can tolerate." She dangled the golden charm that refused to be discarded. "And all this bad luck is ruining my plans. What am I supposed to do? My life is in shambles, and my tips are down."
"You could try to be nice," Sheyenne said.
"How is that going to help anything?" Esther shrieked.
Robin gave her legal opinion. "We're in a gray area here. The bad-luck medallion was a tip, freely given and freely accepted, and because the wizard made no guarantees, implied or explicit, it'll be hard to impose damages. We could file a letter of complaint with the Better Business Bureau, which would then be available for Glenn's potential customers to see. As a long shot, we might succeed in getting him censured in magical professional organizations, and there would be a fine, but you'd probably receive very little in monetary damages."
"Damages?" Esther cocked her head. "I don't want damages, I want revenge! I'm stuck with this thing!"
I lied again. "Because he felt so sorry for what he did to you, Glenn told me how you can get rid of the bad-luck charm. It might help your situation."
She perked up. "Who do I need to kill? What sort of sacrifice do I have to make?"
"Glenn gave it to you, and you accepted it. The obvious solution is for you to give it away to someone else, who will also accept it."
"Give it away?" She clutched the medallion in her talons. "But it's mine! Mine!"
"It could be the only solution," Robin said.
"That's the stupidest thing I've ever heard." Annoyed, Esther spun about. "You've been no help at all."
When she left, all three of us let out a sigh of relief, even though two of us didn't need to breathe.
CHAPTER 42
By that afternoon, an uneasy anticipation was building in the air around the Unnatural Quarter. In any normal month, part-time werewolves would anticipate the rise of the full moon and revel in their temporary bestial freedom. A few months ago, I remembered how Miranda herself had been far more interested in the full-moon party with her friends than she was in the major crackdown on Jekyll Lifestyle Products and Necroceuticals.
This time, however, the two types of werewolves planned more than frenzied running and howling like hyperactive puppies on their first visit to a dog park.
Miranda and Hirsute stopped by our offices. "Just a few matters to wrap up, sweethearts. Legal specifics, dotting the i's and crossing the t's." She had a shine in her eyes, and she licked her lips more often than usual. Hirsute also had a simmering energy; his fists were clenched, and his arm muscles bulged far more than was appropriate.
"Ms. Jekyll, I finished the paperwork to convert your werewolf sanctuary into a not-for-profit corporation," Robin said. "A straightforward matter, and I'll have MLDW list it among the monster-supportive charity organizations." Sheyenne brought out the case folder as well as the notary stamp and record book, ready to formalize the documents so they could be filed with the clerk.
"Good to wrap that up—Hirsute and I plan to be busy this evening. The rumble is going to be quite spectacular. It might appear to be slumming, but those bastards asked for trouble. You saw the cockatrice shit all over me. Who's going to pay for the assault? Who'll pay for my dress? That was a brand-new Vera Wang—and not off the rack!"
Hirsute growled deep in his throat. "I'll make them pay, my dear."
Miranda giggled. "I know you will—you're so chivalrous it makes my blood run hot!" Then she snapped her attention back to Robin. "That brings up another subject. Hirsute is going to defend me during the gang violence, but we'll need some sort of disclaimer or hold-harmless document that clears me of legal responsibility if either of us inflicts damage upon those pathetic Hairballs. And I also want to be able to recoup any dental expenses I might sustain if, for instance, I chip a tooth while sinking my fangs into a victim."
"That won't be possible, Ms. Jekyll," Robin said in a warning voice. "My advice is to steer clear of the violence, for your own safety."
Miranda chuckled. "I'm not worried about my safety, sweetheart. I have Hirsute to protect me. I just don't want legal headaches afterward, when the mangy Hairballs turn out to be sore losers."
Robin shook her head. "I'd like to help you, Ms. Jekyll, but if you attend the rumble, you do so at your own risk, and you accept responsibility for any damages to your dental work or to the throats of your victims."
Miranda was already on the edge of violence this close to moonrise, and her temper flared.
Robin quickly amended, "We do have some recourse, however. Since Dan witnessed the excrement-balloon assault, we can sue Mister Furguson and seek damages for the cost of cleaning your dress."
"It couldn't be cleaned, sweetheart. That cockatrice acid burned right through the fabric."
"Replacement costs, then."
Hirsute flexed and extended his hands, impatient. "Wait until tomorrow—if any of them survive tonight, we file the suit." He nuzzled Miranda's ear. "That dress wouldn't have lasted long, anyway. I was ready to tear it off you."
"Sorry you didn't get the opportunity." She turned and waved farewell. "Hirsute and I have to change into casual clothes. It's a seedy part of town filled with undesirables. I'd hate to get my good jeans dirty."
After they left, the imminent rumble continued to weigh on me. I phoned McGoo to let him know the rumble was supposed to take place at the ruined Smile Syndicate warehouse on the other side of town. He was grateful for the tip. "How the hell did you learn that, Shamble? The department's been trying to find out where we should send reinforcements."
"Werewolves aren't very good at keeping secrets. Ask around. I wouldn't be surprised if people living nearby were selling tickets for good seats."
"Damn scalpers," McGoo said.
"That might not be the best word choice."
"Seemed appropriate to me."
I hung up. Robin had already gone back to her office and now emerged triumphantly holding a folder. She had that look on her face.
"Since the donation receipts didn't match the pieces for sale on the warehouse shelves, we were already convinced Cralo was illicitly obtaining body parts, but we didn't know where he got them. This racket is even bigger than we guessed. One corporation owned by another corporation, but I connected the dots." She smiled at me. "Guess who owns a majority share of Joe's Crematorium?"
"Would I be right if I said Tony Cralo?"
Sheyenne was listening in. "If Cralo owns the Spare Parts Emporium, why would he be burning bodies?"
I grasped the answer. "Because he's not burning them—that's what happened to Adriana Cruz. The bodies come in for cremation, but instead of putting them in the furnace, Joe Muggins loads them into a truck and sends them off to the Emporium for disassembly. Cralo probably has a chop shop in the back room where he can strip out the parts he needs."
Sheyenne drew in a quick imaginary breath. "And then he gives the bereaved family members an urn full of burned hickory or mesquite ashes. Nobody knows the difference."
"Still doesn't explain the scalped werewolves, or the vampire stripped of his organs," I said. "A vampire wouldn't end up in a crematorium."
"So, somebody had to murder Ben Willard to harvest especially valuable body parts," Robin said. "Maybe he liked to go to rough nightclubs, and he picked up the wrong person."
Then something else about Joe Muggins went click inside my head. I'd first seen the crematorium owner in the Voodoo Tattoo parlor, getting a hidden tat of Bite Me on his shoulder. What if Joe had gone out hunting for some rough trade of his own, met Ben Willard, brought him back to the Motel Six Feet Under, set him up . . . ?
"You two put the evidence together and tell McGoo to get an arrest warrant. In the meantime, I'm going to make a quick visit to Joe's Crematorium."
CHAPTER 43
For an edgy vampire who prowled rough nightclubs sporting a Bite Me temporary tattoo, Joe Muggins struck me as awfully wimpy.
The crematorium receptionist had already gone for the day (since she was a Monthly werewolf, this was a big social night for her). Business hours were almost over, so I walked past the front desk and found the meek-looking vampire in his office.
"Sorry, we're not accepting any more—oh, it's you, Mister Chambeaux." Joe Muggins seemed nervous as he rose from his desk. "My, my, I thought we'd taken care of that . . . that other matter. Has there been more trouble?"
I shambled through the door. "You've been a very naughty boy, Mister Muggins."
He stammered. "Wh-what are you talking about?"
"Does the name Ben Willard mean anything to you?"
He thought hard for a moment, then shook his head. "No. Who is he?"
"Okay, maybe I shouldn't have led with that. Bite Me? The club scene? Hard-partying vampire in search of a good time, maybe a little hardball sex? A room by the hour at the Motel Six Feet Under?" I let that hang.
"Ben Willard—so that was his name! He told me to call him 'Frisky.' " Now alarm crossed his face. "But the fact that I have a tattoo—had a tattoo, because it's worn off now—doesn't prove anything."
"He was murdered, all his organs removed, and the vampire body parts leave a trail back to Tony Cralo's Spare Parts Emporium. Also, Cralo is a majority owner of Joe's Crematorium, and we have proof that you aren't actually cremating the bodies you receive—you sell them to the Emporium for disassembly and redistribution."
The vampire was on the verge of panic. "What are you, some kind of detective?"
"I thought you knew that already."
I loomed in the doorway of his small office. He looked for a place to run, but I had him cornered. If I'd known it was going to be this easy to get him to confess, I'd have called McGoo ahead of time, but my BHF was too busy preparing for the gang war.
The crematorium's back door burst open, and two burly forms shouldered their way in, guffawing. "Hurry up, scrotum-face!" Scratch shoved his companion aside. "Let's make this delivery so there's time for a beer before the rumble starts."
The two biker werewolves saw me and pulled up short. Sniff demonstrated his extensive powers of observation. "It's that zombie detective guy again."
Scratch let out a giggle and a snort. "Hey, does anybody call you a stiff dick?"
"Gee, never heard that one before," I said.
Desperate, Joe Muggins wailed, "Get him! He knows everything!"
The two Monthlies bristled. "What do you mean, he knows everything?"
"That's an exaggeration," I said. "I'm a smart guy, but I can think of dozens of subjects that I know nothing about. Like sports teams, for instance. Couldn't care less." I was thinking fast, trying to figure out what to do next.
I could handle Joe Muggins, but I hadn't counted on these two coming in. Having seen Scratch and Sniff roar away from the loading dock of the crematorium, their choppers piled with wrapped packages—for organ deliveries?—I should have guessed they were involved.
"He figured out about the bodies and the wood ash, and the corpses we deliver to the Emporium—even the vampire we killed and culled!"
"I hadn't figured it all out, Mister Muggins, but you're doing a good job of filling in the blanks." I reached for my .38, but Scratch seized me before I could draw my weapon.
"What's he think he's going to do about it?" Sniff asked.
"Not very damn much." Scratch lifted me off my feet. Sniff joined in the fun.
"You said you were in collections," I said to the biker werewolves.
"We collect organs," Scratch said. "Just the high-value pieces, take them right over to the Emporium. The rest of the bodies come over later in the big wagon."
Now that I was safely contained, Joe Muggins was brave and full of himself. "Vampire organs are a niche market—very hard to obtain."
Scratch and Sniff both chuckled. "But fun to get."
"So you went out clubbing," I said to Joe, "found a horny young vampire, lured him back to the motel?"
"I told him we were playing a bondage game," Joe said. "He didn't figure out that the handcuffs were silver until too late."
"After that," Scratch said, "we played a long, slow game of doctor."
"I won," Sniff said.
I was pretty sure that Ben Willard had lost.
"And so all the corpses that show up for a nice clean cremation become part of Cralo's organ and body-part scam?"
"Not a scam," Joe said. "It's being environmentally aware, a form of recycling. We can't let all those perfectly good parts go to waste."
I couldn't move in their grip. I looked from Scratch to Sniff. "I hate to dampen your excitement, boys, but I'm pretty sure there's no market for already-used zombie parts."
"Not much. Bottom of the barrel," Joe agreed from his desk. "But we've got to do something with you, even if we only get scrap prices."
Scratch looked at his watch. "You two gab like a couple of teenage girls going to the bathroom. Let's get a move on! Sniff and I have big plans tonight."
Joe looked at the two as-yet-untransformed werewolves. "Wrap him up and deliver him to Mister Cralo over at the bathhouse. He'll decide what to do."
"How the hell am I going to get a zombie on the back of my chopper?" Sniff said. "Shouldn't we cut him into portable pieces first?"
"He could wrap his arms around your waist and hold on tight," Scratch suggested with a malicious snicker.
"Don't be girlie. He can snuggle up next to you!"
Joe cut the argument short. "I've got rolls of butcher paper in the side room. Wrap him up like a package. Add enough twine, and he's not going anywhere."
That didn't sound pleasant to me, but neither did snuggling my arms around Sniff's waist.
"We've already got the proof of your activities," I said. "I informed the police."
"Oh, I'm soooo scared," Scratch said in a falsetto voice.
Even Joe did not seem worried. "That's what the last four people said. We'll risk it."
They relieved me of my .38 and my cell phone. Joe placed the gun in his desk drawer while Sniff smashed my phone on the floor, then stomped on it for good measure.
"That was my good phone," I said.
"Time to upgrade," Joe said, then supervised as the two wrapped me in sheets of white coated paper like a mummy (or a fresh-cut roast) until I couldn't see.
The biker werewolves picked me up—one by the shoulders, one by the feet—and carried me out of the crematorium.
CHAPTER 44
Choppers aren't made for passengers, especially not for passengers wrapped up as packages, and it was a long, bumpy, unpleasant ride. Scratch and Sniff had propped me with my paper-shrouded face close to the blatting exhaust pipe. Not exactly "easy riding."
Since the full moon would rise soon, the two biker werewolves were anxious to get rid of me so they could get to the rumble in time for the opening salvo. I could tell, however, that they were intimidated by Tony Cralo and wouldn't cut corners on an important job—such as disposing of me.
The choppers pulled up at their destination, but I couldn't see a thing through the wrapping. Scratch and Sniff manhandled me off the bike, straightened me—a relief to my aching joints—and carried me away. I felt them climb a set of stairs and wrestle open a creaking door. They bumped my head against the door frame (the butcher paper provided very little cushion) and took me down a hall, where I heard voices, dripping water, a splash. After they propped me like a garden rake against a wall, somebody tore the butcher paper off my face so I could see.
We were outside a sauna/steam room at the Zombie Bathhouse, with Cralo's two business-suited guards regarding me. One bodyguard glanced at a clipboard and scolded the two thugs. "This isn't the delivery we were expecting. Look at your paperwork—it says two kidneys and a gallbladder, pronto."
"Joe told us to bring this over." Scratch self-consciously wiped his hair back. "We don't have time to argue."
"Possible problems," Sniff said. "Mister Cralo will want to know about him."
The bodyguards looked at me without interest. "What's so special about this one? He's just dead meat." They sniffed in unison. "Not even fresh."
"He figured out the whole operation—even the part about the vampire organs," Scratch said. "Claims he already told the police."
The bodyguards rolled their eyes. "That's what the last four said." The one on the left sighed. "All right, unwrap him and bring him into the steam room. Mister Cralo needs to have words with him."
The other guard lowered his voice. "I bet the boss will be glad for the distraction from that harpy in there, even if it's about a threat to his operation."
After liberating me from the twine and butcher paper, the bodyguards opened the sauna door, and a waft of noisome steam burbled out—not invigorating steam from water ladled over hot rocks, but foul-smelling fumes like the armpit of a stagnant swamp.
Esther stood in front of the sauna door, a white towel wrapped around her body, her harpy face pinched in a disapproving expression. Her moist plumage drooped. I heard a loud burring sound, like a dragonfly drowning, then an extended belch from Cralo's other end. Esther called back to a hulking form on one of the benches. "Tony, I love you, but sometimes your outgassing is disgusting." She gave me her usual look of disapproval. "And you have a visitor."
The bodyguards shoved me into the miasmic room. The obese zombie sat surrounded by the steam, a towel over his shoulders and another one covering his lap; both were the size of bedsheets. "Come to file another complaint, Mister Chambeaux? Just how many dissatisfied customers do you represent?"
"Same one, and he's still dissatisfied," I said. "But I'd like to expand my complaint to include your spare-parts gathering from Joe's Crematorium, and the murder of a vampire for his organs."
Cralo did not look amused. He turned to the two biker werewolves. "What's all this about?"
Scratch and Sniff were eager to take credit. "He was roughing up Joe, but we captured him and brought him here."
Cralo looked to me, and the sunken eyes in his round face blazed brighter. "You think you're smart?"
"Above average," I admitted. "We've already delivered our evidence to the police. The law is closing in on you, Mister Cralo. They'll arrive soon enough."
Scratch and Sniff laughed. "Cops are going to be busy with the werewolf gang war, boss. You don't need to be in any hurry."
The fat zombie shook his head. "You're a pain in the ass, Chambeaux . . . as if I needed any extra bodily aches. Maybe I should leave town after all, lie low for a while—fortunately, the rumble gives me time to pack up and get away without leaving any loose ends."
When he rose to his feet, the towel around his waist fell off. I turned away as quickly as I could, as did Esther, the two bodyguards, and Scratch and Sniff. The guards came forward, eyes screwed shut, carrying what looked like a white terrycloth camping tent, but it was merely a Tony Cralo–sized spa robe. "I'll go underground, deep underground—sewer level. Esther, sweetie-magpie, I want you with me. Don't worry about your clothes or your jewels, I'll get you pretty new things. We'll find a hideout just for you and me, like a little love nest."
Esther tightened her own towel and winced at the stench in the air. I wondered if even sewer dwellers would want someone like Cralo around. She daintily waved her taloned hand. "Oh, my poor thing! That sounds delightful and romantic, but I can't go with you. I have my job, my career." She flounced to the door of the steam room. "I'll wait for you, baby—so long as you bring me expensive things. Call me when you get back."
She hesitated at the sauna door, and I watched her expression brighten. "Wait, I have something for you." From the folds of her feathers, she pulled the gold medallion Glenn the wizard had given her. "Here, take this as a token from me."
Cralo reached out a pudgy hand. "What is it?"
"It's for luck." He accepted the charm, and Esther beat a hasty retreat out of the bathhouse.
Although they loved the possibility of having to dispose of my body, Scratch and Sniff were impatient. "Believe me, boss, the Monthlies and Hairballs will be all the diversion you need to get out of town. You should start packing."
"I'll put a bonus in your next paycheck." Cralo wrapped the thin chain of the charm around his pudgy wrist. I doubted I'd survive long enough to benefit from Cralo's newly acquired bad luck. "In the meantime, what do we do with you, Chambeaux?"
"If you're looking for suggestions, I vote for letting me go with a stern warning."
"I'll take that under careful advisement." He paused for a nanosecond. "No, I'm afraid not. We should bury you under cement. You're already dead, so it won't kill you, but you'll be awfully bored encased in concrete under the foundation of a building." Cralo looked at his two bodyguards. "Do we have any building foundations being poured tonight?"
The two looked at each other, shook their heads. "Not until next week, boss."
Sniff suggested, "Could put him in a pothole in the parking lot of the Emporium and cover him over. That lot really does need repaving."
"We don't have that kind of equipment right now. Besides, I'm still taking bids for the job," Cralo said. "Damned contractors, they're all corrupt. That'll take too long."
"It's a shame," I said. "All I really wanted was a brain. If you had a better customer service department, I wouldn't have had to keep digging."
"I appreciate the feedback. We should improve that part of our business." Cralo smiled at me. "If it's any consolation, I did receive the troll brain you returned, and I already disassembled the customer service rep who gave you so much trouble. I sent a high-priced replacement brain directly to Dr. Victor, and he was very happy to receive it. You can rest in peace."
"Thanks for that, at least," I said. "But is it the brain he wanted?"
"Better—I gave him the vampire brain. Those organs are genuine die-hard quality. It'll perform far beyond your client's expectations. Besides, after that transplant debacle, the rest of the vamp organs were too hot to move through regular channels."
I feared what would happen when Archibald installed the vampire brain into his newly constructed body. If the murderous creature in the heart-transplant ward was any indication, I doubted the coroner's house trailer could contain the violence. I would have rushed off to save him, but at the moment I had bigger problems.
Cralo clipped on a gold Rolex as he got dressed, glanced at the time. "I've got to start packing. Just wrap him up again, take him to the crematorium, and put him in the furnace." The fat zombie laughed close to my face, and his breath smelled as bad as his farts. "You're about to make an ash of yourself, Chambeaux."
CHAPTER 45
Wrapped up again and unable to move, I endured another noisy and bumpy ride back to Joe's Crematorium. This time, fortunately, I was propped away from the exhaust, though I doubt Scratch and Sniff did it out of consideration for me.
Private investigators are solo types, and we work on our cases without an entourage. It's not that we don't play well with others, but detective work isn't exactly a team sport.
I usually chided McGoo when he went off half-cocked without calling for backup, but here I was, tied up, relieved of my weapon, and heading for a cozy evening in a couple-thousand-degree furnace. It was damned poor planning. This was going to be very unpleasant, and in a very unpleasant way.
But live and learn—or if that didn't work, die and learn.
The choppers pulled around the back of Joe's Crematorium with double-barreled muffler blasts. Scratch and Sniff manhandled me off the bike and hauled me inside.
When Joe Muggins saw the burly pair, he scowled. "My, my, why did you bring him back here?"
Scratch ran a hand through his slicked-back hair, which seemed even more fluffy now in anticipation of the full-moon transformation. "Mister Cralo says it's time for a barbecue. Extra crispy."
The little vampire shuffled his feet. "All right, if that's what he thinks is best."
"Could I put in a dissenting vote?" I said.
"Shut up," Sniff said; his forearms were growing noticeably hairier. He shoved me toward Joe Muggins. "Hurry up and take care of this. We've got places to be."
"No waiting in furnace number four," said Joe. "Take him over there."
I struggled. With my legs and arms still wrapped up, I looked like a drunken caterpillar who couldn't break free of a cocoon. As the two werewolves carried me—Scratch by the shoulders, Sniff by the feet—Joe scuttled into a back room and returned with a cardboard box from toilet paper rolls sold by the gross. "We were about to break down these boxes for recycling. This one will do for a makeshift coffin."
"A toilet-paper box?" I asked. "Really? You can't get me something of higher quality?"
Joe pursed his lips. "Why go to the added expense? Should I send a bill to Chambeaux and Deyer?"
"No, that probably wouldn't work," I said.
Robin had paid for my fine embalming job the first time, as well as a quick-release casket and a nice burial plot in the Greenlawn Cemetery. This time, though, it would be a cardboard box, disposed of after hours in a crematorium with no one else watching.
Scratch and Sniff lifted me up and placed me in the box. With the butcher paper and twine bindings, I couldn't move my arms, couldn't separate my elbows, couldn't free my hands.
Grunting and growling, the two lugged the cardboard box toward one of the crematorium furnaces. Hurrying ahead, Joe swung open a metal hatch like a doorman at a fancy hotel. Through the open box top I could see a black-encrusted chamber built to accommodate the largest coffins.
The two werewolves tipped the box onto a rattling line of rollers that would feed me into the furnace. At the controls, Joe turned a knob to start the jets, but nothing happened. "Damn pilot light went out again. Sometimes the breezes come in and—"
Scratch snarled. "Hurry up! Moon's rising in half an hour!"
Joe fiddled with the pre-burner, found a box of long wooden fireplace matches, struck one, and leaned into the furnace. Tiny blue gas jets lit, and the vampire straightened, nodding. "There, that should work just fine."
"Good. We're outta here!" The two headed out, annoyed that they wouldn't have time for a beer before the rumble.
As he rattled my cardboard box along the rollers into the soot-caked furnace, Joe leaned close. "I have slow-cooker settings or broiler—do you have a preference?"
Neither option sounded particularly enjoyable. "Bite me," I said. It seemed clever at the time.
Joe seemed antsy. "Broiler it is, then." He gave my box one last shove into the furnace. "I've had to go to the bathroom since before you arrived. Sorry for the rush." He nudged my head down farther into the toilet-paper box, made sure I was situated in the center of the cremation chamber. "Thank you for your patronage." Squirming as if his bladder were about to burst, he swung the vault door closed. I heard the clunk of the latch.
A few seconds later, the flame jets spurted higher, and fire gushed out to lick the sides of the box. I could smell the cardboard crisping, the acrid smoke curling around. Joe Muggins hadn't even bothered to add hickory or mesquite for seasoning. I was going to be plain bland ash . . . not that it made much difference. The fire crackled around me. My hair singed.
There was only so much the Wannovich sisters' restoration spell could accomplish. I hoped the witches wouldn't be too disappointed at the abrupt end of their potentially successful Shamble & Die series. (Linda Bullwer could probably concoct more adventures on her own, however.)
Smoke filled the chamber. The cardboard box was in flames now, and the butcher paper wrapping my body had turned brown and started to smoke. I wouldn't have minded so much if the cremation had happened before I came back to life, but now I'd had a change of heart. I could understand why Adriana Cruz didn't want to demand completion of the promised services.
The first time around, I hadn't known I was going to die, hadn't heard the killer approach, barely had time to react to the cold gun barrel pressed against the back of my head before the quick pop. It was a complete surprise. Now, as flames consumed the box around me and black smoke closed in, the anticipation was downright terrifying.
I still had cases unsolved, and I was especially upset that Archibald Victor would find himself in mortal danger the moment he installed a vampire brain in his do-it-yourself body kit. Nobody would be able to warn him.
Worse, I knew how much I was going to make Robin grieve, again. And I would be separated from Sheyenne, too . . . unless I came back as a ghost, but the odds of that happening were lottery-jackpot small. As I had demonstrated by my current situation, I'm not that lucky.
Or maybe I was.
Unexpectedly, the gas jets died down, and the bright flames retreated into their nozzles, leaving only tiny pilot lights. The cardboard box was on fire, collapsing into black ash and embers. The butcher paper still burned, which had damaged my sport jacket, among other things.
I heard the latch slam aside, and the heavy vault door swung open. A blast of fire-extinguisher mist filled the furnace chamber, chasing out the smoke. I couldn't see a thing, but at least I wasn't roasted.
"Beaux! You in there?" Sheyenne's voice sounded wonderful.
"I'm sure glad to see you." The paper wrapping had burned and crumbled, and this time when I strained, I managed to snap the cords binding me. When the smoke and fire-extinguisher fumes cleared away, I saw Sheyenne hovering outside the furnace. Her face looked more translucent than usual, her expression distraught. She blasted the metal rollers, cooling them enough that I could scramble, slip, then scramble again until I reached the oven door.
"I'm glad you didn't wait one minute longer," I said. Ten minutes earlier would have been even better, but I wasn't going to complain. I clambered out and straightened myself, brushing my singed jacket, my singed hair, and my singed grayish skin. "I'm thinking a big hug at you right now."
"You told us you were going to the crematorium, but you took way too long. Glad I was impatient."
I looked around the crematorium. "We've got to arrest Joe Muggins. I want that weasely vamp to spend a few decades in a bright, sunny cell. I hope he didn't get away—"
"Oh, Joe's not going anywhere," she said. I heard a muffled shout and a pounding; a door rattled. Sheyenne had jammed a chair under the doorknob of the small employee bathroom, trapping the little vampire inside.
"One more reason to offer you a raise," I said.
"You know you don't pay me, Beaux."
"Then I'll double your salary—tomorrow. I've got a few more fires to put out tonight."
CHAPTER 46
With Joe Muggins locked in the crematorium's employee bathroom, I had more urgent concerns. The Monthlies and the Hairballs were about to start World War Werewolf. By now, Tony Cralo would be making his escape, if he hadn't already skipped town. And Archibald Victor was about to install a vampire brain into his build-your-own body kit, and he was bound to unleash a wild monster whether or not he intended to reanimate it.
Sheyenne rushed off at her spectral speed to let McGoo know about the zombie crime lord. The UQPD would be preoccupied by the werewolf rumble, but if Cralo escaped into the underworld, they might never find him again.
Meanwhile, I retrieved my .38 from the drawer in Joe's desk and headed off to the mad scientist's trailer, hoping I could get there before he installed the brain and inadvertently unleashed a homicidal monster.
Dark was just falling, and the full moon hadn't risen yet. A brewing violence hung in the twilight, and the Quarter felt deserted, as if everyone were getting ready for High Noon . . . or Full Moon.
I made my best speed toward the trailer park. Because the Parlour (BNF) was right on my way, I burst through the door, hoping that Harriet Victor could get in touch with her husband before it was too late.
Not surprisingly, the beauty shop was devoid of customers. Rova was barefoot and doing a contortionist act in one of the chairs, trying to give herself a pedicure, but not doing a good job. When she glanced up at me, she scowled as if she had just gargled with sour-lemon mouthwash. "You're not allowed to come within fifty yards of me—didn't you read the restraining order?"
"What restraining order?" I asked.
"My attorney drew it up. He was going to serve it on you, for harassment."
I didn't have the patience for this. "First of all, a lawyer can't serve his own papers, and second, I haven't had time to receive any restraining orders. Where's Mrs. Victor? This is an emergency!"
The bearded lady came out from the back room, her arms around an open cardboard box filled with crème rinses, shampoos, and skin lotions to display on the Parlour's shelves. The hair on her head, face, hands, and arms was freshly permed, curled, and adorned with colorful ribbons. She said in a chirpy voice, "Mister Chambeaux! Archibald was delighted to receive the surprise package—he's been waiting for that new brain. He called me from home an hour ago, couldn't wait to get to work. I think he puts in more hours in his home lab than at the morgue."
"We have to get in touch with him, Mrs. Victor! He can't use that brain—it's a vampire brain, very dangerous. If he installs it, he could create a monster."
"Well, of course he's creating a monster. That's the whole point."
"Not like this. A similar thing killed a dozen people at the hospital. We'd better stop him before he gets hurt. Can you call him?"
The hair all over her body began to uncurl. "My poor Archibald! When he's trying to concentrate, he always turns the phone off so it doesn't interrupt him."
No time to lose. I turned around and lurched toward the door. "I'm heading over there."
Harriet grabbed her purse. "Then I'll come with you."
We must have been a strange sight, a singed zombie detective and a bearded lady moving like two bats out of hell. In the trailer park, we rushed to the Victors' double-wide with its pretty flower boxes and wind chimes. Harriet removed the house keys from her purse, fumbled with the lock, and pushed the door open. "Archibald, are you all right? Mister Chambeaux says you're in danger."
I entered, hot on her heels. "Step away from the brain, Dr. Victor! It's not safe."
In the back lab, I could see that Archibald had gone to work with the fervor of a truly mad scientist. His experiments filled the cozy trailer: Bubbling chemicals and strange mixtures covered the dinette table, the kitchen counter, the coffee table, the sofa. The brawny build-it-yourself body lay on the hobby table, with all the main pieces now put together.
I spotted the gift-wrapped box from the Spare Parts Emporium, the lid set aside, the box empty—then I saw with weak-kneed relief that the brain was still in its transparent jar, next to the empty cranial cavity of the mostly assembled monster. Archibald hadn't had a chance to install the brain yet. We were safe!
In the meantime, he had been working on something else.
As the mad scientist emerged from the bedroom laboratory, Harriet and I stared in astonishment. We barely recognized him. His voice was supercharged with excitement. "It works, sweetie! We're going to be rich!"
Archibald Victor had given up his toupee, for good. The little coroner no longer needed it—not by a long shot. Except for his ping-pong-ball-sized eyes, I would not have recognized him. He looked like a Neanderthal hippy.
Hair sprouted from every square inch of his pale skin. His eyebrows had turned into ferns. Hair poured from his scalp, growing visibly as we watched. His jaw and chin sported a full beard that rivaled his wife's. Fur sprang out from his neck. His chest hair was a forest sprouting from the open collar of his lab coat. He swiped at his face with hair-covered hands to push his overgrown eyebrows out of the way so he could see. His mustache ran over his lips and continued all the way down to his chest.
"It works, it works!" he repeated. "No one ever has to be bald again!"
"And this is preferable?" I asked.
"It should stop growing soon," Archibald said. "I designed the formula that way."
Harriet grabbed a pair of scissors from the chemistry-strewn counter and raced to her husband. "You need a trim, Archibald. This is alarming!" She began to shear handfuls of hair from his face, clearing his lips so he could speak plainly, cropping in front of his eyes so he could see.
"But look how effective it is, Harriet—I never dreamed of results like this! Maybe it has something to do with the full moon."
"At least we're in time to stop you from using the replacement brain," I said, getting back to the main emergency. "We've got to destroy it."
Archibald looked distracted (at least as far as I could tell, under all that hair). Harriet fussed over him, stroked his wild locks. "We were so worried you might be in danger."
"Oh, a silly body-building competition doesn't matter anymore! Yes, I was about to install the brain, but then I had a breakthrough idea. I had to try the new tonic, and it works! Did I mention we're going to be rich? This is the most important discovery in the history of human civilization, and only I have the formula!"
Harriet nudged her husband into a chair by the kitchen table. "And you've created plenty of business for barbers and stylists everywhere, but you might still need to fine-tune the recipe," she said. "Let me give you a full haircut, make you presentable before we release any announcements." She snipped and clipped, trimming him in all directions. By the time she finished one circuit of his overly enthusiastic follicles, she had to go back and trim again.
While she fussed over the no-longer-hairless coroner, I went to the body-building table to take care of the vampire brain. One disaster averted. Now all I had to worry about was stopping the werewolf gang war. I could leave Harriet and Archibald to their own tangled mess.
Then I spotted something I wanted to see even less than I wanted to see a hair-covered Archibald Victor.
On a separate table, a row of large petri dishes was connected to bottles and flasks that were bubbling over Bunsen burners. Each petri dish was the size of a blue plate special, filled with a greenish liquid. In the petri dishes, like round hunks of sod, sat four burbling werewolf scalps connected to electrodes, twitching and writhing in a chemical bath. They looked as fresh as the day they'd been removed from their original owners.
CHAPTER 47
When Rusty the werewolf had hired me to track down his attacker, I never expected to blunder into this. Archibald Victor was also my client, and I got involved in the conflict only because he received some bad brains. It should have been a simple customer service issue, but I found myself knee-deep in shady organ dealings.
In their jumbo nutrient-filled petri dishes, the gruesome living scalps throbbed. I blurted out, "You're the one who's been scalping werewolves?"
Archibald saw where I was standing and tried to squirm out of the chair. Harriet, still trimming the persistent strands streaming down the coroner's face, gave her husband's bangs a last snip. "Archibald, why can't you clean up after yourself?"
"I have to report this, Dr. Victor." After being caught unawares at Joe's Crematorium, this time I drew my .38 without giving them the benefit of the doubt. "Why would you stun and scalp werewolves?"
"To find the secret, of course." The mad scientist grimaced, spitting out hair that had gotten into his mouth. "Have you seen the thick head of hair on a full-time werewolf? I wanted to reproduce that! Werewolf hair growth is nature's best, and I'm not one to think small. I wanted to develop a werewolf-based hair-follicle treatment that could be marketed to normal balding humans. We'd capture the entire market, and Harriet and I would earn millions! I'd quit my job as the coroner and just do autopsies as a hobby."
Harriet said, "And I could keep the Parlour (BNF) afloat for months."
"As diabolical schemes go, that one's way out there." I remembered the bloody gang violence that was about to start. "Do you realize what you've caused? You provoked a werewolf war!"
"It's all in the name of science and personal styling," Archibald said. "The donors didn't feel a thing. I knocked them out before I took their scalps, and they'll heal. Werewolves are good at that."
I thought of Professor Zevon recovering in the hospital, poor Ernesto and Arnie in their shantytown, covering their heads with old stocking caps, Rusty with his bandanna wrapped around his scabbed head. I held the gun steady. "You didn't think there was anything wrong with assaulting werewolves? Committing grievous injury?"
"Not grievous injury, Mister Chambeaux. It was surgery," Archibald said. "And I needed living scalps to test the hair follicles, to study the growth of each strand, to separate and identify the specific hormones—in particular the chemical changes that promote such persistent hair growth, enhanced by the light of the full moon. They weren't going to give up their scalps just because I asked nicely."
The mad scientist truly was mad. "Sorry to shatter your dreams, Dr. Victor, but you've got to come clean about this— right now. I'll take you over to where the rumble's due to start. We can still stop the werewolf violence and save a lot of lives. I've got a friend on the police force—maybe he'll cut you a break. But we have to spread the word before the gang war breaks out!"
During our tense conversation, Harriet had stopped cutting her husband's hair, and his face was completely covered again. He pulled the strands away from his mouth and eyes once more, just so he could meet my gaze.
"No, they'll confiscate my formula! No one can have it. I worked and sweated and concocted—this is my life's mission. Besides, I haven't applied for a patent yet. I'm not going to let anyone take away our nest egg."
Harriet moved so quickly that she surprised me. She threw herself upon me—and was damned lucky she didn't get herself killed (and not just because she was running with scissors). I wasn't about to shoot a lady, bearded or otherwise. She knocked my gun hand aside, and as I wrestled with her, Harriet yelled, "Get rid of the stuff, Archie! Destroy the evidence—you can always recreate it!"
She was a sweet hairdresser, former circus attraction, and caring business owner, but when she saw her beloved mad-scientist husband threatened, she turned into a mama grizzly (and looked the part). Harriet was strong, too, as she grappled with me.
The Neanderthal-hairy coroner leaped from the chair and scuttled over to his test tubes and beakers, trying not to trip on his own hair. I thought he would try to dispose of the incriminating scalps, but apparently his top-secret hair formula was more important to him.
Before I could break free from the bearded lady, Archibald grabbed the flask of his hair-growth solution and stumbled to the bathroom. Barely able to see where he was going, he upended the flask and poured the solution down the shower drain.
I wrenched my gun hand free, extricated myself from Harriet's clinging grasp, and lurched away, swinging up my .38 again. "Look, I really don't care about the formula! I just want to stop the werewolf war." I glared at both of them. "Remember, I came here to save you from a murderous vampire brain."
I eased myself over to the hobby table where the mostly constructed monster body lay, picked up the transparent jar with the offending brain, and smashed it on the floor. Just to be safe.
"That'll stain my carpeting!" Harriet objected.
"Don't worry, love," Archibald said. "With my formula, I'll be able to buy us a castle."
"When you get out of prison," I said. "Right now, we need to stop the werewolf war. You're coming with me, Dr. Victor—so you can do the right thing."
"I can't let poor Archibald go to jail," Harriet wailed.
"I only did it for you, sweetie," the coroner said, "so I could have hair just like yours."
"Oh, Archibald, I didn't care about your hair. And I wish you'd never worn that silly toupee. Can't you just look like yourself, the man I fell in love with?"
My urgency increased now that the full moon had risen and was shining through the trailer's windows. "Very heartwarming, but can we put a pin in that? We've got to get to the Warehouse District before the fur starts flying!" I hoped we could get there in time.
It's always something, though.
I can't remember the last case where I solved a mystery and closed a file without any complications. Even in Penny Dreadful's fictionalized versions of my cases, nothing was ever so simple.
Before I could march the Victors out of their trailer, the shower drain began to groan and burble, like a dragon with severe intestinal distress. Archibald lifted his long eyebrows away from his face so he could see. Harriet covered her mouth with a hand.
Hair sprouted from the shower drain. Maybe sprouted isn't the right word; exploded is more accurate. Tangled auburn strands writhed out of the drain, growing at an astonishing rate and lashing like tentacles.
I realized that Archibald's amazing hair-growth tonic had been derived from werewolf hormones, and with the added impetus of the full moon, the chemical became supercharged. Within seconds, the shower stall was filled with a writhing, angry sargasso of hair that burst out of the shower, streamed across the bathroom—then went on the attack.
"Do something!" Harriet cried. "Stop it!"
The hair tsunami kept rolling forward, some strands curled in a perm, others stringy and straight. I fired three rounds into the maniacal hairy outburst . . . which, not unexpectedly, did nothing.
Harriet tried to stop it. She grabbed her haircutting scissors and dodged past me to dive into the ever-growing mass.
I tried to stop her. "Stay away from there!"
But the hair kept growing, and the tentacles wrapped around the bearded lady and pulled her into the thicket. Harriet lashed with the scissors, snipping and trimming as quickly as she could, but not fast enough.
"Harriet!" Archibald cried. Unable to see because of his hair problems, he stumbled forward to rescue his wife. The eerily sentient hair reached out, as if knowing the mad scientist was to blame for scalping the werewolves, or maybe it was instinctive affection since the detached hairs came from his adoring wife. Whatever the reason, the python-like clumps of hair wrapped around Archibald and lifted him up, twisting him in a tight, murderous snarl. He flailed, but couldn't escape the constrictive grasp; the living hair was lustrous and full of body.
One of the hairy tentacles knocked me sprawling into the sofa. I couldn't possibly reach them.
Harriet's hand worked its way free, questing for the bottles of hair product on the countertop. She snagged a squeeze-tube of detangler, but it was much too little and too late.
"Harriet!" Archibald wailed, and he struggled, trying to break free and get to her. The hair surrounded them in an impenetrable clump, squeezing them tight. It drew them into the dense heart of the thicket, squeezing, crushing. The couple fell silent.
Deadly tresses continued to boil out of the shower drain, pushing into the main room, where they engulfed the brainless but otherwise complete monster body. The mad scientist had said he designed the formula to expire after a certain point, but I had no idea when that might be. As the awful strands quested toward me, I jumped out the front door. There was no telling how much more the demonic tangle would grow.
Trailer-park neighbors emerged to stand on their porches and watch the spectacle. "What's Dr. Victor doing now?" asked an old toothless gargoyle who sat on a folding lawn chair.
"Just more of the same," said a witch from two trailers over.
Backing away, I saw the sides of the trailer bulge. The windows shattered, and hairy strands burst outward, curling around the roof of the double-wide. I worried we might have to call in an air strike to stop the tangled monstrosity, bombers dumping barrels of high-end conditioner designed for controlling unruly hair.
But then, with a sigh, the twitching strands quested outward one last time, until, exhausted, the hair became limp and lifeless. The mad scientist's trailer sagged and fell silent.
I stared for a long moment, thinking about the Victors. Then it was time for action. I had to get over to the rumble and stop the massacre.
In the last hour, I had survived a crematorium furnace, stopped the activation of a murderous vampire brain, and barely escaped a wild tangle of killer hair.
But the night was still young.
CHAPTER 48
As I ran from the trailer park, the bright moon was just above the skyline, which meant that the Monthlies would be fully transformed by now. I hoped I wasn't too late. Police reinforcements had already converged out at the ruined Smile Syndicate warehouse, but I doubted an entire riot squad would be enough to stop the violence if all the werewolves in the city were howling mad. If I could let the two hairy gangs know that it was actually a well-intentioned mad scientist who had done the scalpings, not a rival werewolf, maybe they would shake paws and go home as friends.
Yeah, I know how that sounds.
The Warehouse District was on the other side of town, but I headed off at my best possible speed, hoping to find an open business where I could make a call. I had to pass by the Spare Parts Emporium, and I realized the zombie crime lord might be making his escape even now . . . but Tony Cralo was not highest on my most-loathed-persons list right now. Funny how things can change in only an hour.
As I neared the skeletal bridge under which the homeless unnaturals huddled, to my surprise, I heard a chorus of howls, and a rival set of growls, yips, and barks echoed out into the night. It sounded like someone training the inhabitants of an animal shelter for barbershop quartets. Barbershop? Not a good image at the moment. I suppressed a shudder and started to shamble faster.
I hesitated. Something wasn't right. Larry had made a point of telling me exactly where the rumble would take place. Why would so many werewolves be here, if the big clash was supposed to be taking place across town at the Smile Syndicate warehouse? Maybe the ruined Smile Syndicate warehouse was already booked for another event?
Taking a chance—although my gambles hadn't paid off lately—I entered the persistent fog bank that hung around the Emporium. The full moon looked like a blurry yellow streetlight filtered through the mist. Shadowy figures moved among the battered transportainers and the large delivery trucks parked on the lot. The cardboard-box houses and tar-paper shacks under the bridge were abandoned—no trolls or vampires or even the forlorn ogre.
Just an entire furry army.
Two armies, in fact—werewolves of one form or the other. They faced off like football teams across a scrimmage line. The Hairballs and Monthlies snarled curses at one another.
Larry was the first figure I blundered into: hunched over, flexing his muscles, ready to fight. He whirled when he heard me, claws extended for a mauling, then he relaxed. "Hey, it's Dan Shamble! How'd you find the party? We weren't supposed to invite non-werewolves—better check with Rusty."
"You lied to me, Larry. You said the rumble was happening in the Warehouse District."
He showed his fangs, but it might have been a grin. "Not exactly . I said the Warehouse District would be a good place for it. You filled in the rest. A little bait-and-switch. Rusty knew you'd blab to your cop friend, so we used you to get them out of our fur, send them to the other side of town."
I felt exasperated. "That's not fair—I'm trying to do the right thing here! I have proof that the Monthlies aren't responsible for the scalpings. It was a mad scientist collecting specimens . . . and he was killed when his own experiment backfired. You don't need to go to war!"
Larry sniffed. "It's a free country. We werewolves can tear each other apart if we want to. It's our right."
"It might be your right, but all the reasons are wrong," I said. "I've got information that affects the entire basis for the rumble. Can I borrow your phone?" I needed to let McGoo know where the real clash was taking place.
"Shamble, I live in a cardboard box. You think I have a monthly cell plan?"
"I suppose not. Sorry."
Rusty saw me and lurched forward. In the light of the full moon, some of his scalped fur had grown back, though it looked like a different color and poked up in all directions. "Out of the way, Mister Shamble! You had your chance to solve the case—now we're taking matters into our own paws."
Furguson came up, all gangly arms and legs, bouncing from side to side like a kid on too much caffeine. "Ready to go, Uncle Rusty!"
"But the attacks had nothing to do with either gang," I insisted. "The Monthlies didn't scalp you!"
The nearby werewolves grumbled at this, but ultimately they didn't care. Rusty shrugged. "We've got enough reasons to fight, even without that. Don't rock the boat now, Shamble."
"Chambeaux," I said, knowing it wouldn't do any good.
I also saw Professor Zevon, who had recovered enough to join the fray. Patches of fur sprouted from his scalped crown, but the appearance was like the aftermath of a clumsy backroom hair transplant, rather than his original lavish head of silver fur. As I continued to shout about Archibald Victor and his hair-tonic experiments, the professor said, "An interesting hypothesis, sir, but irrelevant at the moment." He fumbled with his necktie, tossed it aside, then unbuttoned the top button of his white shirt. He did not, however, remove the nice tweed jacket he had worn for the occasion. A group of his students stood beside him, growling, ready to fight for their teacher's honor—strictly in theory, of course; as academics, they were there only to chronicle the event and preferred to stay back from the fight.
On the other side of the parking lot, the Monthlies were gathering, shoulder to shoulder, baring fangs. The howls grew louder.
Hoping I'd have better luck with the other team, I raced across the open area. Monthlies did spend more time acting human on the nights when the moon wasn't full. Maybe I could get through to them.
Now transformed, the gathered Monthly combatants looked the same as the Hairballs, and I wondered how the two sides would tell each other apart in the middle of the fray. Different colognes? Then I recalled that the gang members had specific tats, courtesy of Antoine Stickler and Voodoo Tattoo.
At the forefront of the Monthlies, Scratch and Sniff were now snarling man-beasts, holding their rusty meat cleavers in clawed hands and ready to do damage. Even covered by all that fresh fur, their myriad tattoos were visible. Sniff's beard had grown much longer.
When the two biker werewolves saw me, they growled in unison. "You're supposed to be dead!" Scratch said.
"Still am—and I'll deal with you two later."
On the edge of the angry lupine crowd, I saw the transformed Miranda Jekyll next to an enormous brute who could only have been Hirsute. While Scratch and Sniff roared challenges and brandished their little choppers, Miranda and Hirsute were content to be spectators.
Hoping Miranda would be more reasonable, I approached her first. She waved at me. "Oh, look, we have a zombie cheering section! Thank you for coming, sweetheart, but stay clear so you don't get hurt."
Hirsute drew in a huge breath to swell his chest. "Can you smell it in the air? Mayhem and bloodletting . . . quite an aphrodisiac." He stroked Miranda's muzzle, and she licked his paw, growling low in her throat.
"I'm tempted to drag you into a dark alley and have my way with you, Hirsute," she said. "Or do you really want to stay for the rumble?"
I interrupted her cooing. "Miranda, can I borrow your phone? I have to make an urgent call."
"Of course, sweetheart. My purse is right over there, with my dress."
Miranda's phone was in a sparkly diamond case, and I had to close out a game of Curses with Friends before I found the dialing pad and called the office. Robin answered right away; I could tell she was worried about me. "Dan! Sheyenne told me—"
"I'm not done yet," I said. "Dr. Victor and his wife are dead, and I'm at the werewolf rumble now, but they won't listen. Tell McGoo the werewolf gangs have gathered by Cralo's Emporium, not at the warehouse. Get the riot squad down here right away. I need some muscle."
Robin sounded concerned. "I'll call—but by the time the cops get there, it'll be too late. Sheyenne's the only one who's fast enough."
I heard her voice in the background. "Already on my way, Beaux!"
"She's a ghost! Sheyenne can't touch anybody—how can she help during a brawl?" Regardless, I knew I'd do better with Sheyenne's support.
Robin was on edge. "I'm not going to be the one to keep her away. And I'm coming, too. Be there as fast as the Pro Bono Mobile can carry me. Calling Officer McGoohan first." She hung up. Robin's old Ford Maverick couldn't go very fast, though, so I had time.
I would try to defuse tensions meanwhile, although nobody seemed interested in hearing the real story about the scalpings. The two sides began howling and roaring even louder challenges, only seconds away from declaring open war.
I pulled my .38, as if that would threaten anyone, and bolted into the parking lot, waving my hands. "Stop—the rumble has been canceled."
As the two sides snarled, I suddenly felt like a meat-flavored rag doll about to be fought over by two very hungry rat terriers. I swept my gun from side to side. "You all need to calm down. The first person who comes forward gets shot."
Rusty stood there in his overalls, burly and angry. "You got silver bullets in that thing, Mister Shamble?"
The question took me by surprise; I couldn't remember, so I bluffed. "Usually."
"He doesn't," Scratch called from the other gang of werewolves.
"Didn't think so." Rusty stepped forward and plucked the gun out of my hand. He threw it to one side, where it bounced on the gravel, skittered, and vanished forever into one of the crater-sized pothole puddles. "Furguson, make yourself useful. Help me out here."
Rusty and his nephew seized me and carried me over to the bridge and the shantytown. Despite my struggles, they lifted me up and shoved me feet-first into one of the rusty barrels half-full of moist ash from old fires. They wedged me down with my hands trapped at my sides, and I was stuck like a cork in a bottle (only much less dignified).
"Stay there, Shamble. This fight is for members only," Rusty growled, and they bounded off. Like a wrestling announcer, Furguson yelled to all the werewolves, "Are you ready to rumble? "
The Monthlies surged forward. The Hairballs roared and pounded their chests, ready to meet them.
Just then Antoine Stickler appeared.
The Jamaican vampire sauntered into the midst of the two groups, yin-yang tattoos and peace signs visible on his arms—freshly re-inked. The dreadlocks looked like the tentacles of a baby kraken that was suckling on his scalp.
"Now, you all done heard Mister Chambeaux," he said, in a calm tone, but his accent was more pronounced, probably due to the tension. "There won't be no peace and happiness in the world unless we listen to each other. I've been impartial—I put gang ink on both sides. I tried to help you be mellow and show you that violence is never an answer. I made voodoo dolls of every one of you, put balm of Gilead all around you and on you, spelled you all to make you think peaceful thoughts—but damn, you are stubborn!" He shook his head and made a disgusted sound. "Now you're making me haul out the Big Ink."
The rival werewolves stared at him. Antoine's lecture was getting an even worse reception than mine had. I struggled in the barrel, rocking back and forth, trying to dump myself over.
From a clip at his waist, Antoine removed a small battery-powered tattoo gun, made a fist of his right hand, and held out his forearm. A new tattoo stood out on his vampire flesh, an intricate mandala pattern. "You all have gang tats, but I loaded every one with special magic—a variation on an old impotence spell."
Rusty growled from across the parking lot.
"Sorry about that, Rusty." Antoine nodded toward him. "The first one was a mistake. This one's on purpose. It's a nonviolence spell."
Werewolves on both sides were snarling now. "Don't ruin our fun!"
"We had plans for tonight!"
"You got no right!"
"Oh, I got no right? You think so?" Using the gun, Antoine completed a pattern, connecting dots on his mandala. "There! Now don't worry, be happy."
Suddenly, all the Hairballs and Monthlies who had been snarling looked disoriented and confused, but very relaxed about it.
"Whoa," said Furguson, swaying, looking around at the crowd. "With all these wolves, let's have a block party!" He called across to the Monthlies. "Hey! You guys could bring a keg."
Some of the Monthlies scratched themselves, perplexed. "We'll need some chips, too."
Antoine Stickler let out a sigh. "Now you see the light." He inhaled deeply, as if imagining a long toke, then he sauntered away, his work done. He called over his shoulder, "Now, resolve your difficulties like cool people."
I finally managed to tip over the rusty barrel, which crashed to the ground. My shoulders hurt, but I wriggled and began to work my way out of my embarrassing confinement. Not exactly the most dignified way to solve a case.
The groups of combatants began to break up, unsure. Though the voodoo spell hadn't affected Miranda and Hirsute, who had not sullied their bodies with tattoos, the pair simply lost interest and ran off to the nearest dark alley together. Which was probably what they wanted anyway. The professor and the students also backed away, relieved not to have to fight, although Professor Zevon had been hoping to derive an academic paper from the material.
Other rumble participants did not have voodoo gang ink, however, and their anger remained unslaked. Scratch and Sniff were having none of the curious détente—and neither was Rusty.
As the first Hairball to get a voodoo tattoo, Rusty had been afflicted with the impotence spell—but that was before Antoine had added his fail-safe trick on the other gang tats. As Rusty looked around at his sickeningly mellow companions, he let out a roar with even more bestial fury than usual—and I knew we were in trouble.
He stalked over to two large tarpaulin-covered cages like the ones I had seen at the cockatrice fights. He wrestled away the fabric covering . . . and I suddenly recalled the mysterious blacked-out coop in his backyard.
Rusty threw open the cage doors and turned away, covering his eyes with a furry forearm. He shouted in a loud guttural voice, "Loyal Hairballs—remember the drill! Squint!"
I heard a shrieking, hissing caterwaul and knew what it was. Not domesticated cockatrices this time. Real ones.
CHAPTER 49
Three angry, snapping, writhing shapes exploded out of their containers. Since I was flat on the ground trying to squirm out of a rusty barrel, I didn't get a good look—fortunately.
Rusty kept his eyes pressed into the crook of his elbow and staggered out of the way. Though the Hairballs were stunned and lethargic from the peace tattoos, they were well trained and reacted by closing their eyes, averting their heads, and slinking away.
At the forefront of the Monthlies, Scratch and Sniff were spoiling for a fight. Their tattoos were healing spells, and not of the peace-inducing variety. Since Rusty was vulnerable with his eyes covered, the two troublemakers bounded in for the kill, their pelt jackets flying behind them. They raised their meat cleavers, bared their long fangs, and let their tongues loll out.
Scratch and Sniff sprang in front of the purebred cockatrices—and froze. Their muscles petrified, their eyes widened and then whitened. I heard a cracking sound, like a sheet of ice shattering on a frozen pond. The two troublemakers turned into perfect ferocious-looking statues, as if some sculptor had captured the essence of lycanthropy.
Hearing the warning, Professor Zevon dutifully covered his eyes and stumbled about, calling to his students, "Can someone gather data? It would make a most interesting paper!"
One student said, "I'll do it, Professor! I've got my lab notebook." He raced forward—and also turned to stone.
Even though the werewolf gangs had been mellowed by Antoine's peace tattoos and no longer wanted to claw out clumps of their rivals' fur or sink fangs into their throats, they could still panic. They began to scatter.
The last thing in the world I wanted was to catch an eyeful of ugly. I finally managed to squirm myself free of the barrel and climbed back to my feet, turning my face away.
The beautiful blond, blue-eyed Sheyenne appeared before me, hands on her spectral hips. "You are really having a bad night, Beaux."
"Tell me about it, but it's better now that you're here with me." I was covered in ash. "Don't go near the cockatrices. One glimpse—"
Sheyenne chuckled. "I've already had a look, Beaux. Yes, they are hideous, but what's a cockatrice going to do to a ghost, turn me into ectoplasmic stone? I'm safe—it's you I'm worried about. I don't dare let you sneak a peek."
Rusty stumbled over to us, still pressing the flat of his paw against his eyes. He was chuffing with laughter. "Did you see those two morons? Scratch and Sniff—serves them right!"
"Yeah, great job, Rusty," I said, keeping a hand over my eyes as well. It was an odd way to have a conversation. I cocked my head in the direction where I thought he was standing. "You've caused the deaths of three people, and your deadly cockatrices are on the loose. How do you plan on catching them?"
"I've got burlap bags right by the cages. We just snag 'em and bag 'em."
"And how exactly are you going to do that?" I asked. "One glimpse, and we turn to stone."
"Oh." Rusty paused. "Maybe I'd better have Furguson do it."
Sheyenne retrieved the necktie that Professor Zevon had discarded prior to the fight. "Use this as a blindfold, Beaux."
I lined it up over my eyes and cinched it tight, putting the knot up against the bullet hole in the back of my head, to hold it in place.
"No peeking," she said.
"I feel safer, but how does that help us? I can't see a thing."
I should have known Sheyenne already had a plan. "But you and I are a team. We're going to take the burlap sacks and catch the cockatrices. I'll guide you to them—like in a game of Marco Polo."
"Never was much good at that game," I said.
She guided me to the cages, where she picked up a burlap sack and pushed it into my hands. I fumbled with the rough sack and pulled it open. After the cockatrice fights, Rusty and Furguson had snagged two of the domesticated creatures and stuffed them into sacks without too much trouble. But Rusty and Furguson hadn't been blindfolded at the time, and these purebred cockatrices seemed larger, meaner, and—judging from the stony effect they had on their victims—even uglier.
Amid the panicked turmoil of fleeing werewolves, I heard someone let out a grunt of surprised disgust, then that crackly ice sound again as he turned to stone.
"This way, Beaux," Sheyenne called. I followed her voice, stumbling across the gravel parking lot. "Hurry! They've split up, but this one's close." I heard hissing, a clacking beak, a slither of scales, and the clatter of talons on gravel. I remembered how vicious even their kinder, gentler cousins had been in the fighting ring.
"It's right in front of you, Beaux. Three steps, now hold the sack open!"
I lurched forward, flailing with the open sack. The cockatrice let out a squawk and scuttled in the opposite direction.
"To your left, your left!"
I swooped with the bag and dove forward—only to land in one of the big muddy potholes. At least it washed some of the ash from my slacks. I stumbled back to my feet.
Sheyenne was close. "I'll drive it to you. There—at one o'clock, Beaux! Lunge!"
I dove forward with the burlap sack and, to my astonishment, snagged the creature. It flapped and flailed inside the sack, but I yanked the mouth shut, twisted the burlap around, and made a quick, solid knot.
"One down, Beaux! Good work!"
She whisked away for just a moment, then returned and thrust another burlap bag into my hands. "Come on, two more to go—no time to rest. I've got a spare sack."
"Which way now?" I asked.
"Turn right a little. Then straight ahead. One of them is perched on the new Sniff statue."
"Proud of its work, no doubt," I muttered. I could hear it cawing and hissing like a rooster guarding its territory.
Sheyenne was so intent on chasing the creature herself that she forgot to warn me about obstacles. I barked my shin against a parked truck's bumper. She was ahead of me, calling back. "Come on, follow my voice!"
The cockatrice hissed and squawked, and Sheyenne yelled, "Hey, ugly!" The creature sounded offended at being unable to turn a ghost to stone.
I smelled a foul stench and guessed the cockatrice had opened its bowels to leave a hefty, smoking decoration on the statue's shoulder. I lurched forward and distracted the thing, and Sheyenne managed to throw the sack over it, catching the thrashing, angry creature by surprise. The flailing cockatrice nearly tore free of the burlap, but I helped Sheyenne wrestle it off the statue and down to the ground.
"My poltergeist powers are wearing thin," she said. "I don't usually exert myself for so long."
I fumbled blindly until I tied off the sack and tossed the annoyed cockatrice to the side. It accidentally landed in one of the crater mud puddles. (I would have done it on purpose if I'd been able to see.)
"The third one's getting away, Beaux! Quick—it's almost to Cralo's warehouse!"
I stood up again, holding the sack. "Just guide me to it, Spooky. I'll take care of this one."
I hoped the Emporium delivery doors were closed; I didn't want to imagine a madcap blindfolded chase through the aisles of the spare parts showroom.
"We've got to head it off! Hurry—it's going around back!" Sheyenne shouted.
It's not easy to hurry when you can't see where you're going; nevertheless, I shambled forward, stumbling across the uneven parking lot, tripping in a deep tire rut. I heard a car engine start somewhere behind the Emporium.
The cockatrice squawked and hissed like a turkey being chased by a poodle. I followed, picking up speed; at this point I didn't care whether or not I damaged myself. I was going to have to ask Mavis and Alma Wannovich for an early refresher spell this month. I certainly had some good stories to share with them by way of payment.
Sheyenne floated beside me, telling me to dodge left or right, offering loving encouragement while also urging me to greater speed. Then her voice changed as I heard the engine roar, the tires squeal. "It's Tony Cralo's limo—that fat zombie's getting away!"
I staggered blindly ahead, flailing my hands. Considering the singes from the crematorium, the ashes from the barrel, and the mud from the parking lot puddles, I must have looked like a frightful decomposing shambler.
Hearing the cockatrice, I ran recklessly ahead and sensed the large shape of a limousine coming toward me. I stopped. So did the limo driver—right after the front bumper thumped into me and knocked me flat on my back. The limo windows were down, as if Cralo wanted one last breath of fresh air from the Quarter before he fled underground forever.
Tony Cralo's voice yelled, "What are you stopping for, Rudy? Just run him over, whoever it is!"
"Can't see, boss," said the hunchbacked chauffeur. "We were in such a rush I forgot my seat cushion, and I'm sitting too low. I'd better step out and take a look."
"I'll do it—you move too slow, and we're in a hurry!"
The limo's back door opened, and I heard Cralo emerge, cursing, grumbling, and outgassing. Gravel crunched, and he stood over me. "Wait, that's Chambeaux! How the hell did he get away from the crematorium?"
The last cockatrice was close by. I heard Sheyenne's voice whisper, "Shoo! Shoo!"
Cralo paused, and I heard the cockatrice prowling outside the limo. It came around the car and let out a challenging hiss. "What is that ugly—?"
The crackling-ice petrification sound came louder than I'd heard before as the fat zombie turned to stone with a final disgusted groan and a last toot. Next, it sounded like a felled tree tottering and a loud crash—accompanied by a surprised squawk and a wet squelch.
Then silence.
"Spooky! What just happened?"
"The good news is you can take off the blindfold now, Beaux."
I touched the necktie blindfold, then paused. "We didn't catch the third cockatrice yet."
"Tony Cralo did."
I yanked the necktie from my eyes and blinked.
Near me, the enormous form of Tony Cralo had turned a chalky grayish-white, and the stone zombie had fallen flat on his face. A few feathers and a lumpy claw of the last cockatrice protruded from beneath the toppled statue.
Rudy, the small hunchbacked driver, was slung low in the driver's seat of the limo. His hands clutched the steering wheel, but he could barely peer over the dash, much less see the ground. "So, is it taken care of, Mister Cralo? Should I be driving now?" Rudy strained to raise his head above the steering wheel.
"Your boss isn't going anywhere," I said. In fact, I had no idea how we'd even move him.
CHAPTER 50
By the time McGoo and Robin arrived at the scene—followed by truckloads of armored police loaded for werewolf—the mayhem was all over. The riot squad had been ready to crack down on a rumble, and they were disappointed to find the crisis wrapped up before they found the right address.
McGoo shook his head. "We wasted hours over by the Smile Syndicate warehouse. Nothing was happening. An old security guard tried to shoo us away, but he hadn't heard anything about a rumble."
Seeing how battered I looked, Sheyenne intangibly fussed over my stained jacket and singed hair while Robin checked me over. "You need a good cleaning, Dan, but there's no severe damage that I can see. Nothing that can't be fixed."
Sheyenne hovered next to me. "Every once in a while, Beaux just needs to be reminded that he can't do without me."
"I'd rather not be reminded again soon," I said. "But I'm glad to have your company anytime." I flexed my arms, winced. "Maybe I should try one of those new zombie energy drinks."
McGoo finished pacing around the parking lot crime scene, taking notes for his report. I asked, "Have you been to the crematorium yet? Sheyenne locked Joe Muggins in the bathroom over there. He's another one of the bad guys—stuck me in a furnace."
"I rescued Dan," Sheyenne said. "As usual."
"Attempting to cremate a zombie against his wishes is a serious crime," Robin said. "I can find several relevant statutes."
McGoo nodded. "Sheyenne tipped me off, and I had a team arrest him at the crematorium. I think he's confessed to anyone who'll listen by now. He was just relieved somebody let him out of that bathroom—it has a big, barred east-facing window that's not blacked out. He would've fried in the sunlight if nobody came before dawn."
I snorted. "That might have been the only real ash the crematorium ever produced." I found it hard to feel sympathy for the guy who had shoved me in a cremation furnace and adjusted the settings to Extra Crispy.
I told them the full story about how Archibald Victor was the one scalping werewolves to gather specimens for his experiments, and also how I'd tracked down and destroyed the vampire brain before it could be transplanted. I still felt saddened by the deaths of the mad scientist and his wife, but they had tangled their own fates and there were no more split ends.
We stood together outside the limo, where the enormous stone Cralo lay facedown in the parking lot. A uniformed cop drove a forklift out of the Emporium loading bay and rumbled over to the obese statue.
Lowering the tines and easing forward, the forklift driver raised the heavy stone figure off the ground. The diesel engine groaned, and exhaust fumes curled into the air; the forklift teetered, unbalanced, as the operator drove away with its heavy burden.
"Better not look down at the smashed cockatrice," I said. "Just in case."
"It's fine. The thing is pulped beyond recognition," Sheyenne said. So we all stared anyway. The tangle of feathers, claws, scales, and blood would have been ugly in any configuration, but now at least it wasn't fatally hideous.
McGoo shuddered. "Compared to that thing, the Tony Cralo statue is practically a work of art." He followed the departing forklift as it rolled into the warehouse loading dock.
Robin stared at the big showroom building. "I don't know if the Spare Parts Emporium can be made into a legitimate business, but I'll bring the full force of MLDW to bear. The new management will have to keep very careful records from now on."
"Same with Joe's Crematorium," I said.
Robin was already running possibilities through her mind. "Pending investigation, all the Emporium's inventory will have to be returned to the original owners or their estates. Lacking any legal claimant, maybe we can get the pieces donated to the Fresh Corpses Zombie Rehab Clinic."
"It would be nice to have some good come of this," Sheyenne said.
Near the limo's open door, I caught a glimpse of gold chain on the ground, a flash of yellow metal that Tony Cralo must have dropped when he stepped out of the vehicle.
Before I could bend to see what it was, a voice interrupted me. "Hey, um, Mister Chambeaux?"
I turned to find Larry the werewolf. "It's all over, Larry. You can go back home to your box now."
"I've got something for you—and for Ms. Deyer." The werewolf handed both of us folded packets of paper bound in blue. "You've been served." He backed away quickly. "Sorry, I needed the money." He vanished before Robin could tear open her envelope.
Sheyenne hovered close to see for herself, and Robin looked at me as if she had bitten into a spoiled pickle. "It's a temporary restraining order—we're both required to stay away from Rova Halsted."
"That is correct," came a man's voice, brittle and professional sounding. Don Tuthery stepped out of a nearby shadow, his wire-rimmed glasses glinting, not a gray hair out of place, his conservative business suit painfully immaculate in a setting of so much mayhem. He looked as if he meant to smile in triumph, but couldn't figure out how to make his lips curve in that direction. "There shall be no direct contact. Neither of you is allowed to approach within fifty feet of the Parlour (BNF), and you're forbidden from getting your hair cut there."
I chuckled. "I can live with that. I've taken enough risks already for one week."
Robin recovered her cool and aloof manner. "When you get to your office tomorrow, Mister Tuthery, you'll find a little surprise of your own—I've already received a preliminary ruling."
From the expression on his face, Tuthery was not aware of any ruling, so Robin was happy to explain it to him. "Since our client, Mister Halsted, has expressly clarified his wishes, the judge ruled that fifty thousand dollars in insurance monies—or the equivalent amount in liquid, tangible, and/or capital assets of the Parlour (BNF)—are to be immediately placed into trust in the name of Jordan Halsted until the proceedings are complete. The Court has appointed a Guardian Ad Litem to represent Jordan's best interests and elect potential trustees. And Rova Halsted cannot be a trustee."
Tuthery stared at her. "I'll contest that."
"Go right ahead," Robin said. "The judge has already looked at the Better Business Bureau complaints about the Parlour (BNF) and is not convinced that the beauty salon is a good investment for the child's future."
"Complaints against a business cannot be used in custody issues," Tuthery sputtered. "You know that."
"The haircut snapshots were offered for character reference only," Robin countered.
"Ms. Halsted is a fully qualified stylist!"
Robin smiled. "I invite you to have your client retake the Boards for her license, just to prove her competence in the field."
"Or you could just have her cut your hair," I suggested.
The attorney flushed red. "Your client is not getting visitation rights, and we'll fight him for child support."
"Already arranged, Mister Tuthery—you really should spend more time at your desk," Robin said. "Mister Halsted has agreed to pay reasonable child support, now that he is gainfully employed. And so long as his payments continue as promised, the judge grants him access to the boy."
"But he's a . . . zombie!" Tuthery looked at me as if I'd actually help his cause.
"Not sufficient grounds for discrimination," Robin replied. "Especially not to deny a father the right to see his son."
Though I had been through several ordeals in a row, I felt good as I turned to walk away; Robin slipped her arm through one of mine, and Sheyenne pretended to hold my other one. The three of us strolled across the parking lot, heads held high, leaving a livid Don Tuthery standing by the limo.
I glanced back at him and saw that something had caught his eye. The lawyer bent down beside the limo and picked up a gold chain with a small gleaming medallion that had fallen to the ground—Esther's bad-luck charm. He furtively glanced around to see if anyone had noticed, and slipped the charm "for luck" into his pocket.
I thought about warning him, but decided that wasn't my place. Instead, we walked away, letting Don Tuthery accept the luck that he deserved.
CHAPTER 51
Two days later, Robin had wrapped up and filed the paperwork to establish a charitable werewolves-only wilderness preserve in Montana. When she came into our offices, Miranda Jekyll was pleased, impatient, and aloof—typical Miranda. Robin handed her the documents, already stamped and notarized by Sheyenne, in a nice black binder. "I am proud to announce the Miranda Jekyll Memorial Werewolf Preserve, as you requested."
Perplexed, I pointed out, "You don't usually name a place 'Memorial' something-or-other until after the person is dead."
Miranda laughed and raised her hand, which the ever-present Hirsute seized, kissed, then nibbled, much to her delight. She struggled to continue, "Sweetheart, what good does it do me if people don't remember me now?"
And so the name stayed.
Miranda walked over to the fly-specked window that looked out upon the dingy streets and ramshackle buildings. She sounded wistful. "While I was in the middle of nowhere, I longed to come back to the Quarter. A person can take only so much fresh air, sunshine, and scenery." She turned around and looked at us. "But I forgot what a pain in the ass monsters can be. Good thing Hirsute and I left that unpleasant rumble when we did." She shook her head. "I hear cockatrices are attracted by perfume."
"They would have been drawn to your sparkling personality, as well, my dear," Hirsute said.
"I'm still a city girl at heart, but I may get used to my role as a country bumpkin." Miranda toyed with her pearls. "Scratch and Sniff are going to make adorable statues out at the gateway to the preserve. A matched set! We're having them crated up and delivered in a few days. In fact, Mister Chambeaux, I understand your friend, Steve Halsted, is a reliable truck driver?"
I smiled. "I believe he is."
Miranda said, "I have to think Scratch and Sniff would like to be up at the preserve. And now at least we can control those boys. Sometimes they were too unruly even for werewolves. They deserved what happened to them."
One of Professor Zevon's students and two other werewolf bystanders had also been turned to stone, however, and they hadn't deserved such a fate. Glenn the wizard was now working to develop an anti-petrification spell; someday, maybe they would all be cured.
Anxious to bring the criminals to justice, McGoo was particularly hoping to restore Tony Cralo so the zombie crime lord could face charges, alongside Joe Muggins. Personally, I figured it was best to let the statues stand for themselves.
And Rusty had to make restitution as well for his part in the tragedy.
Miranda said her farewells, took Hirsute's arm, and left our offices with a last invitation. "If you ever want to take a long walk through the deep, dark woods, we've got what you need in Montana. We're even opening the sanctuary to Girl Scout troops for campouts. It gets chilly out there in the forest, so we're providing all of them with nice, warm red riding hoods. It'll be marvelous."
We said we'd consider visiting, but made no promises.
Back at my desk at last, I reviewed pending cases and enjoyed the quiet for about five minutes before I felt the need to go out again. That "restless zombie" thing again. I was not made for a desk job. Real private investigation involves meeting people, keeping ears and eyes open, being aware of when something just doesn't smell right. Like cockatrice poop.
Sheyenne had closed out the case file for Archibald Victor and was now boxing up the mountain of documents from Cralo's Spare Parts Emporium for delivery to the district attorney. She had taken the time to highlight the most incriminating entries with colorful tape flags.
As I walked the streets, I bumped into McGoo out on his beat. He was writing a citation for an illegally parked hearse outside of Bruno and Heinrich's Embalming Parlor. "See you at the Tavern tonight, Shamble?"
"You know I'll be there."
He finished writing the ticket and placed it on the windshield of the hearse. "Do you know what kind of monster is machine washable?"
"You got me, McGoo."
"A wash-and-werewolf."
"Right. See you tonight." We parted company.
I stopped by Voodoo Tattoo to thank Antoine Stickler for his attempt at stopping the werewolf gang violence. When I entered the tattoo parlor, I saw that he had a customer—a hairy-backed red-furred werewolf who was sprawled facedown on the padded table. Rusty turned his muzzle, glanced at me, and let out a long contented sigh.
"Mister Shamble, so nice to see you! I need to pay my bill, don't I?" His voice was languid, drifting, not at all like the gruff and growling werewolf I knew.
"Did you give him valium, Antoine?" I asked.
The Jamaican vampire was humming as he worked with his needle gun. "Even better." The needle whirred as he continued drawing his pattern. "Permanent peace-and-mellowness tattoo—better than antidepressants. Rusty's not gonna let the stress get to him anymore. He'll be mellow all the time."
After taking a deep, slow inhale, Rusty let out the breath. His tongue lolled as he seemed to melt into the padded bench. "I can even tolerate my nephew now. . . ."
Antoine tossed his dreadlocks and grinned. "I can do one for you, too, Mister Chambeaux. Special discount rate for friends."
Permanent peace and mellowness? "There are times I'd like that, but I need to keep my edge. You don't see many mellow private detectives."
"Don't see many zombie detectives either," Antoine said. "Takes all kinds in the Quarter."
"How did you convince Rusty to submit to that?" I asked. "After your peace tattoos ruined his rumble, I thought he'd be angry with you."
"Court order for anger management and restitution," Antoine said. "And I made him a deal. I figured out how to modify the impotence tat. No way to get rid of it—one hell of a persistent spell—but I inked on an extra design . . . makes the impotence impotent, but only for a couple days each month." Antoine winked. "At full moon, he'll be as virile and hairy as ever."
"But won't he be too mellow to go on the prowl?"
"Don't need to go on the prowl." Rusty closed his eyes and relaxed. "They come to me. I've always been a bitch magnet."
"I don't suppose there's an anti-clumsiness tattoo you can use on Furguson?" I asked.
"I've got some ideas cookin'." Antoine smiled and finished connecting a line on the shaved patch on Rusty's shoulder. As I watched, the fur began growing back into place, covering up the lines.
"Glad you stopped by, Mister Chambeaux. You and I, we're men of the same mind-set. I've got something for you, something special—and I don't do this for just anybody." Antoine stood up, patted Rusty on the back. "You relax there awhile."
Rusty nodded slowly. "Relax . . ."
With great pride, the Jamaican vampire walked to the effigy dolls on his shelves. He displayed the lifelike werewolf figures he had so painstakingly made before the rumble. Scratch and Sniff, Rusty, Furguson, a dozen more, both Monthlies and Hairballs. From the top shelf, he took down a gaunt pale-skinned doll with a fedora, a sport jacket complete with tiny stitched-up bullet holes, even a mark in the middle of the forehead.
"Had to work from memory, but this is sure to be effective. Special keepsake with a thousand uses. Made it with angelica root to ward off evil, bring good luck. I even got the skin color right."
I didn't know what to say. I wasn't even sure I wanted the thing. "That's . . . very kind of you, Antoine. Would it be too much trouble to unmake it? I'm not comfortable having that around. What if it fell into the wrong hands?"
Antoine chuckled. "Aww, who would want to do harm to you, Mister Chambeaux?"
"I've already been murdered once, remember?"
He looked disturbed. "I thought that was a one-time thing. Well, I can keep this doll for you here, put it in my portfolio—it's some of my best work."
Though I wasn't sure what use I'd have for my own personal voodoo doll, I didn't like Antoine's solution, either. "I'll take it with me after all, since you did all that work. We'll keep it in our offices." Locked up in the safe where no one else can get at it.
I put the doll in my pocket—gently—and headed out.
CHAPTER 52
When my dirt buddy Steve Halsted came back to the offices, he was all smiles. Accompanying him, his ex-wife Rova was not smiling, but she looked less bitter than before. In fact, she seemed shaken and resigned. Even after all the unpleasantness she had shown toward so many people, we could at least share grief over the death of Harriet Victor.
"Rova and I have business to wrap up before I start my drive," Steve said. "Got a new rig and a long-haul delivery to Montana."
Robin looked uncertainly at Rova Halsted. "Where is your attorney? The restraining orders are still in effect, and we're not allowed to talk with you."
"I had the orders withdrawn this morning. Mister Tuthery wasn't very happy about it. Can I see your copies?" Sheyenne brought them to her, and Rova tore them in half. "We don't need these anymore. Time to settle this, just me and Steve."
"You shouldn't be here without counsel," Robin warned. "I can't vouch for—"
"Don Tuthery tried to grab my ass, then he tripped on a desk mat and fell face-first into a stapler. Broke his nose." Rova snorted.
"Bad luck," I said innocently.
"He always seemed so suave, but now I see him for what he really is. Needless to say, I terminated my relationship with the Addams Family Practice. I'm representing myself as of now. Steve and I have decided to be adults about this."
"That's always the best course of action," Robin said, "though we rarely see it in this business." She led us into the conference room, while Sheyenne brought in the case files. Even for a zombie, Steve had more of a jaunty step.
Robin said, "Ms. Halsted, I need to remind you that a trust must be set up in the name of your son. The liquid and capital assets of the Parlour (BNF) are, per court order, under the control of a trustee until such time as the full amount of fifty thousand dollars is reached."
"We have the money now," Rova said. "Harriet Victor named me the beneficiary of her insurance. It's more than enough to fund Jordan's trust and also keep the business running. I think I'm going to install two more tanning beds."
I doubted that was a wise business decision, since the original tanning beds were never used, but what did I know about running beauty salons?
Steve said, "We both want to make sure Jordan's taken care of. I understand how hard this is for him, but I still want to see my son once a month or so."
"I've decided to accept the situation and make the best of it," Rova said. "A boy needs his father, even if he's a zombie. And Steve has agreed to make himself as presentable as possible before each visit, so it isn't so . . . jarring."
Robin looked back and forth, surprised and relieved. "I'm so happy for all of you."
Steve's eyes were bright. "I saw Jordan yesterday, and we played catch, threw a ball back and forth a few times." He shook his head in wonderment. "I'm not as nimble as a dad should be anymore, but it's still our time together. Jordan even wants to take me to school, for show-and-tell. There's a day where the students get to describe what's special about their dads. Apparently they're all anxious to see me."
Sheyenne said, "I bet they don't usually have a zombie show up in class."
Steve made an embarrassed gesture. "Not because I'm a zombie—Jordan wants to show me off because I'm a truck driver."
These days, Mrs. Saldana spent much of her time at the zombie rehab clinic—and she had extra work to do, with so many body parts from Cralo's Emporium being donated for charity work. But the old woman's first love was her mission.
When I stopped in at Hope & Salvation, I was surprised to see the patchwork but energetic Adriana Cruz working as a volunteer. Despite her scars and awkward movements, her mood was upbeat. She helped Jerry, the mission's undead assistant, fold and stack metal chairs against the wall.
When I entered, Jerry looked up and smiled, showing his rotten teeth. He had been much more lively company since Chambeaux & Deyer tracked down his lost heart and soul.
Mrs. Saldana had just finished her sermon, having given the monsters their breakfast, and everyone left satisfied for another day of existence. Cheerful as usual, the old woman gathered the hymnals to put them away on a library book cart.
"Good to see you back at the mission, Mrs. Saldana," I said.
She brightened. "That'll happen more often now, since I have Adriana to assist over at the rehab clinic. She's a born administrator."
"I've decided to do something with myself," Adriana said. "I can still make a difference. I'm glad I wasn't cremated after all."
"Good to hear. I wanted to let you know that the crematorium is shut down. Joe Muggins is in prison. All of their crimes unraveled."
"You're an inspiration to all of us, Mister Chambeaux," Mrs. Saldana said.
"I am? What did I do?"
Jerry slid out a wooden piano bench and eased himself down upon it. He cracked his knuckles loudly and prepared to play.
Adriana said, "You made me see that I can keep being useful. Just because I'm dead doesn't mean I can't live my life the way I want to."
I was embarrassed, but no longer capable of blushing, thanks to the embalming fluid. "Glad to hear it," I said.
Mavis and Alma Wannovich also called me an inspiration, and they arranged for Linda Bullwer to join us when I stopped by their apartment for the much-needed refresher on the restoration spell.
Mavis bustled about, pleased to have company. "Happy to do it, Mister Chambeaux. We owe you a great deal."
"In royalties?"
"In favors," Mavis corrected.
The plump vampire ghostwriter sat on the sofa with a notebook propped on her lap. "I need to get to work on the next Shamble and Die novel," Linda Bullwer said. "I've decided to frame it around Senator Balfour and his Unnatural Acts Act, tie it together with the Smile Syndicate, even the troubles at the Full Moon Brothel. Readers love titillating storylines. Throw in a little sex, and you increase your sales."
"Sales are quite fine," Mavis said, "from the initial release."
Alma circled the apartment, snuffling along the baseboard. She made a series of loud grunts to remind her sister of something. Mavis said, "Oh, yes, I almost forgot! The reception at the Worldwide Horror Convention was quite positive. The fans loved the book. There's even been talk of awards."
"Awards?" I asked. "Really?"
Linda Bullwer tapped her pen on the notepad. "I'm campaigning behind the scenes, so we'll see. Penny Dreadful may have a long literary career ahead of her." She took off her cat's-eye glasses and wiped them on her fuzzy sweater. "So long as you have a long and interesting career, Mister Chambeaux."
"I'll do my best," I said.
Back in the office, Robin had mounted a certificate of appreciation in a standard black document frame. The award acknowledged her charitable work for the Monster Legal Defense Workers. "It's good to be appreciated," she said.
"I certainly appreciate you," I said. "And Sheyenne. We make a great team."
Sheyenne drifted in with a stack of papers, which she placed on Robin's much-cleaner-than-usual desk. "Now that the Cralo paperwork is gone, this office seems almost clean for a change. Time to start a new pile. Here's the sewer documentation you asked for."
"What sewer documentation?" I asked. "Did we get a new case?"
"MLDW has called a meeting. Zoning troubles with the sewers. The underdwellers are complaining about the amount of effluent."
"Too much or too little?" I asked. In the underworld, it could have been either.
"That's the heart of the dispute," Robin said.
"I'll leave you to it."
I went into my own office, and Sheyenne followed, wearing her flirtatious smile. "I've been resting up my poltergeist energies since the werewolf rumble, Beaux. I think I've got my strength back. I could try on the body suit again tonight . . . if you're interested."
"If it's you, Spooky, how could I not be interested?" I leaned back in my chair as she drifted out of the office toward her desk, where the phone was ringing.
Even with all the murder and mayhem in the Quarter, all the treachery and corruption I exposed, all the slime (or effluent) I had to deal with, this wasn't so bad. You take whatever life—or afterlife—hands you, and you make the best of it.
The office door crashed open, and another monster staggered in, looking distraught—as they always do.
Time to get back to work.
ACKNOWLEDGMENTS
Going over rough draft manuscripts is a grisly job and not for the faint of heart. Rebecca Moesta, Louis Moesta, Deb Ray, Nancy Greene, and Melinda Brown dug down to the bones of this story and offered invaluable commentary and support. My Kensington editor, Michaela Hamilton, sinks her teeth into every Dan Shamble novel, and everyone at Kensington has shown great enthusiasm for the series. Shamble on!
Want to enjoy more zany adventures with Dan Shamble, Zombie P.I.?
Don't miss the previous entries in the series . . .
DEATH WARMED OVER UNNATURAL ACTS STAKEOUT AT THE VAMPIRE CIRCUS
Available from Kensington Publishing Corp.
Keep reading for sample excerpts....
From Death Warmed Over
CHAPTER 1
I'm dead, for starters—it happens. But I'm still ambulatory, and I can still think, still be a contributing member of society (such as it is, these days). And still solve crimes.
As the detective half of Chambeaux & Deyer Investigations, I'm responsible for our caseload, despite being shot in the head a month ago. My unexpected murder caused a lot of inconvenience to me and to others, but I'm not the sort of guy to leave his partner in the lurch. The cases don't solve themselves.
My partner, Robin Deyer, and I had built a decent business for ourselves, sharing office space, with several file cabinets full of pending cases, both legal matters and private investigations. Although catching my own killer is always on my mind, paying clients have to take priority.
Which is why I found myself sneaking into a cemetery at night while trying to elude a werewolf hit man who'd been following me since sunset—in order to retrieve a lost painting for a ghost.
Just another day at work for me.
The wrought-iron cemetery gate stood ajar with a Welcome mat on either side. These days, visiting hours are round-the-clock, and the gate needs to stay open so that newly risen undead can wander out. When the gates were locked, neighbors complained about moaning and banging sounds throughout the night, which made it difficult for them to sleep.
When I pulled, the gate glided open on well-oiled hinges. A small sign on the bars read, MAINTAINED BY FRIENDS OF THE GREENLAWN CEMETERY. There were more than a hundred ostentatious crypts to choose from, interspersed with less prominent tombstones. I wished I had purchased a guide pamphlet ahead of time, but the gift shop was open only on weekends. I had to find the Ricketts crypt on my own—before the werewolf hit man caught up with me.
The world's a crazy place since the Big Uneasy, the event that changed all the rules and allowed a flood of baffled unnaturals to return—zombies, vampires, werewolves, ghouls, succubi, and the usual associated creatures. In the subsequent ten years, the Unnatural Quarter had settled into a community of sorts—one that offered more work than I could handle.
Now the quarter moon rode high in the sky, giving me enough light to see the rest of the cemetery. The unnatural thug, hired by the heirs of Alvin Ricketts, wasn't one of the monthly full-moon-only lycanthropes: He was a full-time hairy, surly beast, regardless of the time of month. Those are usually the meanest ones.
I moved from one crypt to the next, scrutinizing the blocky stone letters. The place was quiet, spooky . . . part of the ambience. You might think a zombie—even a well-preserved one like myself—would feel perfectly at ease in a graveyard. After all, what do I have to be afraid of? Well, I can still get mangled, for one thing. My body doesn't heal the way it used to, and we've all seen those smelly decomposing shamblers who refuse to take care of themselves, giving zombies everywhere a bad name. And werewolves are experts at mangling.
I wanted to avoid that, if possible.
Even undead, I remain as handsome as ever, with the exception of the holes left by the bullet—the largish exit wound on my forehead and the neat round one at the back of my head, where some bastard came up from behind, pressed the barrel of a .32 caliber pistol against my skull, and popped me before I got a good look at him. Fortunately, a low-slouched fedora covers the big hole. For the most part . . .
In the broader sense, the world hasn't changed much since the Big Uneasy. Most people go about their daily lives, having families, working jobs. But though a decade has passed, the law—not to mention law enforcement—still hasn't caught up with the new reality. According to the latest statistics by the DUS, the Department of Unnatural Services, about one out of every seventy-five corpses wakes up as a zombie, with the odds weighted heavily in favor of suicides or murder victims.
Lucky me to be on the interesting side of statistics.
After returning to life, I had shambled back into the office, picked up my caseload, and got to work again. Same as before . . . sort of. Fortunately, my zombie status isn't a handicap to being a private detective in the Unnatural Quarter. As I said, the cases don't solve themselves.
Days of investigation had led me to the graveyard. I dug through files, interviewed witnesses and suspects, met with the ghost artist Alvin Ricketts and separately with his indignant still-living family. (Despite Robin's best mediation efforts in the offices, the ghost and the living family refused to speak to each other.)
Alvin Ricketts was a successful pop-culture painter before his untimely demise, attributable to a month's worth of sleeping pills washed down with a full bottle of twenty-one-year-old single malt. (No sense letting it go to waste.) The ghost told me he would have taken more pills, but his insurance only authorized a thirty-day supply, and even in the deep gloom of his creative depression, Alvin had (on principle) refused to pay the additional pharmacy charge.
Now, whereas one in seventy-five dead people returns as a zombie, like myself, one in thirty comes back as a ghost (statistics again heavily weighted toward murder victims and suicides). Alvin Ricketts, a pop-art genius, had suffered a long and debilitating creative block, "artistic constipation" he called it. Feeling that he had nothing left to live for, he took his own life.
And then came back.
His ghost, however, found the death experience so inspirational that he found a reawakened and vibrant artistic fervor. Alvin set about painting again, announcing he would soon release his first new work with great fanfare.
His grieving (sic) family was less than enthusiastic about his return to painting, as well as his return from the dead. The artist's tragic suicide, and the fact that there would never be more Alvin Ricketts paintings, had caused his existing work to skyrocket in value—until the ghost's announcement of renewed productivity made the bottom fall out of the market. Collectors waited to see what new material Alvin would release, already speculating about how his artistic technique might have changed in his "post-death period."
The Ricketts family sued him, claiming that since Alvin was dead and they were his heirs, they now owned everything in his estate, including any new or undiscovered works and the profits from subsequent sales.
Alvin contested the claim. He hired Robin Deyer to fight for his rights, and she promptly filed challenges while the ghost happily worked on his new painting. No one had yet seen it, but he claimed the work was his masterpiece.
The Ricketts heirs took the dispute to the next level. "Someone" broke into Alvin's studio and stole the painting. With the supposed masterpiece gone, the pop artist's much-anticipated return to the spotlight was put on hold. The family vehemently denied any involvement, of course.
That's when the ghost hired me, at Robin's suggestion, to track down and retrieve the painting—by any means necessary. The Ricketts heirs had hired a thug to keep me from succeeding in my investigation.
I heard a faint clang, which I recognized as the wrought-iron cemetery gate banging shut against the frame. The werewolf hit man wasn't far behind me. On the bright side, the fact that he was breathing down my neck probably meant I was getting close.
The cemetery had plenty of shadows to choose from, and I stayed hidden as I approached another crypt. BENSON. Not the right one. I had to find RICKETTS.
Werewolves are usually good trackers, but the cemetery abounds with odors of dead things, and he must have kept losing my scent. Since I change clothes frequently and maintain high standards of personal hygiene for a zombie, I don't have much of a smell about me. Unlike most unnaturals, I don't choose to wear colognes, fancy specialized unnatural deodorants, or perfumes.
I turned the corner in front of another low stone building fronted by stubby Corinthian columns. Much to my delight, I saw the inhabitant's name: RICKETTS. The flat stone door had been pried open, the caulking seal split apart.
New rules required quick-release latches on the insides of tombs now, so the undead can conveniently get back out. Some people were even buried with their cell phones, though I doubted they'd get good service from inside. Can you hear me now?
Now, if Alvin Ricketts were a zombie, he would have broken the seal when he came back out of the crypt. But since ghosts can pass through solid walls, Alvin would not have needed to break any door seals for his reemergence. So why was the crypt door ajar?
I spotted the silhouette of a large hairy form loping among the graves, sniffing the ground, coming closer. He still hadn't seen me. I pulled open the stone door just enough to slip through the narrow gap into the crypt, hoping my detective work was right.
During the investigation into the missing masterpiece, the police had obtained search warrants and combed through the homes, properties, and businesses of the Ricketts heirs. Nothing. With my own digging, I discovered a small storage unit that had been rented in the name of Gomez Ricketts, the black sheep of the family—and I was sure they had hidden the painting there.
But when the detectives served their warrant and opened the unit, they found only cases and cases of contraband vampire porn packaged as sick kiddie porn. Because the starlets were actually old-school vampires who had been turned while they were children, they claimed to be well over the legal age—in real years if not in physical maturity. Gomez Ricketts had been arrested for pedophilia/necrophilia, but he was out on bail. Even Robin, in her best legal opinion, couldn't say which way the verdict might go.
More to the point, we didn't find the stolen painting in the storage unit.
So I kept working on the case. Not only did I consult with Alvin's ghost, I also went over the interviews he'd given after his suicide. The ghost had gone into a manic phase, deliriously happy to put death behind him. He talked about awakening to find himself sealed in a crypt, his astral form rising from the cold physical body, his epiphany of throwing those morbid chains behind him. He had vowed never to go back there.
That's when I figured it out: The last place Alvin would ever think to look for his painting was inside his own crypt, which was property owned by the Ricketts family (though a recent court ruling deemed that a person owned his own grave in perpetuity—a landmark decision that benefitted several vampires who were caught in property-rights disputes).
Tonight, I planned to retrieve the painting from its hiding place.
From Unnatural Acts
CHAPTER 1
I never thought a golem could make me cry, but hearing the big clay guy's sad story brought a tear to my normally bloodshot eyes. My business partner Robin, a lawyer (but don't hold it against her), was weeping openly.
"It's so tragic!" she sniffled.
"Well, I certainly thought so," the golem said, lowering his sculpted head, "but I'm biased."
He had lurched into the offices of Chambeaux & Deyer Investigations with the ponderous and inexorable gait that all golems have. "Please," he said, "you've got to help me!"
In my business, most clients introduce themselves like that. It's not that they don't have any manners, but a person doesn't engage the services of a private investigator, or a lawyer, as an ordinary social activity. Our visitors generally come pre-loaded with problems. Robin and I were used to it.
Then, swaying on his thick feet, the golem added, "And you've got to help my people."
Now, that was something new.
Golems are man-sized creatures fashioned out of clay and brought to life by an animation spell. Tailor-made for menial labor, they serve their masters and don't complain about minimum wage (or less, no tips). Traditionally, the creatures are statuesque and bulky, their appearance ranging from store-mannequin smooth to early Claymation, depending on the skill of the sculptor-magician who created them. I've seen do-it-yourself kits on the market, complete with facial molds and step-by-step instructions.
This golem was in bad shape: dried and flaking, his gray skin fissured with cracks. His features were rounded, generic, and less distinctive than a bargain-store dummy's. His brow was furrowed, his chapped gray lips pressed down in a frown. He tottered, and I feared he would crumble right there in the lobby area.
Robin hurried out of her office. "Please, come in, sir. We can see you right away."
Robin Deyer is a young African American woman with anime-worthy brown eyes, a big heart, and a feisty disposition. She and I had formed a loose partnership in the Unnatural Quarter, sharing office space and cooperating on cases. We have plenty of clients, plenty of job security, plenty of headaches. Unnaturals have problems just like anyone else, but zombies, vampires, werewolves, witches, ghouls, and the gamut of monsters are underrepresented in the legal system. That's more than enough cases, if you can handle the odd clientele and the unusual problems.
Since I'm a zombie myself, I fit right in.
I stepped toward the golem and shook his hand. His grip was firm but powdery. "My partner and I would be happy to listen to your case, Mr. . . . ?"
"I don't actually know my name. Sorry." His frown deepened like a character in a cartoon special. "Could you read it for me?" He slowly turned around. In standard magical manufacturing, a golem's name is etched in the soft clay on the back of his neck, where he can never see it for himself. "None of my fellow golems could read. We're budget models."
There it was, in block letters. "It says your name is Bill."
"Oh. I like that name." His frown softened, although the clay face was too stiff to be overly expressive. He stepped forward, disoriented. "Could I have some water, please?"
Sheyenne, the beautiful blond ghost who served as our receptionist, office manager, paralegal, business advisor, and whatever other titles she wanted to come up with, flitted to the kitchenette and returned with some sparkling water that Robin kept in the office refrigerator. The golem took the bottle from Sheyenne's translucent hands and unceremoniously poured it over his skin. "Oh, bubbly! That tingles."
It wasn't what I'd expected him to do, but we were used to unusual clients.
When I'd first hung out my shingle as a PI, I'd still been human, albeit jaded—not quite down-and-out, but willing to consider a nontraditional client base. Robin and I worked together for years in the Quarter, garnering a decent reputation with our work . . . and then I got shot in the back of the head during a case gone wrong. Fortunately, being killed didn't end my career. Ever since the Big Uneasy, staying dead isn't as common as it used to be. I returned from the grave, cleaned myself up, changed clothes, and got back to work. The cases don't solve themselves.
Thanks to high-quality embalming and meticulous personal care, I'm well preserved, not one of those rotting shamblers that give the undead such a bad name. Even with my pallid skin, the shadows under my eyes aren't too bad, and mortician's putty covers up the bullet's entry and exit holes in my skull, for the most part.
Bill massaged the moistened clay, smoothed the cracks and fissures of his skin, and let out a contented sigh. He splashed more water on his face, and his expression brightened. "That's better! Little things can improve life in large ways." After wiping his cheeks and eyes with the last drops of sparkling water, he became more animated. "Is that so much to ask? Civil treatment? Human decency? It wouldn't even cost much. But my people have to endure the most appalling conditions! It's a crime, plain and simple."
He swiveled around to include Robin, Sheyenne, and me. "That's why I came to you. Although I escaped, my people remain enslaved, working under miserable conditions. Please help us!" He deepened his voice, growing more serious. "I know I can count on Chambeaux and Deyer."
Now that the bottle of sparkling water was empty, Sheyenne returned with a glass of tap water, which the golem accepted. She wasn't going to give him the expensive stuff anymore if he was just going to pour it all over his body. "Was there anyone in particular who referred you to us?" she asked.
"I saw your name on a tourist map. Everyone in town knows Chambeaux and Deyer gives unnaturals a fair shake when there's trouble." He held out a rumpled, folded giveaway map carried by many businesses in the Quarter, more remarkable for its cartoon pictures and cheerful drawings than its cartographic detail.
Sheyenne flashed me a dazzling smile. "See, Beaux? I told you our ad on the chamber-of-commerce map would be worth the investment." Beaux is Sheyenne's pet name for me; no one else gets to call me that. (Come to think of it, no one had ever tried.)
"I thought you couldn't read, Bill," I said.
"I can look at the pictures, and the shop had an old vampire proofreader who mumbled aloud as he read the words," Bill said. "As a golem, you hear things."
"The important thing is that Mr., uh, Bill found us," Robin said. She had been sold on the case as soon as the golem told us his plight. If it weren't for Sheyenne looking out for us, Robin would be inclined to embrace any client in trouble, whether or not he, she, or it could pay.
Since joining us, postmortem, Sheyenne had worked tirelessly—not that ghosts got tired—to manage our business and keep Chambeaux & Deyer in the black. I didn't know what I'd do without her, professionally or personally.
Before her death, Sheyenne had been a med student, working her way through school as a cocktail waitress and occasional nightclub singer at one of the Unnatural Quarter high-end establishments. She and I had a thing in life, a relationship with real potential, but that had been snuffed out when Sheyenne was murdered, and then me, too.
Thus, our romance was an uphill struggle.
While it's corny to talk about "undying love," Fate gave us a second chance . . . then blew us a big loud raspberry. Sheyenne and I each came back from the dead in our respective way—me as a zombie, and Sheyenne as a ghost—but ghosts can never touch any living, or formerly living, person. So much for the physical side of our relationship . . . but I still like having her around.
Now that he was moisturized, Bill the golem seemed a new person, and he no longer flaked off mud as he followed Robin into our conference room. She carried a yellow legal pad, ready to take notes. Since it wasn't yet clear whether the golem needed a detective, an attorney, or both, I joined them. Sheyenne brought more water, a whole pitcher this time. We let Bill have it all.
Golems aren't the smartest action figures in the toy box—they don't need to be—but even though Bill was uneducated, he wasn't unintelligent, and he had a very strong sense of right and wrong. When he started talking, his passion for Justice was apparent. I realized he would make a powerful witness. Robin fell for him right away; he was just her type of client.
"There are a hundred other disenfranchised golems just like me," Bill said. "Living in miserable conditions, slaves in a sweatshop, brought to life and put to work."
"Who created you?" I asked. "Where is this sweatshop located? And what work did you do?"
Bill's clay brain could not hold three questions at a time, so he answered only two of them. "We manufacture Unnatural Quarter souvenirs—vampire ashtrays made with real vampire ash, T-shirts, place mats, paperweights, holders for toothpicks marketed as 'stakes for itsy-bitsy vampires.' "
Several new gift shops had recently opened up in the Quarter, a chain called Kreepsakes. All those inane souvenirs had to come from somewhere.
More than a decade after the Big Uneasy brought back all the legendary monsters, normal humans had recovered from their shock and horror enough that a few tourists ventured into the Quarter. This had never been the best part of town, even without the monsters, but businesses welcomed the increased tourism as an unexpected form of urban renewal.
"Our master is a necromancer who calls himself Maximus Max," Bill continued. "The golems are mass produced, slapped together from uneven clay, then awakened with a bootleg animation spell that he runs off on an old smelly mimeograph. Shoddy work, but he doesn't care. He's a slave driver!"
Robin grew more incensed. "This is outrageous! How can he get away with this right out in the open?"
"Not exactly out in the open. We labor in an underground chamber, badly lit, no ventilation . . . not even an employee break room. Through lack of routine maintenance, we dry out and crumble." He bent his big blunt fingers, straightened them, then dipped his hand into the pitcher of water, where he left a murky residue. "We suffer constant aches and pains. As the mimeographed animation spell fades, we can't move very well. Eventually, we fall apart. I've seen many coworkers and friends just crumble on the job. Then other golems have to sweep up the mess and dump it into a bin, while Maximus Max whips up a new batch of clay so he can create more golems. No one lasts very long."
"That's monstrous." Robin took detailed notes. She looked up and said in a soft, compassionate voice, "And how did you escape, Bill?"
The golem shuddered. "There was an accident on the bottling line. When a batch of our Fires of Hell hot sauce melted the glass bottles and corroded the labeling machine, three of my golem friends had to clean up the mess. But the hot sauce ruined them, too, and they fell apart.
"I was in the second-wave cleanup crew, shoveling the mess into a wheelbarrow. Max commanded me to empty it into a Dumpster in the alley above, but he forgot to command me to come back. So when I was done, I just walked away." Bill hung his head. "But my people are still there, still enslaved. Can you free them? Stop the suffering?"
I addressed the golem. "Why didn't you go to the police when you escaped?"
Bill blinked his big artificial eyes, now that he was more moisturized. "Would they have listened to me? I don't have any papers. Legally speaking, I'm the necromancer's property."
Robin dabbed her eyes with a tissue and pushed her legal pad aside. "It sounds like a civil rights lawsuit in the making, Bill. We can investigate Maximus Max's sweatshop for conformance to workplace safety codes. Armed with that information, I'll find a sympathetic judge and file an injunction to stop the work line temporarily."
Bill was disappointed. "But how long will that take? They need help now!"
"I think he was hoping for something more immediate, Robin," I suggested. "I'll talk to Officer McGoohan, see if he'll raid the place . . . but even that might be a day or two."
The golem's face showed increasing alarm. "I can't stay here—I'm not safe! Maximus Max will be looking for me. He'll know where to find me."
"How?" Sheyenne asked, sounding skeptical.
"I'm an escaped golem looking for action and legal representation—where else would I go but Chambeaux and Deyer? That's what the tourist map says."
"I've got an idea," I said. "Spooky, call Tiffany and tell her I'll come to her comedy improv show if she does me a quick favor."
Sheyenne responded with an impish grin. "Good idea, Beaux."
Tiffany was the buffest—and butchest—vampire I'd ever met. She had a gruff demeanor and treated her life with the utmost seriousness the second time around. But she had more of a sense of humor than I originally thought. Earlier that afternoon, Tiffany had dropped in, wearing a grin that showed her white fangs; she waved a pack of tickets and asked if we'd come see her for open-mic night at the Laughing Skull, a comedy club down in Little Transylvania. Maybe we could trade favors....
I knew Tiffany from the All-Day/All-Nite Fitness Center, where I tried to keep myself in shape. Zombies don't have to worry about cholesterol levels or love handles, but it's important to maintain muscle tone and flexibility. The aftereffects of death can substantially impact one's quality of life. I worked out regularly, but Tiffany was downright obsessive about it. She said she could bench-press a coffin filled with lead bricks (though why she would want to, I couldn't say).
Like many vampires, Tiffany had invested well and didn't need a regular job, but due to her intimidating physique, I kept her in mind in case I ever needed extra muscle. I'd never tried to call in a favor before, but Sheyenne was very persuasive.
Tiffany the vampire walked through the door wearing a denim work shirt and jeans. She had narrow hips, square shoulders, no waist, all muscle. She looked as if she'd been assembled from solid concrete blocks; if any foolish vampire slayer had tried to pound a stake through her heart, it would have splintered into toothpicks.
Tiffany said gruffly, "Tell me what you got, Chambeaux." When Bill emerged from the conference room, she eyed him up and down. "You're a big boy."
"I was made that way. Mr. Chambeaux said you can keep me safe."
After I explained the situation, she said, "Sure, I'll give you a place to stay. Hang out at my house for a few days until this blows over." Tiffany glanced at me, raised her eyebrows. "A few days—right, Chambeaux?"
Robin answered for me. "That should be all we need to start the legal proceedings."
Bill's clay lips rolled upward in a genuine smile now. "My people and I are indebted to you, Miss Tiffany."
"No debt involved. Actually, I could use a hand if you don't mind pitching in. I'm doing some remodeling at home, installing shelves, flooring, and a workbench in the garage, plus dark paneling and a wet bar in the basement den. I also need help setting up some heavy tools I ordered—circular saw and drill press, that kind of thing."
"I would be happy to help," Bill agreed.
"Thanks for the favor, Tiffany," I said.
The vampire gave me a brusque nod. "Don't worry, he'll be putty in my hands."
From Stakeout at the Vampire Circus
CHAPTER 1
The circus is supposed to be fun, even a monster circus, but the experience turned sour when somebody tried to murder the vampire trapeze artist.
As a private detective, albeit a zombie, I investigate cases of all sorts in the Unnatural Quarter, applying my deductive skills and persistent determination (yes, the undead can be very persistent indeed). Some of my cases are admittedly strange; most are even stranger than that.
I'd been hired by a transvestite fortune-teller to find a stolen deck of magic cards, and he had sent me two free tickets to the circus. Gotta love the perks of the job. Not one to let an opportunity go to waste, I invited my girlfriend to accompany me; in many ways her detective skills are as good as my own.
Sheyenne is beautiful, blond, and intangible. I had started to fall in love with her when both of us were alive, and I still like having her around, despite the difficulties of an unnatural relationship—as a ghost, she can't physically touch me, and as a zombie I have my own limitations.
We showed our passes at the circus entrance gate and entered a whirlwind of colors, sounds, smells. Big tents, wild rides, popcorn and cotton candy for the humans, more exotic treats for the unnaturals. One booth sold deep-fried artichoke hearts, while another sold deep-fried human hearts. Seeing me shamble by, a persistent vendor offered me a free sample of brains on a stick, but I politely declined.
I'm a well-preserved zombie and have never acquired a taste for brains. I've got my standards of behavior, not to mention personal hygiene. Given a little bit of care and effort, a zombie doesn't have to rot and fall apart, and I take pride in looking mostly human. Some people have even called me handsome—Sheyenne certainly does, but she's biased.
As Sheyenne flitted past the line of food stalls, her eyes were bright, her smile dazzling; I could imagine what she must have looked like as a little girl. I hadn't seen her this happy since she'd been poisoned to death.
Nearby, a muscular clay golem lifted a wooden mallet at the Test Your Strength game and slammed it down with such force that he not only rang the bell at the top of the pole, he split the mallet in half. A troll barker at the game muttered and handed the golem a pink plush bunny as a prize. The golem set the stuffed animal next to a pile of fuzzy prizes, paid another few coins, and took a fresh mallet to play the game again.
Many of the attendees were humans, attracted by the low prices of the human matinee; the nocturnal monsters would come out for the evening show. More than a decade had passed since the Big Uneasy, when all the legendary monsters came back to the world, and human society was finally realizing that unnaturals were people just like everyone else. Yes, some were ferocious and bloodthirsty—but so were some humans. Most monsters just wanted to live and let live (even though the definition of "living" had blurred).
Sheyenne saw crowds streaming toward the Big Top. "The lion tamer should be finishing, but the vampire trapeze artist is due to start. Do you think we could . . ."
I gave her my best smile. With stiff facial muscles, my "best smile" was only average, but even so, I saved it for Sheyenne. "Sure, Spooky. We've got an hour before we're supposed to meet Zelda. Let's call it 'gathering background information.'"
"Or we could just call it part of the date," Sheyenne teased.
"That, too."
We followed other humans through the tent flaps. A pudgy twelve-year-old boy was harassing his sister, poking her arm incessantly, until he glanced at me and Sheyenne. I had pulled the fedora low, but it didn't entirely conceal the bullet hole in my forehead. When the pudgy kid gawked at the sight, his sister took advantage of the distraction and began poking him until their mother hurried them into the Big Top.
Inside, Sheyenne pointed to empty bleachers not far from the entrance. The thick canvas kept out direct sunlight, protecting the vampire performers and shrouding the interior in a pleasant nighttime gloom. My eyes adjusted quickly, because gloom is a natural state for me. Always on the case, I remained alert. If I'd been more alert while I was still alive, I would be . . . well, still alive.
When I was a human private detective in the Quarter, Sheyenne's ghost had asked me to investigate her murder, which got me in trouble; I didn't even see the creep come up behind me in a dark alley, put a gun to the back of my head, and pull the trigger.
Under most circumstances, that would have put an end to my career, but you can't keep a good detective down. Thanks to the changed world, I came back from the dead, back on the case. Soon enough, I fell into my old routine, investigating mysteries wherever they might take me . . . even to the circus.
Sheyenne drifted to the nearest bleacher, and I climbed stiffly beside her. The spotlight shone down on a side ring, where a brown-furred werewolf in a scarlet vest—Calvin—cracked his bullwhip, snarling right back at a pair of snarling lions who failed to follow his commands. The thick-maned male cat growled, while the big female opened her mouth wide to show a yawn full of fangs. The lion tamer roared a response, cracked the whip again, and urged the big cats to do tricks, but they absolutely refused.
The lions flexed their claws, and the werewolf flexed his own in a show of dominance, but the lions weren't buying it. Just when it looked as if the fur was about to fly, a loud drumroll came from the center ring.
The spotlight swiveled away from the lion tamer to fall upon the ringmaster, a tall vampire with steel-gray hair. "Ladies and gentlemen, naturals and unnaturals of all ages—in the center ring, our main event!" He pointed upward, and the spotlight swung to the cavernous tent's rigging strung with high wires and a trapeze platform. A Baryshnikov look-alike stood on the platform, a gymnastic vampire in a silver lamé full-body leotard. He wore a medallion around his neck, a bright red ribbon with some kind of amulet, and a professional sneer.
"Bela, our vampire trapeze artist, master of the ropes—graceful, talented . . . a real swinger!" The ringmaster paused until the audience realized they were supposed to respond with polite laughter. Up on the platform, Bela lifted his chin, as if their applause was beneath him (and, technically speaking, it was, since the bleachers were far below).
"For his death-defying feat, Bela will perform without a safety net above one hundred sharpened wooden stakes!" The spotlight swung down to the floor of the ring, which was covered with a forest of pointy sticks, just waiting to perform impalement duties.
The suitably impressed audience gasped.
On the trapeze platform, Bela's haughty sneer was wide enough to show his fangs; I could see them even from my seat in the bleachers. The gold medallion at his neck glinted in the spotlight. Rolling his shoulders to loosen up, the vampire grasped the trapeze handle and lunged out into the open air. He seemed not to care a whit about the sharp wooden stakes as he swung across to the other side. At the apex of his arc, he swung back again, gaining speed. On the backswing, Bela spun around the trapeze bar, doing a loop. As he reached the apex once again, he released, did a quick somersault high in the air, and caught the bar as he dropped down.
The audience applauded. Werewolves in the bleachers howled their appreciation; some ghouls and less-well-preserved zombies let out long, low moans that sounded upbeat, considering. I shot a glance at Sheyenne, and judging by her delighted expression, she seemed to be enjoying herself.
Bela swung back, hanging on with one hand as he gave a dismissive wave to the audience. Vampires usually have fluid movements. I remembered that one vamp had tried out for the Olympic gymnastics team four years ago—and was promptly disqualified, though the Olympic judges could not articulate a valid reason. The vampire sued, and the matter was tied up in the courts until long past the conclusion of the Olympics. The vampire gymnast took the long view, however, as she would be just as spry and healthy in the next four-year cycle, and the next, and the next.
A big drumroll signaled Bela's finale. He swung back and forth one more time, pumping with his legs, increasing speed, and the bar soared up to the highest point yet. The vampire released his hold, flung himself into the air for another somersault, then a second, then a third as the empty trapeze swung in its clockwork arc, gliding back toward him, all perfectly choreographed.
As he dropped, Bela reached out. His fingertips brushed the bar—and missed. He flailed his hands in the air, trying to grab the trapeze, but the bar swung past out of reach, and gravity did its work. Bela tumbled toward the hundred sharp wooden stakes below.
Someone screamed. Even with my rigor-mortis-stiff knees, I lurched to my feet.
But at the last possible moment, the vampire's plummeting form transformed in the air. Mere inches above the deadly points, Bela turned into a bat, stretching and flapping his leathery wings. He flew away, the medallion still dangling from his little furry rodent neck. He alighted on the opposite trapeze platform, then transformed back into a vampire just in time to catch the returning trapeze. He held on, showing his pointed fangs in a superior grin, and took a deep bow. On cue, the band played a loud "Ta-da!"
After a stunned moment, the audience erupted in wild applause. Sheyenne was beaming enough to make her ectoplasm glow. Even I was smiling. "That was worth the price of admission," I said.
Sheyenne looked at me. "We didn't pay anything—we got free tickets."
"Then it's worth twice as much."
With the show over, the audience rose from the bleachers and filed toward the exit. "The cases don't solve themselves," I said to Sheyenne. "Let's go find that fortune-teller."
KENSINGTON BOOKS are published by
Kensington Publishing Corp.
119 West 40th Street
New York, NY 10018
Copyright © 2013 by WordFire, Inc.
All rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.
Kensington and the K logo Reg. U.S. Pat. & TM Off.
ISBN: 978-0-7582-7738-1
eISBN-13: 978-0-7582-7739-8
eISBN-10: 0-7582-7739-3
First Kensington Electronic Edition: May 2013
Notes
starring Dan Shamble, Zombie P.I.
|
{
"redpajama_set_name": "RedPajamaBook"
}
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\section{Introduction}
Given the unrenormalizability of the 4D Einstein-Hilbert action, the semi-classical description has been widely used in the black hole literature. (See, e.g., \cite{Birrell} and \cite{muk} for reviews.)
This description led to the discovery of Hawking radiation and many other useful results, and we believe it is essential to go beyond it to solve Black Hole Information (BHI) problem and surrounding issues.
The validity of the semi-classical description is one of the four postulates of black hole complementarity (BHC) \cite{Susskind:1993if}. However,
the postulates' mutual compatibility has been questioned in the recent work of \cite{Almheiri:2012rt}, according to which one or more of the four postulates of BHC must break down. In particular, an infalling observer will experience a firewall when crossing the event horizon of a sufficiently old black hole in `violation'\footnote{What is being challenged might not be Equivalence Principle itself but the conventional lore that Equivalence Principle implies a smooth horizon. We will come {back} to this in the conclusion.} of the Equivalence Principle.
In this work, we analyze a 2D theory to study the behavior of the stretched horizon as observed by an infalling observer.
The 2D theory that we consider describes the quantum fluctuations of the selected 2D hypersurface of the BTZ spacetime \cite{Banados:1992wn}.\footnote{Recent works that tackle black hole information problem by analyzing concrete models include \cite{Kawai:2013mda,Berenstein:2013tya,Silverstein:2014yza}.} (See \cite{Carlip:1998qw} for review of the BTZ black hole.) It was proposed in \cite{Hatefi:2012bp} that it should be possible to tackle the BHI problem without directly dealing with the quantization issue of the 3D or 4D gravity. One may tackle the BHI after first reducing the theory to a selected 2D hypersurface through the procedure called dimensional reduction to a hypersurface of foliation (or ADM reduction for short).
The ADM reduction is a variation of (but significantly different from) the standard Kaluza-Klein reduction.
The procedure was motivated by the endeavor to derive AdS/CFT from the first principle. To be specific, let us take the prime example, AdS$_5$/CFT$_4$.
The CFT, ${\cal N}=4$ SYM in this case, can be viewed as the theory of the hypersurface of AdS$_5$ at $r=\infty$.
By applying the procedure in the Hamilton-Jacobi formulation, it was shown that the 5D
AdS gravity admits a class of solutions with a ``moduli field", which in turn was identified as the abelian worldvolume (i.e., the hypersurface at a fixed $r$) gauge field \cite{Sato:2002kv}\cite{Hatefi:2012bp}. This may be viewed as the way in which the actual dualization of the bulk theory to the boundary theory should generally work.\footnote{ With the result in \cite{GonzalezRey:1998uh} and conjecture in \cite{Schwarz:2013wra} combined, the derivation of AdS$_5$/CFT$_4$
seems within close reach. The author thanks H.-S. Yang for discussion on this point.}
The ADM reduction has subsequently been applied to 4D Einstein-Hilbert action \cite{Park:2013iqa}\cite{Park:2013vpa} and 3D AdS gravity action \cite{Park:2013bma}. In this work, we build on the latter case. There has been expectation in the literature \cite{Coussaert:1995zp,Frolov:1999my,Krasnov:2000ia,Giacomini:2003cg,
Chen:2003si,Yuan:2011gq,Nakatsu:1999wt} that the BTZ geometry should be associated with Liouville theory \cite{Belavin:1984vu}.\footnote{{As well known, 3D pure gravity does not have propagating degrees of freedom. Then there should be two possibilities regarding what is responsible for the Liouville degrees of freedom. Firstly, it could be non-perturbative degrees of freedom. Secondly, they could be associated with some type of "boundary" degrees of freedom. Out work is in line with the second possibility as we further contemplate in the conclusion.} }
This expectation has been confirmed in the ADM setup \cite{Park:2013bma}: the theory reduced along the $\varphi$ (respectively, $r$) direction has turned out to be curved-space Liouville (respectively, `flat-space' Liouville) theory. (Liouville theory was obtained in \cite{Solodukhin:1998tc} by the standard dimensional reduction in the 4D Einstein-Hilbert case.)
In this work, we take the Liouville theory resulting from the $\varphi$-reduction and analyze its implications for black hole physics. In particular, we study its implications for the behavior of the surface that we identify as the stretched horizon. We compute the zero-mode contribution to the expectation value of the 2D stress-energy tensor. Although the Liouville theory is super-renormalizable, various issues pertaining to detailed renormalization procedure are still present. We outline the procedure for computing the rest of the contributions (i.e., nonzero-mode contributions).
In order to examine the behavior of the stretched horizon, we consider the coordinate-invariant quantity that involves the 2D stress-energy tensor $T^S_{ab},\; a=(t,r)$:
\begin{eqnarray}
<S|T^S_{ab}|S> U_S^a U_S^b \label{mo}
\end{eqnarray}
where `$S$' denotes the Schwarzschild coordinates and $U_S^a$ is the timelike geodesic.
(A recent firewall-related analysis of the vacuum expectation value of the stress tensor appeared in \cite{Lowe:2013zxa}. See, e.g., \cite{Unruh:1976db,Fulling:1977zs,GP}\cite{Birrell} for early discussions.)
This quantity is the energy density as measured by an infalling observer.
As wellknown, Liouville theory has a central charge.
We show that the central charge term induces the invariant quantity above to generate the Planck scale energy density at the stretched horizon, a behavior consistent with the idea of the unsmooth horizon \cite{Braunstein:2009my}\cite{Almheiri:2012rt}.
\vspace{.3in}
The rest of the paper is organized as follows. In section 2, we review the $\varphi$-reduction of the BTZ spacetime.
In section 3, we compute the zero-mode contribution to \rf{mo}.
Conceptually, narrowing down to the zero-mode contribution means an additional ADM reduction along the $r$ direction, and one gets to deal with the quantum mechanics of the `point stretched horizon', i.e., stretched horizon of zero spatial dimension.
Once the zero-mode result gets contracted with the {timelike} geodesic, one gets the Planck scale energy. The computation of the genuine 2D contributions is outlined. We end with discussion and future directions.
\section{Review of $\varphi$-reduction}
In the next section which is the main body of the work, we identify the stretched horizon and analyze its dynamics by computing the zero-mode contribution to the
vacuum expectation value of the 2D stress-energy tensor.
To set the stage for the next section, we obtain the 2D theory by applying the ADM reduction technique along the angular direction \cite{Park:2013bma}.\footnote{Here we do not carefully trace the terms that arise from the virtual boundary contribution because they will not play a role in the 2D stress-energy tensor. (Such terms were carefully dealt with in \cite{Park:2013vpa} and \cite{Park:2013bma}.)}
\vspace{.2in}
Let us consider the 3D action
\begin{equation}
S=\int\, d^3x\, \sqrt{-\tilde{g}^{\sst{(3)}}}\left[\tilde{R}^{\sst{(3)}}+\fr2{l^2} \right]
\label{AdS3act}
\end{equation}
where $l$ is the AdS length scale.
By employing the ADM formalism, the metric can be put in the form,
\begin{eqnarray}
ds_3^2=(\tilde{n}_\varphi^2+\tilde{h}^{ab}\tilde{N}_a \tilde{N}_b)d\varphi^2+2\tilde{N}_a d\varphi dx^a+\tilde{h}_{ab}dx^adx^b,
\quad a=t,r
\end{eqnarray}
In this formalism, the 3D action takes
\begin{eqnarray}
S&=&\int d^2x d\varphi \, \sqrt{-\tilde{h}}\;\tilde{n}_\varphi\left[\tilde{R}^{\sst{(2)}}+\tilde{K}^2-\tilde{K}_{ab}\tilde{K}^{ab}
-2\L\right] \label{3drescorphi}
\end{eqnarray}
$\tilde{N}_a$ can be {gauge-fixed} to $\tilde{N}_a=0$.
Let us rescale the 3D metric $\tilde{h}_{\m\n}$ (where $\m=(a,\varphi)=(t,r,\varphi)$) by
\begin{eqnarray}
\tilde{h}_{\m\n}=e^{2\tilde{\rho}(t,r,\varphi)}h_{\m\n}
\end{eqnarray}
The field $\tilde{\rho}$ is related to the lapse function associated with $h_{\m\n}$: $n_\varphi$ is defined as ${{n}}_\varphi(t,r,{ \varphi}) \equiv e^{{ -}\rho(t,r)}$ {where $\rho(t,r)$ is a dimensionally reduced $\tilde{\rho}(t,r,\varphi)$ field.}
This leads to
\begin{eqnarray}
S=\int d^2x d\varphi \, \sqrt{-h_{\sst{(2)}}}\;\Big[R^{\sst{(2)}}+K^2-K_{ab}K^{ab}
-4{\nabla}_{\sst{(3)}}^2 \tilde{\rho}-2({\nabla}_{\sst{(3)}} \tilde{\rho})^2+\fr2{l^2}e^{2\tilde{\rho}}
\Big] \label{3drescorphi3}
\end{eqnarray}
After reduction to 2D and by
setting $\tilde{\rho}(t,{ r},{ \varphi}) =\rho_0(r)+ \rho(t,r)$ with {$e^{\rho_0(r)}=r$}, one gets
\begin{eqnarray}
S=\int dtdr \sqrt{-h_{\sst{(2)}}}\;{r}\left[R^{\sst{(2)}}
+\a_\varphi(r) \rho -2(\nabla_a\rho)^2+(K^2-K_{ab}K^{ab})
+\fr{2{r^2}}{l^2} e^{2\rho}\right] \label{3drescor3q}
\end{eqnarray}
where $\a_\varphi(r)=4\nabla^2\rho_0$. $\a_\varphi(r)$ can be adjusted to our needs by renormalization procedure. (See below \rf{3drescor3q3}.)
Let us gauge-fix the 2D metric $h_{{\sst{(2)}} ab}$ to
\begin{eqnarray}
ds^2_2=\tilde{\g}_{0ab} dx^a dx^b
\label{adsansr2q2}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{\g}_{0ab}\equiv
\left(
\begin{array}{cc}
-{\fr1{r^2}}f(r) & 0 \\
0 & \fr{1}{{ r^2}f(r)} \\
\end{array} \label{gamz2q}
\right)
\end{eqnarray}
Let us rescale the 2D metric of \rf{3drescor3q} to the original 2D part of the 3D metric so that
\begin{eqnarray}
\tilde{\g}_{0ab}\equiv \fr{1}{r^2}\g_{ab},\quad\g_{0ab}=
\left(
\begin{array}{cc}
-f(r) & 0 \\
0 & \fr{1}{f(r)} \\
\end{array} \label{gamz2q2}
\right)
\end{eqnarray}
The action now takes
\begin{eqnarray}
S&=&\int d^2x \sqrt{-\g_0}\left[
-2{ r}(\nabla_a\rho)^2+\fr{\a_\varphi(r)}{ r} \rho
+\fr{2{ r}}{l^2} e^{2\rho}\right] \label{3drescor3q2}
\end{eqnarray}
where the field-independent terms have been removed.
This can be rewritten
\begin{eqnarray}
S&=&\int d^2x \left[
\fr{{ r}}{8\pi f}(\pa_t\rho )^2-\fr{{ r}f}{8\pi}(\pa_r\rho )^2
+\fr{\a_\varphi(r)}{16\pi { r}} \rho +\fr{{ r}}{8\pi l^2} e^{2\rho}\right] \label{3drescor3q3}
\end{eqnarray}
where the action has been numerically rescaled. The linear term may be omitted as part of renormalization procedure (namely, by the freedom to choose the starting action in renormalization procedure) \cite{Distler:1988jt}\cite{Seiberg:1990eb}\cite{Park:2013iqa}.
(Alternatively, the linear term does not appear at all if one rescales the 3D metric twice separately, first with $e^{2\rho}$ and next with $e^{2\rho_0}$.)
\section{Behavior of horizon }
To examine the behavior of the horizon (or, more precisely speaking, the stretched horizon to be identified below), let us consider the following coordinate invariant quantity: the energy density measured by a free-falling observer, given by
\begin{eqnarray}
<S|\;T_{ab}^S \;|S>U_S^a U_S^b \label{2dse}
\end{eqnarray}
Recall that the stress-energy tensor has a constant term that originates from the presence of the exponential interaction. The constant term plays an important role: the $<S|\;T_{ab}^S \;|S>$ term would vanish without it.
The geodesic, $U_S^a$, was worked out in \cite{Cruz:1994ir}; the timelike geodesic is given by
\begin{eqnarray}
r^2\dot{r}^2 &=& -(r^4-r^2)+c_0^2 r^2 \label{rg}
\end{eqnarray}
\begin{eqnarray}
\dot{t} &=& \fr{c_0 r^2}{r^2(r^2-r_H^2)} \label{tg}
\end{eqnarray}
where $c_0$ is an integration constant.
\subsection{1D description of stretched horizon}
Before we get to outline the 2D perturbative computation of \rf{2dse}, let us narrow down to the ground state sector of the theory. This sector pertains to the zero modes, $(q,p)$, of the mode expansion of {$\rho$} and its canonical momentum $\Pi$. The mode expansion should take a form analogous to the flat case \cite{Seiberg:1990eb}\cite{Ginsparg:1993is}:
\begin{eqnarray}
\rho(t,r)&=&q(t)+\sum_{n\neq 0}\fr{i}{n}(a_n(t)e^{-inr^*}+b_n(t)e^{inr^*})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\Pi(t,r)&=&p(t)+\sum_{n\neq 0}\fr{1}{4\pi}(a_n(t)e^{-inr^*}+b_n(t)e^{inr^*})
\end{eqnarray}
where $r^*$ is a tortoise-type coordinate \cite{Banados:1992gq}.
The zero-mode system may also be viewed as the 1D system that results from the additional radial reduction of \rf{3drescor3q3}:
\begin{eqnarray}
S&=&\int dt \left[
\fr1{8\pi }(\pa_tq )^2
+\fr{{f}}{8\pi l^2} e^{2q}\right] \label{1Dact}
\end{eqnarray}
where we have rescaled the action by $f/r$. By choosing the location of the hypersurface at $r=\rho_H+\d_P$ where $\d_P$ is a Planck scale distance, the $r$-reduction naturally leads to the notion of the stretched horizon degrees of freedom. In other words, the resulting 1D field $q(t)$ should be interpreted as the degree of freedom of the stretched horizon.
The full non-perturbative contribution from the ground sector can be found by solving the 1D Schrodinger equation with the Hamiltonian given by
\begin{eqnarray}
H_0=\fr12\Big(\fr{\pa}{\pa q}\Big)^2+\fr{{ f} }{8l^2}e^{2 q}+\fr{1}{8} \label{zh}
\end{eqnarray}
It can be noted that the sign of the kinetic term has flipped when it is compared with the corresponding sign in \cite{Seiberg:1990eb}. This change originates from the ``wrong" sign of the kinetic term in \rf{3drescor3q3}.
The Schrodinger equation $H_0\psi=E\psi$ takes
\begin{eqnarray}
\left[\Big(Z\fr{\pa}{\pa Z}\Big)^2 +\fr{{ f}}{4l^2}Z^2+\fr{1}{4} \right]\psi_e=
\Big(\fr{1}{4}+e\Big)\psi_e \label{scheq}
\end{eqnarray}
where $Z\equiv e^{ q}$. The energy $E$ is $E=\fr{1}{8}+\fr{e}{2}$ and the solution is a Bessel function. (In the case of \cite{Seiberg:1990eb} the solution was a modified Bessel function.)
{For the lowest energy state in the ground sector, let us set $e=0$.}
The Hamiltonian of \cite{Seiberg:1990eb} was positive-definite whereas the Hamiltonian \rf{zh} is not. Therefore, the energy, in particular, the value of $e$ can be negative in the present case. (See \cite{Kim:2013caa} for the recent discussion on appearance of negative energy in the context of Firewall.) We interpret
the appearance of the negative-$e$ states as a signal of instability \cite{Park:2013iqa}, and
will put the negative energy states aside. (An indication that this is justified is that in the the positive definite case of \cite{Seiberg:1990eb}, the negative energy case $e<0$ yields solutions that do not satisfy the boundary
condition. We are simply viewing the negative-$e$ states as associated with
the instability although the solutions satisfy the boundary condition in the present case.) Relatedly, with the positive-definite Hamiltonian, the energies of the $r$-dependent states will be higher than those of the $r$-independent states, and this pattern should remain true even in the present case once the negative energy sector is put aside.
For the interacting theory, the contribution of the zero-modes to the energy density as measured by the infalling observer should be
\begin{eqnarray}
H_0 UU
\end{eqnarray}
where $U=\dot{t},\; H_0= T$ (the 1D stress-energy tensor) and $\dot{t}$ is given in \rf{tg}. For $r=r_H+\d_P$ it becomes
\begin{eqnarray}
\dot{t} \simeq \fr{c_0 r_H^2}{2r_H^3}\fr1{\d_P}
\end{eqnarray}
The Planck scale behavior\footnote{Observations on diverging stress-energy at the event horizon were made in the past as well \cite{Birrell}. The present work may therefore be viewed to some extent as reinterpretation of those observations.} comes from multiplication with $U$:
\begin{eqnarray}
H_0 UU
\simeq \fr{c_0^2}{32r_H^2}\fr1{\d_P^2}
\end{eqnarray}
If there were no interaction term, this contribution could not arise:
for the branch of positive energy, $E>0$, the solution in the absence of the $Z^2$ term in \rf{scheq} diverges as $l\rightarrow \infty$ and should be abandoned. We take this as an indication that the
free theory does not display the Planck scale energy at the stretched horizon.
In passing, let us also note the stress-energy measured by the
stationary Schwarzschild observer,
\begin{eqnarray}
<S|\;T_{00}^S \;|S>
\end{eqnarray}
The zero-mode contribution should be just 1/8 according to the computation
above. (The expected $r$-dependence will appear once the nonzero mode contributions are taken into account.)
\subsection{2D approach}
It should be possible to carry out the perturbative analysis by treating the exponential term as a perturbation.
The loop divergences are expected and renormalization should be carried out.
As part of the renormalization procedure, the starting point of the 2D action can be taken as \rf{3drescor3q3}, which we quote here for convenience (the linear term has been omitted):
\begin{eqnarray}
S&=&\int d^2x \left[
\fr{{ r}}{8\pi f}(\pa_t\rho )^2-\fr{{ r}f}{8\pi}(\pa_r\rho )^2
+\fr{1}{8\pi l^2} e^{2\rho}\right] \label{3drescor3q3q}
\end{eqnarray}
Once the renormalization is complete, one can compute the vacuum
expectation value of
\begin{eqnarray}
T_{\pm \pm}&=& -\fr12 (\pa_\pm \rho)^2
+\fr{{ 1}}{8l^2}e^{2 \rho}+\fr{1}{8}
\end{eqnarray}
in some appropriately redefined 2D coordinates.
This computation will involve construction of the Green function among other things, and to that end it will be useful to make use of the Kruskal-type coordinates obtained in \cite{Banados:1992gq}\cite{Carlip:1995qv} in the intermediate step. The advantage for employing the Kruskal type coordinates is clear: the Laplace equation takes the same form as the flat case.
It should be possible to express the Green function in terms of the sum over the mode functions. Then the stress-energy tensor can be computed along the lines of the corresponding flat space computation.
\section{Conclusion}
In this work, we have computed the ground state contribution to the energy density as measured by a free-falling observer in the 2D theory. The additional ADM reduction
along $r$ has led to the natural identification of the stretched horizon.
The ground sector is spanned by the zero modes of the field $\rho$ and its conjugate momentum $\Pi$. We have shown that the quantum gravity interaction is essential for the stretched horizon to display the Planck scale excitations observed by a free-falling observer.
The result of this work supports the unsmooth horizon proposals in the literature.
We believe that the result is at odds with the conventional lore -which is based on semi-classical physics -that Equivalence Principle implies smooth horizon (rather than Equivalence Principle itself).
{In one of the footnotes, we have briefly addressed the origin of the Liouville degrees of freedom. The idea of the virtual boundary associated with ADM reduction seems to be in line with the proposal in \cite{Carlip:1999db} to view a horizon as a boundary. There is an interesting investigation that needs to be done to fully justify the view of a horizon as a boundary surface. Carrying out the ADM procedure, one will obtain a Liouville type theory even if one chooses the radial location to be outside of the horizon, i.e., a bulk point. As well known, 3D gravity does not have any propagating degrees of freedom. The Liouville theory in this case should not be taken
as genuine degrees of freedom since it should be possible to remove all of its dof by gauge symmetry. However, if one chooses the radial location at the event horizon, it is expected that the gauging procedure would presumably encounter some type of singularities due to quantum effects.
}
One obvious future direction is to compute the $r$-dependent sector's contribution to the stress-energy; renormalization procedure must be completed prior to this task.
Once one reaches this point it would be possible to see the additional role of the interaction: information would be coded on the stretched horizon in the manner that reflects the interaction.
Another direction is with regard to the teleological nature of an event horizon \cite{tele}. The teleological nature of the stretched horizon was discussed in \cite{Susskind:1993if}. We believe that the teleological nature
of the stretched horizon may well play a role in the proposed mechanisms of blackening and bleaching \cite{Park:2013rm}.
It should be possible to use the 2D setup that was reviewed in section 2 to study
the potential presence of a bleaching mechanism. For that, it would be useful to construct a wave-packet and follow its time evolution. Some of the ingredients in \cite{Schoutens:1993hu} will be useful. An interacting QFT approach that shares a certain spirit with the semi-classical description would be required; most textbooks on QFT focuses on computing S-matrix. In our view, the best setup should be what is called the Schrodinger approach of QFT \cite{Hatfield}.
\vspace{.2in}
We will report on some of these issues in the near future.
\vspace{.7 in}
\noindent {\bf Acknowledgments}
\\
\noindent I would like thank the members of CQUeST Sogang University, the physics department of Hanyang University, Kyung Hee University, and KIAS for useful discussions.
I especially thank B.-H. Lee for his hospitality during my visit to CQUeST.
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{
"redpajama_set_name": "RedPajamaArXiv"
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