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Photo: Now That I've Found You by Kristina Forest.. Image Courtesy Macmillan Publishing Group
Are you ready to transform yourself into a LEGO Minifigure? by Cristine Struble
Charmed season 2 episode 7 review: Ghosts of Parker past by Molly Catherine Turner
See the cover of Kristina Forest's Now That I've Found You and read an exclusive excerpt!
by Lacy Baugher 1 year ago Follow @LacyMB
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Photo: Now That I've Found You by Kristina Forest.. Image Courtesy Macmillan Publishing Group
Kristina Forest's second contemporary YA novel, Now That I've Found You, will hit shelves in June. We've got an exclusive look at the cover – and an excerpt to tide you over!
For those of you who love contemporary YA stories and need a new romantic comedy to obsess over in your lives, well. Have we got the book for you.
Author Kristina Forest's upcoming second novel, Now That I've Found You, is a quirky, YA rom-com about a disgraced starlet named Evie who enlists the help of a cute delivery boy when her eccentric grandma goes missing in New York City. It's a story of adventure, discovering your self-worth and, of course, finding love along the way.
🎉🎉🎉Book 2 news! My second YA novel, NOW THAT I'VE FOUND YOU will be available on June 2, 2020!
It's about an up and coming actress named Evie & an aspiring musician named Milo spend a wild week together in NYC looking for Evie's famous grandmother after she goes missing 💕🌇 pic.twitter.com/hGqy6PsCIT
— Kristina Forest (@KristinaForest) September 16, 2019
Forest herself describes the story as Where'd You Go Bernadette meets Don't Tell Mom the Babysitter's Dead, and basically? Sign us up right now.
Also, just look at how gorgeous this cover is! The colors! The subtle shading! The obvious fact that these two are going to be in love forever!
The official synopsis for Now that I've Found You positions the story as intriguing mystery, a story of self-discovery and an adorable romance, all rolled into one.
Eighteen-year-old Evie Jones is poised to be Hollywood's next big movie star, following in the footsteps of her über famous grandma, Evelyn Conaway. That is, until a friend's betrayal ultimately leads to Evie being fired and blacklisted.
A public appearance with Evelyn Conaway, AKA Gigi, is just the thing to save Evie's floundering career. But the week Evie plans to present Gigi with a major award in front of Hollywood's elite, Gigi, a recluse who's been out of the limelight for almost 20 years, disappears.
With time running out and her comeback on the line, Evie reluctantly enlists the help of the last person to see Gigi before she disappeared: Milo Williams, a cute musician whom Evie isn't sure she can trust. As Evie and Milo spend a wild week together in New York City searching for Gigi, romance and adventure abound, and Evie makes some new discoveries about Gigi, but most importantly, about herself.
And that's not all – Culturess has an exclusive excerpt of Now That I've Found You, to give you a feel for the novel's characters and story.
The book is due to hit shelves in June of 2020. And that's probably going to feel like an awful long wait after you get a peek at this story.
Next: Read an exclusive excerpt now!
Culturess 1 yearBlood Heir is worth the wait, and one of the best YA fantasy novels of 2019
Entertainment Weekly 1 yearM. Night Shyamalan's 'Servant' gets an early season 2 renewal from Apple
Culturess 1 yearA River of Royal Blood gifts readers with an intricately crafted world full of political intrigue
Culturess 1 yearAHS: 1984 is over: Here's every moment from the finale that'll make us miss it
Culturess 1 yearAHS: 1984 season finale live stream and preview: Watch online | {
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<?xml version="1.0" encoding="UTF-8"?>
<?xml-stylesheet type='text/xsl' href='../../../../../../../../../../test.xsl'?>
<!--
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-->
<!DOCTYPE Test SYSTEM '../../../../../../../../../../test.dtd'>
<Test
ID="ST01Test"
date-of-creation="2005-04-19"
timeout="1"
>
<!--
-->
<Description>
Check ability of debuggee VM to do steps into source lines.
</Description>
<Keyword name="functional"/>
<Source name="ST01Test.java"/>
<Modification date="2005-01-20" />
<Runner ID="Runtime">
<Param name="toRun" value="org.apache.harmony.test.func.jpda.jdwp.scenario.ST01.ST01Test">
</Param>
</Runner>
</Test> | {
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Q: Regex for a string with a command inside it I have managed to create a command that takes an argument and does substitutions on it with l3regex. However, if the argument contains itself a command, the command name is printed in the output as if it was a string.
\documentclass{article}
\usepackage{l3regex}
\ExplSyntaxOn
\newcommand\mycommand[1]{
\str_set:Nn \l_temp_str {#1}
\regex_replace_all:nnN {[abc]} {\c{textbf}\cB\{\0\cE\}} \l_temp_str
{\l_temp_str}
}
\ExplSyntaxOff
\begin{document}
\mycommand{wax\textit{b}ycz}
\end{document}
How should I correct the code so that the argument string can include any command?
A: Try using a token list rather than a string:
\documentclass{article}
\usepackage{xparse,l3regex}
\ExplSyntaxOn
\NewDocumentCommand{\mycommand}{m}
{
\tl_set:Nn \l_tmpa_tl { #1 }
%\str_set:Nn \l_temp_str {#1}
\regex_replace_all:nnN {[abc]} {\c{textbf}\cB\{\0\cE\}} \l_tmpa_tl
\tl_use:N \l_tmpa_tl
}
\ExplSyntaxOff
\begin{document}
\mycommand{wax\textit{b}ycz}
\end{document}
This should treat commands as commands.
| {
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{"url":"http:\/\/physics.stackexchange.com\/tags\/weak-interaction\/new","text":"# Tag Info\n\n3\n\nThis is an interesting reference, but it seems to me that the measurements are controversial enough that they are certainly not, at this point, at the stage where they force any re-think of nuclear and solar physics as currently understood. The paper in question is Power spectrum analyses of nuclear decay rates. D. Javorsek II et al., Astropart. Phys. ...\n\n1\n\nNo, a weak decay doesn't imply a change of $S$. For example, the decay of the neutron \u2013 the basic part of the beta-decay \u2013 has $S=0$ both in the initial and final state. So the first proposition is false and only the second one is true.\n\n0\n\nIn QM most stuff boils down to particles. For example, electromagnetism is mediated by photons and EM radiation is literally just beams of photons. We should then have a couple of obvious candidates for strange and new kinds of radiation. The strong force is mediated by gluons and there are 8 types of gluons. At a larger scale (like with atomic nuclei) ...\n\nTop 50 recent answers are included","date":"2016-07-24 14:48:49","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\": 0.8068796992301941, \"perplexity\": 483.2897554726904}, \"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-2016-30\/segments\/1469257824109.37\/warc\/CC-MAIN-20160723071024-00249-ip-10-185-27-174.ec2.internal.warc.gz\"}"} | null | null |
«Останній гарем» () 1999 Італія режисер Ферзан Озпетек.
Зміст
Початок XX століття. Османська імперія доживає останні дні.
Юна наложниця Сафійе потрапляє в розкішний палац султана Абдул-Хаміда II. Вона мріє стати фавориткою султана і народити йому спадкоємця.
У гаремі Сафійе знайомиться з чорношкірим євнухом Надіром. Взаємна симпатія кастрата і наложниці переростає в пристрасну любов, яка долає станові заборони і фізичні перешкоди.
Примітки
Нагороди
«Золотий Помаранч» Анталійського кінофестивалю Нагорода (Найкраща актриса другого плану Серра Йїлмаз)
Посилання
на сайті IMDB
Фільми Італії 1999
Фільми Франції 1999
Фільми-драми Туреччини
Фільми-драми Італії
Фільми-драми Франції
Фільми турецькою мовою
Фільми французькою мовою
Фільми італійською мовою
Фільми Ферзана Озпетека
Фільми про Туреччину
Фільми про Османську імперію
Фільми про Середньовіччя | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,683 |
/ Home / Tags / sustainability-education
Pages tagged with: sustainability-education
Find out more about the Sustainability Hub at the University of Plymouth
University of Plymouth academic named 'Sustainability Champion'
Paul Murray, a well-respected academic in the field of education for sustainable development, has won a Green Gown Award in recognition of more than a quarter of a century of teaching and learning and research.
University wins national newspaper award for sustainability
The School of Nursing and Midwifery scoop a Guardian University Award for their pioneering European project, NurSusTOOLKIT.
Find out more about the Centre for Sustainable Futures Team
The Plymouth A-Z of sustainability in the curriculum
Resources to help disciplines to embed sustainability into the curriculum
Staff training and professional development in sustainability education
Courses, workshops, and events in Plymouth University for staff seeking to develop their knowledge and skills in sustainability education
Global elite collaborate to address surfing industry's sustainability challenges
Plymouth University news: The surfing world's most powerful figures and practitioners have been brought together for a new book, published by University of Plymouth Press, and exploring how the industry is grappling with the challenge of sustainability
Masters programmes in sustainability
Sustainability themed programmes at Plymouth
University committed to embedding sustainability in post-16 education
The University has signed the 'SDG Accord' - the higher education sector's response to the United Nations' Sustainable Development Goals (SDGs)
European project to teach sustainability in nursing
University launches the NurSusTOOLKIT – a new European resource to embed sustainability into nursing and health education
Last updated: 5 May 2017
Marine science meets photography at student exhibition in the Ocean City
Plymouth University news: Students on the marine science and photography courses work together on a series of pictures that will go on display at the Sea and Me exhibition
Last updated: 6 February 2017
Enhancing knowledge crucial to improved energy-saving behaviours
Plymouth University news: Increasing public knowledge and understanding about energy issues is vital if improved energy-saving behaviours are to be encouraged among individuals and organisations, a study conducted at Plymouth University suggests | {
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Q: regex to allow one decimal number or a range of decimal number I need a JavaScript regular expression which matches a single decimal number or a range of decimal number:
8,4-10 true
8,4-10,5 true
8,4-10,5-3 false
8,4- false
25 true
25,5 true
24.1 false
1a false
abc false
I have come up with this pattern, but it doesn't give correct result:
(([0-9])+(,([0-9])+)?)?(-(([0-9])+(,([0-9])+)?))?
A: Your regex is correct, you just need to anchor it to make sure you're matching the whole string, not just part of it.
Simplifying what can be, it becomes:
^\d+(?:,\d+)?(?:-\d+(?:,\d+)?)?$
\d is short for [0-9], (?:...) is a non capturing group, ^$ are anchors matching the beginning and end of the string (making sure there is nothing other than what you want).
See demo here.
| {
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In March, 2011, two hackers, "Ne0h" and "TinKode" compromised MySQL.com and posted the site's customer usernames and passwords. According to the pair, they used the site's SQL Injection (SQLi) vulnerabilities to launch the attack.
Just over a year later, D33Ds Company, a hacker group posted passwords of 450,000 Yahoo users. Like the Oracle owned MySQL.com website attack, the D33Ds Company hacker group said it had taken advantage of Yahoo site's SQL Injection vulnerability to attack.
And a little over two years ago, it was widely reported that over a million websites that use WordPress were at risk of a security compromise from a SQL Injection. The WP-Slimstat plugin critical security flaw was blamed for the increased risk of possible hijacking of websites using the WordPress content management system.
These are just few of the many SQLi vulnerability cases that have left web developers scratching their heads in search for quick and effective security solutions. These and other incidences gave people a taste of the magnitude of damage that SQL injection vulnerabilities can cause a company if they are not addressed.
While the discussions around SQL Injection have been ongoing for over a decade now, some developers still have little understanding of what it is all about and even fewer have seen the and understood the reality of the dangers it presents. Therefore, in this article, you will learn what SQLi is, why you should fix it and how to go about fixing it.
SQL Injection vulnerability: what is it?
SQL Injection is one of the top 10 OWASP vulnerabilities in the category "Injection." This vulnerability is extremely easy to take advantage of. Additionally, it is quite common and its impact is awfully severe. A website developer is responsible for recognizing such a vulnerability and patch it.
The above line of code is all that an attacker requires to get complete access to a website's entire database. Do you still doubt that? Let us examine the exact way the hacker would use this to own your entire database.
Sending the ') part of the payload will set the query so that it returns no results; email returns as blank: email=".
In the second part 1=1 will always equal true, resulting in the returning of the first entry in the table of users.
Using the ') or admin='t'– payload, the hacker has found his way into the system enough to return an administrator user. This gives him information about a website administrator within the site's database.
As you have seen in the above demonstration, it is extremely dangerous to fail to fix an SQL Injection vulnerability. The process of fixing SQL queries requires the use of parameterization. This is the most secure way of handling unsafe inputs by users.
Whichever ORM(Object-Relational Mapping) you choose, you should be able to use its facilities to parameterize queries.
Here are two common types of queries that are unsafe and how you can fix them. The examples are based on Active Record ORM.
This is the most common for queries in Ruby.
As you can see above, line 8 and line 3 look similar. However, they are quite different in that unlike line 8, line 3 is using string formatting instead of using parameterization. String formatting is unsecure and will not provide adequate protection against possible SQL injection.
The other important lesson we can learn from taking a closer look at the Safe vs Unsafe examples above is that you know you are vulnerable to an SQL Injection if you find yourself having to add surrounding quotes to queries. This is a rule of thumb.
As you can see, this is not the prettiest approach, but it makes sure the parameters pass in separately from each query. As a result, parameterization takes place, keeping your site safe from possible SQL injection. | {
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\section{Introduction}
Recurrent novae (RNe) are ordinary novae systems for which the recurrence time scale happens to be less than roughly one century. That is, RNe are binary stars where matter accretes from a donor star onto the surface of a white dwarf, where the accumulated material will eventually start a thermonuclear explosion that makes the nova eruption. The ordinary nova systems for which only one eruption has been detected are labeled as `classical novae' (CNe), even though some of them might really be RNe for which all-but-one eruption from the last century has been missed. The nova systems have a continuum of recurrence time scales, ranging from roughly a decade to perhaps 100,000 years or longer, with the division at roughly one century being purely based on observational expedience.
Why do some nova systems have a recurrence time scale ($\tau_{rec}$) of less than a century? Here, theory provides a confident answer that two conditions are required to get short inter-eruption times. The first requirement is that the white dwarf mass must be near the Chandrasekhar mass. The reason is that the high white dwarf mass makes for a high surface gravity which allows even a relatively small amount of matter to reach the kindling temperature for thermonuclear runaway. The limit is that the mass of the white dwarf ($M_{wd}$) must be greater than something like 1.2 $M_{\odot}$. The second requirement is that the mass accretion rate ({$\dot{M}$) must be high. The reason is that a high $\dot{M}$ is the only way to get enough material onto the white dwarf to reach the runaway condition during the short inter-eruption time interval. Indeed, the RNe have an accretion rate higher than most cataclysmic variables, with values close to $10^{-7}$ $M_{\odot}$ yr$^{-1}$ (Hachisu \& Kato 2000; 2001; Webbink et al. 1987; Patterson et al. 1998; Knigge, King, \& Patterson 2000; Hachisu et al. 2000; Hachisu, Kato, \& Schaefer 2003).
So the RNe have white dwarfs near the Chandrasekhar mass and matter is being piled on at a very high rate. With this situation, it is possible that the white dwarf will soon be driven to the Chandrasekhar mass and then collapse as a Type Ia supernova. As such, RNe are a likely answer to the perennial Type Ia progenitor question. Often in the literature, discussion does not go past this simple argument. But there is a major loophole in that the mass ejected during each nova event has not been accounted for. If the white dwarf ejects more mass during an eruption than it has accreted during the previous inter-eruption interval, then the white dwarf will not be gaining mass and will not collapse as a Type Ia explosion. So the question really comes down to evaluating the recurrence time scale ($\tau_{rec}$), the mass accretion rate ($\dot{M}$), the mass ejected by the nova event ($M_{ejecta}$), and then whether $M_{ejecta}>\tau_{rec}\dot{M}$. It turns out that the ejecta mass has a very large uncertainty (over two orders of magnitude) from both observational and theoretical work. Also, my recent work (see below) has demonstrated that $\tau_{rec}$ generally has errors of a factor of three and $\dot{M}$ has large uncertainties. Until we can answer the question of whether $M_{ejecta}>\tau_{rec}\dot{M}$, RNe can only remain a good idea as a solution for the progenitor problem.
Another potential problem for the RNe answer is whether there are enough RNe in our galaxy and Local Group to produce the Type Ia events. That is, is the RN death rate equal to the Type Ia supernova rate? Published answers to date appear to say `no' both from a theoretical population synthesis (Branch et al. 1995) and an observational basis (della Valle \& Livio 1996). (Unfortunately, the answer appears to be `no' for all candidate classes of progenitors, see Branch et al. 1995) But the theoretical basis for the `no' to RNe is suspect as population synthesis analysis has very large uncertainties when dealing with rare evolutionary paths, especially after common envelope binary evolution has occurred. This distrust of population synthesis for evaluating rates in such cases is further emphasized by studies of the fraction of 3-8 M$_{\odot}$ stars that must undergo Type Ia explosions is orders of magnitude higher than theory would give (Maoz 2008). And the observational `no' is also suspect because the realities of discovery statistics were ignored and these can easily increase the RN death rate by a factor of 100 or more. The way to answer the RNe death rate question can only be to get better demographics for RNe (including their numbers, light curves, accretion rates, and recurrence time scales) as well as to determine their real discovery statistics.
The Type Ia progenitor question is an old one, with major reviews from each of the last four decades appearing in Livio (2000), Branch et al. (1995), Trimble (1984), and Whelan \& Iben (1973). Even today, extensive debates rage as to which of many systems account for most of the Type Ia events. The primary dividing line is whether the progenitors have one or two white dwarfs in the binary. The double degenerate idea has two white dwarfs in close orbit for which angular momentum loss by gravitational radiation grinds the orbit down until the two white dwarfs collide and exceed the Chandrasekhar mass resulting in a supernova explosion. No double white dwarf system with adequate combined mass and a short enough orbital period has ever been identified, despite massive efforts, so we cannot point to any star in the sky as being a progenitor of this class (Robinson \& Shafter 1987; Napiwotzki et al. 2001). For example, even the star KPD 1930+275 with a white dwarf in a binary of total mass 1.47$\pm$0.01 M$_{\odot}$ (Maxted et al. 2000, but see Geier et al. 2007) will lose enough mass so as to avoid exceeding the Chandrasekhar mass (Ergma et al. 2001). The single degenerate model has a mass donor star feeding matter onto a white dwarf which eventually exceeds the Chandrasekhar mass also resulting in a supernova explosion. For this model, we can point to several populous classes of stars which are candidate progenitors including the RNe, symbiotic stars, and super-soft sources. These classes have substantial overlap within their populations. With these many possibilities all of which presumably work, the question comes down to which is the dominant progenitor. And with the possibility of two roughly equal populations of Type Ia supernovae (Sullivan et al. 2006), there might even be two dominant progenitor classes.
The progenitor question has recently come to the forefront as being of high importance. The reason is that Type Ia supernovae have become a premier tool for cosmology, for example with the Hubble diagram providing the first and strongest evidence for the existence of Dark Energy. But supernova cosmology is based on the idea that the luminosity-decline relation has no evolution with redshift. If $z\sim1$ supernovae systematically have lower metallicity (appropriate for their long lookback time) then their envelopes will have systematically lower opacity with an inevitable change in the luminosity-distance relation and a systematic distortion of the derived distances. Such systematic effects will lead to a change in the shape of the Hubble diagram, with this error perhaps being interpreted as evidence for Dark Energy. Dom\'inguez et al. (2001) have demonstrated that the range of evolutionary effects (from systematic changes in redshift of both the metallicity and the main sequence mass of the progenitor) can lead to typical errors of $\sim$0.2 mag out to $z=1$. The supernova researchers only argue against evolution with their evidence that the high-z events have similar spectra to low-z events (Hook et al. 2005), although this is not proof of the lack of evolution at less than the ~0.2 mag level. The question of evolution is critical as the amount of evolution at $z\sim1$ is comparable to the differences between cosmologies with and without Dark Energy. Without knowing the identity of the progenitor, evolution calculations are not possible and the effect can significantly change the shape of the Hubble diagram. So in principle, the progenitor problem is critical for the entire supernova cosmology enterprise. And it will only become more important, as future experiments (like the SNAP satellite) are seeking to measure the Hubble diagram with an accuracy of ~0.01 mag, a level at which evolution effects can dominate and perhaps make the goal unreachable. So a solution to the progenitor problem has wide importance.
The solution can come only with a broad investigation of the various candidate progenitors. A part of this will be the study of RNe, with particular attention to questions relating to whether $M_{ejecta}>\tau_{rec}~\dot{M}$ and the value for the RN death rate. These studies are essentially demographic investigations that must examine many fine details of all the known RNe systems.
The only comprehensive review or summary of RNe is the influential and superb paper by Webbink et al. (1987). But a lot has been learned since that time. Unfortunately, all of the many subsequent observations are widely scattered. For later reviews, we have a conference proceeding on one specific eruption (Evans et al. 2008), two brief overviews in conference proceedings (Anupama 2002; 2008), and one short summary (Sekiguchi 1995). A deeper problem is that all of the photometry from before 1970 or so has systematically wrong magnitudes due to universal errors in the magnitude scale, and thus the majority of RNe events have only poorly measured properties.
Over many years, I have been collecting a large database of observations of all the known galactic RNe, both in quiescence and in eruption. For example, I have measured comparison star magnitudes and sequences so that old and new magnitudes can be correctly placed onto a uniform modern magnitude scale. I have also remeasured all the archival photographic plates of all the pre-1970 eruptions, so that the light curves are directly comparable. I have made exhaustive searches of the archival material for previously undiscovered eruptions. I have extensive long-term light curves for the RNe at quiescence. I have 47 and 80 eclipse times on both sides of the latest eruptions of U Sco and CI Aql respectively. The easy-to-point-to results are the discovery of one new RN system, the identification of six previously unknown eruptions, the discovery of five of the orbital periods, the confident measure of the orbital period change across two eruptions, and the prediction of the dates of the next eruptions of U Sco and T Pyx.
Now is the time for a comprehensive review of RNe photometry because of (a) the importance of the progenitor problem, (b) the lack of recent comprehensive reviews, (c) the scattered nature of the data in the literature, (d) the now-corrected calibration problems with the old magnitudes, and (e) the new data. This paper seeks to fill the need by providing a comprehensive set of new data, remeasured old data, and data from the literature. The goal is to provide a uniform analysis on the various properties of all known galactic RNe and the demographics of the group.
\section{The Known Galactic Recurrent Novae}
What are the known RNe in our own galaxy? This is an important question because the best census is needed for valid demographic studies. The question of which stars are RNe has sometimes been uncertain. One source of confusion is whether a particular eruption is a nova event (involving thermonuclear runaway on the surface of the white dwarf) or a dwarf nova event (involving a brightening of the accretion disk). Dwarf novae with rare high-amplitude eruptions can be easily confused with RNe, as can CNe that also display dwarf nova eruptions.
There is a long history of claims for stars being recurrent novae. One of the first discovered variable stars is Mira, and it was called a nova when its first peak was recognized and then a second eruption was spotted. The modern idea of `recurrent nova' had to await the separation of nova events as based on spectroscopy. At various times, several dwarf novae (including WZ Sge, VY Aqr, RZ Leo, and V1195 Oph) have been identified as RNe (Webbink et al. 1987). The two eruptions of V616 Mon (A0620-00) pointed to an unusual RN (Webbink et al. 1987), but we now know this system actually has a black hole as the compact star. V529 Ori has a ``remarkably checkered history" and it is now unclear whether the reported events are of the same star or even were nova eruptions (Webbink et al. 1987).
V1017 Sgr is an interesting case, with eruptions in 1901, 1919, 1973, and 1991, with a history and details in Webbink et al. (1987). The eruption in 1919 has a peak of seventh magnitude (or somewhat brighter) and a light curve shape that is typical for novae events. But the other three eruptions all peak at fainter than tenth magnitude and have light curve shapes typical of dwarf novae events. The small brightenings are confirmed as being dwarf nova events by their spectroscopic development (Webbink et al. 1987; Vidal \& Rogers 1974). The hybrid nature of V1017 Sgr has the strong precedent from a few other hybrid systems. Most famously GK Per had a prototypical nova eruption in 1901 and has now had 14 dwarf novae events starting in 1966. So V1017 Sgr is not a RN.
As old novae (labeled as being CN) are having their second nova event discovered, the numbers of known RNe are steadily increasing. Thus, V394 CrA, V745 Sco, V3890 Sgr, CI Aql, and IM Nor have been recognized as RNe with second eruptions in 1987, 1989, 1990, 2000, and 2002. I expect that this discovery rate will speed up as more old novae are eligible to get their second eruption. These newly discovered systems usually involve stars where the first eruption is poorly observed. The known RN population is now dominated by systems with only moderate coverage and little recognition by researchers.
The currently known RN are T Pyx, IM Nor, CI Aql, V2487 Oph, U Sco, V394 CrA, T CrB, RS Oph, V745 Sco, and V3890 Sgr. I have ordered these systems by their orbital period (as finalized later in this paper), even though the period of V2487 Oph is only approximately known. With one of the RN systems being proven only a few months ago (V2487 Oph), I realize that this list will soon have more additions.
Table 1 provides a list of these RNe, along with their primary properties. The second and third columns give the peak and quiescent brightness in the V-band ($V_{max}$ and $V_{min}$) as taken from Section 5. The fourth column gives the time it takes for the V-band light curve to fade by three magnitudes from the peak brightness ($t_3$) as taken from Section 6. The next column gives the orbital period for the binary system as taken from Section 10. The final column gives the years of all known eruptions as taken from Section 3.
Several properties seen in Table 1 demonstrate a stark contrast between RNe and CNe. The RNe median amplitude is 8.6 mag (and none larger than 11.2 mag), in strong contrast with the CNe with most having amplitudes of $>$12 mag. The RNe are shorter in eruption duration (median $t_3$ of 11 days) than the CNe (median $t_3$ of 44 days). Almost all the classical novae have orbital periods ($P_{orbital}$) of less than 0.3 days, whereas 80\% of the RNe have longer orbital periods (with four of them having values of near one year). On average, the recurrent and classical nova populations have substantially different properties even if there is much overlap, with the RNe being generally low-amplitude, fast-declining eruptions in systems with long-orbital periods.
\subsection{Subdivision of the Recurrent Novae}
The RNe present us with a wide diversity of properties. It is tempting to seek divisions and categories that make sense for organizing this diversity. The divisions might be constructed for two purposes: first to identify the range of properties, and second to seek bimodality (or trimododality) which will indicate separate physical processes or histories. In the past, workers have tried to make divisions based on the orbital period (or equivalently, the luminosity class of the companion), the eruption duration (or equivalently, the $t_3$ value), and the light curve shape (specifically, the presence of a plateau in the tail of the eruption).
Schaefer (1990) and Schaefer \& Ringwald (1995) pointed to the fact that the orbital periods come in three separate groups: very short periods with low-mass main sequence star companions (like almost all other cataclysmic variables), periods comparable to one day (with subgiant companions), and very long periods comparable to one year (with red giant companions). Such a division might be relevant for pointing to separate evolutionary paths leading to RNe, but no discussion or models have been presented to explain the three separate paths that lead to a three-fold division. A three-fold division might have consequences relating to the interaction of the nova with the circumstellar environment, for example where a red giant companion might have a wind that can produce hard x-rays when the nova shell plows through it. But even for this, the RNe with red giants have widely varying wind densities, while there is no distinction between the two shorter period classes because both have no sensible winds. The size of the companion star will determine how much hydrogen appears in the spectrum of Type Ia supernovae, should these systems end with such an explosion. Finally, a three-fold division will produce differing spectra in quiescence, as the different luminosity classes for the companion star will be revealed by their spectra. Despite the tempting nature of this three-fold division, I do not know how to evaluate the significance of a tri-modal period distribution (see Table 1 and Figure 1), and with the small number of known RNe, I suspect that the significance is low. In principle, the size of the companion star is irrelevant to the eruption physics (which likely depends only on the white dwarf mass, the composition of the accreted material, and perhaps on the accretion rate). Most importantly, the three-fold division does not correlate with any of the eruption properties. That is, the eruption light curves, the peak absolute magnitudes, the spectral developments, the elemental abundances, the amplitudes, the accretion rates, the speed classes, the recurrence time scales, and the galactic positions all have no relation or correlation with the size of the companion star. In all, the three-fold division is relevant only for peripheral issues concerning RNe.
Most RNe are very fast in decline, so it might be tempting to split off the relatively slow decline events (T Pyx, IM Nor, and CI Aql) into their own class (Duerbeck et al. 2002). Kato et al. (2002) suggested a variant on this by separating out just IM Nor and CI Aql into a subdivision based on similarities in the light curve and spectra. Some division based on $t_3$ would be relevant for the eruption physics, but I see no evidence for anything other than a continuous distribution of RNe durations. Figure 1 displays a plot of $P_{orbital}$ versus $t_3$ for the ten RNe, and I see no significant groupings or bimodality.
Hachisu et al. (2000a; 2002; 2009) point to a plateau in the tail of light curves as indicating a RN, with the plateau caused by an extended phase of hydrogen burning. No demographic study of nova plateaus has been made, although I will use my templates in Section 6 to quantify the situation for plateaus in RNe. The result will be that 60-90\% of the RNe are seen to have plateaus. Most critically, the RNe with plateaus (T Pyx, IM Nor, CI Aql, V2487 Oph, U Sco, and RS Oph) creates a group that has no correlation with any other categorization ideas.
Kenyon (1986) has separated out T CrB and RS Oph as being in a separate class of symbiotic nova. Symbiotic stars are defined by him to be those with absorption features of a late type giant star plus hydrogen or helium emission lines from some hot region. T CrB, RS Oph, V745 Sco, and V3890 Sgr certainly meet this definition. Nevertheless, Miko{\l}ajewska (2009) points out that these RNe are at the extreme edge of the range of properties of symbiotic stars, with short orbital periods, high mass white dwarfs, and very small mass ratios. Symbiotic novae are defined by Kenyon to be a {\it very slow} eruption involving thermonuclear burning of hydrogen on a symbiotic star. While the RNe might have nova eruptions on (technically) symbiotic stars, they are not symbiotic novae. Iben (2003) identifies the critical distinction between classical novae and symbiotic novae as the donor star fills its Roche lobe for classical novae and does not fill its Roche lobe for symbiotic novae. With this as the critical distinction, RS Oph and T CrB are not symbiotic novae (Schaefer 2009). But the exact definition is not important, because the eruptions of T CrB, RS Oph, V745 Sco, and V3890 Sgr are tremendously different from those of the symbiotic novae. Symbiotic novae have amplitudes from 2-6 mag and $t_3$ values from years to decades. Certainly, all symbiotic novae are fundamentally different from all RNe. That is, the fast eruptions of T CrB and RS Oph have no useful resemblance to the symbiotic novae, so any definition that lumps the two together has no utility for understanding either class.
The RNe can plausibly be divided up into groups based on amplitude (perhaps separating out IM Nor, V394 CrA, and U Sco), or recurrence time scale (separating out U Sco, RS Oph, T Pyx, and likely V2487 Oph). An extreme example of the chaos and instability in RN subdivisions is in Rosenbush (2002) wherein {\it four} categories were defined. The trouble with all these ideas is that the divisions are neither bimodal nor physically motivated, so it looks to me like all the groups are simply the arbitrary selection of some dividing line as applied to some favored property. Perhaps more importantly, these various divisions do not break up the RNe into consistent groups, which is to say that any one selected dividing line does not have any predictive power for other properties.
\subsection{A New Division}
I propose a new division based on the mechanism by which the RN has achieved its high accretion rate. The short-period systems (T Pyx and IM Nor) cannot have steady mass transfer at the high required rate, so their accretion is being driven by a declining supersoft x-ray source which heats the companion star (Knigge et al. 2000; Schaefer et al. 2009). The other RNe all have evolved stars, for which the high accretion rate is being driven by the evolutionary expansion of the donor star. (Most classical novae do not have the high accretion caused by an evolving secondary star, and those few that do, for example GK Per, do not have large enough white dwarf mass to become recurrent.) This physically motivated distinction has the advantage of dividing the RNe into two groups based on the underlying cause for the fast recurrence time scale.
A further advantage is that this division identifies a stark distinction in the evolutionary paths of the systems. The short period systems with accretion driven by a supersoft x-ray source will have cyclic evolution going from slow accretion, to a normal nova eruption that starts a supersoft source, to a high accretion rate system with recurrent nova eruptions, to a hibernation state, and then starting the million year cycle over again (Schaefer et al. 2009). The long period systems with accretion driven by the secondary star evolution will be steady as the donor star continues its expansion, with the RN state being long living.
An important advantage of this new division is that it divides the RNe into two groups based on whether the systems will ultimately collapse as a Type Ia supernova. The short period systems (driven by the supersoft source) will not become supernovae, the reason being that the RN phase only lasts a few centuries out of a million years while the dynamically dominant event (the regular nova eruption that starts the supersoft source) will eject more mass than is accreted (Schaefer et al. 2009). The long period systems (driven by expansion of the secondary) will become supernovae as the steady accretion will pour more mass onto the white dwarf than is ejected by eruptions (Hachisu \& Kato 2001; Hachisu et al. 2000a; 2002; Hachisu, Kato, \& Schaefer 2003). Thus, people interested in Type Ia progenitors should only look at the systems with evolved companions, while the rate calculations for the death of RNe should not include the short period systems.
In summary, this new division of the RNe into short-period and long-period systems has come up with a division that produces the same groups as based on {\it four} widely-different and important physical bases. The short-period RNe (T Pyx and IM Nor) have short orbital periods (less than a third of a day), have their accretion driven by heating from the white dwarf, have long evolutionary cycles (between RN eruptions and hibernation), and will {\it not} become Type Ia supernovae. The long-period RNe (the other 8 RNe) have long orbital periods (longer than a third of a day), have their accretion driven by expansion of the companion star, have a steady evolutionary phase as the companion expands, and apparently will become Type Ia supernova.
\section{All Known Eruptions}
What are the known eruptions from the galactic RNe? This list is important for establishing the recurrence time scale, the inter-eruption intervals, and the demographics for discovery statistics. Many of the eruptions are well known and need no particular discussion. However, eight eruptions (including one possible eruption) are little known or unpublished, and these will be discussed in detail here. Table 2 has a detailed listing of all eruptions and full information on the discovery.
\subsection{Specific Eruptions}
The 1941 eruption of CI Aql was discovered with a directed search through all the archival plates at Harvard College Observatory (Schaefer 2001b). The importance of this discovery is that CI Aql is one of the systems for which I have made a good measure of its period change across the 2000 eruption (hence yielding a value for $M_{ejecta}$) so that a confident value for $\tau_{rec}$ can be used to test whether the white dwarf is gaining mass. Indeed, the recurrence time scale has changed from its original 83 years (1917 to 2000) to 24 and 59/N years, where N might be 1, 2, or 3 to represent any missed eruptions from 1941 to 2000. The detailed light curve for this eruption is presented for the first time in Section 5.
The 1900 eruption of V2487 Oph was discovered as a result of intentionally testing a prediction that the system (known as a CN from its 1998 eruption) is actually a RN (Pagnotta, Schaefer, \& Xiao 2008; Pagnotta et al. 2009). That the star might be a recurrent nova was recognized by Hachisu et al (2002) on the basis of a plateau in the light curve of the 1998 eruption as well as by a model estimate that the system has a high mass white dwarf. We independently identified V2487 Oph as being a strong candidate RN based on its very high expansion velocity and the presence of high excitation lines in its outburst spectra. In 2004, we had searched many of the Harvard plates for prior eruptions, but had not been exhaustive in the search. In the summer of 2008, we again looked through all remaining Harvard plates (and all Sonneberg archival plates) and found one eruption. This eruption was evidenced by a single Harvard plate (AM505) from June 1900. As always, when an event is detected on only one plate, we have to be cautious about the eruption. In this case, the reality of the eruption was obvious immediately. One reason was that the image was several magnitudes above the plate limit, so the significance of the image is high. But the immediately obvious clincher was that all star images on the plate had a characteristic dumbbell shape (a result of a double exposure of equal duration with a small offset between exposures) and this exact same image shape was seen in the V2487 Oph image. This proves that the light recorded by the image came through the telescope and completely rules out any possible plate defect situation as well as most of the other possible alternatives. The lack of trailing (of the basic dumbbell shape) also eliminated the possibility of an asteroid, while this is further ruled out by a comprehensive search of all asteroids by Brian Marsden. The position of the transient image was measured to be within a few arc-seconds of the position of V2487 Oph, and this makes the possibility of some other nova extremely unlikely. In all, V2487 Oph is known to have had an eruption in 1900 with very high confidence. Other plates in 1900 were also examined, but none were recorded at a time to sufficient depth to have any hope of catching the eruption. That is, only one of the plates could possibly have recorded the 1900 outburst. This is actually the usual case for Harvard plates of that era for very fast novae. (V2487 Oph has a very fast decline, with it fading by 3 magnitudes below peak within 8 days.) That is, with ordinary gaps in the plate schedule, an eruption most of the time would be completely missed and most of the remainder would be found on only one plate. This provides a strong lesson to us that most of the early eruptions of fast novae would be completely lost.
The 1917 eruption of U Sco was also discovered with an exhaustive search through the Harvard plates (Schaefer 2004a). The pre-1940 plates at Harvard had already been examined by Thomas (1940), but she had missed the one plate (AC18624) that shows this eruption, likely because the plate had U Sco only in the extreme corner. This circumstance allowed for the reality of the image to be confidently made because the normal point spread function for stars at that position has a large and characteristic shape shared by the U Sco image as well as all surrounding stars. The image of U Sco was bright and roughly three magnitudes above the plate limit, so there is no chance of random grain enhancements being mistaken for a real object in the sky. Also, the position of the image matches that of U Sco to within a few arc-seconds. With the perfect image shape, brightness, and position, the identity of this transient object with U Sco is certain. The magnitude of U Sco in this image corresponds to U Sco being within one day of peak. All plates around the time of peak were examined, with none others showing U Sco, however, this is not surprising as the poor time coverage and poor limiting magnitude of the closest plates all come nowhere near to being able to show this eruption (see the light curve in Section 5). Out of the entire month around the peak, a U Sco eruption would have been missed on 20 days of that month. So it is sheer luck that this U Sco eruption was discovered at all, and the most likely case was for it to have been recorded on only one plate. (The identical situation arises for the 1906 eruption of U Sco [Thomas 1940 and see Section 5], the 1890 eruption of T Pyx, and for the 1900 eruption of V2487 Oph [Pagnotta, Schaefer, \& Xiao 2008; Pagnotta et al. 2009], all of which are only known by a single plate.) A strong lesson from this eruption is that it is easy for a U Sco eruption to slip through even the best old coverage in the world. The light curve information for this eruption (little though there is) is presented in this paper for the first time.
The 1945 eruption of U Sco was also discovered as part of an exhaustive examination of all Harvard plates showing U Sco (Schaefer 2001a). However in this case, 37 plates show U Sco in outburst. Nevertheless, the peak was missed as the earliest plates in the light curve only show U Sco at tenth magnitude (roughly 10X fainter than peak). The light curve is published here for the first time. This eruption is after the plate search of Thomas (1940), as otherwise she would have easily found it. The first four eruptions of U Sco in the last century (in 1906, 1917, 1936, and 1945) are known only retrospectively and only from Harvard plates. Indeed, the Harvard plates provide the {\it only} record for 11 of the 37 RN eruptions, plus the primary record for 8 more eruption, and thus this one archival collection provides roughly half of all known RN eruptions. The number of plates that record each U Sco eruptions is 1, 1, 12, and 37 respectively, which demonstrates the increase in frequency of Harvard plates. This rise in frequency explains why the expected ~1896 eruption was missed and it readily suggests that many other RNe could easily have missed eruptions in the early twentieth century.
The 1969 eruption of U Sco was only recorded late in its tail by two observers and the data have never been published. However, one of the observers is Albert Jones, who is widely recognized as one of the greatest visual observers of all time. He is the discoverer of many nova eruptions (including two {\it other} RN events) as well as a co-discoverer of SN1987A. Janet Mattei has identified him as being the best observer in the world and as having ``photometric eyes", while Daniel Green has identified Jones as the best observer in the world of cometary magnitudes. It would be hard to question even one positive detection of U Sco by Albert Jones, but he has ten positive observations all of which define a fading tail. And the second observer is the legendary Frank Bateson, who was the longtime leader of the RASNZ variable star section. (Bateson told me about this 1969 event during one of his visits to America.) There is no chance that these two skilled observers would be mistaken about the light curve. Given that U Sco is always around 17-18 mag with no excursions brighter than that (other than eruptions), the 1969 event can only have been a nova eruption missed by the rest of the world. This is not surprising as the U Sco peak was in early February and the position was just coming out from behind the Sun deep in the dawn sky. This illustrates that the solar gap (with U Sco being only 3.6$\degr$ from the Sun every 28 November) will force the missing of some U Sco eruptions. Again, we have the lesson that the discovery of U Sco eruptions is hit-or-miss and depends on random happenstance such that a significant fraction of the events will be missed entirely.
The 1842 eruption of T CrB has been published by Sir John F. W. Herschel (1866), although this might simply be a report of a nearby constant star. John Herschel (son of William Herschel) is one of the greatest visual observers of all time, so his report must be taken seriously. He writes that he had been mapping the stars in 1842 by means of measuring the positions of all stars inside triangular regions which cover the entire sky. Upon the 1866 discovery of T CrB, he had examined his notes from 9 June 1842 and found that he had recorded a 6.7 magnitude star at the exact same position! McLaughlin (1939) claims that Herschel's position is one degree off and hence the observation was not of T CrB but rather of BD+25$\degr$3020 (a normal G-type main sequence star). However, Herschel was confident that his new star was at the same position as the 1866 event and that no mistake was made. The case depends on the interpretation of the identification of the new star in Herschel's chart. If Herschel indicated his new star with an asterisk then the position is that of T CrB, while if the editor had added the asterisk to Herschel's chart and the 1842 star is indicated by the 6' notation next to it then the indicated position is indeed far from T CrB and consistent with BD+25$\degr$3020. The case is not clear from the old article, so Herschel's original charts or the {\it Monthly Notices} records should be examined if they still exist. A reason to doubt McLaughlin's hypothesis is that BD+25$\degr$3020 is very faint (at a constant $V=7.06$), so that it is unlikely (but not impossible) that Herschel could detect it. In all, I do not know whether to accept the 1842 eruption as real. Nonetheless, the confidence in this event is sufficiently low that I should not include it in the rest of this paper.
The 1907 eruption of RS Oph is only known from its post-eruption dip visible in the Harvard data (Schaefer 2004b). After {\it all} eruptions, RS Oph fades below its normal quiescent magnitude by over one magnitude for a duration of about half a year. Such fading is {\it only} seen after a known eruption, other than on one occasion in 1907. In 1907, when RS Oph came out from behind the Sun, it was faint, getting down to B=13.1 mag and fainter. (Before the solar gap, RS Oph was averaging B=11.8 mag.) RS Oph remained in this faint state for at least 148 days until the start of the next seasonal gap, after which it was back to normal with an average magnitude around 11.8 mag. The obvious idea is that this is a post-eruption dip and that the outburst itself was lost due to conjunction with the Sun. This is a reasonable and inevitable conclusion, but it is not `proven'. While it is always possible that RS Oph faded for half a year in an unprecedented dip unrelated to a nova event, this possibility is contrived and has zero positive evidence. As such, I take the evidence to point with good confidence to the fact that RS Oph had a nova eruption in the winter of 1907.
The 1945 eruption of RS Oph is only known from AAVSO data, where the star was seen fading from 9.6 mag down into a typical post-eruption dip (Oppenheimer \& Mattei 1993). Again, the post-eruption dip was just as the star was coming out from the solar gap in the dawn sky. However, in contrast with the 1907 eruption, the 1945 eruption also shows a definite fading tail during which RS Oph was certainly brighter than at any other time outside of a nova outburst. The combination of the fading tail and the classical post-eruption dip makes the case for a 1945 nova so strong as to be certain.
\subsection{Overview of Eruptions}
In all, we have 37 eruptions from the ten known galactic RNe. Half of the RNe have only two known eruptions, although they might have many more in the last century that were missed. U Sco has a total of nine known outbursts while RS Oph has a total of eight nova events.
The 37 eruptions are listed in Table 2. The first two columns are the star and the year of the maximum. The third column gives the Julian Date of the day of peak brightness (as taken from Section 6). The fourth column gives the time between eruptions in years ($\Delta$T). The next three columns give the name of the first discoverer (with co-discoverers listed as footnotes), the source or means of that discovery, and a reference. Columns eight and nine give the Julian Date of the discovery and the time difference (in days) between the peak and the discovery. The last column indicates whether the eruption is recorded on the Harvard plates, and if so then whether the eruption is recorded exclusively on the Harvard plates.
The shortest known inter-eruption interval is 7.88 years (for U Sco between 1979 and 1987) while the longest is 97.98 years (for V2487 Oph between 1900 and 1998). A dozen eruptions were discovered with the Harvard plate stacks, while another three were discovered with other plate archives, all long after the eruption had come and gone. The Harvard plates provides the only known data for 11 eruptions and the bulk of the data for 15 eruptions, such that without the Harvard plates we would not even have realized that the RN phenomenon existed until the first example would have been discovered in 1946 (T CrB). Only four events were discovered by professional astronomers near peak. All of the other eruptions (exactly half at 19 events) were discovered by amateur astronomers or with amateur astronomers' data. (In this count, I have chosen to take the three PROBLICOM discoveries of William Liller as being that of an amateur, despite his having been a professional astronomer of high repute who worked intensively with the Harvard plate collection, because he has retired and no one was paying for his time or the equipment that he was operating from the patio of his house in Vina del Mar; see Liller 1992 and Liller \& Mayer 1990.) For the 23 non-archival discoveries (i.e, those with a discovery delay of less than 1000 days), 18 were made within four days of the peak brightness, while 15 were made within one day of maximum light. Eight of the RN events have more than one independent discoverer, while three of the events have three or more discoverers.
\section{Comparison Stars}
The photometric histories of the RNe are always determined by comparing the target system with nearby comparison stars of known magnitudes. In preparing for this paper, I have had three distinct applications which require comparison stars. First, I need accurate magnitudes in many bands for just a few relatively faint stars near to the RN to serve as comparison and check stars for differential photometry with CCD images. Second, I need magnitudes of many stars spread over the entire magnitude range of the outburst to serve as sequences for the eruption light curve on archival photographic plates. Third, I need to know the real magnitudes on a modern magnitude scale for various stars used as comparison stars by olden astronomers who have reported eruption light curve magnitudes by direct comparison with these stars.
The magnitude scale was systematized by Pogson (incidentally, the discoverer of the first known RN outburst) in the nineteenth century. By roughly a century ago, the standard magnitude scale was the `North Polar Sequence' defined with the Harvard plates. The sequence was extended to faint magnitudes by various methods such as comparing image diameters on double exposures where the ratio of exposure times were known precisely. We now know that reciprocity failure and other non-linearities in photographic emulsions make for substantial systematic errors that accumulate as fainter and fainter stars are calibrated in this fashion. Magnitudes for target stars were determined by comparing star image sizes on photographs of fields double exposed on the pole region plus the target field. By the 1930's, the North Polar Sequence had been transferred to many of the Harvard-Groningen Selected Areas for more convenient placement on the sky. The trouble with these transfer sequences was that substantial and then-unknown errors were always present. In general, all magnitude sequences from before the 1950's were systematically distorted (from the modern scale) with the errors getting larger as the stars get fainter. Typical errors for a tenth magnitude star are from 0.5-1.5 magnitudes. Sandage (2001) has made a modern study of magnitudes in the primary Selected Areas and finds systematic distortions of 0.6 mag at B=8, 0.8 mag at B=10, 1.0 mag at B=12, and 1.3 mag at B=14. These systematic errors in the primary standard stars were then transfered to all magnitudes in the literature. I have made detailed comparison of many old sequences versus modern magnitudes, and I find that there is much scatter on top of this systematic problem. For comparison stars for use with Pluto, the errors were 0.7 mag in 1934 for a photographic sequence by Walter Baade and 0.01 in 1954 for a photoelectric sequence by Robert Hardie (Schaefer, Buie, \& Smith 2008). For comparison stars for use with SN1937C, the errors ranged from 0.3 to 0.9 mag in the B-band and 0.7 in the V-band (Schaefer 1994). For comparison stars for use with SN1960F, the errors ranged from 0.3 to 0.9 mag in the B-band and 0.7 in the V-band (Schaefer 1996a). For comparison stars for use with SN1974G, the errors ranged from 0.3 to 0.9 mag in the B-band and 0.7 in the V-band (Schaefer 1998). For comparison stars for use with SN1981B, the errors ranged from 0.3 to 0.9 mag in the B-band and 0.7 in the V-band (Schaefer 1995). The point of this recitation is to emphasize that any magnitude before $\sim$1950 certainly has big errors while any magnitude before ~1970 or so likely has big errors. Given this point, a comprehensive photometric history of the old eruptions must somehow overcome this big problem. This can be done either by remeasuring the magnitudes on the original plate material or by rescaling the old reported magnitudes based on the corrected magnitudes for the comparison stars.
The creation of modern accurate magnitude sequences had to await the widespread usage of linear detectors; the photoelectric photometer and later the CCD. Early photoelectric photometers could not go faint enough to be relevant, so it was only in the 1950's that comparison stars could be calibrated deep enough to be useful. Even then, it took many years for useful and accurate standard stars to be determined (Landolt 1973; 1983; 1992) and for RNe sequences to be measured (e.g., Kilkenny et al. 1993; Henden \& Munari 2006). Many deep sequences had to await the availability of CCD cameras and the increase of telescope availability. By the 1970's or so, magnitude sequences and RNe magnitudes are all on the modern magnitude scale. For purposes of this paper, any magnitude in the literature after 1970 is taken to be correct.
The archival photographic plates (when used with modern comparison stars) are exactly on the modern B-magnitude scale. That is, the color terms are effectively zero. Another worry arises from the realization that the RNe have emission lines while the comparison stars do not, so some systematic variations could conceivably arise. But any such effects are miniscule for two reasons. First, any offset will be constant and thus not effect the shape of the light curve. For all applications of the photographic plates, we only care about the light curve shape, so an offset is irrelevant to the conclusions. This includes the comparison between photographic, photoelectric, and CCD magnitudes, with the reason being that these all have essentially constant central wavelengths and equivalent widths of the spectral sensitivity (by design). Second, the RNe have typically only one percent of their blue flux in emission lines (cf. Williams 1983). So even in the false case where slight changes in a passband allow the line light to be recorded or not, the variations will only be of order 0.01 mag. For realistic passbands with nearly constant width and roughly Gaussian shapes, the errors will always be much smaller than 0.01 mag. Thus, we can have good confidence that the old archival plate RN magnitudes have no measurable systematic bias when compared to ordinary field stars or when compared with photoelectric or CCD magnitudes.
For my three applications (see beginning of this section), I have created two sets of comparison stars for each RN. The first set is three faint stars near to the RN, which I have measured in the bands BVRIJHK. These stars are designated as `Comp' (typically a few magnitudes brighter than the normal quiescent magnitude for the RN), `Check' (typically close to the quiescent RN in both position and magnitude), and `Check2'. These stars are all chosen to be well isolated from neighboring stars, and all have been found to be constant over the many observations that I have made. The observations to calibrate these comparison stars have been made many times over on many telescopes (the 0.9m, 1.0m, and 1.3m telescopes at Cerro Tololo and the 0.8m and 2.1m at McDonald). These observations have been made with the usual full program of observations of standard stars (Landolt 1992) at high and low airmass on photometric nights. The typical measurement uncertainty in my measures is likely 0.01-0.02 mag. A number of these stars have been measured by other workers, including, for example, the T Pyx stars by Arlo Landolt (2005 private communication). I have JHK calibration for the stars, but the magnitudes that I will adopt are those taken from the 2MASS survey (Cutri et al. 2003\footnote{http://irsa.ipac.caltech.edu/applications/Gator/}). The magnitudes for these stars are presented in Table 3. The specific positions of these stars will be listed in Table 4.
My second set of comparison stars consists of sequences in B and V for stars spanning the range from the quiescent to peak magnitude. The reason for restricting myself to B and V for this application is that essentially all of the old literature, archival records, and photographic plates are in one of those two magnitude systems. Also, the availability of the B-V color then allows for seeking color terms in the conversion to modern magnitudes. The stars that I have selected are single, isolated, constant, and evenly spaced in magnitude as best as possible. I have also selected as many stars as possible that have been used by previous workers for their sequences. With this last point, I then have a modern and a literature value for the same sequence, and thus a reasonable conversion formula can be determined. Faint stars in the sequences are always near to the RN, and I thus have measured them as part of the calibration program described in the previous paragraph. Bright stars in the sequence are necessarily far from the RN, so I could independently calibrate only a few of them. For stars brighter than tenth magnitude or so, well-measured photometry is already available from the literature, which I have accessed and averaged from the SIMBAD database\footnote{http://simbad.u-strasbg.fr/simbad/}. For many of the RNe, the AAVSO has already constructed well-measured sequences\footnote{http://www.aavso.org/observing/charts/vsp/}, often going quite deep. A variety of other sources have been used to fill in my sequences, including the Tycho2 catalogue\footnote{http://www.rssd.esa.int/hipparcos} (Hog et al. 2000), the GSC2 catalog\footnote{http://www-gsss.stsci.edu/Catalogs/GSC/GSC2/GSC2.htm} (Lasker et al. 2008), and the ASAS3 catalogue\footnote{http://archive.princeton.edu/~asas/} (Pojmanski 2001).
My magnitude sequences are presented in Table 4. The first and second columns are a designation for the field and reference name for the star. My star names are just sequential letters, with a few small gaps because some stars in the series were later identified as being possibly variable or otherwise unsuitable. The third column lists the positions (in J2000 coordinates) accurate to roughly one arc-second as useful for the unique identification of the star. The fourth column lists the formal catalog name for the star (if any). The fifth column gives the star names from the AAVSO name, which is always equal to the V-band magnitude expressed to the nearest 0.1 mag with the decimal dropped, so for example, a V=12.34 mag star would be designated `123'. The next two columns give the B and V magnitudes of the stars. The uncertainties in these magnitudes are likely to be 0.01-0.02 mag, although occasionally larger uncertainties are possible. In some cases, where the magnitude is known to be uncertain by roughly 0.1 mag or larger, the value is only quoted to the nearest tenth. The eighth column cites the source of the quoted magnitudes. The notation K1993 refers to Kilkenny et al. (1993) and H\&M 2006 refers to Henden \& Munari (2006). The last column gives the designations as used earlier in the literature, with a single or double letter indicating the reference followed by a colon and the designation in that reference. For T CrB, we have A (Ashbrook 1946), W (Wright 1946), B (Bertaud 1947), and Pe (Petit 1946). For RS Oph, we have S (Shapley 1933), Pr (Prager 1940), and F (Fleming 1907). For V745 Sco, we have Pl (Plaut 1958). For V3890 Sgr, we have M (Miller 1991). Most of the lines in the comparison sequences table are only presented in the on-line version only.
Some of the extant magnitudes for old eruptions are now only available as specific reports in the literature. Most of these explicitly identify their comparison stars and report their adopted magnitudes for these stars. In general, these old sequences have substantial errors and these are propagated into the RN magnitudes. Fortunately, it is not too late to correct the old magnitudes. The general solution is to plot the old adopted magnitudes versus the modern magnitudes for these stars, and to fit some smooth curve so as to establish the relation between the old and modern magnitudes. With this relation, I then correct the reported RN magnitudes onto a modern magnitude scale. To this end, I have chosen many of the comparison stars in my sequences as being those relevant for correcting old sequences.
For the U Sco sequence of Pogson (Thomas 1940), we have five stars with old and new magnitudes. The RMS deviation of Pogson's sequences is 0.39 mag. Nevertheless, the best fit linear relation is that the Pogson magnitudes are close to the modern magnitudes.
Various series of magnitudes for the eruptions of T CrB have been published that need correction to the modern magnitude scale. For the 1866 eruption, we are often not told the comparison stars (nor the adopted magnitudes for the stars used), so I can only take the reported magnitudes at face value. Distortions in the light curve are likely to be minimal for times when the nova is brighter than seventh magnitude or so as the old magnitude standards were reasonable for bright stars. (Indeed, the 1866 light curve will be seen to closely match the light curve shape from the 1946 eruption down to around eighth magnitude.) Vertical compression in the light curve tail is possible, but this will be confused with the effects of the variations in the red giant companion. The comparison stars of Payne-Gaposhkin \& Wright (1946) are also used by Wright (1946), with a correction of $B=0.8 \times B_{Wright}+1.9$ for $B_{Wright}<11$ and $B=2 \times B_{Wright}-11.3$ for $B_{Wright}>11$. The magnitudes for the comparison stars of Pettit (1946) have a constant offset from the modern magnitudes, such that $V=V_{Pettit}+0.12$. The comparison stars of Shapley (1933) were also used by Weber (1961) and are well represented by a constant offset with $B=B_{Shapley}+0.36$. The comparison stars of Ashbrook (1946) have magnitudes that deviate from those in Table 4 with an RMS scatter of 0.12 mag and no systematic shifts, so I will adopt his visual estimates with no corrections. Similarly, the comparison stars from Bertaud (1947), except for $\epsilon$ CrB, have an RMS error of 0.11 mag with no evidence for systematic errors. The AAVSO magnitudes are also correct. The calibration of the comparison stars in Gordon \& Kron (1979) is modern.
RS Oph has a variety of magnitude sequences published for the early eruption with the Harvard plates (Fleming 1907; Prager 1940; Shapley 1933). These B-band sequences have substantial differences from each other and from the modern magnitudes. For Fleming, the correction is $B=B_{Fleming}+1.5$ for $B_{Fleming}<10$ and $B=1.6 \times B_{Fleming}-4.5$ for $B_{Fleming}>10$. For Prager, the correction is $B=B_{Prager}+0.07$ for $B_{Prager}<10$ and $B=1.4 \times B_{Prager}-3.93$ for $B_{Prager}>10$. For Shapley, the correction is $B=1.13 \times B_{Shapley}-1.0$ for $B_{Shapley}<13$ and $B=0.8 \times B_{Shapley}+3.3$ for $B_{Shapley}>13$.
Five of the dozen magnitudes for the 1949 eruption of V745 Sco are from Plaut (1958) with Leiden plates which I have not examined. Plaut explicitly states his comparison stars (although his stars h, I, and k cannot be identified with his published chart) and his adopted magnitudes, thus his RN magnitudes can be converted to a modern B-band magnitude. Plaut's magnitudes ($B_{Plaut}$) have a best-fit linear relation with my final magnitudes (see Table 4) such that $B=1.4 \times B_{Plaut}-5.0$.
Most of the magnitudes for the 1962 eruption of V3890 Sgr are from the plates at the Maria Mitchell Observatory. These magnitudes were originally estimated by H. Dinerstein, but Miller (1991) has gone back and remeasured the magnitudes. I have not examined the Maria Mitchell plates, but we can place the magnitudes from Miller onto a modern magnitude scale by comparison of the sequences. From her comparison stars, I find that $B=1.02 \times B_{Miller}-0.52$ for $B_{Miller}>14$ and $B=1.23 \times B_{Miller}-3.46$ for $B_{Miller}<14$. This correspondence will be applied to the Miller magnitudes for V3890 Sgr. Wenzel (1990) only quotes the magnitude for one of his comparison stars, but this is in the center of the range, so I adopt $B=B_{Wenzel}+0.24$.
\section{Eruption Light Curves}
The light curves for the 37 eruptions have a wide variety of coverage. Four of the eruptions have only one plate to form their light curve, while many of the older eruptions have at most a few dozen plates recording the event. The more recent eruptions have extensive coverage, with the best one being the 2006 outburst of RS Oph.
Here, I collect all the existing data for most of the eruptions. (However, for a few of the eruptions with large amounts of data, my tabulations may not be complete, as some data sources may have been missed. Fortunately, in these cases, the magnitudes collected will provide a full light curve anyway.) I will present all these magnitudes explicitly in tables in this paper, because most of the magnitudes have never been published, while the published light curves are widely scattered in the literature and have been extensively revised onto a modern magnitude scale as part of this paper.
A large source of data is from the visual observations in the databases of the American Association of Variable Star Observers (AAVSO), Royal Astronomical Society of New Zealand (RASNZ), Variable Star Observers League of Japan (VSOLJ), and the Variable Star Network (VSNET). These observations are almost entirely in the visual band. (Almost all AAVSO work is performed with the color vision, that is with the cones in the retina of the eye for which the effective wavelength equals that of the Johnson V-band, as the target is more than 0.8 mag brighter than the limiting magnitude, see Schaefer 1996b.) The modern AAVSO magnitude sequences are all accurate, so the only question is with the older AAVSO data. For estimates where the star is brighter than roughly ninth magnitude, the deviations in the comparison star magnitudes from their modern values are relatively small. For cases where it matters, I have gotten the sequences used at the time and have made the corrections if needed. The typical uncertainty in these visual magnitudes is 0.15, although these will change with circumstances and observer, from roughly 0.1 to 0.3 mag.
The AAVSO database contains $\sim83\%$ of all magnitudes of RNe in eruption. I will not include the AAVSO magnitudes in my ten tables in the printed version of this paper, because the tabulation would take many pages of journal space and because the AAVSO provides a reliable long-term archive that is freely available to everyone. Nevertheless, the complete list of AAVSO eruption magnitudes will be presented in Table 5, with most of the 12,115 magnitudes only visible in the on-line version of this paper. In this table, the first column identifies the RN, the second column gives the Julian Date of the observation, the third column gives the observing band, the fourth column gives the observed magnitude with its error bar (taken to be 0.15 mag in general), and the last column cites the source.
For many of the eruptions (also during quiescence) the primary data source is the archival photographic plates at Harvard College Observatory. This collection has roughly 500,000 plates from 1890 to 1953, with a smaller collection covering around the 1980's. Each plate is a thin glass rectangle (usually 8.5x10 inches in size) with a photographic emulsion on one side. Almost all plates have blue sensitive emulsion. The Harvard plates provided the original definition of the B-band, and later scales for photoelectric and CCD magnitudes kept the same spectral sensitivity. So if the comparison stars that set the magnitude scale are in modern B magnitudes, then the resultant RN magnitude will also be on the modern magnitude scale. (The old magnitudes taken from the plates have been called 'photographic' magnitudes, with the difference to the modern B magnitudes being only that the old comparison star magnitudes have offsets compared to modern measures, see Section 4.) Thus, with my modern comparison star sequences, the Harvard plates provide a wonderful `time machine' to get RNe light curves as far back as 1890.
The plates are taken in series, with one series for each telescope used. The plates from each series are numbered sequentially, with a capital letter series identifier followed by the sequential number. For example, one of the deeper plate series (taken with the 24-inch Bruce Doublet telescope) is designated with the letter `A', such that plate A1 was taken in 1893 and the last A plate (A27504) was taken in 1950. The better A plates have a limiting magnitude of roughly 18, with of order 30 useable plates that show the field of any one RNe. The primary deep series (with limits of 16-18 mag) are the A, MC, and MF, while there are many tens-of thousands of plates with less deep limits (with limits of 13-16 mag) in the B and I series. Harvard also has several series (chiefly RH, RB, AM, AC, and the Damons) that are patrol plates, where large areas of the sky are regularly photographed to moderate limits (typically 12-15 mag). The entire sky is covered roughly equally, with typically 1000-3000 plates covering any particular RN. Further details are available in Collazzi et al. (2009) or at the Harvard archive web site\footnote{http://tdc-www.harvard.edu/plates/}.
The one-sigma uncertainty in the magnitude depends substantially on the plate material, with various field effects increasing the uncertainty towards the edge of the plate. For a well exposed field with a good sequence of comparison stars, measurements with iris diaphragm photometers can give a median accuracy of 0.08 mag, and can get as good as 0.04 mag (Schaefer, Buie, \& Smith 2008). However, most of the plates were measured by direct visual comparison of image sizes by means of a view through a loupe (a handheld magnifier). I have a very large amount of experience in this, and I also have performed a variety of experiments over the years for the accuracy of these comparisons. The median accuracy is 0.15 mag, with typical accuracy ranging from 0.1 to 0.2 mag. It is difficult to evaluate the uncertainty for any one individual plate, so I have used a default one-sigma error of 0.15 mag unless my notes point to some other value as quoted in the tables. While the measurement uncertainty for plates is roughly a factor of ten worse than usually obtainable with photoelectric or CCD techniques, this uncertainty is comparable or smaller than the usual random flickering. Thus, for many purposes, the moderate accuracy of photographic magnitudes is perfectly fine and just as good as more 'modern' measures.
The eruption light curves are collected together into one table for each RN (Tables 6-15). The format for each line will come in one of two forms, depending on whether the B-band and V-band magnitudes were made effectively simultaneously. For most observations, the columns will list the Julian date of the observation, the band, the observed magnitude (along with its uncertainty), and the source of the data. If B and V magnitudes are from the same time, then the second and third columns will instead report the B-band and V-band magnitudes in that order. (This is simply to make the tables more compact.) Each eruption will have a header specifying the year of the peak. The source of the data might be the Harvard plate number (e.g., B5289, AM1172), the organization reporting the data, or the literature reference. The observations are listed in time order. The complete eruption light curves consist of those magnitudes from each of the tables for individual RNe (Tables 6-15) plus the AAVSO magnitudes for many individual eruptions collected into Table 5 (most of which is only visible in the on-line table). For the next ten paragraphs in this section, I will describe the data sources specific to each of the ten galactic RNe.
T Pyx has five known eruptions , with photometry for the first four known almost entirely from the Harvard plates. I have remeasured the brightness of all the eruption plates at Harvard with the comparison stars from Table 4. The 1966 eruption has three large data sources, including the AAVSO, Landolt (1970), and Eggen et al. (1967). These data essentially stopped at day 80 after the peak. However, within the last year, the original data logbooks of Albert Jones (see Section 3.1) have been transcribed and added to the AAVSO data collection. This `new' data has a wonderful coverage from day 80-300, and suddenly reveals a sharp drop-off in the light curve followed by a plateau, never previously suspected. All data sets show a day-to-day variation of 0.5-1.0 mag. These variations are all much larger than any uncertainty (with the best illustration of this being the time resolved light curve of Landolt 1970), and are certainly intrinsic to T Pyx. T Pyx is the only RN to display such flares/oscillations. The light curves for the eruptions are presented in Tables 5-6 and Figures 2-5.
IM Nor has two eruptions, with the first only known from eight Harvard plates. A light curve for the 1920 outburst was never published, although Elliot \& Liller (1972) remeasured the magnitudes and plotted a low resolution curve with no numbers. Neither J. Elliot nor W. Liller could locate their original magnitudes, and I could not find the original notebook with Ida Woods' measures at Harvard. So the measures of the 1920 eruption light curve are entirely from my own viewing of the original plates along with the comparison star sequence in Table 4. One of the plates has a V-band spectral sensitivity. For the 2002 eruption, Liller's first two magnitudes are in a red bandpass and they represent the initial rise to peak. Almost all the remaining light curve is in the V-band, taken from data reported either in the {\it IAU Circulars} or by the AAVSO. The exception is the average magnitudes on five nights late in the tail as measured with CCD photometry by Woudt \& Warner (2003), for which I only quote the nightly average. Significantly, the long times series by B. Monard (taken from the AAVSO database) well covers the late tail of the eruption, although in a non-standard band-pass (see Table 22). The light curve for both eruptions are presented in Tables 5, 7, and 23 plus Figures 6-7. The top panel in Figure 7 shows just V-band observations, while the lower panel adds in the unfiltered-CCD magnitudes of Monard with a 0.6 mag offset to make the them match in the large time intervals of close overlap. By this combination of V and unfiltered magnitudes, we get a single light curve that covers from before the peak all the way until the nova returns to quiescence. A potential trouble with this combination is that the differential color might change over time (say, due to increasing relative strength of the H$\alpha$ line).
CI Aql has three known eruptions, with the first in 1917 being covered primarily with the Harvard plates plus a few from Heidelberg. Williams (2000) reports a modern measure of the Harvard plates from the 1917 eruption, although in this paper I will report on my independent measures based on the comparison sequence in Table 4. The system had its second discovered eruption in 2000, with a fairly well observed light curve and the realization that yet another little-known nova was actually a RN. The 2000 eruption has wonderful coverage by the AAVSO observers, giving good measures of the light curve fluctuations in the visual band from just before peak until very late times. Additional observations are reported in the {\it IAU Circulars} and Matsumoto et al. (2001). The light curve for the 1941 eruption is entirely from the Harvard plates and is reported here for the first time. The light curves for the three eruptions are presented in Tables 5 and 8 plus Figures 8-10.
V2487 Oph has two eruptions, with the first in 1900 being known only from a single Harvard plate. In 1900, the discovery plate shows the nova near its peak magnitude, which only lasts a day or two. There are a variety of other Harvard plates taken that month, but none of them go deep enough to come anywhere near recording this eruption. The light curve from 1998 reveals one of the fastest known nova eruptions. The light curves for both eruptions are listed in Table 9, while the 1998 eruption light curve is plotted in Figure 11.
U Sco has a total of nine known eruptions. The first observed eruption was in 1863, where only one observer recorded the event. This was one of the earliest novae that was recorded in modern times. The light curve observations by Pogson have been collected by Thomas (1940). Pogson's comparison star magnitudes are consistent with the modern magnitudes, so his magnitudes are taken exactly. The next four eruptions (in 1906, 1917, 1936, and 1945) are only known from the Harvard plates, and the magnitudes given in Table 10 are entirely from my own measures of the original plates with the modern magnitude sequence from Table 4. The 1917 and 1945 eruption light curves are published here for the first time. The 1906 and 1917 eruptions have only one magnitude, while there are many limits from around the same time for which these are all fully consistent with the usual light curve. I will not give a figure plotting these one-point eruption light curves. The 1945 eruption is the only one to record the sudden drop in B-band brightness starting 33 days after peak. Other than the serendipitous archival plates from the 1945 eruption, only four magnitudes have been recorded for U Sco in eruption after the sharp drop at 33 days. All four magnitudes are in the V-band and they do {\it not} show any evidence for the drop. It might be possible that the drop did not occur in the 1979 and 1999 events, or it might be that the drop is only prominent in the blue light. The 1969 eruption is known only from 16 visual observations, now residing in the archives of the {\it Royal Astronomical Society of New Zealand} (RASNZ). The observations are published here for the first time. I have examined the magnitude sequence used by Jones and Bateson (courtesy of P. Loader) and find that their magnitudes are close to the modern magnitude scale. Almost all of the magnitudes from the 1979, 1987, and 1999 eruptions are from the AAVSO. A variety of sources in the literature have reported a few additional magnitudes, and I am presuming these to be on the modern magnitude scale even though they do not report on their comparison stars. The magnitudes from Budzinovskaya et al. (1992) are not useable because their detector was a TV system with unknown color sensitivity, and their quoted comparison star magnitudes deviate from Table 4 unsystematically with errors ranging from -1.02 to +0.22 magnitudes. U Sco has been seen with certainty on its rising branch only three times; once on a Harvard plate in 1936 at B=10.75 mag 0.25 day before maximum, once by Dr. N. W. Taylor (of the RASNZ) in 1987 at V=14.0 mag in the day before maximum, and once by P. Schmeer in 1999 at V=9.5 mag 0.3 day before maximum. The final rise to maximum in the 1936 event is at a rate of 19 magnitudes per day. With this rate for the entire rising branch, U Sco will go from minimum to maximum in roughly 6-12 hours. In the 1999 eruption, we know that U Sco rose from below V=14.3 to peak in less than 0.46 days. U Sco goes from minimum to peak to one magnitude below peak all in under one day, and this is a daunting constraint for fast response in the upcoming U Sco eruption. The light curves for all nine eruptions are listed in Table 10, while most of these are plotted in Figures 12-18.
V394 CrA has two poorly observed eruptions. The 1949 eruption has a few magnitudes appear scattered in the literature, and seven Harvard plates have detections. The 1987 eruption has a few AAVSO magnitudes, a variety of magnitudes presented in the {\it IAU Circulars}, plus 14 B and V magnitudes from Sekiguchi et al. (1989). The photographic observations in the late tail of the 1949 eruption have a large scatter, while the visual observations early in the 1987 eruption have a large scatter. In both cases the scatter might indicate very large oscillations in the light curve, but I do not think so. The good photoelectric observations of Sekiguchi et al. (1989) do not show such oscillations. Instead, I take the scatter in 1949 to be the usual problems with faint comparison stars in the 1940's and I take the scatter in 1987 to be the usual problems with observers early in an eruption using makeshift comparison stars before standardized sequences are available. I have not been able to personally examine the non-Harvard plates for the 1949 event, but my broad experience suggests that such problems are common. The light curves for the two eruptions are given in Table 11 and Figures 19-20.
T CrB has two eruptions (or maybe three if we accept the 1842 event). The two eruptions were very well observed, likely due to its brightening up to the second magnitude. The first was in 1866, where T CrB was the earliest nova event that was well studied. The quiescent nova was identified as a tenth magnitude variable source, so various groups kept track of the variations. In 1920, the great Leslie Peltier started regularly monitoring the brightness. (Peltier and Albert Jones are justifiably considered as the greatest visual observers of the last century.) Based on a generic suspicion that old nova might erupt again, he started to regularly monitor several old novae with bright quiescent levels, including T CrB, RS Oph, and GK Per. At the time he started, no RN had been established and the concept was not known. As told in a famous passage from his inspirational book {\it Starlight Nights} (Peltier 1965), he watched T CrB for decades, only to miss the one night when it finally erupted, so that another person made the anticipated discovery in 1946. Peltier says ``We had been friends for many years; on thousands of nights I had watched over it as it slept" and then after the eruption that ``There is no warmth between us anymore". An important product of Peltier's work is a good long-term light curve before the eruption, which displays a unique pre-eruption dip in the light curve (Peltier 1945). The 1946 eruption was heavily observed by amateurs and professionals, in both B-band and V-band. The post-eruption light curve displays a unique brightening that lasted a hundred days, starting after the light curve had already returned to minimum and been there for fifty days. A check of the old light curve from 1866 showed people that a similar post-eruption event occurred previously and hence was not some disconnected happenstance. I have heard a variety of theoretical models (e.g. Webbink et al. 1987, Selvelli et al. 1992; Hachisu \& Kato 2001), but I have also heard convincing counter-arguments to these various ideas, so at this time I do not think that we know the cause of the re-brightening. The light curves are given in Table 12 and Figures 21-23. As the time of interest for the light curves is so long and the AAVSO database records so many magnitudes, I have presented the AAVSO magnitudes away from the main peak as averages over 0.01 year time bins.
RS Oph has eight recorded eruptions in the last century. The first two (in 1898 and 1907) are only recorded on Harvard plates. The remaining six eruptions have almost all their light curve information coming from the AAVSO database, with only occasional other magnitudes from the professional literature (mainly scattered in the {\it IAU Circulars}). The AAVSO has roughly 8000 observations covering the times of the eruptions, and to save space in the tables and to reduce the usual observational scatter, I have constructed 3-day averages for time intervals where the light curve is not changing rapidly and where there are typically $>$4 values per interval. The unique property of the RS Oph light curve (not seen in any other RN or classical nova) is the post-eruption dip. That is, from day 100 to 500, or so, RS Oph is significantly fainter by over one magnitude than its normal quiescence level. This dip appears after all eruptions. The exact level of the dip varies slightly from eruption-to-eruption, and this is easily attributed to the known long-period variations of the red giant companion star. This same effect makes for small variations in the late tail of the decline from the nova event. Similarly, the pre-eruption level changes somewhat with the normal oscillations of the companion. The light curves are given in Tables 5 and 13 plus Figures 24-31.
V745 Sco has two poorly-observed eruptions. For the 1937 eruption light curve, seven magnitudes come from my measurements of the Harvard plates. I have not examined the five archival plates at Leiden (Plaut 1958), but I have corrected the magnitudes reported by Plaut based on the his magnitudes for comparison stars (see Section 4). The 1989 eruption was not well observed, with few visual observations (and no CCD measures), likely due to the faintness and speed of the decline. The visual magnitudes in the 1989 decline show scatter at about the half-magnitude level, likely due to the usual problems with comparison stars and color terms. The light curve is presented in Table 14 and Figures 32 and 33.
V3890 Sgr has two eruptions. For the 1962 eruption, we only have B-band magnitudes from archival plates at the Maria Mitchell Observatory (Dinerstein 1973; Miller 1991) and the Sonneberg Observatory (Wenzel 1990). The 1990 eruption has a V-band light curve that is well defined by visual estimates. Note the occasional large scatter, which I take to be the usual systematic problems. Buckley et al. (1990) report on UBV photometry, with this providing our only color information, and also provides the way to compare the two eruptions. The light curve for both eruptions is given in Table 15 and Figures 34 and 35.
\section{Light Curve Shapes}
The light curves based on individual data points show the usual amount of scatter and gaps. A smoothed light curve can better represent the real variations of the eruption. Such a template light curve is also better for display as well as deriving light curve properties (like $t_3$). In this section, I will describe the templates for the ten known galactic RNe.
\subsection{Construction of Templates}
The templates are simply smoothed light curves with the gaps filled in. A fundamental problem (at least in principle) concerns the best time scale for the smoothing, as too short a time scale will result in random measurement noise being taken as a real variation in the nova, while too long a time scale could smooth out real changes in slope of the decline. In practice, this is not a problem, as the data frequency is high enough to well-sample the relatively slow variations in the decline. That is, with one exception, the underlying variations are smooth on time scales that are a fairly small fraction of the time since the peak. The one exception is the rapid oscillations in the light curve of T Pyx, which we see from Landolt (1970) to be $>0.8$ mag on time scales of under one day. In this case where we only have brief snips from the oscillations, all we can do is to create a template that tracks through the middle of these oscillations.
Another fundamental problem is whether a separate template should be created for each eruption. As discussed in the next subsection, the eruption light appears to follow the same template for all eruptions, and this justifies the creation of a single template applicable to all eruptions. This means that gaps in one eruption can be filled in with observations from other eruptions. Gaps in one band can also be filled in with information from another band, provided that some reasonable color can be adopted.
For my templates, I created piecewise-linear light curves that tracked through the middle of the data. I used Occam's Razor to try to have the templates follow the observations with a minimal number of points. The templates were constructed to minimize the deviations for all the eruptions of each individual RN. I also made small adjustments (within the photometric uncertainties) so as to keep a fairly smooth B-V color curve.
The templates are expressed as a light curve with B and V magnitudes as a function of time from the peak for many specific times. For times between the given times, a simple linear interpolation should be used to give the magnitude at that time. The templates are presented for all ten RNe in Table 16. The first column gives the time in days from the peak, while the last two columns give the B-band and V-band magnitudes at the specified time. The templates all start with the average quiescent magnitude for a time nominally 999 days before the peak. Generally, there are no data late in the tail of the eruption, so the templates stop at the times of the last available data in these cases.
\subsection{Are the Templates Constant?}
For a given RN, does its light curve change from eruption-to-eruption? That is, does the template change with each eruption? This can be tested by looking to see if there are any significant deviations from my average templates. For this, we can examine the eruption light curves in each figure to see where the observations deviate from the template. A substantial problem with this task is that we have to worry about the usual photometric errors. Another problem relates to the variability of the additive brightness of the companion star, as this component changes greatly for T CrB and RS Oph and dominates late in the tail of the eruption light curve. Finally, we have to worry about the flickering, eclipses, and daily random fluctuations displayed by various RNe. Here is a nova-by-nova evaluation of whether the eruption light curves are constant:
For T Pyx, we have the substantial problem that its light curve has large amplitude oscillations on the time scale of a day (Landolt 1970). This will create an ambiguity in that deviations from the template could arise either from the oscillations, from eruption-to-eruption changes, or from ordinary photometric errors. Given the known oscillations, I see no significant deviations from the average template for any of the first four eruptions, which are all based on Harvard data. For the 1966 eruption, the pre-maximum magnitudes from AAVSO observers are an average of one magnitude below the template. These early magnitudes are certainly incorrect as seen by comparison with the photoelectric magnitudes of Eggen et al. (1967), as is likely a simple problem with the comparison stars available in the first days of the eruption. In all, T Pyx appears to have an unvarying eruption light curve, although with fast oscillations superposed.
For IM Nor, we have the problem that the 1920 eruption is known only from B-band magnitudes (except for one plate) while the 2002 eruption is known only with V-band magnitudes. So all I can do is construct the two templates and see whether the color curve is reasonable. The templates have a B-V color of 0.7 mag throughout, and this is consistent with a reasonable color and extinction. In all, as far as we can tell, the two eruptions had the same light curve.
For CI Aql, we have the problem that the first two outbursts have no V-band magnitudes while the third eruption has little B-band data. In addition, the light curves for the first two eruptions have no overlap in time after peak. Thus, we can only make poor tests for constancy of the light curve shapes. In all, it looks like the light curves for all three eruptions are identical to within the normal measurement problems. The one possible exception to this concerns the two brightest magnitudes from the 1917 eruption. These magnitudes are B=8.7, while the 2000 eruption has B=9.5 at the same phase in the eruption. We cannot blame a single plate flaw for this difference as we have two plates in 1917. And the 2000 B-band template is unlikely to be so far off as this would require an extreme B-V color that changes fast. Nor can we simply move the date of maximum in 1917 as this is well-constrained by the faint first point on the rising branch. With CI Aql not showing fast oscillations, the two 1917 plates make a plausible case for a substantial difference in light curve shape between the 1917 and 2000 eruptions. Nevertheless, I am not convinced by such a conclusion. The problem is that the plate quality for these plates is low, the stars are near the edge, and strong field effects (in particular, the changing image size and shape with distance from the plate center) can lead to substantial systematic errors when the comparison stars are relatively far away (as will be the case when the nova is near peak). In all, I do not judge the evidence to be sufficiently strong to force a conclusion that the light curves change.
For V2487 Oph, we only have one magnitude from the first eruption, so we cannot make any real comparison between light curves.
For U Sco, we have many eruptions, but the overlaps for particular times and bands is such that only several of the events can be well compared. The constructed template provides a reasonable representation for all the light curves. We have the usual photometric systematic problems, for example with the V-band magnitudes of Kiyota (1999) being systematically fainter than those of the AAVSO from 8-25 days after the 1999 peak. Small deviations from the template can be pointed to (e.g., days 29 and 30 in the 1936 eruption and around day 10 in the 1987 eruption). But I judge these to be too small to be significant given the usual photometric systematic problems, eclipses, and flickering. In all, I take the U Sco light curve to be constant from eruption-to-eruption.
For V394 CrA, we have a relatively small amount of poor data. The problems with the scatter (likely due to ordinary problems with comparison star calibration) means that the derived template will have a larger uncertainty than for other RNe. Within this larger uncertainty, there is no evidence that the light curve shape was different in 1949 from 1987.
For T CrB, I have constructed the templates from the very well observed 1946 outburst. The 1866 eruption fits the 1946 template closely up until the star gets close to minimum. Around the time that the RN returns to the quiescent level, we see a series of magnitudes from a single observer that are bright (compared to everyone else) by up to one magnitude from day 21 to 80. This is certainly some usual error (perhaps a poor sequence of comparison stars or perhaps a wrong star identification) by Bird (1866). Another error is the report that T CrB was seen at V=5.1 on JD 2431856 (i.e., 4 days before anyone else saw any rise) as recorded in the AAVSO database. This report would have constituted the real discovery of the 1946 outburst if correct. But discussions at the time (e.g., Pettit 1946) give the discovery to N. Knight (Knight 1946) even though the Siberian observer A. Kamenchuk saw the rise earlier (Kamenchuk 1946), and so there must have been some known reason to discount the observation. I expect that the quoted date should have been reported as JD2431865, with the last two digits switched. During the secondary maximum, the 1866 light curve is consistently a third of a magnitude brighter than at the same time in the 1946 light curve. Could this be a significant difference between the two eruptions? Possibly. A possible resolution might simply be due to errors in the early comparison stars. Alternatively, the red giant companion star has long term variations with an amplitude of roughly one magnitude, and the two secondary maxima could well simply be at different phases in the red giant cycle, resulting in the total system having some apparent offset in the light curve when the system is near quiescence. That is, the differences between the light curves are likely just due to a different luminosity of the red giant. Either explanation would have the entire nova eruption luminosity history plus any accretion disk brightening history being identical. Likely the luminosity of the red giant would be considered an incidental and confusing light source that can be ignored for purposes of understanding the RN phenomenon and for purposes of understanding the specific eruption physics. As such, the light curve for the nova event itself (plus any possible accretion disk phenomenon) is apparently constant from eruption-to-eruption.
For RS Oph, the template provides a good fit to all eight light curves. But the match is not perfect. When RS Oph is faint, the light curve is often somewhat bright or dim compared to the template. The deviations from the template vary slowly with respect to the duration of the eruption, and the deviations are the largest during the post-eruption dips. During the post-eruption dips, the nova eruption light has faded away, and the accretion has stopped with an empty disk (Worters et al. 2007), so the only significant optical light source at the bottom of the post-eruption dip is the red giant. So variations from the template must come from the red giant. Such slow variations of the red giant are already known, for example from the modulations on the orbital period (Gromadzki et al. 2008) and from the power density spectrum of the variations in quiescence (see Section 12.3). These variations have a typical amplitude of half a magnitude, and this is exactly what we see in the deviations from the template. So we have a consistent picture that the nova light always follows the same template while the red giant varies slowly with moderate amplitude, such that the nova-plus-red-giant brightness will show deviations as observed from the average template. (After the end of the post-eruption dip, when accretion resumes, the quiescent light will be from both the varying red giant plus the varying accretion disk and hot spot.) Therefore, the nova part of the light curve is consistent with being constant from eruption-to-eruption.
For V745 Sco, any comparison of the 1937 and 1989 eruption light curves has the problem that the first was only measured in B-band and the second was only measured in V-band. So all I can do is to construct templates for the two colors and then evaluate whether the resultant color curve is reasonable. With this, B-V varies from 1.1 near peak to 1.9 later in the tail, and I judge this to be reasonable because other RNe have similar color behavior. The V-band light curve has substantial problems due to scatter, so there is a large uncertainty in the derived color curve. As such, the two light curves are apparently consistent with each other.
For V3890 Sgr, we have little direct comparison between the two eruption light curves. Nevertheless, a simple template can be constructed with a constant B-V color that agrees with the light curves from both events. As such, it appears that the light curve shape is constant for the two outbursts of V3890 Sgr.
In all, I have found no significant change in the nova light curves from eruption-to-eruption. The only confident exception to this rather strong statement is that T Pyx has fast oscillations which likely do not repeat exactly from eruption-to-eruption. Other possible exceptions, as noted above, all arise from obvious or likely errors in the photometry. Also, note that the non-nova light, in particular the light from the red giant companion star can make small eruption-to-eruption changes in the total system brightness. My conclusion that the nova light always has identical light curves is a fairly strong statement, involving 33 eruptions. What this is telling us is that the eruption light curve depends on system parameters (like the white dwarf mass and the composition) which do not vary from eruption-to-eruption. Also, given that the accretion rates vary greatly over time, the constancy of the eruption light curves suggests that the eruption depends little on the accretion rate.
\subsection{Light Curve Properties}
The templates can be used to estimate the properties of the RNe light curves. That is, we can get a better value for peaks and durations from the templates than can be gotten by considering individual light curves with all their various gaps and the scatter of individual magnitudes. With a template, we are averaging over all nearby magnitudes in a single light curve and averaging over all light curves.
From the templates in Table 16, I have evaluated a variety of light curve properties for all ten known galactic RNe. These are in presented in Table 17. The first two properties ($B_{peak}$ and $V_{peak}$) are the brightest magnitudes in the light curve for the B- and V-bands, while the third line gives the V-band magnitude 15 days after peak. Lines 4 and 5 give the B-V color at peak ($B-V_{peak}$) and at a time when the V-band light curve has faded by 2 magnitudes from peak ($B-V_{Vpeak+2}$). The next two lines give the amplitude (the difference between the peak and quiescent magnitudes) in the B- and V-bands. The next eight lines give the time in days from the peak until when the V-band light curve has faded by some integral number of magnitudes (as indicated in the subscript). So for example, $t_2$ and $t_8$ are the times from the peak until when the template has faded by 2 and 8 magnitudes from the peak. The value of $t_{-3}$ is the rise time from when the nova is 3 magnitudes fainter than peak until the time of the peak. Some of the values are not given for the cases where the amplitude of the light curve is too small for the `t' value to be defined or for the cases where the light curve is not followed late enough to allow a measure. The next dozen lines give the time (in days) for which the light curves are brighter than the magnitude indicated in the subscript (e.g., $D_{B=9mag}$). These eruption durations are based on the apparent magnitude (instead of the magnitude relative to the peak). The purpose in defining these durations is for use in discovery statistics, where it matters how long the eruption is brighter than some discovery threshold. The first half of these magnitudes are for the B-band, while the last half are for the V-band. The last two lines give equivalent durations for the optical light curves ($D_B$ and $D_V$) in the B- and V-bands. These are calculated with the integral under the light curve. The optical flux is used (not the magnitude) in this integral, so the output will be proportional to the optical energy. The flux has the quiescent flux subtracted out, so the output is referring to the energy of the nova eruption. The flux has been normalized to that of the peak, so the values are distance independent. The quoted values have units of `days', and they are like an effective duration. The total optical energy of the nova eruptions are equal to that of an idealized square-shaped light curve with the same peak and the given equivalent duration. These equivalent durations are useful for calculating the energetics of the outburst.
\subsection{Comparison of Light Curves}
How do the light curves of the RNe compare? In Figure 36, I have simply plotted all the templates from Table 16. We see an apparent wide range of peaks, durations, and quiescence levels. By construction, the peak times all come in the same position. The peak magnitudes are governed in part by distances and extinctions (poorly known) that are irrelevant to the eruption physics. So a better comparison might be to vertically shift all the templates so that they match at the peak magnitude, as shown in Figure 37. We still see a wide variety of durations and decline rates. To compare the {\it shapes} of the light curves, we can scale the horizontal axis by $t_3$ so as normalize out the durations, as in Figure 38. Now, all the light curves are starting to look similar. Part of this is by construction, as the scaled templates must match at two points. Nevertheless, the light curves are indeed not greatly different.
Are RN light curves different from classical nova light curves? This question cannot be answered with confidence, as I know of no comparable study of light curve shapes for classical novae. (Payne-Gaposhkin 1964 and Duerbeck 1981 give a fairly broad categorization of nova light curve shapes, but there is little quantification or plots for a real comparison.) Nevertheless, from perusing various papers with light curves, I think that the RN light curves are similar to those of many novae. A possible difference is that the RNe do not have the diversity of CNe, for example there are no RNe with DQ-Her-like dips nor any RNe with slow symbiotic-like light curves.
A comparison of the B-band and the V-band templates for an individual RN will give the B-V color curve. The color curves for the ten RNe have been collected in Figure 39. I am somewhat surprised by the small color evolution for many of the RNe. I would have expected that RNe would behave like CNe, starting out with a negative B-V before peak, B-V$\sim$0 around peak, and substantial reddening as the nova fades. (Of course, interstellar extinction will redden this whole idealized color curve.) Instead, T Pyx, IM Nor,V2487 Oph, U Sco, V394 CrA, and V3890 Sgr have nearly constant colors. And the color evolutions for T CrB and RS Oph are apparently the simple transition from a constant blue nova color to a constant red companion color as the nova light fades. (The color curve for V745 Sco is poorly determined, see Figures 32 and 33.) With this, it appears that the color of the nova light is roughly constant (or at least does not vary much) throughout the eruptions for 90\% of the RNe. The one exception is CI Aql, which has a brief reddening interval followed by a return to a blue color followed by a slow reddening. This is neither the nearly-constant color pattern of the other RNe nor the monotonic-reddening pattern of CNe.
\subsection{Broken Power Law Light Curves}
Hachisu \& Kato (2006; 2007) present a general model for nova light curves and conclude that they have found a 'universal decline law' based on the ordinary physics of the expansion of the nova shells. They derive that the flux should fall off proportional to $t^{-1.75}$ when the nova is declining from roughly 2 to 6 magnitudes below peak, then break to a decline proportional to $t^{-3.5}$, followed soon by shifting to a decline proportional to $t^{-3.0}$. This is a tremendously useful generalization. The break from the $t^{-1.75}$ behavior occurs at a time which varies somewhat with composition, but mainly with the mass of the white dwarf. The break time varies from one year or more for white dwarf masses of 0.6 $M_{\odot}$, to $\sim100$ days for 1.0 $M_{\odot}$, to $\sim25$ days for 1.3 $M_{\odot}$. This result provides a practical means of estimating the mass of the white dwarf from the light curve alone. The RNe (necessarily with $M_{WD}> 1.2$ $M_{\odot}$) must have a fast break.
This result has three limitations. First it does not account for the light curve before or near peak. Second, a variety of effects can be superposed on the universal decline. Thus, flares and oscillations prominent in the light curve do not have their rapid variability described, and we can only fit power law models through the middle of the light curves. Also, dust formation is not addressed, and we can only ignore the portions of the light curve dominated by dust absorption. The ordinary luminosity at quiescence is not part of the universal decline, so this light must be subtracted out to get the eruption light.
The third limitation of the universal decline law is that it is addressing the continuum flux alone. That is, the emission line flux is not included, and this flux rises rapidly and starts dominating during the transition phase of the nova event. Hachisu \& Kato point out that the Stromgren `y' band filter is narrow and does not include many of the prominent emission lines, so y-band light curves should follow their universal decline law. But the broader V-band filter includes emission lines which start becoming prominent before the break and raise the observed V-band light curve above the universal decline law. They point out that the V-band light curves have a transition from one universal decline law to a parallel light curve with identical slopes and break times yet brighter so as to include the emission flux. This transition will make the application of the universal light curve into a messy fit for V-band data. Unfortunately, all we have for the recurrent novae are V-band light curves, so we can only make do. Nevertheless, I will report here on fits of the RN templates to broken power laws. Despite the messy transition caused by the emission lines, the V-band light curves should start out with a $t^{-1.75}$ slope, have a break around 25 days, and end up declining as $t^{-3.0}$.
When constructing a power law for the light curve, we must specify some zero for the time. That is, the flux will be proportional to $(t-t_0)^{-1.75}$, where $t$ is the time of the observation and $t_0$ is the zero time. Hachisu \& Kato take the zero time to be the instant that the eruption expansion starts. Operationally, I will take this as the time when the nova brightness starts its rise from quiescence. The uncertainties in the power law slope will not be significant for the times after the nova has faded by two magnitudes from peak. With the usual conversion from flux to magnitudes, the power law segments should obey an equation of $V=V_{break} - 2.5 \times \alpha \times \log[(t-t_0)/(t_{break}-t_0)]$, where $t_{break}$ is the break time, $V_{break}$ is the V-band magnitude at that break time, and $\alpha$ is either -1.75 (for $t<t_{break}$) or -3.0 (for $t>t_{break}$). Again, given the complexity of the emission line flux rising during the transition period, the details around the time of the break should not closely follow the universal decline law. Also, the quiescent flux must be subtracted to give the V-band magnitude of the nova light alone.
I have fitted the RN light curve templates to a simple broken power law. The fit is described by four parameters; the slope for times before the break ($\alpha_{early}$), the $t-t_0$ time of the break ($t_{break}$), the V-band magnitude of the break ($V_{break}$), and the slope for times after the break ($\alpha_{late}$). These values are presented in Table 17. With my use of the V-band, we can only acknowledge that the fits are only a messy version of the universal decline law. For example, with T Pyx, the break apparently occurs before the peak and the plateau is a significant bump that is blindly fitted over. A substantial problem is how to handle the plateaus (which are claimed to be unique to RNe), as some choices in how to handle them can improve the comparison between theory and observation. Nevertheless, the median values are $\langle \alpha_{early} \rangle = -1.45$ and $\langle \alpha_{late} \rangle = -2.9$. I take these median values (despite their substantial scatter) to be in reasonable agreement with the theoretical universal decline law.
The universal decline law provides a nice and specific theoretical prediction to be tested. (It also provides a nice organizing principle for understanding the declines and for estimating the white dwarf masses.) I have plotted the declining portion of the RN templates (after subtracting the quiescent fluxes) on a logarithmic time axis in Figures 40 and 41. The first of these figures has the light curves without any vertical shifting so that individual RNe can be picked out. The second of these figures has all the light curves shifted vertically and horizontally so that they all their fitted breaks at the same point. This figure also has line segments representing $\alpha=-1.75$ and $\alpha=-3.0$ which pass through the origin, as a representation of the universal decline law. We see that there is the usual amount of scatter, but that the light curves generally are close to $\alpha=-1.75$ for early times and are close to $\alpha=-3.0$ at late times. That is, the universal decline law of Hachisu \& Kato gives a good description of the behavior of RN light curves. Indeed, it is astonishing that the complexity and variety of light curves presented in Figure 36 can be reduced to a single uniform shape as in Figure 41. This match with theory is by no means perfect. For example, the late-time slope for CI Aql is $\alpha_{early}=-1.8$ or else $t_{break}>200$ days, while the apparent break time for IM Nor is at 235 days. Other than these two cases, the RNe have fast break times (much faster than ordinary classical novae) as appropriate for their higher mass white dwarfs. (RS Oph has a break time at 77 days and is within the range allowed for RNe masses.) And four of the RNe have break times apparently $\sim6$ days, which indicate that the white dwarf is very close to the Chandrasekhar mass. On this basis, I would identify V2487 Oph, U Sco, V745 Sco, and T CrB as the systems most likely to collapse as a Type Ia supernova the soonest.
\subsection{Sharp Drops}
In the last year, three of the RNe have been recognized to have sharp drops in their light curves. Sharp drops in nova light curves (whether CN or RN) have not been anticipated, have no precedent, have no prior suggestion, and have no prior theoretical prediction. What has happened in the last year is that three sets of archival data have finally been examined. The first case is that the 1966 T Pyx data from Albert Jones has finally been taken out of his log books and made available through the AAVSO data base. The second case is that I have taken B. Monard's recent unfiltered CCD data from the AAVSO data base and combined them with the regular AAVSO V-band data (see Figure 7). The third case is that I have measured the modern magnitudes for all plates that show the 1945 U Sco eruption and the final light curve had the sharp drop apparent only in the preparation of the template for this paper. It is a sobering thought for our community that there had been no other effective photometry after day 80 for T Pyx and after day 30 for U Sco.
T Pyx has a sudden decline starting on day 85 in both the B-band and V-band. In 20 days, the brightness declines by about 2.0 magnitudes. Then the light curve goes nearly flat for roughly 65 days in the V-band and apparently for only 30 days in the B-band.
IM Nor has a sudden decline starting on day 235, fading by two magnitudes in 40 days. The slope of the light curve is substantially flatter both before and after the sharp drop. For this conclusion, the primary evidence for the drop is a comparison of the slope soon after day 235 with unfiltered CCD light versus the slope soon before day 235 with visual observations. A worry is that the unfiltered magnitudes might have a substantial color correction term (to get to V magnitudes) that changes greatly with time, and that this change might produce the sharp drop as an artifact. In particular, the unfiltered CCD magnitudes would include emission lines (primarily the $H\alpha$ line) that will start rising in prominence relative to the continuum around the start of the transition region. This worry can be largely refuted by two arguments. First, after a constant offset, the long light curve from the unfiltered CCD magnitudes matches closely the V-band magnitudes at the same time. This includes 6 nights of unfiltered magnitudes from the peak up until the sharp break matching the high-time-resolution V-band light curve plus the V-band photometry of Woudt \& Warner (2003) matching the light curve from the unfiltered CCD magnitudes after the sharp drop. Based on this, it appears that any color correction term must be nearly constant. Second, the proposed changes in the color correction term go in the opposite direction to that needed to produce the sharp break. That is, as the $H\alpha$ flux increases relative to the continuum during the transition region, the unfiltered CCD magnitudes will include the growing $H\alpha$ flux and will relatively brighten. In all, there are good reasons for taking the sharp drop at face value.
U Sco sudden decline starting on day 33 and declining in the B-band by 1.6 mag over the next two days. This decline is attested to by many good quality plates measured on the modern B-band magnitude scale, so there is no doubt about the existence of the sharp drop. The V-band light curve after day 33 has only four measures (one in 1979 and three in 1999), so it is poorly defined. Nevertheless, the V-band light curve apparently does not have the same sharp drop so apparent in the 1945 B-band data. Possible reconciliations include that U Sco has the sharp drop only in blue colors (from dust formation?), the V-band data is not accurate so as to mask the drop, or U Sco had a sharp drop in 1945 that was not repeated in 1979 or 1999.
Similar sharp drops could possibly occur late in the light curves of V394 CrA, V745 Sco, and V3890 Sgr plus maybe V2487 Oph; although the available light curves are not adequate to see these drops should they exist. Sharp drops do not occur for CI Aql, T CrB, and RS Oph. Perhaps it is significant that the sharp drops tend to occur in the systems with the shorter orbital periods.
What is the cause of the sudden drops? Dust formation is not consistent with the largely achromatic dimming in T Pyx, nor with the prominence of the IM Nor sharp drop in such a red band, nor with the consistency between the Woudt \& Warner (2003) V-band magnitude and the unfiltered CCD magnitudes at the bottom of the drop. Within the framework of the Hachisu \& Kato model for plateaus (see the next subsection), another possible explanation is that the sharp drops correspond to the end of the supersoft phase (with nuclear burning near the white dwarf) such that the reprocessed light is suddenly lost from the light curve. Presumably, we will soon have a detailed theoretical model to explain the sharp drops.
\subsection{Plateaus}
Do the RNe light curves have plateaus? This is important because plateaus are being identified as a diagnostic of the RN status. Specifically, this has been claimed for CI Aql (Hachisu, Kato, \& Schaefer 2003), V2487 Oph (Hachisu et al. 2002), and U Sco (Hachisu et al. 2000a). The claim is that the plateau phase is characteristic of the U Sco subclass of RNe. And recently the plateau idea has been extended to RS Oph (Hachisu et al. 2009). The idea is that the plateau is caused ``by the combination of a slightly irradiated [companion star] and a fully irradiated flaring-up disk with a radius $\sim$1.4 times the Roche lobe size" (Hachisu et al. 2000a). The irradiation of the disk by the supersoft emission (from the nuclear burning near the white dwarf after the wind has stopped) leads to a fairly constant optical flux added with the steadily declining light from the shell leads to a flattening of the light curve until the time when the nuclear burning turns off. The claim is that these `true plateaus' will only occur under the conditions that also produce a recurrent nova. As such, only RNe can have true plateaus, although not all RNe need to have plateaus.
How can we recognize a plateau in a light curve? Presumably, the plateau will be a substantial time interval during which the light curve has near zero slope followed by a relatively steep decline. In this definition, I do not know of any formal limits for my words ``substantial", ``near", and ``relatively", although we should be able to recognize a plateau when we see it. A real problem is that all nova events start out with a steeply falling brightness that slows down, and this could be taken as a plateau. A practical problem is that many eruptions do not have sufficiently long coverage to reveal either the plateau or the sudden drop after the plateau. Unfortunately, I do not know of any test with optical data (say, from a spectrum) to identify a plateau, so we are only left with looking for a stretch with a near zero slope followed by a steep drop. (If true plateaus are caused by the end of the supersoft phase, then presumably we can check the x-ray spectrum to identify the plateaus.) We can look through Figures 2-35 and Table 16 to see if we can spot any plateaus.
A substantial problem is that `false plateaus' might also occur in nova light curves. These `false plateaus' (so-called by Hachisu \& Kato) arise from a different mechanism from true plateaus. False plateaus are when a broad bandpass (like the V-band) includes emission line flux which increases in brightness (compared to the continuum) such that the the continuum-plus-line flux is relatively constant. The false plateaus can be in CN or RN. How can the two types of plateaus be distinguished? One possibility is to use a relatively narrow bandpass (such as the Stromgren y-band) which has little contamination from emission lines, such that a false plateau will not appear. Another possibility is to compare the light curve shape around the plateau for different filters (with presumably different contributions from emission lines), such that a false plateau will have substantial differences from band-to-band.
For T Pyx, we have good coverage going to near the quiescence level. The light curves display an obvious plateau starting around 105 days after the peak. This plateau comes immediately after a sharp drop in the light curve, and this is different from the other plateaus, as these have the flat portion starting during an apparently normal decline. Another unique feature of the T Pyx plateau is that the duration is greatly different between B-band ($\lesssim$30 days) and V-band (65 days). Within the framework of the Hachisu \& Kato model, we have an inconsistency because the supersoft phase should end when the plateau stops (around day 170) and when the sudden drop starts (around day 85). Another problem is that the V-band and B-band light curve shapes are substantially and significantly different, with different durations and strong color evolution, with this implying that T Pyx has a false plateau.
The 2002 eruption of IM Nor has an interval from day 120-235 with a flatter slope. This interval has a slope of 0.008 mag/day (as compared to 0.033 mag/day before and 0.048 mag/day after). This slope is small compared to the other plateau slopes (see below), but this is deceptive since IM Nor has such a long duration that the before and after slopes would also qualify as being ``near zero". A plateau from 120-235 days is consistent with the sharp drop at 235 days being the turn off time for the supersoft phase. Unfortunately, we do not have any color information to distinguish between true and false plateaus. In all, I conclude that IM Nor has a plateau, and the sharp drop suggests that this is a true plateau.
CI Aql has {\it rises} over days 14-17 and 36-55, but these are neither near-zero slopes nor substantially long. Hachisu, Kato, \& Schaefer (2003) point to the plateau as being at $V\sim14$ mag from days 150-300. Indeed, my templates show the CI Aql light curve to go nearly flat with a slope of 0.003 mag/day for days 80-190. After this, there is a somewhat steeper slope (0.005 mag/day), even though this slope is still very flat (see Figure 10). Color information is not adequate to distinguish between the true and false classes. However, CI Aql does have x-ray observations which show it to be a supersoft source during the plateau phase, so it is a true plateau (Hachisu, Kato, \& Schaefer 2003). In all, CI Aql does appear to have a true plateau in its light curve.
V2487 Oph has a definite plateau from days 14-22. The slope during this plateau is 0.013 mag/day, while the slope before was 0.100 mag/day and the slope after was 0.067 mag/day. Thus, the plateau is sharply distinct from the general decline and there is a definite steepening of the decline. As such, I accept this as a plateau. Unfortunately, we do not have color information or x-ray information from the time of the plateau, so the cause of the plateau is not known. Hachisu et al. (2002) point to a plateau phase from days 10-30 after peak, and my analysis confirms this result.
U Sco has a definite flattening during days 10-33 after peak. The slope during the plateau is 0.05 mag/day, which is still fairly steep compared to other plateaus. Before the plateau the slope is around 0.4 mag/day, while after the plateau the slope is 0.8 mag/day in the B-band. This shift is large enough and sharp enough while the slope is sufficiently flat that the plateau designation seems more natural than a simple slowing of the decline. The plateau is identical in shape between B-band and V-band, which argues that flattening is a true plateau. U Sco has been examined in the X-ray regime only one time (20 days after peak) by {\it BeppoSAX} from 0.2-2.0 keV, where it was found to be a supersoft source (Kahabka et al. 1999), which further points to the plateau being `true'. With this, the case for a true plateau for U Sco is confident.
V394 CrA is claimed to have a plateau from 10-30 days (Hachisu \& Kato 2000). Its light curve has a definite flattening around day 20, going to a slope of 0.026 mag/day in the V-band and 0.018 mag/day in the B-band. This might be a plateau, but it could also be an ordinary slow-down in the decline. A decline at the last observed rate would put V394 CrA back to the quiescent magnitude around 320 days after peak. (This is consistent with V394 CrA being back at minimum by July 1989; Schaefer 1990.) The coverage of the light curve does not go past 64 days after peak, so we cannot know whether there was any sudden drop at the end of a plateau. In all, only a weak case can be made for the flattening in the light curve being a plateau, and I judge the existence of the plateau to be possible but not conclusive.
T CrB shows a smooth decline with no evidence of any plateau. (The secondary maxima are not related to plateaus.)
RS Oph has a smooth decline on its way to the post-eruption dip, but there is a definite flattening from days 50-76. The slope is 0.015 mag/day, while the slopes are 0.052 and 0.041 mag/day before and after. So there is a distinct flattening followed by a distinct drop-off. But the slope is definitely not zero, and it is unclear whether 0.015 mag/day is near enough to zero to count as a plateau. The lack of color changes throughout the plateau and the supersoft phase during the plateau points to this being a `true plateau'. Hachisu et al. (2009) identify this flattening as a plateau with the same mechanism as for the U-Sco-like RNe.
V745 Sco has a poorly observed break in its decline starting at around 10 days after peak. The slope changes from 0.35 mag/day to 0.035 mag/day. This could well be a plateau, but doubts can arise from the sparseness of the post-break observations, the photometric problems with the light curve, and the lack of an observed drop-off after the plateau. The RN would return to its quiescent level around 170 days after peak if it continued to decline at the rate of 0.035 mag/day, and this is plausible for the lack of any drop-off. As such, the existence of a plateau is possible but not conclusive.
The V3890 Sgr light curve has no near-flat segment, and the coverage gets close to the quiescent level. Some might point to the lesser slope from days 4-17 (0.12 mag/day) as being a plateau, followed by a steepening (0.24 mag/day). We are left with a situation much as for RS Oph, which Hachisu et al. 2009 claims to have a plateau. The question is how flat does a plateau have to be? Is the important feature of a plateau the fact that the time interval has a flat light curve or that the time interval has a distinctly flatter slope than the intervals before and after? Until this ambiguity can be resolved, I will consider a plateau in the V3890 Sgr light curve as `possible'.
So we are left with six RNe (T Pyx, IM Nor CI Aql, V2487 Oph, U Sco, and RS Oph) all showing definite plateaus, three RNe (V394 CrA, V745 Sco, and V3890 Sgr) have possible-but-inconclusive plateaus, and only one RN (T CrB) certainly has no plateau. Of those with plateaus, apparently all are `true' plateaus other than that for T Pyx. These conclusions are tabulated in Table 17. Among the U-Sco-like systems (i.e., those with $P_{orbital}\sim1$ day), all four have or might-have plateaus. For RNe, $60-90\%$ have plateaus.
Do classical nova light curves display plateaus? Unfortunately, this question cannot be answered as there is no study of the shapes of CN light curves. Without such a study, we do not know if CN plateaus are common, occasional, or non-existent. It might be that all systems with small $t_3$ have plateaus or that a substantial fraction of all nova events of all types have plateaus, in which case the RN plateaus would not be diagnostic. Or it might be that the occasional plateaus might point to RNe hiding out as CNe because of missed eruptions.
\section{The Missed Eruptions}
A nova eruption can be missed for many reasons. Perhaps the eruption occurred around the time of its conjunction with the Sun (during the solar gap). Perhaps the eruption peaked and faded around the time of Full Moon when no one was looking (during the lunar gap). Perhaps the eruption peaked and faded during a time when no one was watching (during an observational gap). Perhaps the eruption occurred during a time when someone was watching but the watchers did not go deep enough to catch the event. Perhaps the eruption occurred at a time when people were looking deep but the eruption never got bright enough to be detected. Perhaps the eruption occurred with people looking deep enough, but no one recognized the event. A missed eruption can come from one of these reasons or from some combination of reasons.
We have a variety of reasons for knowing that many RN eruptions in the last century have been missed. Four of the known RN eruptions are known from only one Harvard plate, and the light curves and distribution of shapes demonstrate that these systems have low likelihoods for detection of the events, with the implication that other events have been missed. The RNe are fast eruptions that easily fit into the solar and lunar gaps, so again we know that a significant fraction of RN eruptions will be missed. In the last century, U Sco has eruption intervals of 10.8, 19.3, 8.9, 23.7, 10.4, 7.9, and 11.8 years, so it appears that it has a regular eruption cycle of $10\pm2$ years with missed eruption around 1927 and 1957. When a concerted effort was made to examine archival data worldwide, I found six missed eruptions while another group found another missed eruption. Few of the eruptions had independent discoveries (see Table 2), and this implies a fairly low discovery efficiency.
An understanding of the missed eruptions is necessary for measuring the recurrence time scale, and for the questions of the number of RNe in our galaxy as well as for whether the white dwarf is gaining mass. Indeed, these are the questions that have motivated my extensive time commitment to searching through old archives worldwide.
Even though many eruptions were missed, we can quantify how many eruptions were missed in a statistical manner. For example, even though V2487 Oph was only seen to go off in 1900 and 1998, we can use the discovery efficiencies to estimate the number of missed events. So we must evaluate the discovery efficiencies for each RN. And this must be done both for undirected nova searches (when the observer is scanning large areas of the sky for any nova that might appear) and for directed searches (when the observer is looking for a nova event from some particular star). These efficiencies will be estimated in the next two subsections. The last subsection will use the efficiencies to estimate the likelihood and number of missed eruptions.
\subsection{Undirected Search Efficiency}
The most general nova search is where the observer is looking for new stars anywhere in a large field. Typically, this will be an amateur scanning the sky with binoculars, the blink comparison of plates, or the automated comparison of CCD images for variables. All first-discovered RN events will be discovered with undirected searches. And, since no one is monitoring old novae with any regularity, all second-discovered RN events were discovered as undirected searches, with the exception of the discovery of the 1900 outburst of V2487 Oph. (Peltier was monitoring T CrB with only one then-known eruption, but the actual discovery was made by an amateur spotting a bright star in the sky.) The discovery efficiency will depend on the peak brightness of the nova, its position, and its duration.
The undirected discovery efficiency as a function of peak magnitude can be evaluated with a calculation by Shafter (2002). He has constructed a theoretical peak magnitude distribution from a detailed model of a realistic galaxy (including the space distribution and galactic extinction) and a typical peak absolute magnitude. I have constructed an observed peak magnitude distribution from 182 cataloged events (Downes et al. 1997) while Shafter (2002) has constructed a similar distribution. The model magnitude distribution is normalized for classical novae that peak brighter than second magnitude (with an implicit assumption that {\it all} bright nova will be discovered). With this normalization, the model distribution reflects what the observed distribution should be provided all events are discovered. As such, the ratio of the number of observed events to the model events will reveal the discovery efficiency. Thus, with Shafter's model, I can calculate the discovery efficiency as a function of magnitude, $f_{disc}(V_{peak})$. I find values of 1.0, 0.35, 0.22, 0.14, and 0.09 for peak magnitudes of 2, 4, 6, 8, and 10 respectively. This result might be startling to some people, as it shows that most novae are missed. Indeed, as most novae have faint peaks, something like 90\% of outbursts are lost. Also, three-quarters of well-placed naked-eye events are missed. But such poor efficiencies are not surprising to me, based on detailed examination of gaps and threshold from several nova hunters, AAVSO data, and Harvard plates.
The discovery efficiency as calculated by Shafter does not include the effects of the solar gap. How long is the solar gap? More usefully, what is the probability that a randomly timed nova eruption will be lost due to the solar gap? I will evaluate this by two means. (a) From detailed examination of dates of observations from AAVSO and Harvard plates, I have quantified the gap duration in days for many RNe for many years. The median solar gap is 150 days. For a median duration for the RN being brighter than the discovery threshold of 30 days, there will be 120 days out of the year for which the solar gap would make for a missed eruption. This gives a discovery efficiency of $(365-120)/365=0.67$ from the solar gap alone. (b) For randomly timed nova peaks, we would expect them to be evenly distributed throughout the year. Also, we would expect an even distribution in terms of the number of days from opposition. To evaluate the distribution of nova peaks with respect to the time from opposition, I have tabulated all first and second-class data in Table 1.1 of Payne-Gaposhkin (1964). I have created histograms for novae with the days from opposition divided into six equal bins (close to one month in size). For all 72 novae, the numbers are 20, 15, 22, 9, 3, and 3 for the six bins. We see a sharp drop off in numbers with most discoveries in the half-year centered on opposition. I will take the uniform discovery rate to equal the average of the first three bins, 19 in this case. With this uniform rate, we would expect $6\times19=114$, whereas only 72 were discovered for an efficiency of $72/114=0.63$ (with a one-sigma uncertainty of 0.08). I find that this efficiency does not depend on the decade of the nova. I also find that this efficiency has only a small dependency of the peak magnitude, being constant (to within the Poisson statistics) for all novae with peaks fainter than fourth magnitude. (For bright novae peaking above second magnitude, the efficiency is $0.80$, but the statistics are so poor that this could still be 0.63 with no significant magnitude dependency of the efficiency.) So, from both means, I find the solar gap to lead to a discovery efficiency of close to 2/3. This means that the number of missed novae events is about half of those observed.
One implication of this result is that the nova rate in our Milky Way galaxy (as calculated by Shafter 2002) must be increased by 25-50\%. The reason is that Shafter's normalization of the theoretical magnitude distribution is based on equating the observed and theoretical rates for novae brighter than second magnitude, and this presumes that {\it all} bright novae are discovered. But the solar gap is large and even very bright novae are missed due to solar proximity. For a discovery efficiency of 67-80\%, the nova rate must be increased by 25-50\%, and this would increase the nova rate in our Milky Way by 25-50\%.
The discovery efficiency also depends strongly on the duration of the nova event. A nova with a large $t_3$ will have many chances to be discovered, whereas a fast nova can likely fit into the gaps and be missed. To a good approximation, the discovery efficiency will be proportional to $t_3$. The discovery efficiencies from the Shafter calculation and for the solar gaps are both based on ordinary classical novae, for which the median $t_3$ value is 44 days (Shafter 1997). Thus, we have a multiplicative factor for the discovery efficiency of an individual nova of $(t_3/44)$ with units of days.
The observed nova rate does not vary greatly over time within the last century, as can be seen in plots of the nova discovery rate in Shafter (1997) and Della Valle \& Livio (1996). From Table 2, I have constructed a similar histogram for the discoveries of RNe, and again find that the rate does not vary significantly with decade. As such, I will take the undirected search efficiency to not vary with time between 1890 and present.
We can now put these results together to get a formula for calculating the discovery fraction in an undirected search for any particular nova. We have $F_{disc}=f_{disc}(V_{peak})\times 0.67 \times (t_3/44)$. This can now be applied to each of the ten RNe for all of the years 1890 to 2008. The results of this are presented in Table 18, with one line for each RN. The median efficiency is 4\% (with a full range of 0.6-19\%). In other words, the general all-sky searches for novae eruptions will only discover around 1-out-of-25 RN events. This means that most RNe remain undiscovered.
\subsection{Directed Search Efficiency}
When a system is known to be recurrent, searchers will specifically monitor the star and examine old plates, with these procedures substantially increasing the chance of discovering an old or new eruption. Fortunately, we can usually get good lists of the dates and limits of all the observations. For example, the AAVSO, RASNZ, and VSNET databases record the full observation details. Additionally, the various plate archives also have dates for all plates and I have recorded the limiting magnitudes. These can then be used to calculate the probability an eruption can slip through the directed search.
In Table 19, I have tabulated the gaps and limits for all of the searches with archival plates that have been made by myself and coworkers. These constitute all of the archival plate searches. (Earlier searches have been made, but these are superseded by the searches reported in this paper, as evidenced by the three eruptions discovered despite previous searches of the same archives.) I have also tabulated the gaps and limits from the AAVSO and RASNZ databases, as well as Liller's PROBLICOM search published for CI Aql. In Table 19, I have ordered the searches by the RN (given in the first column) in my usual order by orbital period. The second and third columns give the source of the archival data and the applicable years. The next column gives the typical limiting magnitude for whether the RN would have been detected had it been in outburst ($m_{lim}$). These thresholds vary substantially, typically over a range of one-or-more magnitudes, however plates with poor limiting magnitudes are generally not counted even though the plates were examined. These limits were not recorded for all plates, but instead were recorded for many hundreds of uniformly sampled plates for each target. The next column gives the approximate number of observations or plates used in the search ($N_{obs}$), with an accuracy of about 10\%. Columns 6 and 7 give the average length (in days) for the solar gap ($S_g$) and other gaps ($S_o$), as taken from detailed tabulations of observation dates from many years. The gaps vary substantially in number and size from year-to-year, so the quoted number and duration of the gaps can only represent the average case. The next-to-last column gives the number of days for which the RN is brighter than $m_{lim}$ ($T_{vis}$). These values are tabulated from the brightest value of $m_{lim}$ (so as to provide conservative limits) and for the durations used in Table 19.
We can now calculate the discovery efficiency. This is taken as the fraction of a year where the event would have been discovered for a randomly placed peak. Let me give some example to illustrate how this works. If deep observations are made every day in a year (for a circumpolar target) then all eruptions would have been detected and the fraction detected ($F_{disc}$) will be unity. Now let us take the case of a RN that is brighter than tenth magnitude for 30 days and a set of observations with a 120 day solar gap plus no other gaps as long as 30 days. Then, for 365 days as possible peak dates throughout the year, the fraction of peak dates for which an eruption will be discovered is $F_{disc}=[365-(120-30)]/365=0.75$. For the general case, the solar gap is $G_s$, the other gaps have average length $G_o$ and number $N_g$, and the RN has a time of $T_{vis}$ brighter than the discovery threshold. In this case, we have $F_{disc}=[365-(G_s-T_{vis})-N_g(G_o-T_{vis})]/365$, where the subtractions are not allowed to go negative. With this, my calculated $F_{disc}$ values are tabulated for each search in the last column of Table 19.
How should we combine the $F_{disc}$ values for multiple searches in the same year? Unfortunately, we cannot simply add the $F_{disc}$ values as the observations are largely duplicative rather than complimentary. That is because all data sources have largely the same gap structure. Both amateurs and patrol plate series tend to have a much larger solar gap than could be possible if the observers have pushed hard to fill the gaps. Also, both amateurs and patrol plates avoid times of the Full Moon. For example, this last summer at the Sonneberg Observatory, I saw that the currently ongoing photographic patrol program did not try to observe for any time when the moon is up. (The Sonneberg program is currently the only all-sky year-round program that is recording deep images for many uses.) The implication is that the combined $F_{disc}$ should be approximately equal to the largest value from the searches for that year.
With this, we can construct the values of $F_{disc}$ for every year from 1890 to present for each of the ten RNe. These are tabulated in Table 20. The last line gives the discovery efficiency averaged over the years 1890-2008 ($<F_{disc}>$).
The number of missed eruptions can be estimated from $<F_{disc}>$ and the observed number of eruptions ($E_{obs}$). The actual number of eruptions in the time interval should be $E_{total}\sim E_{obs}/<F_{disc}>$. The number of missed eruptions is $E_{missed}=(E_{total}-E_{obs})$. For cases with low $<F_{disc}>$, we can have a substantial $E_{missed}$, with the best example being V2487 Oph ($<F_{disc}>=0.30$ and $E_{obs}=2$) with likely 4-5 missed eruptions. With extra information, we have to take each RN on a case-by-case basis.
\section{Recurrence Time Scales}
The recurrence time scale ($\tau_{rec}$) is important for RN demographics and for knowing whether $M_{ejecta}>\tau_{rec}\dot{M}$. A naive means to evaluate $\tau_{rec}$ would be to simply take the eruptions tabulated in Table 2, and divide some time interval by the number of eruptions. But we have to be careful in choosing the time interval and we have to account for the missed eruptions. This must be done on a case-by-case basis.
For T Pyx, the time intervals between eruptions are 11.9, 17.9, 24.6, and 22.1 years, for an average of 19 years. This led to the common expectation that T Pyx would have its next eruption around 1986 (i.e., 1967+19), so many people were keeping close track trying to make the discovery. There has been increasing frustration as T Pyx did not erupt. The time since the last eruption is now 42 years and counting. For T Pyx, the duration of the eruption is sufficiently long that the eruption would be discovered even if it went off at the start of a seasonal gap. In such a case, just a few observations per year will discover all eruptions. As T Pyx has been under close observations every year since 1890, we can be confident that no eruptions have been missed. We now realize that what has happened is that the T Pyx accretion rate dropped substantially soon after the time of the last eruption, so it will be a long time until the next eruption (Schaefer 2005). Indeed, a more detailed accounting that includes the recent declines since 2005 plus the associated larger trigger mass implies that T Pyx won't erupt for many centuries (Schaefer et al. 2009). Also, with the likely continuing decline in accretion, T Pyx will soon be going into hibernation, and thus will not suffer any further RN events for almost a million years (Schaefer et al. 2009). That is, T Pyx has stopped being a recurrent nova. So how do we define the 'recurrence time' in such a case? A further realization is that the recurrence time scale changes from eruption-to-eruption (from 11.9 to $>$42 years in the case of T Pyx). I think that a reasonable way to characterize the T Pyx recurrence time scale is to take the average inter-eruption interval over the time when T Pyx was a RN (that is, to not include the time after the 1967 eruption when the accretion rate dropped to the point where T Pyx no longer was a RN), and that is 19 years.
For IM Nor, we have two known eruptions (1920 and 2002) for a simplistic recurrence time scale of 81.5 years. But there could easily have been missed eruptions in the many decades between. For the interval with a plausible missed event (1930-1991), $<F_{disc}>=0.85$, with most of the chances for a missed eruption being from 1955-1977. As such, the most likely case is for no missed eruption, although one missed eruption is a real possibility. With this, the average $\tau_{rec}$ is 82 years, or perhaps 41 years.
For CI Aql, we have inter-eruption intervals of 23.9 and 59.1 years. This suggests that one-or-two eruptions were missed, resulting in a fairly constant interval. Experience with other RNe (particularly T Pyx and RS Oph) shows that factor-of-three variations are common, which removes the necessity to invoke missed eruptions to have a near-constant interval. Additionally, the coverage from 1950-1990 (with the Harvard, Sonneberg, Maria Mitchell, and PROBLICOM photographs) is pretty good (86\%), so a missed eruption is unlikely (but possible). However, the quiescent magnitude from 1917-1941 is close to that between 1941-2000, so with a relatively constant accretion rate, we must have missed one or two eruptions between 1941 and 2000. The interval before the 2000 eruption is critical for CI Aql, as I have measured the change of eclipse period across the last eruption (hence giving $M_{ejecta}$) so we need to know $\Delta$T to evaluate whether $M_{ejecta}>\Delta T~\dot{M}$. With one missed eruption (say, around 1970) we have $\Delta T\approx 30$ years, while with two missed eruptions we have $\Delta T\approx 20$ years. The most likely average $\tau_{rec}$ is something like $(2009-1890)/5=24$ years.
For V2487 Oph, we have a case where many eruptions certainly were missed. In this case, we cannot use the simplistic recurrence time scale (98 years), but must instead correct for the missed outbursts. The $\langle F_{disc} \rangle$ is 0.30, so the total number of eruptions in the interval 1890-2008 should be approximately $2/0.30=6.7$. With this, the best estimate is that V2487 Oph had 6-7 eruptions (4-5 missed), for average recurrence time scale of roughly $(2009-1890)/6.7=18$ years.
For U Sco, we have nine eruptions, with five inter-eruption time intervals of near ten years. Two of the inter-eruption time intervals are close to two-times-ten years, and this suggests that one eruption was missed in these intervals. This is the exact situation expected since a significant fraction of U Sco eruptions must be lost due to the solar gap. This would mean missed eruptions around 1927 and 1957. The inter-eruption interval varies at least from 7.9 to 11.8 years. This variation has been shown (Schaefer 2005) to be a simple result of secular changes in the mass accretion rate resulting in differing times required to accumulate the trigger mass on the surface of the white dwarf. The average recurrence time scale is $(1999-1906)/9=10.3$ years.
For V394 CrA, we have two eruptions so a simplistic recurrence time scale is the time between the two eruptions, 38.3 years (see Table 2). The $\langle F_{disc} \rangle$ is 0.48 with two eruptions discovered, which suggests that $E_{total}$ is four (for two missed eruptions). With this, the best estimate average recurrence time scale is $(2008-1890)/4=30$ years.
For T CrB, the nova peak is short, but the eruptions are recognizable for a long time by the post-eruption rebrightening. Between the Harvard plates and the AAVSO observers, T CrB has been closely monitored since 1890, and there is no possibility of any missed eruption. From 1866 to 1890, T CrB has been monitored with reasonable frequency, so there is likely no missed eruptions in that interval. So in all, we can be confident that T CrB had only two eruptions from 1866 to present. The recurrence time equals the one observed interval of 80 years.
For RS Oph, the post-eruption dip is recognizable for almost a year after the short peak. This means that just a few observations per year are sufficient to recognize an event. Between the Harvard plates and the AAVSO observers, RS Oph has been closely monitored since 1890. So we can be confident that no eruptions have been missed since 1890. The inter-eruption intervals vary from 8.6 to 26.6 years. The average recurrence time is $(2009-1890)/8=14.7$ years.
For V745 Sco, we have two eruptions with an inter-eruption time interval of 52.2 years. But many eruptions have likely been missed due to the faint peak ($V_{peak}=9.4$) and fast decline ($t_3=9$ days). With the average $F_{disc}=0.36$ and $E_{obs}=2$, we have $E_{total}=5.6$ (for 3 or 4 missed eruptions). As such, our best estimate of the average recurrence time scale is $(2009-1890)/5.6=21$ years.
For V3890 Sgr, there are two eruptions with an inter-eruption interval of 27.9 years. But there are likely a number of missed eruptions. (For example, with a recurrence time scale of 27.9 years, we would have missed eruptions in 1934 and 1906.) With $F_{disc}=0.43$ and $E_{obs}=2$, we have $E_{total}=4.7$ (for 2 or 3 missed eruptions). With this, we have the best estimate for the average recurrence time scale to be $(2009-1890)/4.7=25$ years.
The results from this section are collected in Table 21. The second column gives the inter-eruption time intervals ($\Delta$T) as expected for missing events. It is important to realize that the inter-eruption intervals can change significantly. For well-known cases, the ratio of the longest-to-shortest $\Delta$T is $\geq$3.4 for T Pyx, $\geq$1.5 for U Sco, and 3.1 for RS Oph. The third column gives the average recurrence time scale.
\section{The Next Eruption}
With the results from the last section, we can make some predictions as to the dates of the next eruptions for each of the known RNe. This task has some utility in planning upcoming observations and for promoting close monitoring as appropriate.
Schaefer (2005) has made predictions for the next eruptions of T Pyx and U Sco. The physical basis is that the eruptions should be triggered when some constant amount of mass has accumulated on the surface of the white dwarf, and this constant can be calibrated for each RN by looking at the B-band flux, which is a measure of the accretion rate. The eruption will be triggered when the accumulated material ($<\dot{M}>\Delta T$) reaches some critical ignition mass ($M_{ign}$). The accretion rate averaged over the pre-eruption interval ($<\dot{M}>$) will be proportional to a known power of the B-band flux ($F_B^{1.5}$), which is directly observable on many nights in the pre-eruption interval. We then get the time from the previous eruption to the next eruption to be $\Delta T \propto M_{ign} F_B^{-1.5}$ with $M_{ign}$ being a constant for a particular RN so $\Delta T \propto F_B^{-1.5}$. The constant of proportionality can be directly determined by looking at previous inter-eruption intervals where $\Delta T$ is known. This method was confirmed by the good correlation between the inter-eruption interval and the average magnitude during that interval. That is, when the quiescent star is bright (i.e., with high accretion rates) the next eruption comes fast, while when the RN is faint between eruptions the wait is long. This provided a simple explanation for why T Pyx has already gone 42 years since its 1967 eruption whereas its previous average $\Delta$T was 19 years, because the B-band brightness dropped by a magnitude around the time of the 1967 eruption. With the T Pyx accretion largely turned off, we will have to wait a long time for the white dwarf to accumulate enough mass to trigger the next thermonuclear runaway. The quantitative analysis gives the next eruption date of 2052$\pm$3 (Schaefer 2005), while a more detailed analysis shows that the next eruption is either a millennium or a million years away (Schaefer et al. 2009). Similarly, for U Sco, the upcoming eruption date is 2009.3$\pm$1.0. The primary uncertainty in this result is that the accretion rate could easily change after the date of 2005, whereas the predicted dates are assuming that the accretion continues at the average rate for the earlier portions of the current inter-eruption time interval. My continued and frequent monitoring of the two systems (with the SMARTS 1.3m telescope on Cerro Tololo) shows that U Sco continues from 2005-2009 with the same B-band brightnesses.
Selvelli et al. (2008) have revisited the question of the predicted date for the next eruption of T Pyx. Their analysis is based on a very detailed reconstruction of all system parameters, with a premier place for the ultraviolet observations with the {\it International Ultraviolet Explorer}. They too are using the basis that $\Delta T \propto M_{ign} / \dot{M}$. Their $\dot{M}$ is evaluated from the accretion luminosity, which is dominated by flux in the ultraviolet as they well measure. Their $M_{ign}$ is taken from theoretical models as a strong function of the white dwarf mass and the accretion rate. Their conclusion is that the next eruption will be in the year 2025 (for a recurrence time of 58 years), without stating any uncertainty. Unfortunately, their real error bar on the next eruption date is very large. Their $M_{ign}$ depends strongly on the uncertain mass of the white dwarf, such that for their stated model results of 1.36-1.38 $M_{\odot}$, their ignition mass changes by a factor of two and hence the $\Delta T$ changes by a factor of two. Other researchers are not so confident in the claimed accuracy for the determination of the white dwarf mass, so the associated uncertainty in $\Delta T$ will be even larger than a factor of two. Another problem comes from the strong inverse-square dependancy of $\dot{M}$ on the distance. Even for their optimistically small range of distances ($3500\pm500$ pc), their associated uncertainty in $\dot{M}$ and hence $\Delta T$ is nearly a factor of two. Again, other researchers evaluate the distance uncertainty to be even larger (see below), so the real uncertainty from this one source is likely substantially larger than their implicit factor of two. With two independent sources of error providing one-sigma ranges of a factor of two (or more) each, the overall range of uncertainty should be roughly a factor of three (or more). As such, the one-sigma uncertainty range for the $\Delta T$ estimate of Selvelli et al. (2008) is really $\sim 33-100$ years (or more), so that their predicted date for the next eruption is anytime from 2000-2067. As such, the Selvelli et al. (2008) estimate is fully consistent with the much more accurate estimate of $2052\pm3$ from Schaefer (2005). The reason for the much greater accuracy of my estimate is simply that all the unknowns are collected into one constant of proportionality which is evaluated empirically (from many prior eruptions) with no presumptions on models or distance.
van Loon (2009) has pointed to a pattern in the inter-eruption intervals of RS Oph to make a prediction that the next eruption should occur around the year 2015. The idea is that the $\Delta T$ values apparently follow a cycle of roughly 9-22-22 years, so the next eruption after 2006 should be nine years later. There is no physical basis for any such cycling and it is hard to conceive how the system could keep track of where it is in a cycle of three, so I take this idea to be essentially numerology. A convincing refutation is simply to recognize that the 1945 eruption was overlooked, so the real series of inter-eruption times (from Table 2) is 8.6, 26.6, 12.3, 12.6, 9.3, 17.3, and 21.0, with no pattern apparent.
For the eight RNe without the accurate physics-based predictions, we can estimate the next eruption date by simply adding the average recurrence time scale to the latest eruption date. Such predictions are neither reliable nor accurate. I have listed predicted dates for the next eruptions in Table 21. Five of the ten RNe have expected outbursts in the next decade. This provides an urge for our community to keep these RNe under intense monitoring.
With U Sco, we have a confident case where we know well in advance exactly which star will suffer a nova eruption and approximately when. This allows for the possibility of planning a detailed observing program to take advantage of the prediction. Indeed, for this opportunity, coworkers and myself have a set of plans and target-of-opportunity observations in place, with coverage in x-ray/ultraviolet/optical/infrared with photometry and spectroscopy. Goals are to get hourly monitoring for an early discovery as well as to resolve the rise, daily BVRIJHK photometry for many months, fast photometry to catch eclipse times in the tail, and the spectral energy distribution from the x-ray to the infrared on many nights. A wonderful opportunity arises with U Sco's total eclipses with an accurately known ephemeris, so we can get an eclipse mapping of the supersoft source with the XRT detector on {\it Swift}. A bad case will be if U Sco happens to rise during the solar gap. With good circumstances, we hope to make the imminent U Sco eruption into the best observed nova of all times.
Other than the inevitable and imminent eruption of U Sco, the next RN eruption could well come from a `classical nova' whose previous eruption was only poorly observed. In recent years, we have been surprised by second outbursts from V394 CrA (in 1987), V745 Sco (in 1989), V3890 Sgr (in 1990), CI Aql (in 2000), and IM Nor (in 2002). Soon enough, one of the many forgotten old-novae will pop off and reveal a new RN system.
We can estimate the rate at which new RNe will be discovered. From earlier, I estimated that $\sim$80 RNe are hiding in the catalogs as ordinary CNe, and these are the systems that will be identified when a second eruption is discovered. From Table 21, the median recurrence time scale for RNe is 24 years, so there should be of order 80/24=3.3 RNe eruptions every year. With a typical discovery efficiency of 1-in-25 (for RNe in an undirected nova search), we will only catch a second eruption every $25/3.3=7.5$ years. This is comparable to the observed rate for new RNe based on undirected searches of once per 4.2 years (5 new RNe since 1987). Indeed, with the observed rate for discovering second eruptions from old `classical novae' being larger than predicted, it looks like there might be many more than 80 RNe hiding in the CN lists.
\section{Photometric Variations on the Orbital Period}
So far in this paper, I have been primarily giving the photometric history of the RN eruptions. To be comprehensive, I must also deal fully with the photometric histories during quiescence. For the next three sections, I will in turn give comprehensive new data sets plus analysis for the photometric behavior during quiescence of all ten RNe, first for their variability associated with the orbital period, then their fast variations, and finally for their long-term secular changes.
Part of a comprehensive history is to present the detailed photometry, partly to display the basis for the results and partly to allow later investigators access to the original data for further investigations. To this end, in Table 22, I am presenting complete lists of all known magnitudes during quiescence for V2487 Oph, V394 CrA, V745 Sco, and V3890 Sgr. (The columns are the RN, the heliocentric Julian Date of the middle of the observation, the band for the quoted magnitude, the magnitude with its one-sigma error bar, and the source for the measure.) This table is long, so only a stub is presented in the print version of this article, while the whole table is available in the on-line version. The T Pyx quiescence light curve is not included as it has already been exhaustively presented in Schaefer et al. (1992), Patterson et al. (1998), and Schaefer (2005). IM Nor is not included in Table 22 because the only existing data are from the old sky surveys (see below), my three BVRIJH measures (see Table 25), and Monard's fast time series photometry (see Table 5 and 23). I have vast amounts of data in quiescence for CI Aql and U Sco, with these being aimed at eclipse timings, and these magnitudes being reserved for a later paper on the change of their orbital periods. I have not included (in Table 22) the list of the incredibly large number of quiescent magnitudes for T CrB and RS Oph, as these are available in the AAVSO data archive (but see Figures 62 and 63). The data reported in this paper and as part of my long-term program now constitute virtually all of the known quiescent magnitudes for IM Nor, CI Aql, V2487 Oph, U Sco, V394 CrA, V745 Sco, and V3890 Sgr, plus the large majority of the published T Pyx quiescent magnitudes.
Most cataclysmic variables show periodic photometric modulations, with the period arising from the orbit. These can arise from eclipses, ellipsoidal variations, irradiation effects, and asymmetric emission from the hot spot. As such, the modulations can provide information of a wide variety for the properties of the system. The single most important property of any cataclysmic variable system is the orbital period. As such, we can learn a lot from seeking and measuring the orbital modulations. This is especially true for RNe, as they uniquely have an incredibly wide range of periods (from 0.076 to 519 days; over a factor of 6000). In this section, I will present the case for each of the galactic RNe. In summary, periods are now confidently known for nine systems (five discovered as a part of the programs reported in this paper), of which three systems are eclipsing (CI Aql, U Sco, and V394 CrA), one system apparently has shallow eclipses (V3890 Sgr), and another system has likely eclipses (IM Nor).
\subsection{T Pyx}
Early work on T Pyx gave claimed orbital periods of $100\pm5$ minutes (0.069 days, Szkody \& Feinswog 1988) and 3.439 hours (0.1433 days, Barrera \& Vogt 1989). Schaefer (1990) reports on 372 B-band magnitudes from 1988-1989 and found a highly significant and stable modulation with a period of 2.3783 hours (0.099096 days). The problem with this was that the Fourier transform had severe daily alias structures, so it was hard to pick out the correct alias and other aliases were possible (specifically including the alias at 0.076 days). Schaefer et al. (1992) collected 1713 magnitudes from 1966 to 1990, and found that the true period is $0.07616\pm0.00017$ days. The modulation was found to be variable in time, with no explanation suggested. Patterson et al. (1998) collected a wonderful set of time series observations (185 hours total on 38 nights) on telescopes in Chile and South Africa from 1996-1997. Their excellent time coverage allowed them to see the more complicated picture with multiple periodicities. The dominant periodicity is at $0.0762264\pm0.0000004$ days with high stability, and they identify this as the orbital period. Their averaged light curve folded on this period is shown in Figure 42. Collecting all the data together, they find that the orbital period is increasing with $\dot{P} = 6\times 10^{-10}$ and $P/\dot{P} = 300,000$ years. (This period change reconciles the two accurate periodicities in Schaefer et al. 1992 and in Patterson et al. 1998.) Patterson et al. also recognized weaker signals with periods of 0.1098 days and 1.24 days, with these varying in amplitude and of unknown stability. With these transient periodic signals beating with the orbital modulation, they provide a ready explanation for why previous workers had reported night-to-night changes in the light curve shape.
The underlying cause of these modulations is not certain. The 0.076 day periodicity dominates and is very stable, so this seems very likely to be the true orbital period. Patterson et al. suggests that the 1.24 day periodicity might arise from precession in the accretion disk. They have no suggestion for the origin of the 0.1098 day period, although they mention the white dwarf rotation period in a related context. The shape of the light curve (see Figure 42) is a flat top lasting half the orbit and then a dip that lasts the other half of the orbit. The duration of the dip is much too long for it to be an eclipse. Patterson et al. (1998) suggest that the dip is caused by heating effects on the companion star for a fairly low inclination ($\sim 10\degr-20\degr$) system. They point out that the observed change in the orbital period could easily arise from conservative mass transfer in the binary at a rate of $\dot{M}=2\times 10^{-6}~M_{\odot}$ yr$^{-1}$. However, they caution that cataclysmic variables have a long history of deception by period changes (which often is dominated by some cause other than conservative mass transfer) and they point out that any theoretical understanding of the high accretion rate (in such a short-orbit system) is lacking. An obvious observational task is to check the T Pyx orbital period to see if it is indeed increasing linearly (with a quadratic O-C curve) as predicted by their ephemeris.
With T Pyx having a distinct orbital modulation, it is a RN system for which we can now get a pre-eruption orbital period, and then also get a post-eruption orbital period after the next eruption. A substantial problem is that the modulations on the orbital period are not distinct when viewing just a few cycles (due to the usual flickering plus other periodicities being superposed), so many cycles are needed to give a good epoch. Also, even with a well-measured averaged light curve from around one epoch (e.g., Figure 42), the phase of minimum light is poorly defined when compared to the minimum of an eclipsing system, and this will then require long runs of observations to determine the orbital period to adequate accuracy. Another daunting problem with this task is that T Pyx is expected to next erupt $10^3$ to $10^6$ years from now (see Section 9), and that is a long time to wait.
\subsection{IM Nor}
IM Nor was largely ignored until it erupted for a second time in 2002. I then realized that a good run of time series photometry was highly desirable so as to get an orbital period, and even had telescope observations scheduled from Cerro Tololo. But Woudt \& Warner (2003) were the first, and they did a nice job at the Sutherland site of the South African Astronomical Observatory. In February 2003, late in the tail of the 2002 eruption, they discovered an orbital period of 2.462 hours (0.1026 days) manifested as periodic dips in brightness (see top panel of Figure 43). At first glance, the dips look like they are eclipses, but Woudt \& Warner point out that the duration of the dips is too long for being eclipses of either the accretion disk or the (possibly bloated) companion. They consider the possibility that the dips are caused by superhumps, but they reject this possibility due to the shape and amplitude of the dips not matching those of superhumps in other systems. Then, taking CI Aql as an analogy, they conclude that IM Nor has a light curve where the dips are primarily caused by a reflection effect from the heated face of the secondary star, plus partial eclipse of the accretion disk. With their CI Aql analogy, Woudt \& Warner might expect that IM Nor in quiescence would have only an eclipse of the normal width, as the irradiation of the companion star will be greatly reduced.
The AAVSO database has a wonderful collection of fast time series photometry of IM Nor by Berto Monard of Bronberg Observatory (Pretoria South Africa). He reports 6256 measures on 17 nights with typical intergration times of 30 seconds, plus 15 measures on scattered nights from JD2452460-2453360 (all given in Table 5). These measures are taken with an unfiltered CCD camera and the zero point for the magnitudes are taken for a red band. As such, we cannot place these magnitudes onto any of the standard magnitude systems, but it is nevertheless highly useful for defining the shape of the late tail of the light curve and for tracking the periodic modulations. The nightly magnitudes (or the nightly average magnitudes) are given in Table 23. We see the usual slowing-decline, with the nova reaching its quiescent steady state around JD2453050 (761 days after the peak).
The Monard light curves display a significant modulation with the 0.1026 day period (see bottom panel of Figure 43). When the nova is bright, the modulation has a low amplitude. The amplitude steadily increases as the system fades. The amplitudes for each night where Monard has a long time series are presented in Table 23. To the best of my knowledge, this is the first time that anyone has measured the change of eclipse amplitude throughout the tail of a nova eruption. Presumably, these amplitudes should be used by theorists and modelers to test their models concerning the optical depth all the way to the center of the nova shell. Also, it might be possible to translate these optical depths as a function of time into a measure of the total mass ejected by the nova event.
The dip in the folded quiescent light curve (Fig. 43) lasts roughly 0.3 in phase, so it is impossible that it is entirely caused by an eclipse. The light curve shape has not changed greatly from the time of Woudt \& Warner (2003) until quiescence. With the suggestion of Woudt \& Warner that the light curve shape is caused in part by irradiation, then apparently the degree of irradiation has changed little from the time in the tail of the light curve until a time 129 days after quiescence has been reached. One possible explanation is that IM Nor has a persistent supersoft x-ray source which heats its companion star so as to drive the high mass accretion rate (like for T Pyx, see Knigge et al. 2000, Schaefer et al. 2009). That is, IM Nor has a short orbital period (inside the period gap), so ordinary angular loss mechanisms cannot drive the accretion at the high rate required to power the fast recurrence time scale, and the supersoft source hypothesis provides a ready explanation. And this supersoft source can provide the fairly constant irradiation to account for the light curves in Figure 43.
The Monard light curves can also be used to produce a very accurate set of eclipse times. I have made folds and fits and have derived a set of heliocentric JDs for the photometric minima for 12 night (see Table 23). The ephemeris of Woudt \& Warner (2003) has the minima at HJD $2452696.53+N\times0.1026$, while the accuracy of this is not adequate to cover the whole time interval. With the minima from Monard, I have improved the ephemeris. The integer for the cycle count ($N$) and the $O-C$ are given in Table 23, with the one-sigma scatter in the minima times equalling 0.002 days. The better period is $0.10263312\pm0.00000027$ days and the epoch is HJD $2452696.52538\pm0.00055$. A substantial trouble with taking this accurate period as the precise orbital period is that the center-of-light in the system (which corresponds to the phase of minimum) shifts from the tail of the eruption (where the center-of-light is near the white dwarf) to quiescence (where the center-of-light is near the hot spot). Another question relates to whether the minima are associated with eclipses or with the irradiated side of the companion star. Until this question is answered, we cannot make any real corrections to my accurate period.
With IM Nor possibly showing eclipses (or at least some dip tied to the orbital period), this system can be used to measure the orbital period change across the next eruption. This would require, starting soon, a long series of eclipse timings. The disappointing likelihood is that the next IM Nor eruption will be a long time coming ($\sim$2084 or perhaps $\sim$2043), and this is a long time to wait.
\subsection{CI Aql}
CI Aql was identified as an eclipsing binary (Mennickent \& Honeycutt 1995) long before its second known eruption in 2000. The orbital period was 0.62 days, and their folded light curve shows deep partial eclipses superposed on prominent ellipsoidal variations. With the eruption in 2000, it was obvious that an accurate measure of the post-eruption orbital period could lead to a good measure of the period change caused by the mass loss due to the ejected shell of material. The pre-eruption period can be well-measured because the Mennickent \& Honeycutt light curves go back to 1991 and because an accurate post-eruption O-C curve can be extrapolated back to the date of the eruption so as to give a good effective eclipse time at the end of the pre-eruption interval. With this in mind, in 2001, I started a prolonged program to get eclipse timings of CI Aql. I used various telescopes at McDonald Observatory and at Cerro Tololo. I have also found eclipses going back as early as 1926 as taken from the Harvard plates. I now have a total of 80 eclipse times based on 4500 individual magnitudes, with this program ongoing. My best orbital period is $0.6183609\pm0.0000005$ days. The analysis of the full O-C curve is beyond the scope of this paper, and will be reserved for a later paper.
Lederle \& Kimeswenger (2003) used the Mennickent \& Honeycutt data, along with their own eclipse timings, to claim a measure of the orbital period change, which they then ascribed to be caused by the usual conservative mass transfer in a binary. This conclusion has many grave problems. First, they ignore the effects of the mass loss during the eruption, so their derived $\dot{M}$ can only be a limit. Second, both of their eclipse timings are on the tail of the eruption, and this means that the center of light was shifted (to near the white dwarf) from its normal quiescent position (near the hot spot) which introduces a systematic offset in the time of eclipse minimum. The size of this shift is up to 0.006 days for CI Aql and around 0.015 days for U Sco, with these effects completely swamping the claimed effect. Third, the claimed period shift is based on eclipse times over small baselines, so they must be poor in accuracy, and indeed the claimed period change is only roughly 2-sigma in significance (even ignoring the systematic problems). In all, the claimed period change is certainly wrong.
The light curve of CI Aql in the late tail of its 2000 eruption changes its shape as it fades. The peak V magnitude in the light curve varies as 14.7 mag on JD 2452019 (347 days after peak), 15.3 mag on JD 2452125 (453 days after peak), 15.9 on JD 2452224 (552 days after peak), 15.9 mag on JD 2452426 (754 days after peak), and has fallen to the quiescent level with a maximum of 16.0 mag on JD 2452769 (1097 days after peak). The secondary minimum is not visible before day 500 or so. The depth of the primary eclipse is 1.0 mag on day 453, 0.8 mag on day 552, 0.8 mag on day 754, and 0.7 mag in quiescence. The marked asymmetry between the brightness at phases 0.25 and 0.75 is first apparent in the light curve on day 754.
The light curve of CI Aql in quiescence is shown in Figure 44. A number of points can be drawn from this figure: First, the light curve repeats from orbit-to-orbit with fair accuracy, with the exception of around the phase 0.1-0.2. I would guess that these variations are associated with the usual changes in the hot spot brightness, where sometimes the hot spot is bright and sometimes it is dim. Second, the system is brighter at phase 0.25 than at phase 0.75 by 0.12 mag. I know of no precedent for such an asymmetry. The maximum deviation from a symmetric light curve is around phases 0.1-0.2, and this suggests that the asymmetry is caused by the extra light from the hot spot being preferentially beamed in one direction. But I know of no theoretical expectation for the phasing of this extra light. Third, the secondary minimum is apparent with a depth of 0.1 mag or slightly smaller. Fourth, the usual flickering in the light curve has an amplitude of under 0.1 mag, and the flickering has apparently disappeared during the eclipses.
\subsection{V2487 Oph}
V2487 Oph does not have a known orbital period. Hachisu et al. (2002) have likened this system to the U Sco `subclass' and hence suggest a period range between 0.3-3.0 days or so. With this system only discovered as an RN a few months ago, no one has taken a long close look at the system. However, I have taken a brief look back in May/June 2002 and May 2003. With the McDonald Observatory 82-inch and 0.8-m telescopes, I took 68 CCD images in BVRI colors on 8 nights. I have also recently taken photometry on 21 nights during September/October 2008 with the SMARTS 1.3-m telescope in Chile, resulting in 20 BVRI magnitudes plus simultaneously 21 J-band magnitudes. In the spirit of this paper being a comprehensive photometric history and these sketchy observations being all the photometry in quiescence, I have tabulated all my 109 magnitudes in Table 22. Unfortunately, no significant periodic modulation was seen. V2487 Oph is an obvious candidate for a detailed intensive study.
\subsection{U Sco}
U Sco was discovered to have a deep eclipse with an orbital period of 1.23 days (Schaefer 1990). I immediately realized that this provides a good opportunity to get an accurate pre-eruption orbital period leading to the measure of the mass of the ejected shell. With the lure of a big science return, I started to regularly take time series photometry so as to measure eclipse times. The idea was to measure the pre-eruption orbital period, await the next eruption, and then measure the post-eruption orbital period. It would take perhaps a decade to measure the periods with sufficient accuracy, so I knew back in the late 1980's that I was starting on a very long-term program. The next eruption occurred in 1999, and I have since been measuring further eclipse times. I currently have 47 eclipse times based on 2300 individual magnitudes. The analysis of this large data set is beyond the scope of this paper, and will be reserved for a later paper.
Schaefer \& Ringwald (1995) used the early eclipses to derive an orbital period of $1.2305631\pm0.0000030$ days. Matsumoto, Kato, \& Hachisu (2003b) have tried to use these data along with an eclipse time during the 2000 eruption to deduce a change in the orbital period. This procedure gives a completely erroneous result. One primary reason is that there is a shift in the phase of minimum light associated with the expected change in the position of the center of light from the offset hot spot to a region centered on the white dwarf during the eruption. My long series of eclipse timings demonstrates that this effect is 0.015 days (with systematic drifting as the nova fades) that I can see in both the 1987 and 1999 eruptions. That is, the primary measure that goes into their putative period change suffers a systematic error of 22 minutes simply because it is during an eruption, and this completely invalidates their derived period change. Another reason is that their claimed eclipse time based on the alleged detection of a secondary minimum is spurious because they are merely pointing to an insignificant bump in the light curve with the usual flickering (with many such bumps recognizable).
Various light curves for U Sco are presented in Figures 45-47. We see a prominent and deep eclipse. These eclipses appear to be flat-bottomed, and so the eclipse is total. For the primary eclipse, the average contact times have phases $-0.0857\pm0.0036$, $-0.0132\pm0.0014$, $+0.0121\pm0.0021$, and $+0.0915\pm0.0224$ for the first, second, third, and fourth contacts respectively. The phases are with respect to the time of minimum light in the eclipse as based on parabolic fits to the faint half of the eclipse. The fourth contact has a substantial scatter which is larger than the measurement uncertainty. At quadrature (i.e., phases -0.25 and +0.25), the system brightness varies considerably, but the average brightness is B=18.5, V=17.8, and I=17.3 mag. The scatter in the magnitudes during eclipse is substantially smaller (presumably because the variable hot spot is not visible), while the average brightness is B=19.9, V=18.9, and I=18.1 mag. The B-band light curve outside of eclipse is apparently flat with the usual flickering superposed. The I-band light curve displays a prominent secondary minimum (with typical depth of 0.3 mag) and possible ellipsoidal variations. The contact times for the secondary minimum (based on only three secondary minima) are at phases of -0.40 and +0.36. The reason why the secondary minimum is visible in the I-band light curve (but not the B-band light curve) is because the companion star is much redder than the accretion disk, so in red light the loss of light during the secondary eclipse is a substantial portion of the total system light.
\subsection{V394 CrA}
V394 CrA has a photometric period of 0.7577 days with a roughly sinusoidal modulation of amplitude $\sim$0.5 mag (Schaefer 1990). This periodicity is highly significant. But the phased light curve shows much scatter. And my later data (see Schaefer 2009 and Table 22) have years that display a lower amplitude. What appears to be going on is that the periodicity is prominent when V394 CrA is faint and the periodicity displays a lower amplitude when the star is bright. This could arise by there being some extra unmodulated light that is variable on a long time scale, such that when the extra light is bright it damps the periodic modulations. Detailed analysis of the light curves demonstrates a consistency with this idea (that the amplitude decreases appropriately as extra light is added), although the variations are sufficiently large that this is not a fine conclusion. Even during times when V394 CrA is bright, I can see the basic sinusoidal pattern rising and falling. So a folded light curve will always show large scatter even though the basic sinusoidal pattern is stable.
Schaefer (2009) presents a detailed analysis of the 500 magnitudes from 1989 to 2008 for V394 CrA. The photometric modulation is stable, implying that it is related to the orbital period. However, the folded light curve shows an asymmetry between the odd and even cycles, and this is a clear indication that the orbital period is actually twice the photometric period. With this, the orbital period is $1.515682\pm0.000008$ days, which is accurate enough for use in seeking its change across the next eruption (presumably about a decade from now). The ephemeris for the primary minimum is $HJD_{primary}=2453660.81 + N\times 1.515682$. The folded and averaged light curve (Fig. 48) shows what looks like broad primary and secondary eclipses, although (like IM Nor and T Pyx) the durations might be too long to be true eclipses. The depths are 0.40 and 0.28 mag for the primary and secondary eclipses (from the average brightness outside minima). There might be some ellipsoidal modulations with the eclipses superposed. The out-of-eclipse phases display an asymmetry with the system brighter in the elongation after the primary eclipse than after the secondary eclipse. This behavior is similar to that seen in CI Aql.
\subsection{T CrB}
T CrB has a red giant companion star (M3 III), so it must have a long orbital period. Sanford (1949) discovered the period with a radial velocity curve, and this period has been refined by Kraft (1958), Kenyon \& Garcia (1986), and Fekel et al. (2000) to $P_{orb}=227.5687\pm0.0099$ days and an epoch for zero phase (when the red giant is in conjunction in front of the white dwarf) is $2447861.73\pm 4.6$. With a 40-year light curve, Leibowitz, Ofek, \& Mattei (1997) independently find the orbital period to be $227.532\pm0.170$ days. This period implies that the red giant companion star is close to that of a canonical isolated M3 III star if it fills its Roche lobe.
The mass of the T CrB hot star has been a critical question in the history of this system. Kraft (1958) originally determined this mass to be $\ge$2.1 $M_{\odot}$, which is certainly much larger than the Chandrasekhar mass. Webbink (1976) and Webbink et al. (1987) took this to heart, decided that the hot star has to be a main sequence star, and constructed a model for the nova eruption based on an accretion instability. This model has always left many researchers uncomfortable, partly because the nova event is so characteristic of thermonuclear events in classical novae and partly because there is no precedent for Webbink's scenario. And the entire basis for the model (the high mass for the accreting star) is not regarded as confident. Part of the reason is that the mass is even today based on only seven photographic radial velocities (with only two plates near quadrature) for the hot component taken by Kraft in the 1950's, and Kraft has twice cast doubts on his own measures (Selvelli, Cassatella, \& Gilmozzi 1992). Part of the reason is that the radial velocity curves of the hot component are notoriously unreliable because the emission lines from the disk do not accurately follow the motion of the white dwarf (cf. Wade 1985; Robinson 1992; Schaefer \& Ringwald 1995). Part of the reason is that Kraft's mass is only two-sigma from the Chandrasekhar mass. Part of the reason is that a further study by Kenyon \& Garcia (1986) has lowered the limit to simply $>1.6 M_{\odot}$, and only small changes in the mass ratio will get the limit below the Chandrasekhar mass. All of these reasons combine to make it reasonable to think that the mass of the hot component could well be that of a white dwarf. On top of this, Selvelli, Cassatella, \& Gilmozzi (1992) make a strong case that T CrB must have a white dwarf. In particular, T CrB is emitting a very high luminosity mostly in the ultraviolet, its spectrum contains high-excitation emission lines like He II and N V, these emission lines have profiles with very large rotational broadening, the system shows flickering even down to the 10-second time scale, and the system is a bright x-ray source. All of these observed properties are essentially impossible within the main sequence accretor model of Webbink, yet are normal for a white dwarf with an accretion disk. They make many further points showing how the 1946 eruption of T CrB is inconsistent with predictions of the Webbink model and is consistent with the usual thermonuclear runaway model for RNe. In view of all this, one of the authors of the Webbink et al. (1987) paper has already admitted that T CrB has a white dwarf (Livio 1992a). So, despite not being able to point to where Kraft made an error of 8 km/s in his radial velocity on one plate, our community is now confident that T CrB does have a white dwarf and its eruptions are powered by thermonuclear runaway.
The orbital period is strongly manifest in the light curve as ellipsoidal variations on the red giant (Zamanov et al. 2004; Leibowitz, Okek, \& Mattei 1997; Belczy\'nski \& Miko{\l}ajewska 1998). The folded light curve is shown in Figure 49, taken directly from Figure 1 of Zamanov et al. (2004). The folded light curve in Leibowitz, Ofek, \& Mattei is identical in shape (sinusoidal) and average (V=10.1 mag), although it has a smaller full amplitude (0.14 mag instead of 0.33-0.39 mag). The orbital inclination has an upper limit of close to 70$\degr$ based on the lack of eclipses. Belczy\'nski \& Miko{\l}ajewska (1998) perform a complete analysis of the ellipsoidal variations and the radial velocity curve, with their conclusions being that the mass of the white dwarf is $1.2\pm0.2$ M$_{\odot}$, the mass of the red giant is $0.7\pm0.2$ M$_{\odot}$, the mass ratio is $0.6\pm0.2$, the orbital inclination is $60\degr \pm 5 \degr$, and the red giant is rotating synchronously.
Leibowitz, Ofek, and Mattei (1997) use a 40-year light curve to claim the existence of ``an oscillation, possibly periodic with a period of 9840 d, with an amplitude of 0.09 mag." Unfortunately, their 40 years of data only contains less than 1.5 cycles. As such, no claim for such a periodicity should have been made. They undoubtedly have seen ups-and-downs in the light curve at the 0.1 mag level with a time scale of decades, but this is a greatly different case than a ``periodicity".
The spectral type of the red giant companion star has been reported from M0 to M5. In Table 24, I have collected these reported spectral types. A substantial worry for any such compilation is that the various spectral types are based on different diagnostics, with these inconsistencies perhaps producing substantial scatter. For T CrB, with its radial velocity curve, we can reliably phase the measures. In principle, there must be a reflection effect at some level, wherein the hemisphere facing the white dwarf is heated and displays an earlier spectral type than does the outward facing hemisphere. If this effect is detectable in T CrB, we might see the reported spectral types correlated with phase in Table 24. The zero phase for the ephemeris in Table 24 is for when the red giant is in conjunction in front of the white dwarf, so the spectral type will be the coldest around zero phase and hottest around 0.5 phase. However, I do not see any such phase-type correlation, so any reflection effect must produce a heating roughly smaller than the equivalent of 2 subclasses. This fairly weak conclusion is in agreement with the observational evidence that the reflection effect does not appear in the optical light curve and the theoretical evidence that the luminosity of the white dwarf is too small to produce a measurable reflection effect (Belczy\'nski \& Miko{\l}ajewska 1998).
\subsection{RS Oph}
RS Oph has a long orbital period. This is known because the spectral features of an M giant are seen in its spectrum. Garcia (1986) measured 8 points on a radial velocity curve over 479 days and suggested that RS Oph has a period of $230\pm10$ days, but I see no significant periodicity in his noisy data (even with hindsight). Dobrzycka \& Kenyon (1994) measured 47 radial velocities for the absorption features and they got a fine sinusoidal curve with a period of $460\pm5$ days. Fekel et al. (2000) added 15 more radial velocities and substantially extended the time interval with data, and thus improved the orbital period measure to $455.72\pm0.83$ days. Brandi et al. (2009) have added spectra from 1998 to 2008 and fine-tune the orbital period to 453.6$\pm$0.4 days. The JD epoch when the red giant is in conjunction in front of the white dwarf (zero phase) is $2445043.54\pm5$. Significantly, Brandi et al. (2009) also measure the radial velocity curve for the wings of the H$\alpha$ line, and derive a mass ratio (for the red giant and the white dwarf) to be $0.59\pm0.05$.
The orbital period has only weak manifestation in the light curve. Oppenheimer \& Mattei (1993) were only able to find a weak signal with a period of 471 days only in the 1985-1993 time interval. Their Fourier analysis was of the extensive collection of AAVSO magnitudes from 1933-1993. No signal at 230 days was ever seen. The reason for this small modulation on the orbital period is likely because the large non-orbital variations (see Section 12.2) mask the orbital effects and because RS Oph has a moderately-low inclination ($\leq 35 \degr$, Dobrzycka \& Kenyon 1994; $49\degr - 52\degr$, Brandi et al. 2009). Gromadzki et al. (2008) have performed essentially the same analysis, except they have also taken out the long term trends. They find a significant periodicity at $452.9\pm2$ days, and no significant signal at the half-orbital period. They find roughly sinusoidal photometric modulation in the V-band with amplitude of 0.4-0.6 mag from 1935-1998, but an amplitude of less than 0.2 mag from 1998-2003. The lack of any signal at the half-orbital period again rules out any significant ellipsoidal effect. The variations at the orbital period might be from reflection effects or from asymmetries in the emission of light from the disk. The timing of the photometric peaks has the minima occurring when the red giant is in front, and this is consistent with a reflection effect. They point out that the minima are deeper when the star is brighter, with the implication that the companion star is responsible for the brightness fluctuations in quiescence.
The RS Oph system might have a reflection effect that is manifest in the spectral classification of the red giant companion star. In particular, the star should appear colder when the red giant is in conjunction in front of the white dwarf (near zero phase) and should appear hotter when the inward hemisphere is visible (near 0.5 phase). In Table 24, I have collected the many reported measured spectral types for RS Oph, as well as the V-band magnitude for that date (as taken from the AAVSO database). The spectral type has no correlation at all with the orbital phase. This is easy to understand because of the low inclination so that the viewing geometry does not change greatly with orbital phase. The spectral type has been plotted versus the V-band magnitude in Figure 50. (For the vertical axis of the plot, the spectral types are translated to numbers, M0 to 0, M1 to 1, and so on, with the modern convention that eliminates the K6-K9 classes for giants, so K5 is set to -1, and K4 set to -2.) Here we see a strong correlation where the spectral type changes from K5 to M4 roughly linearly as the V-band magnitude changes from 10.7 to 12.2 mag. This effect will be discussed in Section 12.2 as evidence for giant convection cells on the red giant dominating the long-term variations.
The referee has expressed distrust of the three RS Oph spectral measures by Walker (1979), Bohigas et al. (1989), and Sherrington \& Jameson (1983) as based on narrow procedural grounds. This points to a generic problem with collecting measures from diverse sources (as in Table 24). The referee further suggests that only the spectral types from Anupama \& Miko{\l}ajewska (1999) and Dobrzycka et al. (1996) form a homogenous data set of sufficient size and quality to be used for the purposes of seeking correlations. With these rejections and selections, all remaining measures show RS Oph to have only small variability between K5 and M0. In this case, the correlation of spectral type with magnitude goes away, while the spectral types still show no correlation with orbital phase.
In some references, the spectral type of RS Oph is stated to vary from G5 to M4. The apparent basis for the G5 extreme is the measure reported in Adams, Humason, \& Joy (1927). However, this classification has been later denied by one of the original observers after a re-examination of the original spectral plates (Humason 1938). As such, we should now correctly state that the spectral type of RS Oph only varies from K4 to M4, or maybe just K5 to M0.
\subsection{V745 Sco}
V745 Sco must have a very long orbital period because its spectrum shows the companion star to be a M4 III red giant (Harrison, Johnson, \& Spyromilio 1993). Schaefer (2009) reports on a series of 516 R-band magnitudes and 98 J-band magnitudes from 2004 June to 2008 August plus 7 BVI magnitudes, for a total of 621 measures (see Table 22). With this, a highly significant photometric periodicity of 255$\pm$10 days was found. The systematic variations in the light curve between odd and even cycles demonstrates that the orbital period is twice the photometric period. The folded and binned light curve is displayed in Figure 51.
For 510 day ellipsoidal oscillations, we are left with an ambiguity as to what phase corresponds to the red giant being in conjunction in front of the white dwarf. Presumably, this zero phase will correspond to one of the two minima in brightness, as the conjunction will be when the small cross section of the red giant is pointing towards Earth. But which of the minima has the red giant in front? This can be distinguished by looking at the depths of the minima as well as the measured spectral types for the red giant as a function of orbital phase. The conjunction with the red giant in front of the white dwarf (what I am taking as zero phase) should have the heated hemisphere away from Earth so we will see the cool hemisphere and a relatively late spectral type and the deepest minimum. At a phase of 0.5, the heated hemisphere will be pointed towards the Earth and the minimum will not be deep due to the reflection effects, and the spectral type will be relatively early. Taking the deepest minimum to be zero phase, I get an ephemeris for the minima to be JD $2453800+N\times510$. In Table 24, I have collected all the measured spectral types and calculated the orbital phases. Unfortunately, all observations were fortuitously taken near elongation (phase ~0.25 and ~0.75), and indeed the spectral types change little, being M6$\pm$2. As such, the spectral types are not helpful for confirming the zero of the orbital phase.
\subsection{V3890 Sgr}
V3890 Sgr has a M5III red giant companion star discovered with infrared spectroscopy (Harrison, Johnson, \& Spyromilio 1993). Schaefer (2009) reports on 374 optical magnitudes over 138 nights from June 2004 to September 2008 and 105 J-band magnitudes with the SMARTS telescopes in Chile, 465 magnitudes effectively in the I-band on 340 nights with the ROTSE robotic telescope in Australia and Texas, 41 B-band magnitudes and 13 deep limits from 1899 to 1939 with the Harvard plates, 207 B-band magnitudes from 1956-1991 measured from archival plates in the collection of the Maria Mitchell Observatory on Nantucket (Robinson, Clayton, \& Schaefer 2006), and 68 visual magnitudes (not counting limits) from 1995 to 2004 reported by amateur astronomers from around the world that appear in the AAVSO database. This collection of 374+105+465+54+207+68=1273 magnitudes has been used to search for photometric periodicities by means of a discrete Fourier transform. Two photometric periodicities were found.
The first periodicity, at $103.8\pm0.4$ days, was independently identified in the SMARTS J-band and ROTSE I-band light curves. In both data sets, the folded light curve shows a simple sine wave with little scatter (Fig. 52). The full amplitude is 0.14 mag in the J-band, and close to 0.26 mag in the I-band, R-band data, and V-band. The RMS scatter around the best fit 103.8 day sine waves are 0.04, 0.15, 0.23, 0.44, and 0.57 mag for the J, I, R, V, and B bands respectively. In the B-band, the large scatter is dominated by the usual flickering (which is much larger than in redder bands) and not by photometric measurement errors (which are substantially smaller than the observed scatter), so that the presumed amplitude of 0.26 mag is lost in the noise. The spectral energy distribution shows that the light from the red giant companion star dominates by a factor of a thousand in the infrared, so the 103.8 day periodicity must arise from the red giant. Schaefer (2009) attributed this periodicity to pulsations in the red giant, because pulsations of comparable amplitude and period are common in red giants (Fraser, Hawley, \& Cook 2008).
The second photometric period is $259.85\pm 0.15$ days, with strong signals in the SMARTS, AAVSO, and Harvard data sets (Schaefer 2009). This periodicity is stable from 1899-2008, and this demonstrates that this is tied to the orbital period. The J-band and I-band data do not show any significant signal at a period near 259.85 days. I take this to mean that this period is associated with the bluer light and hence with the accretion disk. This is further supported by the systematic rise in amplitude from the red towards the blue. Unfortunately, the rise in amplitude of this signal (towards the blue) is competing with the rise in the flickering towards the blue, so that the signal is most significant around the V-band and R-band. In the V-band, the folded light curve appears as roughly a sine wave with an amplitude of half a magnitude and an RMS scatter of 0.3 mag. Again, the odd and even numbered minima and maxima are systematically different form each other (see Fig. 53), so the orbital period must be twice the dominant photometric period, so $P_{orb}=519.7\pm0.3$ days (Schaefer 2009). The SMARTS light curve, folded on a 519.7 day period and then bined in phase is shown in Figure 53. We see a pronounced difference between the depths of the primary and secondary minima. With this, the ephemeris for the primary minimum is JD $2454730+N\times519.7$.
Schaefer (2009) suggests that the variations seen in Figure 53 arise from ellipsoidal variations (to explain the primary photometric periodicity at half the orbital period) with a shallow eclipse (to explain the substantially deeper primary minimum. Several complications (or problems) cloud this scenario. First, the AAVSO data show the SMARTS primary minimum to be less deep than the secondary minimum. That is, in the V-band, the depths of the two minima are reversed, and this would require some additional effect on top of the eclipse. Second, ellipsoidal modulations should be easily detectable in the I and J bands and they should dominate in the near-infrared, but they are not visible. Third, the flickering shown in Fig. 57 is at an orbital phase of 0.03, where the hot source might be eclipsed. Any final conclusion must await more data and some clear theoretical understanding.
In principle, the irradiation effects on the red giant might be apparent with a systematic change of its spectral class with orbital phase. Indeed, reports of the spectral class vary from K5.5 to M8 (see Table 24). However, both measures involving a K5.5 classification are at orbital phases 0.97 and 0.62, while the latest spectral classification (M8) has a phase near to elongation. As such, there is no correlation between the spectral class and the orbital phase. After finding no phase/spectrum correlation for any of the four RNe with red giant companions, we realize that the principle is too weak to have any utility.
\section{Short-term Variability in Quiescence}
In quiescence, what is the photometric behavior of the RNe on short-time scales? Here, I am taking the term `short-time scale' to mean faster than the orbital period. For comparison, all cataclysmic variables show flickering and some show coherent periodicities and some show quasi-periodicities. The flickering is generally on the time scale of minutes and is always very blue in color, with the likely cause being due to blobs in the accretion stream creating momentary brightenings of the hot spot or possibly due to fluctuations in the inner accretion disk. The coherent periodicities might be associated with the rotation of a white dwarf with a high magnetic field. Quasi-periodicities might be associated with superhumps and eccentric accretion disks. So what is the answer for the ten galactic RNe? In short, they all display the usual flickering and nothing else. Here is a star-by-star account:
T Pyx suffers fast and irregular flares with typical amplitude of 0.1 mag. Two typical examples are displayed in Figure 54. The typical duration of the flares is $\sim10$ minutes. The flares are more prominent in blue light. The flare amplitude is roughly equal to the orbital modulation, so the flaring makes for a large scatter in folded light curves and it often hides the general shape of the periodic signal. For example, compare the average light curve in Figure 42 with the light curve in Figure 54 (with both covering a similar duration). Patterson et al. (1998) find no periodicity (or quasi-periodicity) that is faster then the orbit. Thus, T Pyx has normal flickering only.
The only time series studies of IM Nor is those of Woudt \& Warner (2003) with 14.23 hours of high-speed V-band photometry and of Monard (in this paper) with 135 hours of high speed unfiltered CCD photometry. We see frequent jittering at the 0.03 mag level and occasional flares up to the 0.1 mag level. Perhaps the jittering goes away during the dips in the light curve. There is no apparent periodicity or quasi-periodicity in the variations, although the data stream is too short to make too fine a measure of this. In all, it looks like IM Nor has the usual flickering as seen in all cataclysmic variables.
I have many long time series in the V-band on CI Aql, especially with coverage of the eclipses. Outside of eclipses, CI Aql shows variability on a wide range of time scales faster than the orbital period. This is readily seen in Figure 44, where the deviations from the average curve represent the fast variability. We see flares on time scales of ten minutes with amplitudes of order 0.05 mag, and we see hour-long brightenings at the 0.15 mag level. In all, I judge the flickering to be less than that seen for T Pyx and IM Nor. The flickering is apparently absent during the eclipses.
Little is known about the short-term variability of V2487 Oph. However, my brief time series in 2002 and 2003 displayed significant short term variations. In 2002, V2487 Oph was varying between V-band magnitudes of 17.2-17.4 mag over five nights, but also had a flare of at least 0.6 mag amplitude (getting to V=16.73) with a rise time of twenty minutes. In three nights in 2003 (with 0.5, 0.8, and 3.8 hours respectively), V2487 Oph varied from 17.13-17.53 mag (V-band) with monotonic rises and falls on each night. This is typical of flickering in cataclysmic variables.
I have many long time series of U Sco, with most of these covering the eclipses. Nevertheless, there is substantial coverage with reasonable time resolution outside of eclipses. Some of these time series can be seen in Figures 45 and 47, where the deviations from the average light curve will show the fast variations. In these figures, we can see strings of nearly connected points that were all taken in rapid succession. For example, in the upper left of Figure 47 from phase -0.45 to -0.3, we see a time series which displays the usual flickering with time scales of minutes to hours and amplitudes up to 0.1 mag. The flickering can largely disappear at times, as is apparent in the upper time series of Figure 47 for phases 0.05 to 0.1 and 0.35 to 0.45. The out-of-eclipse level varies substantially from orbit-to-orbit, and we can see up-to half a magnitude changes in both the B-band and the I-band. The brightness at mid-eclipse varies by half a magnitude in the B-band but by less than 0.2 mag in the I-band.
I do not have good long times series photometry of V394 CrA. Part of the reason is that the RN is faint (down to B=20 mag) so that the Cerro Tololo 0.9-m telescope is not big enough for useful short exposures and part of the reason is that I had chosen to sample the light curve on many nights instead of dedicating a few nights. All of my photometry is tabulated in Table 22, and I know of no other photometry. My best time series are from nights in my first run in 1989, and the best of these is displayed in Figure 55. We see brightness changes of 0.07 mag on times scales of a dozen minutes and a flare with duration of two hours and an amplitude of half a magnitude. At other times the light curve appears smooth, for example on JD 2447716 we see no significant variations superposed on a linear decline of 1.0 mag/day lasting for nearly seven hours. This all looks like the normal flickering on a cataclysmic variable.
T CrB shows fast flickering on the time scale of $\sim5$ minutes from peak-to-peak with typical amplitudes of 0.1-0.2 mag in the V-band (Walker 1954; Walker 1977; Oskanian 1983; Dobrzycka, Kenyon, \& Milone 1996; Zamanov et al. 2004). This flickering is more prominent in blue light, with amplitudes typically half a magnitude in the ultraviolet and near-zero in the V-band and R-band. The flickering displays the usual distribution of peak amplitudes with frequent low-amplitude events and less-common high-amplitude events. Zamanov et al. (2004) has constructed a power density spectrum which shows the power to vary as the frequency to the -1.46 power (typical of cataclysmic variables, Yonehara et al. 1997; Bruch 1992) from seconds to hours. No periodicity is seen. The flickering is visible throughout the entire orbit. On occasion, the flickering goes away entirely, even in ultraviolet light. The amplitude of the flickering is linearly proportional to the average brightness (after the red giant contribution is subtracted out). This fast flickering is typical of many cataclysmic variables.
RS Oph shows fast flickering with a typical peak-to-peak time scale of $\sim 8$ minutes with a typical amplitude of 0.1 mag (Walker 1954; Walker 1977; Dobrzycka, Kenyon, \& Milone 1996; Simon, Hudec, \& Hroch 2004). The amplitude of the flickering increases with the average intensity level. Dobrzycka, Kenyon, \& Milone claim that RS Oph has its flickering riding on top of an 82 minute sinusoidal modulation, but this claimed periodicity is certainly not significant given their short time series and this claim is not supported by other data sets. The flickering does not depend on the orbital phase, and this is not surprising given the likely low inclination of the orbit. The flickering light is blue in color. The distribution of flickering amplitudes shows many low-amplitude events and few high-amplitude events. This flickering is not present during the post-eruption dips, presumably because the accretion disk has emptied out during the nova event so neither the accretion disk nor the hot spot is present at such times (Worters et al. 2007).
V745 Sco shows fast flickering (see Figure 56). The flares appears to be continuous, with durations of half-an-hour to one hour, and with amplitudes of 0.1-0.2 mag. For the one day shown (JD2453185, see left panel), the flickering is superposed on a general decline at the rate of 0.6 mag/day. The average magnitude level changes substantially from night-to-night (see right panel), with daily changes of order 0.1 mag. The total range for the mean magnitude of V745 Sco in this observing run is from 15.5 to 14.9 mag. These variations appear to be correlated over a time scale of 5 days. In all, V745 Sco displays typical flickering as seen in all cataclysmic variables.
V3890 Sgr shows fast flickering, as shown in Figure 57. We see continuous flickering with typical durations of half-an-hour and typical amplitudes of 0.03 mag. The night-to-night variations show a rising trend of 0.1 mag/day and a superposed changes of $\sim0.1$ mag. The flickering appears to have a smaller amplitude than is characteristic of other RNe and other cataclysmic variables. This might be due to generally low amplitudes in the R-band, or it might be due to the effectively constant light from the red giant companion star diluting the flare light.
\section{Long-term Variability in Quiescence}
In quiescence, what is the photometric behavior of the RNe on long -time scales? Here, I am taking the term `long-time scale' to mean slower than the orbital period. Long-term variability is important for a variety of reasons, including (a) a test for whether the total mass of material accreted between eruptions is constant, (b) a search for pre-eruption rises as claimed for roughly half of the novae, (c) a test for whether novae slowly decline for decades after the eruption, and (d) a measure of accretion rate variability on long-time scales. These questions play into larger issues relating to the hibernation model. The big advantage of RN for these questions is that the systems can be monitored intensively {\it before} eruptions and they generally have been intensively followed for long times {\it after} eruptions.
\subsection{Individual Systems}
T Pyx has an excellent well-sampled quiescent light curve from 1892 to present. This is possible because the system is bright enough at minimum to be seen on many Harvard plates and because of intense amateur and professional attention ever since its 1944 eruption. I have exhaustively searched the Harvard plates and measured magnitudes during quiescence on a modern magnitude scale (i.e., from Tables 3 and 4). I have also collected and corrected a large number of B-band magnitudes from the literature. This quiescent light curve for T Pyx is displayed in Figure 58. (This figure is updated from that presented in Schaefer 2005.) We see a long term trend for T Pyx to dim, from 13.8 mag in 1892 to 15.7 mag in the last year. This trend is highly significant and is large (nearly a factor of six in luminosity). The trend has no particular or repeated connection to the times of the eruptions. For example, we don't see pre-eruption rises or consistent declines after eruptions. Superposed on the declining trend is the usual flickering which leads to the observed scatter. The scatter from 1925-1953 is definitely much larger than the scatter after 1977, and this effect is too large to be observational error in the Harvard magnitudes. We see that when T Pyx is bright we have short inter-eruption intervals and that when T Pyx is faint we see long inter-eruption intervals. When the system has a high accretion rate, the disk will be bright and the critical mass will collect on the surface of the white dwarf quickly. When the system has a low accretion rate, the disk will be dim and it will take a long time to collect the critical mass. A detailed analysis for T Pyx and U Sco (Schaefer 2005) shows that the average B-band flux (actually, a power of the flux so as to be proportional to the accretion rate) times the duration of the inter-eruption interval (to be proportional to the total mass accreted onto the surface of the white dwarf) is a constant from interval-to-interval. Thus, we see that the drop in brightness (and hence the slowing of the accretion) after 1970 means that T Pyx will take a very long time to accrete a critical mass onto the surface of the white dwarf. If the system continues at the same rate then we have to wait centuries before the next eruption (Schaefer et al. 2009). If the accretion rate keeps falling off as it has over the last century, then T Pyx will be moving into a state of hibernation and it will be almost a million years until its next eruption (Schaefer et al. 2009). In either case, T Pyx has stopped being a RN.
IM Nor has no published photometry during quiescence. (Woudt \& Warner's 2003 light curve and period come from the late tail of the 2002 eruption when the V-band magnitude was at 16.5 mag around 409 days after the peak.) The magnitudes during quiescence that I can pull out are my own BVRIJH sets on three days (see Table 25), two sky survey magnitudes (see below), and Monard's unfiltered magnitudes (see Figure 43 and Tables 5 and 23). From my photometry, I find R-band magnitudes of 17.80, 17.75, and 17.71 on Julian Dates of 2453590.58 (2005 Aug 8), 2453607.52 (2005 Aug 25), and 2453804.81 (2006 March 10) respectively. From the AAO-SES sky survey plate on JD 2448827.9 I find R=17.4, while from the SERC-J sky survey plate on JD 2443960.3 I find V=18.5 mag and hence R=17.9 for the usual V-R color, both with an estimated error bar of $\pm 0.3$ mag. While the sampling is bad, the brightness does not have any significant variations on the one-year time scale or before/after eruption, with the small differences seen being comparable to the orbital modulation and measurement uncertainty. Monard's magnitudes (on a non-standard red magnitude system) show a variation between 17.0-17.7 with an RMS scatter of 0.24 mags (0.12 mag if one outlier datum is ignored) for 8 nights during quiescence. In all, the only information shows a nearly constant star with variations of $\sim0.2$ mag or less.
CI Aql has a variety of difficulties in producing a long-term light curve that is consistent. One problem is that the magnitudes from eclipses will only add noise, so I have systematically deleted the observations within 0.1 phase of eclipse minima. (The two exceptions in the figure will be the old eclipses from the Harvard plates, included to illustrate the old times.) To avoid biasing the long-term average towards those nights on which I happen to have many magnitudes, I have taken nightly averages. We also have to worry about catching tails of eruptions and confusing them for variability during quiescence. Another problem is that all the early magnitudes are in the B-band and most of the later magnitudes are in the V-band, so I have converted all the B-band magnitudes to V-band with an average $B-V=1.03$ mag. A final problem is that the magnitudes from Robocam (Mennickent \& Honeycut 1995) might need a small additive offset applied to all the magnitudes. With these complications, I have constructed a long-term light curve from the Harvard plates, Schmidt sky surveys, the Robocam series, and my own CCD magnitudes from several telescopes at each of McDonald Observatory and Cerro Tololo Inter-American Observatory (see Figure 59). While ignoring the old eclipses and the two eruptions, we see that CI Aql is nearly constant at V=16.1 mag, with the usual flickering causing scatter about this average. Given the gap from 1996-2000 and the uncertainty in the normalization of the Robocam data, this light curve is not useful for determining whether the 2000 eruption had a pre-eruption rise on a time scale of less than five years. Unfortunately, the poor coverage does not allow a real check of whether the accretion rate (as measured by the system brightness) varies inversely with the inter-eruption interval. Nevertheless, we have an apparent inconsistency between a constant quiescent brightness and the varying intervals between observed eruptions (23.6 years from 1917-1941 and 59.3 years from 1941-2000). In particular, with no missed eruptions from 1941-2000, we would expect that the average B-band magnitude in this interval to be 1.0 mag fainter than the earlier time interval, and Figure 59 shows that this is not so (despite gaps in the coverage). This discrepancy could be resolved if there was an eruption around 1970 (or perhaps two eruptions around 1960 and 1980). All throughout this interval the estimated probability for detecting an eruption is $83\%$ (see Table 20), so eruptions could well have been missed. Maybe the best resolution for the inconsistency would have one or two missed eruptions, with this possibility having important consequences for testing $M_{ejecta}>\tau_{rec}\dot{M}$ for CI Aql.
V2487 Oph does not have much useful long-term information on its quiescent magnitude. All I know is that the average V-band magnitude was 17.26 in 2002, 17.38 in 2003, and 17.43 in 2008. This is consistent with normal variations due to flickering on top of no secular changes.
U Sco has a fairly good long-term light curve in quiescence. Most of this is provided by my regular monitoring program started in 1988. Before that time, I have made use of magnitudes from deep Schmidt survey plates. A complete tabulation of my data is presented in Figure 60. This is basically an updated version of a figure in Schaefer (2005). That study demonstrated that the material accreted onto the white dwarf is a constant over each inter-eruption interval, and also made the prediction that the next eruption will be in the year $2009.3\pm1.0$. The updated light curve shows that the average brightness from 2005-2008 is close to the same as from 1999-2004, and this means that the predicted date is still good. We see that U Sco varies substantially, from roughly 18 to 19 mag. But this variation has no apparent correlation with the time of eruption. That is, we see no pre-eruption rise and no systematic decline after eruption.
V394 CrA has many B-band magnitudes from my observing runs at Cerro Tololo (with the 0.9-m and 1.0-m telescopes) in 1989, 1994, 1995, 1996, 2004, and 2005, plus many B-band and R-band magnitudes from many nights throughout the observing seasons of 2004-2008 on the Cerro Tololo 1.3-m telescope. In all, I have 298 and 160 magnitudes in B and R respectively. These can provide a good measure of the long term activity of V394 CrA, and they have been plotted in Figure 61. (For better comparison, the R-band magnitudes have been converted to B-band magnitude using B-R=1.2 mag.) We see a large amplitude long-term variability. V394 CrA goes from 18.3 to 20.5 mag, almost a factor of ten difference in luminosity. This wide variation occurs on time scales of hours (see Figure 55), on time scales of tens of days (see Figure 48), and on time scales of years (see Figure 61).
T CrB has extensive photometry of its quiescent state since 1890. I have my measures of the brightness of T CrB from 1890 to 1953 on the Harvard plates, Leslie Peltier kept frequent track of the visual magnitude from 1920 to 1946, and many AAVSO observers have covered it from the 1946 eruption to present. The AAVSO record is particularly impressive with over 80,000 observations (averaging 1300 magnitudes per year) with essentially no seasonal gap. Leibowitz, Ofek, \& Mattei (1997) have analyzed the AAVSO record for 40 years, from 1956 to 1995, with results already given in Section 10.9. Here, I have extracted the AAVSO magnitudes from 1948 (just after all the rebrightenings associated with the 1946 eruption are over) until 2004. This light curve is displayed in Figure 62, with the AAVSO magnitudes binned into 0.01 year intervals. We see an overall decline from 9.9 in 1948 to 10.25 in 2004 (averaging 0.0062 mag per year), which is small but highly significant. But the structure in Figure 62 is too complicated to characterize it as simply a decline at some rate. Rather, a better description would be that there was a linear decline of roughly 0.014 mag per year from the end of the eruption until 1970, then a flat topped rebrightening by 0.2 mag lasting from 1970 to 1992, followed by a flat light curve from 1992 to 2004. We clearly see that T CrB is varying up and down on decadal time scales by $\sim0.2$ mag. With such variations, we cannot come to any real conclusions about whether the apparent decline after the eruption is some sort of a systematic effect (perhaps associated with a cooling white dwarf). In the face of such up-and-down variations, we can only seek systematic effects after eruptions by a statistical comparison of many events.
RS Oph has an excellent visual light curve from 1933 to present from the AAVSO. I have constructed a light curve from 47,000 individual magnitudes, binned into 0.01 year intervals, from 1934 to 2004, with the eruptions and the post-eruption dips clipped out (see Figure 63). We see a chaotic mess with variability on time scales of months to years to decades where the character of that variability is constantly changing. We see flares from one-month to two-years in duration, trends lasting for a decade, and decades of relative calm. The total amplitude of variation is over two magnitudes, from roughly 9.9-12.4 mag. In no case is there any apparent variation associated with an eruption. (Recall, the post-eruption dips have been clipped out of this light curve.) An imprudent observer might point to the rises in 1967 and 2005 as `pre-eruption rises', but a look at the entire light curve shows that this is just normal variation, and a selection effect where the flat light curves before the 1933 and 1945 eruptions as well as the decline before the 1958 eruption are ignored. I have made a list of 30 significant local maxima in the quiescent light curve and the time between peaks varies from 0.7 to 9.5 years with no preferred value. Oppenheimer \& Mattei (1993) made a period search for each inter-eruption interval and found nothing significant. They also took a Fourier transform of the entire data set and noted a periodicity of 2016 days. I have repeated this analysis on my longer data stream and can reproduce this result with a period closer to 2030 days. However, a look at Figure 63 immediately shows that any such period is poor, being created by a rough alignment of a handful of peaks. In such a case, the significance of a peak in a Fourier transform is not the relevant question, but rather we should be asking at what probability we would get a false periodicity if random flare times are scattered about. (To realize that a Fourier transform can yield an apparently significant peak with no real periodicity, consider a star with occasional long and bright flares at random times, where a Fourier transform of some time interval with just two random flares will always give a strong nominally-significant Fourier peak for a `period' equal to the separation time of the two flares. When more peaks are added, more possibilities for periods arise. After long experience with this question in relation to low-mass x-ray binary systems, a general conclusion is that more than nine peaks have to be seen in a shot-noise situation before a periodicity can be believed. This rule can be overturned with {\it independent} information available, for example if a fairly specific light curve shape is expected and observed.) I have already created a statistical method for exactly this question in the context of alleged periodicities in Gamma-Ray Burst light curves (Schaefer \& Desai 1988), with this method being exactly relevant for the question at hand. I find that the existence of some apparent-but-false periodicity is a likely result of random flare times for the number of flares as seen in Figure 63. That is, when our human pattern-recognition computer is allowed to freely move around the zero phase and period to best match the peaks which we can define by {\it a posteriori} criteria, then we can usually find some periodicity, especially if we are allowed to skip predicted peaks and are allowed to have sloppy matches between observed and predicted peak times. In Figure 63, we see missing peaks at 1954.2 and 1987.6 and sloppy matches at 1948.7 and 1959.8 and at other times. (And the eruptions occur at all phases of this false periodicity.) The model for looking at flare times is very flexible (that is the problem), but let me present one typical example and its calculated probability of chance periodicity. Let us take a peak to be when any small time interval has two or more points brighter than 10.4 mag in Figure 63 (so as to include all the peaks that match the 2030 day claim), which gives us a total of 12 peaks from 1939.6-1992.9. Of these dozen peaks, eight of them match the best-fit 2030 day periodicity to within 1.5 years, even though two predicted times (not in a row) have missed eruptions. With this, the value of $P(12,8,0.03,1)$ is found from interpolation in Table 5 of Schaefer \& Desai (1988) to be near 50\%. That is, it is even-odds that a periodicity as significant as observed will be created by the random placement of flares in the RS Oph light curve. This is exactly as expected for the `random case', and it is certainly too poor to consider the 2030 day `periodicity' as being significant. In all, we can be confident that RS Oph in quiescence does not have any significant periodicity.
V745 Sco has a long-term light curve that consists only of my CCD magnitudes taken at Cerro Tololo from 2004-2008, almost all in the R-band (see Figure 64). What we see is a star that varies by up to 2.5 magnitudes on all time scales, from minutes (see Figure 56 left panel) to days (Figure 56 right panel) to months and years (see Figure 64). The long-term variations show prominent peaks of over one magnitude amplitude with durations of 40-200 days. And there are even longer term variations, as V745 Sco generally faded from 2004 to 2005-7 only to have a fast rise (0.006 mag/day) from 2007 to 2008. With variations like this, I see no way to connect such with any effects of the 1989 eruption.
V3890 Sgr has many magnitudes from 1899 to present, but I have various problems in constructing a consistent long-term light curve for quiescence. One problem is that the various sources are made with different bands, so long-term trends might be confused with color terms. Nevertheless, the colors of V3890 Sgr do not change greatly, so I have converted the B-band magnitudes to the V-band with $B-V=0.9$ mag and I have converted the R-band magnitudes to the V-band with $V-R=1.25$. But for this, I cannot convert the ROTSE magnitudes, as ROTSE uses an unfiltered CCD so that I do not know how to reliably make the conversion. Another big problem is that three of the data sets (Harvard, Maria Mitchell, and AAVSO) have limiting magnitudes such that V3890 Sgr is frequently invisible. In this case, we are only seeing the brighter parts of the light curve and get an incorrect idea of the average magnitude. For V3890 Sgr varying substantially, our derived average magnitude will always be just a bit brighter than the typical limit, and will not be simply related to the true mean brightness level. I do not know how to solve this problem except to point it out to the reader to be wary on this point. With this warning and corrections to V-band magnitude, I have constructed a long-term light curve for V3890 Sgr in quiescence from 1899 to present (see Figure 65). A problem with this composite light curve is that all four data sets are largely disjoint in time, so we cannot get a good idea of any offsets in the magnitudes. The only data set without the limiting magnitude problem is the SMARTS data (after JD 2453180) and this displays a relatively tight magnitude range of mostly 15.2-15.8 (after correction to the V-band). This is in marked contrast with the AAVSO data (JD 2449788-2453068) with a range from 14.1-15.8 even with truncation around fifteenth magnitude, is in marked contrast with the Maria Mitchell plates (JD 2435695-2449573) with a range of around 15.2-16.7 even with truncation around 16.5 mag, and is also in marked contrast with the Harvard plates (JD 2414846-2429465) with a range of 16.1-18.4 even with a truncation of 17.5 mag and deeper. Why do the SMARTS data have a completely different and smaller range than all the other data sets? One possibility is that V3890 Sgr has been `calming down' in recent years, but this seems unlikely as the observed ranges are too closely related to the data sources. Another possibility (my preferred possibility) is that the flickering and variability are relatively small in amplitude in red light and have a relatively large amplitude in blue light. With this, the Harvard blue data will have the largest amplitude of variation as the deep plates can pick up V3890 Sgr when it is faint. My long-term light curve can also be used to address questions of the relation of the quiescent brightness to the two eruptions (marked by vertical lines in Figure 65). The saving circumstance is that both eruptions have one data set (from the Maria Mitchell plates) both before and after. One question is whether the light curve shows any anticipatory rises before the eruptions? Indeed, looking at Figure 65, an incautious observer might suggest a pre-eruption dip before the 1962 event and a pre-eruption rise before the 1990 event. But I judge the existence of a mechanism that alternates rise and dips to be elusive and I judge both possibilities to be consistent with the ordinary scatter in the light curve. The differences in the data sets and the possibility of systematic changes in the limiting magnitudes over time makes problematic any search for secular drifts during quiescence.
\subsection{RN are Highly Variable on All Time Scales in Quiescence}
A default view of novae in quiescence is that they are largely constant (excepting eclipses and orbital variations as well as the usual flickering), and this extends to the long time scales. But a strong result from this section is that seven-out-of-nine RNe are varying by more than a magnitude on time scales longer than their orbital period. That is, on time scales of years to decades to a century, these novae are varying apparently-chaotically with amplitudes from 1-2.5 magnitudes. This is huge, with the luminosity changing irregularly by factors of from 2.5-10 year-by-year and decade-by-decade. Two of the RNe (IM Nor and CI Aql) apparently have only small long-term variations while one RN (V2487 Oph) has little information on long-term variability. Now combine this with the fact that {\it all} RNe show fast flickering (see Section 11) and {\it all} RNe show large amplitude variations on the orbital time scales (see Section 10). What we have is a strong and somewhat surprising conclusion that RNe display large amplitude changes on time scales of minutes, hours, days, weeks, months, years, and decades.
This strong and surprising result needs explanation. As far as I know, no theorist has ever tried explaining most of the variations reported in this paper. A few of the variations have ready explanations (like the ellipsoidal effects and the eclipses), while other can have reasonable old models tried (like having the flickering come from variable accretion onto the hot spot). But how can we explain the large amplitude decadal variations? Why do five of the RNe have an asymmetric eclipse light curve, with it being brightest during the elongation immediately {\it after} the primary minimum? Why does the folded light curve for T Pyx (and perhaps IM Nor) show a dip that is much too long to be an eclipse?
Another question is whether classical novae have similar variability on all time scales as do RNe? I expect that most CN researchers would be surprised to find that many normal nova are behaving as chaotically as the RNe. If CNe are different from RNe (perhaps with less variability), then we need to understand how the difference arises. To answer these questions, we need a comprehensive study of CNe to complement the study of RNe reported in this paper. A quality study along these lines is Duerbeck (1992), with an artful mixture of long-term visual observations, photometry from the literature, plus his own CCD observations, yielding long-term light curves with large numbers of magnitudes for a handful of the best-known novae. I think that this impressive work should be extended to many more CN.
\subsection{Power Density Spectra}
A power density spectrum (PDS) is a depiction of the Fourier transform of the light curve with both the Fourier power and frequency depicted on logarithmic axes. For this to produce useful results, the time series should be uniformly sampled and cover a wide range of time. Generally, the PDS can be represented by power laws, possibly with breaks, superposed periodicities, and quasi-periodicities. Cataclysmic variables (including novae) generally show power laws with indices typically from -1.5 to -2.0 (Bruch 1992; Yonehara et al. 1997), and often there are breaks at low frequencies where the PDS rolls over to a near-zero slope. Roughly, the PDSs show the usual $1/f$ noise that breaks to white noise at low frequency. For cataclysmic variables, the PDS is only ever measured for frequencies corresponding to times shorter than a long night. The PDS has only been published for one RN, and that for T CrB for frequencies corresponding to less than an hour with the index averaging -1.46 (Zamanov et al. 2004; Dobrotka et al. 2009).
The AAVSO has a wonderful collection of data for T CrB and RS Oph that is good for making PDS to extremely low frequencies. I have created uniformly sampled light curves with time bins of 0.01 year duration with each bin usually having a dozen averaged magnitudes. For T CrB, we have essentially complete coverage (i.e., with no solar gaps) from 1947 to present, thus allowing frequencies corresponding to periods of up to decades to be measured. For RS Oph, we have coverage from 1934 to present, however there are yearly solar gaps averaging a quarter of a year in duration plus five eruption gaps (including the post-eruption dips) lasting about one year. The gaps have been filled in by taking adjacent data streams, and for RS Oph this will create unknown distortions (likely small and non-systematic) for frequencies corresponding to periods shorter than a fraction of a year. The Fourier transforms for the uniformly sampled light curves were then calculated and averaged into equal logarithmic bins in frequency. The resultant PDSs are plotted for T CrB and RS Oph in Figures 66 and 67.
The T CrB PDS has a power law index of -0.9 for frequencies above $10^{-4}$ cycles per day, i.e., 7 nanoHertz. (The one high point close to the log-frequency of -2.05 corresponds to the ellipsoidal modulation at half the orbital period.) Below frequencies of $\sim 10^{-3.5}$ cycles per day, the random scatter will inevitably get large due to the few cycles involved in the light curve for these frequencies. That is, the particular realization of the light curve (and hence the PDS) for the time interval 1947-2008 will have ordinary deviations from any long-term average PDS. An obvious idea is that the power law PDS in Figure 66 connects to the power law PDS at much higher frequencies as reported by Zamanov et al. (2004) and Dobrotka et al. (2009). But this idea can only be tested by comparing the absolute normalization of the powers, and the idea is suspect because the amplitude of fast flickering is comparable to the amplitude of the long-term trends.
The RS Oph PDS has a power law index of -1.8 for frequencies above $10^{-3.3}$ cycles per day. The low frequency slope is roughly flat below this frequency (corresponding to white noise), with this whole PDS looking characteristic of other cataclysmic variables. However, the break to white noise is likely not of high significance due to the particular realization of any underlying PDS resulting in expected variations at the lowest frequencies.
For both T CrB and RS Oph, the normal quiescent light appears to have comparable contributions from both the red giant and the accretion disk. With this, either the red giant or the accretion component can be the seat of the long-term variations. One or both of two physical mechanism might causes the long-term variations and the power law PDS spectra for T CrB and RS Oph:
The first mechanism is accretion instability, whether due to instabilities in the accretion disk (Yonehara et al. 1997; Dobrotka et al. 2009) or due to the variations in the rate of material falling of the companion star. We already know that RS Oph and T CrB have variable accretion rates (both because of the observed flickering and because of the precedent for variability on all time scales for short period cataclysmic variables), so the first mechanism must be operating. The accretion instability mechanism is known to produce power law PDS spectra (Yonehara et al. 1997; Dobrotka et al. 2009). Anupama \& Miko{\l}ajewska (1999) have found that T CrB and RS Oph both have a scattered (but significant) correlation between the Balmer line flux and the V-band magnitude, so that some fraction of the long-term brightness changes must be associated with the accretion. The correlation between the spectral type and brightness in RS Oph (see Fig. 50) could arise because the varying luminosity of the accretion component can veil the absorption bands of the red giant. In addition, some fraction of the optical spectrum appears as a cF-type shell absorption spectrum (associated with ongoing accretion), and this fraction of the light cannot have variations associated with the red giant (Gromadzki et al. 2008). One specific type of accretion instability is the dwarf nova eruption, and we know that novae with long orbital periods have dwarf nova events. The flares in the light curve of RS Oph (Fig. 63) have some similarities, yet their shape is not that of dwarf novae events (most are too pointed at the top), the individual flares are not similar to each other (they vary greatly in duration, shape, and amplitude), and the flares are not even moderately evenly spaced in time. So the flares in RS Oph are not dwarf nova events. For a model of PDS power law indices arising in accretion disks, the variability can only occur on dynamical time scales within the disk, which are all shorter than the rotation period (Yonehara et al. 1997). With T CrB and RS Oph both having variability on the time scale of a decade and longer, this implies that the long term variability is not caused by accretion instability.
The second mechanism (to explain the power law PDS) is random variations arising from convection cells bringing hotter material to the surface, with a power law distribution of cell sizes and the larger cells varying over longer time scales (Schwarzschild 1975). We know that the second mechanism is likely to be operating because a wide variety of red giants and supergiants (both isolated and in interacting binaries, both symbiotic and non-symbiotic, both Miras and non-pulsating stars) have ubiquitous power law PDS to very low frequencies (Kiss et al. 2006; Stello et al. 2008; Templeton \& Karovska 2009) which is identified with the convection cells on the surface (Antia et al. 1984; Hayes 1984; Gray 2001; 2008; Bedding 2003; Freytag et al. 2002). The PDS of other red giants is strikingly similar to the PDS of RS Oph and T CrB, so it is natural to expect the the RNe red giants will by themselves produce the long-term variability. These variations are thought to arise from ordinary convection in the red giant, where cells of different size bring hot and bright material from the interior to the surface. Note that the radius of the red giant does not change, so the accretion rate will not fluctuate by a large amount from this cause. The faster lower-amplitude fluctuations will be caused by the small convection cells which cover only a small fraction of the surface and which have a relatively fast turnover time. The longest time scale variations will be caused by the largest convection cells which can provide a large fraction of the star's surface with hot material. The possible break to white noise at the lowest frequencies in the PDS for RS Oph could perhaps be caused by some maximum size to the convection cells. A prediction of this model is that the surface temperature of the red giant should be correlated with brightness. For example, when a large convection cell brings a large area of hot material to the surface, the hot temperature of that large area will make for a hotter spectral classification based on spectral lines as well as a higher luminosity (by Boltzmann's law). This prediction is exactly fulfilled by the spectral class versus V-band magnitude correlation already identified (see Figure 50, Table 24, and Section 10.8). Indeed, a detailed calculation can reproduce the slope observed in Figure 50. A problem with the second mechanism is that it cannot account for the correlation between brightness and the Balmer line flux (Anupama \& Miko{\l}ajewska 1999).
With the arguments presented in the last two paragraphs, I conclude that both mechanisms are operating. A way of estimating the relative contribution is by looking at Figures 5 and 6 of Anupama \& Miko{\l}ajewska (1999), where the Balmer line fluxes are plotted versus the V-band magnitudes. For both T CrB and RS Oph, the confident existence of a correlation reveals the long-term variations associated with the accretion, while the large scatter about these fit lines reveals the long-term variations associated with some separate mechanism (presumably convection cells on the red giant). The variance in these plots associated with the correlation is similar to the variance about the correlation, so that means that the two mechanisms have comparable effects. That is, apparently, both mechanisms have comparable effects (at least in the V-band) and neither dominates.
The other RNe do not have well-sampled long-term light curves, so a PDS cannot be constructed. Nevertheless, the long-term light curves for T Pyx (Fig. 58), U Sco (Fig. 60), V394 CrA (Fig. 61), V745 Sco (Fig. 64), and V3890 Sgr (Fig. 65) all display variations on the longest observed time scales from many years up to a century. With this, it appears that a common characteristic of RNe is to have significant power in their PDS down to very low frequencies. It is reasonable that the very-low-frequency variations for the four systems with red giant companions (T CrB, RS Oph, V745 Sco, and V3890 Sgr) are caused by normal fluctuations in the red giant. However, this explanation does not work for the systems that have high-amplitude very-low-frequency variations that do not have red giant companions (T Pyx, U Sco, and V394 CrA). Presumably, for these three systems, there must be some secular changes in the system that leads to long-term variations in the accretion rate. For T Pyx, with the white dwarf being a long-lasting supersoft source of high luminosity (Patterson et al. 1998), the decline might be related to the supersoft emission. U Sco and V394 CrA are not supersoft sources (other than late in their eruption tails as the wind fails and before the nuclear burning stops), so they do not have any known explanation for their long-term variations.
\subsection{The Secular Decline of T Pyx}
T Pyx has been systematically declining in brightness from 1890 to 2009, fading from 13.8 to 15.7 mag in the B-band. This is highly significant. This secular decline is also completely unprecedented for RNe, CNe, or any cataclysmic variable. This mysterious decline has a variety of implications.
Schaefer et al. (2009) reports on a single image from the year 2007 of the T Pyx shell as taken by the {\it Hubble Space Telescope}. When compared with similar images from 1994 and 1995, the individual knots are seen to be expanding homologously with velocity $\sim600$ km s$^{-1}$. The lack of deviations from this expansion demonstrates that none of the knots has experienced significant deceleration. With this, the nova eruption that ejected the visible knots was in the year $1866\pm5$. The mass in the `1866' ejecta is $\sim10^{-4.5}$ M$_{\odot}$. Given the mass and ejection velocity, the `1866' eruption can only be a regular nova eruption (i.e., not a RN eruption) where the prior accretion rate was around $4 \times 10^{-11}$ $M_{\odot}$ yr$^{-1}$ (as appropriate for an ordinary cataclysmic variable below the period gap driven only by angular momentum losses from gravitational radiation) for a time interval of roughly $\sim$750,000 years. The transition from this low accretion state to the high observed accretion state for the many decades after 1890 ($\dot{M}\gtrsim 10^{-7}$ $M_{\odot}$ yr$^{-1}$) was caused by the ignition of a luminous supersoft source on the white dwarf which heats the atmosphere of the companion star so as to drive the high accretion (as suggested by Knigge et al. 2000).
Schaefer et al. (2009) take the secular decline in T Pyx to demonstrate that the driven accretion is not adequate to keep the supersoft source self-sustained. By 2009, the accretion rate has fallen to only 3\% of its value in 1890. This provides a natural explanation for why T Pyx was not seen by {\it XMM-Newton} to be a supersoft source in 2006 (Selvelli et al. 2008). The secular decline also explains why T Pyx has not had its long-expected eruption in the late 1980's, why T Pyx will not have another eruption for a very long time, and that T Pyx has stopped being a recurrent nova. With the continuing secular decline, the accretion in T Pyx will virtually stop within decades from now, and the system will enter a state of hibernation, calculated to have a duration of order 2,600,000 years.
Schaefer et al. (2009) then puts together a full picture of the evolution of the T Pyx system. It starts out with the system being an ordinary cataclysmic variable (CV) with low accretion rate appropriate for a system below the period gap, until an ordinary nova eruption ejects a massive shell and ignites a supersoft source which drives a high accretion rate so as to power fast RN eruptions, with the secular decline in the accretion stopping the high accretion after one or two centuries, only to have the system fall into a state of hibernation until the relentless losses from gravitational radiation forces the system back into contact and back to the state of an ordinary CV again. Thus, T Pyx will have cyclic evolution, evolving between ordinary CV state, the RN state, and the hibernation state, with a time scale of order 3,300,000 years. Changes in the mass of the white dwarf are dominated by the regular nova eruptions (like the `1866' event) for which more mass is ejected than accreted (Yaron et al. 2005), with the RN episodes too short to make any difference. Thus, T Pyx will {\it not} become a Type Ia supernova.
\subsection{Pre-eruption Rises and Dips}
Robinson (1975) has made the only study of pre-eruption magnitudes for CN, with his data source being only old literature papers. One of his main results is to identify a previously-unrecognized phenomenon as being common, the pre-eruption rise. He finds that 5-out-of-11 CN show a rise in their light curve starting from 1-15 years before the eruption and brightening by 0.25-1.5 mag. This statement and conclusion is echoed in various nova reviews (e.g., Warner 2002) even until recently as part of our general and critical knowledge on CN. The claimed effect is astounding, as theorists have no plausible idea as to how the accretion rate (governed by the companion star) `knows' in advance to anticipate that the eruption trigger (governed by the depth of material on the surface of the white dwarf) is getting near critical. Given this disconnect with theory and the claimed high frequency (roughly half of CNe), I am surprised that no theoretical or observational follow-up has been made on Robinson's work.
To help solve the observational question, during the summer of 2008, I led a group from Louisiana State University to visit the Harvard and Sonneberg archival plate collections to examine the original data for all the claimed pre-eruption rises. The idea was that we would use our modern magnitude sequences and examine all available plates. Our results (Collazzi et al. 2009) are that all of the claimed pre-eruption rises are seen to not be real or significant rises, with a problem being simple errors in the early literature. The one exception is that one of Robinson's pre-eruption rise cases has been confirmed. V533 Her displays a very well sampled rise starting roughly 1.5 years before the fast eruption with a steady rise by up to about 1.5 mags brightening. A very long and well-sampled light curve both before and after eruption shows the pre-eruption rise to be completely unique and far outside the normal behavior of the system. Many other classical novae were also examined for pre-eruption rises, and we found and confirmed one other case, for V1500 Cyg, which has already been well documented (Kukarkin \& Kholopov 1975; Alksne \& Platais 1975; Samus 1975; Duerbeck 1987; Rosino \& Tempesti 1977; Wade 1987). This pre-eruption rise went from B=21.5 to B=13.5 in the month before the eruption. So, again, theorists have the task of explaining how well-observed pre-eruption rises can occur.
RNe light curves should also be examined for pre-eruption rises. This class of nova has the big advantage that we can know in advance which star will explode so pre-eruption magnitudes are readily available. (This is in contrast to the case for CN, where the only way to get pre-eruption magnitudes is from archival photographs.) RNe share the identical system configurations, identical physics in the accretion stream and disk, and identical physics of the nova eruption, so any physical mechanism that might cause pre-eruption rises for CNe should also cause identical pre-eruption rises on RNe. In this case, we should examine the long-term light curves of RNe in quiescence.
Collazzi et al. (2009) have examined the quiescent light curves presented in this paper for pre-eruption rises before 11 eruptions of 5 RNe. In all 11 eruptions, no pre-eruption rise was found. Including 16 CNe and RNe, pre-eruption rises have been found for only two systems (V533 Her and V1500 Cyg).
While not a pre-eruption {\it rise}, the unique pre-eruption behavior of T CrB before its 1946 eruption must be highlighted. Figure 23 shows that T CrB had a prominent pre-eruption {\it dip} starting a year before the fast rise of the eruption itself. T CrB had been systematically monitored by the AAVSO (primarily by Leslie Peltier) since 1919, with scattered observations going back to the 1860's. The dip is unique in T CrB's history now of 145 years, and the observers back in 1945 realized that something special was going on (Peltier 1945). The dip below the normal quiescent magnitude started close to one year before the eruption, with the fairly sharp decline going from roughly 1-2 mag fainter than T CrB's previous normal state. Such dips (by 1-2 mag below the steady level) have not occurred before or after the 1946 eruption (c.f. Figure 62). The faintest ever magnitude occurred 29 days before the eruption, when T CrB was close to two magnitudes faint. This time coincidence (29 days for an event that is unique out of 145 years) is what makes me confident that there must be some causal connection between the dip and the eruption. This dip cannot be associated with ellipsoidal variations because the colors vary strongly in ways contrary to the idea, the duration of the minimum is much longer than possible for ellipsoidal effects, such effects are not visible in the 1919-1945 AAVSO light curve, and the amplitude is much deeper than ever observed. The dip cannot be due to an ejection of gas which forms a shrouding dust shell, because the dip amplitude is comparable in B-band and V-band, while the color evolution is much different from that of an R CrB star. The brightness in the minimum of the dip is comparable to the brightness of the red giant alone, which would suggest that accretion has somehow turned off, but how does the companion star `know' to decrease the rate of material being pushed over the Roche lobe one year before the time when the pressure at the base of the hydrogen layer on the white dwarf reaches the trigger point. The orbital phases (based on the ephemeris of Fekel et al. 2000) of the dip covers 1.3 orbital periods, with the dip starting at phase 0.28, the dip being deepest in the V-band at phase 0.54 (with the white dwarf being in front of the red giant), the dip ending in the V-band at phase 0.16 (after the red giant has passed around in front of the white dwarf), and the dip being deepest in the B-band at phase 0.56. With no reasonable explanations, I think that the pre-eruption dip sets a challenge to theorists.
\subsection{Pre-eruption Dwarf Nova Events}
A small number of novae certainly show dwarf nova outbursts starting long after the nova eruption has faded away. The classic case is GK Per (Nova Per 1901), which has now had 14 small eruptions starting in 1966. Another case is V446 Her (Nova Her 1960), which started showing flares of 5-11 day durations and 1.5 mag amplitudes starting around 1990 (Honeycutt, Robertson, \& Turner 1995). Discussions of these hybrid systems (e.g., Livio 1992b; Vogt 1989) include V3890 Sgr as a hybrid system. The basis for this is a few points on a light curve presented by D. Hoffleit in an addendum to Dinerstein (1973) that show an apparent flare in 1939 with a duration of less than a hundred days and with an amplitude of around 1.5 mag. These data are from the Harvard plates and I have independently examined these plates and many more. I find the same `flare' that Hoffleit describes, even though my magnitudes have a systematic offset from hers due to the usual differences in comparison star magnitudes. Hoffleit merely looked at a relatively small number of plates and noticed one `flare'. Later workers looked at her light curve and declared this event to be a dwarf nova outburst. But, when I look over a large number of plates, I find many other `flare' events in the Harvard and Maria Mitchell data (see Figure 65). With V3890 Sgr always varying chaotically, apparent flares (which could be confused with a dwarf nova eruption if viewed in isolation) are continually happening. This is not the behavior of a dwarf nova, rather it is just the normal variations on all time scales. As such, V3890 Sgr must be taken off the lists of novae with dwarf novae events.
V3890 Sgr was considered important as being one of three putative examples of nova systems that display dwarf nova outbursts {\it before} the nova eruption. With V3890 Sgr now eliminated from this list, we have to consider the other systems. V446 Her certainly shows dwarf nova events long after its 1960 eruption (Honeycutt, Robertson, \& Turner 1995), but the only evidence for outbursts before the nova event is a small section out of a long light curve which shows a rise and a fall (Fig. 4 of Robinson 1975; Fig. 1a of Stienon 1971). By itself, this `flare' has little conviction as it does not rise significantly above the average magnitude. Robinson himself emphatically denies that this `flare' is a dwarf nova event, but later readers have not read closely and so the case is now presented in the literature as being a pre-eruption dwarf nova event (Livio 1992b; Vogt 1989). And it gets worse when the whole light curve (Fig. 3 of Robinson 1975) is seen to display similar ups and downs frequently and on all time scales, with the putative `dwarf nova event' being just one of many examples all happening frequently. So even these data alone merely show that V446 Her does not have pre-eruption dwarf nova events, but rather is just varying in the usual manner. In addition, Collazzi et al. (2009) have examined all the original plates at Harvard and Sonneberg and have measured a much longer light curve in quiescence, both before and after the nova event. We find that V446 Her is varying up and down on all time scales. So again, we see that a normal variation when viewed in isolation was mistaken to be a dwarf nova event. As such, V446 Her should be taken off the lists of novae with {\it pre-eruption} dwarf novae events.
Livio (1992b) mentions a third case of pre-eruption dwarf nova events, for the weird and unique very-slow nova PU Vul. (PU Vul is a symbiotic nova, so it is unclear whether any example from this event has application to classical novae.) The basis for the claimed dwarf nova eruptions is a light curve from the Harvard plates from 1890 to 1979, with this showing definite variability from 16.5 to 15.0 mag, including one brightening by 1.5 mag that lasted roughly 100 days (Liller \& Liller 1979). But such an event is much too long in duration and much too small in amplitude to be a dwarf nova eruption. In addition, the frequency of magnitudes has the star well above its minimum roughly half the time, and this is not consistent with the behavior of dwarf novae. Instead, the pre-eruption light curve is consistent with the usual variability displayed by many novae both before and after the eruption (Collazzi et al. 2009). As such, there is good evidence to discount the possibility that PU Vul displayed pre-eruption dwarf nova eruptions.
So the three examples of novae with pre-eruption dwarf nova events are wrong. But there is another unheralded case; V1017 Sgr. As mentioned in Secion 2, it had a classical nova event in 1919 plus dwarf nova events in 1901, 1973, and 1991 (Webbink et al. 1987). The 1901 event is a good example of a pre-eruption dwarf nova. Likely, it is significant that V1017 Sgr has an unusually long orbital period of 5.7 days (Sekiguchi 1992). (This period is based on relatively few points in a radial velocity curve and has never been confirmed. Indeed, I have made a sustained effort to detect photometric modulations, but these have returned no significant variations with the claimed orbital period.) The long orbital period might be a condition to get dwarf nova events in a system with such a hot central source, with the precedent including GK Per at 1.99 days and Q Cyg at 0.42 days.
\subsection{Post-eruption Declines?}
Long after the nova eruption is over (perhaps 1-10 years later), does the quiescent nova slowly fade in brightness? The standard expectation by nova researchers is that the system will remain essentially constant in brightness. (This is excepting the known effects of orbital modulation, dwarf nova eruptions, and flickering, all on too short a time scale to be important for whether the nova systems fade away over the decades or centuries.) This expectation arises in part because the quiescent brightness should be driven by the mass transfer rate which largely should be unaware of the depth of accreted material on the white dwarf. Also, Robinson (1975) has established that the pre-eruption magnitude generally is close to the post-eruption magnitude, which argues against any fading. Finally, occasional non-systematic magnitude measures have not been realized to show any significant fading in a few systems, so no famous case is known to highlight the question.
The expectation of a constant quiescent level was first questioned about two decades ago. The basis was the hibernation model for the evolution of nova, with long-term systematic changes in the accretion rate. The picture is that the mass loss from a nova event will cause a slight separation in the binary, and the high $\dot{M}$ is maintained by the hot white dwarf irradiating the companion star, puffing up its atmosphere, and driving a non-steady-state high accretion. As the white dwarf cools down from the eruption, the irradiation declines, the accretion rate falls, and the systems fades. Kovetz, Prialnik, \& Shara (1988) made a specific prediction within the hibernation model that a typical slow decline rate in the first century after outburst should be of order 0.012 mag per year (for bolometric magnitudes). We now have a specific prediction to test.
The first search for a systematic decline rate was a paper by Vogt (1990). But the methodology used a one time brightness measure (the magnitude at quiescence minus the eruption peak magnitude) decades after the eruption for many novae, where the measure is a tangle of many effects leading to a huge scatter. The second search (Duerbeck 1992) was a well-planned use of quality long-term data from the literature and contemporary CCD magnitudes for 15 of the best observed novae up to 72 years after the outburst. Well-sampled light curves were only presented for six novae. Duerbeck found that the old novae all showed variability on under one-year time scales with amplitudes ranging from 0.2-2 mags. (This is similar to the case for RNe.) For long-term variations, his set of CN showed all types of variations (i.e., rising, declining, flat, falling-then-rising, and with flares) with amplitudes ranging from near-zero to 1 mag. (This is similar to the case for the RNe, although the RNe systems usually have a higher amplitude of long-term variations.) Despite having the light curves going in all directions, the overall average was for a small decline at a rate of $0.010\pm0.003$ mag per year. I interpret this as a reasonable agreement with the prediction of fading from the hibernation model, but that other mechanisms are also present making for a substantial variation around this average decline rate.
In principle, the hibernation model should apply to the RNe, and so we can test to see if they have a systematic decline after eruption. The short recurrence times will make it difficult to distinguish a secular fading in quiescence from a possibly lingering tail of the eruption. The short recurrence times will also often prevent any long stretch of time for which a slope can be better measured. T Pyx appears to be in a century-long decline at an average rate of 0.016 mag per year (Figure 58) that makes it hard to attribute to any one nova outburst. Such a rate must have started from 1850-1890 and it cannot keep up for long (at least with T Pyx remaining a RN), so we realize that the average decline rate can at best be episodic. CI Aql has a poorly-sampled but flat light curve with zero post-eruption declines (Figure 59). U Sco has what might be a post-eruption decline from 1988-1995 (at a rate of around 0.04 mag per year), although the next inter-eruption interval has a flat average light curve with zero decline (Figure 60). V394 CrA has good coverage from 1989 to 2008 after its 1987 eruption, and there are substantial variations superposed on an overall rise of 0.05 mag per year or so (Figure 61). T CrB started out by declining at a rate of 0.014 mag per year for the first 22 years, then had a flare lasting 22 years with an amplitude of 0.2 mag, and then the light curve has been flat for the last 16 years (Figure 62). RS Oph has a complex light curve (Fig. 63) with chaotic rises and falls on all time scales. The time intervals from 1948-1959, 1969-1982, and 1986-2002 might be claimed to display post-eruption declines (plus many added short-term flares). But these declines are small (about a third of a magnitude) compared to the variations in the light curve, the whole post-eruption time intervals 1969-1985 and 1986-2006 are actually more-or-less flat (with flares), while the other post-eruption intervals do not show declines (1935-1945 is flat and 1959-1967 is chaotic with a rising trend). So the apparent declines in RS Oph are neither significant nor systematic. V745 Sco has only observations from 2004-2008, long after its 1989 outburst, yet we still see rises and falls of up to 2.5 magnitudes (Fig. 64), so there is certainly no simple decline. V3890 Sgr has a problematic light curve (Figure 65), but there is no large decline before or after the 1962 and 1990 eruptions. We do not have any useful information on the post-eruption decline for IM Nor or V2487 Oph.
Just as with Duerbeck's CN study, my RN study finds a very mixed set of behaviors for the post-eruption decline. Some of the RNe have no decline (CI Aql, U Sco after 1999, and V3890 Sgr), others have an apparent decline (U Sco after 1987, and RS Oph after the 1945 eruption), one has a systematic decline that is not affected by eruptions and must be transitory (T Pyx), one has a systematic rise (V394 CrA), and some have chaotic rising and falling (T CrB, RS Oph after the 1958 eruption, and V745 Sco). If I blindly average the decline rates, I get 0.002 mag per year. While there might be a fading due to hibernation in RNe, there must be larger forces changing the decline rate in a chaotic manner.
\section{Colors in Quiescence}
The colors of the RNe are diagnostic of the accretion disk, the companion star, and the extinction. Some colors for most of the RNe have already been published in the literature. However I have colors for UBVRIJHK for all the RNe on many dates each. In this section, I will report on my new colors and add in the colors from the literature. These have been collected in Table 25. The B-I versus I-K colors have been plotted for a single epoch in Figure 68.
A complication is that the colors are changing on all time scales. We have already seen that (a) the blue flickering light (and its color) changes on time scales of minutes and longer, (b) the colors change on the orbital time scale due to eclipses, irradiation of the secondary, and ellipsoidal effects, and (c) the brightness and presumably the color changes on the time scale of years and decades for unknown reasons. This complexity means that the color is ever changing, so we can hardly quote one value. This is the reason that I have included all the individual colors and their dates. A reasonable but risky simplification is to take some average of the individual measures and use this value for model comparisons.
A further complication is that the colors are affected by extinction, whereas we need the intrinsic colors for comparison with models and for using the spectral energy distribution to derive the accretion luminosity. For this, we must have some independent measure of the extinction.
\section{Extinction and Distance}
To convert the observed photometry into luminosity and energy (and hence to answer many physics questions about the system), we need both the distance and extinction. Distances are always hard-to-get for cataclysmic variables. The problem is that there are no standard candles, no redshifts, and no associated stars/clusters to use, while measures involving the intervening interstellar material are always too crude. A further problem is that researchers often select one or two of the various distance estimates while ignoring other valid measures, with biased selections possibly distorting the best estimate. For this paper with its emphasis on collecting observational results, a trap to be avoided is to use distance estimates based on theoretical models, as otherwise we would have later models being tested against earlier models instead of against the observations. For this paper, I will provide new extinctions and distances to all the RNe based on intrinsic colors in the light curve and on the maximum magnitude-rate of decline (MMRD) relation based on my new templates. I will also provide new distances based on the flux of the companion star alone and the assumption that the star fills its Roche lobe, with these providing the best distances for the five stars with such measures. I will also provide an unbiased review of the literature for observational measures of both distance and extinction.
The first subsection will systematically present the evidence for the extinction. The second subsection will systematically use three relations based on the light curve to get the absolute magnitude of the nova at peak and hence the distances for all ten RNe. The third subsection will isolate the flux from the companion star (either by total eclipse or by going to the infrared) so as to get a distance measure to five RNe. The fourth subsection will go through the RNe one at a time, reviewing the literature and deriving a consensus distance estimate. The fifth subsection will use the distance to RS Oph to address the question (critical to system models) of whether the red giant star fills its Roche lobe.
\subsection{Extinction}
Extinction can be measured in various ways, including the depths of interstellar absorption lines, the Balmer decrement, and a comparison of an observed color to some presumed intrinsic color. These methods are notoriously unreliable with large scatter and questionable assumptions. So we have to realize that high accuracy will not be possible.
One method for estimating extinction is to use the observed color of the erupting nova around the time of the peak. van den Bergh \& Younger (1987) demonstrated that the intrinsic $B-V$ color of nova events is $+0.23\pm0.16$ at peak and $-0.02\pm0.12$ when the V-band has faded by two magnitudes from peak. With this, the excess color ($E_{B-V}$) can be derived from the difference between the intrinsic and observed colors. To this end, I have used my light curve templates to evaluate the required colors (see Table 17). These derived extinctions (expressed as $E_{B-V}$ in magnitudes) are collected into Table 26. I have also collected the various published extinctions for inclusion in the table. The bottom line of Table 26 is a concordance extinction, with the error bars roughly covering the spread of estimates. I will use these final extinction estimates in the calculations for distances and spectral energy distributions below.
\subsection{Distances from MMRD Relations}
The `maximum magnitude versus rate of decline' (MMRD) relations connect the light curves to the peak absolute magnitudes of nova events. van den Bergh \& Younger (1987) show that nova eruptions have a nearly constant absolute magnitude at 15 days after peak with $M_V(15)=-5.23\pm 0.39$. The distance modulus (i.e., $\mu=V_{15days}-M_V(15)$) can then be calculated. A substantial worry for many of the RNe is that these results are not for fast nova events, whereas the few fast novae might be exceptions, suggesting that there might be some dependancy on the time it takes for the light curve to decline by two or three magnitudes from its peak ($t_2$ or $t_3$). The MMRD relations that explicitly use the rate of decline come in various versions. I will use two from Downes \& Duerbeck (2000) that make use of $t_2$ and $t_3$ and separate out the fast events (Duerbeck's `A' class). For the fast events they give $M_{Vpeak}=-11.26+1.58\log (t_3)$ and $M_{Vpeak}=-10.79+1.53\log (t_2)$, while for the slow events (T Pyx, IM Nor, and CI Aql) they give $M_{Vpeak}=-8.13+0.57\log (t_3)$ and $M_{Vpeak}=-8.71+1.03\log (t_2)$. The scatter for any one nova about these relations is $\sim0.5$ mag. With the $t_2$ and $t_3$ values from Table 17, we can calculate $M_{Vpeak}$ and then the distance modulus ($\mu=V_{peak}-M_{Vpeak}$) for each of the three relations. These three $\mu$ values are averaged together and then converted to distances with the usual equation $\mu=5\log (d_{pc}) - 5 + 3.1E_{B-V}$ where $d_{pc}$ is the distance in parsecs. These three values are not independent, so the average is an attempt to smooth out the variations from any one relation. The light curve inputs, distance moduli for all three relations, the averaged distance moduli, and the MMRD distances are presented in Table 27 for all ten RNe. The formal error in $\log d_{pc}$ is near 0.2, but the systematic uncertainties are likely much larger.
From Table 27, we see that three of the RNe have calculated distances of $>24$ kiloparsecs. These are not reasonable distances, as it places the systems outside the far edge of our galaxy. This is alerting us to the likelihood of large systematic errors, at least in these cases.
\subsection{Distances from Companion Stars}
The best of the distance methods is to use the properties of the companion star. In particular, with the known size of the companion star and a measured surface temperature, we can derive a blackbody distance to the companion. But this only works when the light from the companion can be isolated. This can be done either by looking in the infrared (where the red giant dominates the flux for T CrB, RS Oph, V745 Sco, and V3890 Sgr) or during {\it total} eclipse (for U Sco). The distances can then be derived by three methods, which are all using the same essential physics, yet which differ in the input measures. I have no reason to prefer any one method or any one set of inputs. So I will take the distance to be the average of all three values. The scatter amongst these three values will be a reasonable measure of the measurement uncertainty
If we can get the brightness, temperature, and size of the companion, then we can use the blackbody surface flux and the inverse-square law of light to derive a distance. For this, the only way to get the size of the companion is to take the size of the Roche lobe as based on the orbital period (with the value having only weak dependency on the stellar masses in the system). For RNe, the Roche lobe filling case is confidently known as the means to get matter accreting onto the white dwarf, yet nevertheless the Roche lobe filling assumption has been questioned for RS Oph (see Section 14.5). The temperature can either be gotten from looking for the peak of the spectral energy distribution or by a classical spectral classification of absorption lines on the companion. To get the brightness of the companion star, we must have some way of isolating its flux from the rest of the system. This can be done by measuring the system brightness during a total eclipse (for U Sco) or by looking in the infrared where the red giant dominates (for T CrB, RS Oph, V745 Sco, and V3890 Sgr). If there happens to be additional light in the system not realized in this calculation, then the real distance should be {\it farther} than the derived distance. The necessary conditions are available for only these five RNe. For these systems, the distances determined with the companion star are much more accurate and reliable than all other distance measures.
The first method for getting distances to the companion stars (see Schaefer 2009 for details) uses the spectral type to get the effective surface temperature of the companion star. Problems with this include the fill-in of lines with disk light, the uncertainties in the spectral class measures, imperfect calibration of the temperatures for a given spectral class, and the non-simultaneity of the spectral class measurement with the magnitude measurement.
The second method (see Schaefer 2009 for details) uses the frequency of maximum spectral flux and Wien's Law to get the effective surface temperature of the companion star. A problem with this is that the companion's light is not a perfect blackbody.
The third method uses the Barnes-Evans relation (Barnes \& Evans 1976). The physics of this is the same as for calculating the blackbody distances, but the calibration is purely empirical and varies with spectral type and luminosity class. An advantage of this method is that it is closely independent of reddening. A useful version of this relation is $F_K = 4.22 - 0.1m_K - 0.5 \log(\phi)$, where $m_K$ is the observed K-band magnitude (as taken from Table 25) and $\phi$ is the angular diameter of the star in milli-arc-seconds. The value of $F_K$ is $3.84\pm0.2$ for normal red giants in the range of spectral types covered by the RNe (Cahn 1980). With the Roche lobe radius (determined with the orbital period), we can convert the angular diameter of the companion star to a distance.
Livio, Truran, \& Webbink (1986) used the first method to get a distance to RS Oph of 3180-3290 pc, while Barry et al. (2006), Brandi et al. (2009), and Schaefer (2009) reproduce the same distance. Belczy\'nski \& Miko{\l}ajewska (1998) also used the first method for T CrB to get a distance of $960\pm150$ pc. Schaefer (2009) has presented detailed calculations for the distances to the four RNe with red giant companions by both the first and second methods. The distance to the companion stars by all three methods are given in Table 27. The RMS scatter in the values produced by the three methods is indicative of the uncertainties associated with the blackbody distances. For the four RNe with red giant companions, the fractional one-sigma uncertainties are 21\%, 27\%, 13\%, and 31\%, with an average of 23\% error. For these four RNe, I will take the best estimate of the distances to be the average from all three methods with a 23\% uncertainty.
The distance to U Sco can be confidently measured because we can determine the temperature and flux from the companion star {\it alone} by looking during the time of its total eclipse. The equations for the first two methods are presented in Schaefer (2009). The input, intermediate results, and final distance are presented in Table 28 in the same format as Table 1 of Schaefer (2009). The mass of the white dwarfs in RN systems must be close to 1.35 M$_{\odot}$. Based on radial velocity information and models for the companion star, $M_{comp}$ is close to 1.0 M$_{\odot}$ (Johnston \& Kulkarni 1992; Schaefer \& Ringwald 1995). The uncertainty on the companion mass is likely around 30\%, and this translates into an uncertainty on $R_{Roche}$ of only 10\%. Thus, we know the size of the Roche lobe to all needed accuracy. For U Sco, the companion was determined to be G3-6 III-IV (Webbink et al. 1987), K2 IV (Anupama \& Dewangan 2000), F8$\pm$2 (Johnston \& Kulkarni 1992), and G0$\pm$5 (Hanes 1985), so I will take the median spectral type to be G5 IV. The effective temperatures for the average spectral type (G5 IV) is taken from Drilling \& Landolt (2000). The range of variation (or uncertainty) in spectral type corresponds to an uncertainty in $T_{comp}$ of 300 degrees or less. The V-band magnitude for the companion star alone in the U Sco system is taken from the depth of the total eclipse (see Figure 46). The bolometric magnitude for U Sco is $m_{bol}=m-A+BC$ with BC being the usual bolometric correction of -0.3 for a G5 IV star (Drilling \& Landolt 2000). For U Sco, the observed colors at minimum are B-V=1.02 and V-I=0.79, and these correspond to intrinsic colors of 0.8 and 0.5 mag respectively. The B-I color suggests that the companion has an effective temperature of around 6000 K (Johnson 1966). For this temperature, $\nu_{max}$ is close to the center of the I-band, so I will use my observation that $I=18.1$ mag during the total eclipse (see Figure 45) and convert this to get an extinction-corrected $f_{\nu}(\nu_{max})$ for use in Table 28. For the two similar methods (with independent input), I get two distances (included in Table 27) which average to 12,000 pc. A comparison of the distances from the first two methods for all five RNe gives an uncertainty of 17\%. So the distance to U Sco is $12000\pm2000$ pc, which places U Sco at the far side of the Milky Way's bulge.
One implication from these distances is to have confident evidence that the MMRD distance for U Sco and T CrB have large errors, by factors of 3.1 and 4.0 respectively. These errors are large, corresponding to 2.5 and 3.0 mag in the absolute magnitude. With this, these two very-fast RNe become by-far the biggest outliers in the MMRD relations (Downes \& Duerbeck 2000). It is unclear to me whether the problem lies with the incredible speed of the decline or some quirk related to the short recurrence time scales. Both possibilities are poor because the other three systems are also fast RNe yet the MMRD works acceptably for them. (Their error ratios are 0.5, 1.9, and 1.3 for RS Oph, V745 Sco, and V3890 Sgr.) I will adopt a factor-of-two error for MMRD distances, and I will be happy if I can find alternative evidence if the MMRD distance appears unreasonable.
\subsection{Distances to Individual Systems}
For T Pyx, with its prominent shell, we could hope to get a distance with an expansion parallax. But this hope is stymied because the shell is expanding at a relatively slow velocity (with no deceleration) from an ordinary nova event (presumably around 1866) for which the date is not independently known and for which the expansion velocity is not known (Schaefer et al. 2009). Webbink et al. (1987), Patterson et al. (1998), and Selvelli et al. (2008) evaluate and summarize T Pyx distance measures. Webbink et al. (1987) could only place limits of $>1050$ pc (from the calcium lines) and $<4500$ pc (from the rate-of-decline relation). Patterson et al. (1998) consider several measures (Ca II lines, the speed of decline relation, the 2175A feature, and the Eddington limit) without providing details, then concluding that the distance is 2500-4500 pc. Selvelli et al. (2008) give the same basic input and relations, along with full details, and conclude that T Pyx has a distance of $3500\pm350$ pc. The MMRD gives a distance of 3200 pc. Given the poor observational constraints and the lack of consensus, I will adopt the middle ground offered by Patterson et al. (1998), with T Pyx having a distance between 2500-4500 pc, or $3000\pm1000$.
For IM Nor, we do not have any good constraints on distance. Duerbeck et al. (2002) used the strengths of the interstellar Ca II, Na I, K I, and diffuse interstellar lines to conclude that $E_{B-V}\ge0.8$ and $d\ge2500$ pc. Orio et al. (2005) used the equivalent width of the Na I line to find $E_{B-V}$ from 0.5-1.1 mag and then the MMRD relation to get distances between 1500-6400 pc. The Orio et al. distance is based on a $t_2$ value that is a factor of two smaller than given in Table 17, and so their distance is superseded by my MMRD distance of 3400 pc (Table 27). So, with nothing better in sight, I will adopt the distance as from Table 27, $d=3400^{+3400}_{-1700}$ pc.
For CI Aql, we again only have moderate quality extinction information and MMRD to get the distance. (Theoretical models of CI Aql (Hachisu \& Kato 2003; Hachisu, Kato, \& Schaefer 2003; Lederle \& Kimeswenger 2003) uniformly give values close to $E_{B-V}=1.0$ mag and $d=1500$ pc. But for this paper, I will only consider model-independent results.) The only observationally-based distance is my MMRD results (Table 27) with $d=5000^{+5000}_{-2500}$ pc.
For V2487 Oph, we again have only the MMRD relation (Table 27) with $d=32,400$ pc. For a galactic latitude of $+7.8\degr$, the nova is 3.2 kpc above the disk and is on the far outer edge of the galaxy. Such an extreme position is worrisome that the distance is greatly over-estimated. Indeed, I would place an upper limit on the distance of perhaps 25 kpc for any plausible association with our Milky Way. I can think of two reasonable suggestions to avoid the implausibly large distance suggested from the MMRD relation. One resolution is that the peak was missed, so that a brighter $V_{peak}$ would give a closer distance. But to get a distance equal to that to the galactic center, we would have to have $V_{peak}=4.5$ mag, and this is not plausible. For the latest upper limit on the pre-eruption light curve, where the nova was not seen 8 days before the discovery, the extrapolated peak must have been fainter than 6.8 mag. For an extinction of 0.5 mag and a peak at 6.8 mag, the shortest possible distance is 14,100 pc. The second resolution is to note that U Sco is apparently a similar system whose MMRD distance is 3.1 times farther than a reliable distance based on the secondary (Section 14.3), so we could try to apply the same correction factor to V2487 Oph. With this, the MMRD distance of 32,400 pc gets reduced to 10,000 pc. In all, the range of plausible distances is from 10,000 to 25,000 pc, with the lower part of this range preferred simply due to the higher star density. I will express this large uncertainty as $d=12,000^{+13,000}_{-2,000}$ pc.
For U Sco, we have many papers that present observationally-based distances. Conclusions include 22-95 kpc (Webbink 1978), $3.5\pm1.5$ kpc (Hanes 1985), 8-29 kpc (Schaefer 1990), 17 kpc (Duerbeck \& Seitter 1980), 5-25 kpc (Webbink et al. 1987), 14 kpc (Warner 1995), and 60 kpc (Warner 1987). (Model-based estimates tend to be with shorter distances, including 4.1-6.1 kpc from Hachisu et al. 2000a; 6-8 kpc from Hachisu et al. 2000b; and 3.3-8.6 kpc from Kato 1990.) My MMRD distance (Table 27) is 37,700 pc. The many large distances in the literature are just versions of this latest MMRD value. Webbink et al. (1987) puts an upper limit on the distance of $\sim 25$ kpc ``if it is plausibly to be a member of our galaxy". In this confusion, we are lucky that the system has total eclipses, as this allows us to derive a reliable and fairly accurate distance, and this is $12,000\pm2000$ pc.
For V394 CrA, we have little information on distance. Duerbeck (1988) gave a distance limit based on the presumption that the peak absolute magnitude is $\leq -7$ (i.e., a fast nova will be super-Eddington) to get a distance $\geq5000$ pc. My MMRD distance from Table 27 is 24,400 pc. For a galactic latitude of $-7.7\degr$, the nova would be 3 kiloparsecs above the plane and on the far edge of the Milky Way. We can place an upper limit on the distance as $\sim25,000$ pc for the system to be associated with our Milky Way. The far out MMRD position can be moderated by an increase in the extinction, but the colors are already too blue. An alternative way to moderate the disturbingly-large MMRD distance is to realize that the similar U Sco system has a MMRD distance that is 3.1 times too large (see above), so perhaps a better MMRD distance for V394 CrA (as calibrated with U Sco) would be 7,800 pc. The observational situation is that we only have limits of 7,800-25,000 pc, while the MMRD distance has substantial calibration questions so that any distance within the allowed range is possible. Within the preferred range, the shorter distance scale is much more likely as the star density gets rather thin for the higher distances. In all, I will take the distance to be $10,000^{+15,000}_{-3,000}$ pc. Hachisu \& Kato (2000) use their theoretical model to conclude that the distance is something like 4200-6100 pc. But again, it is important to realize that such a result is model dependent and is not the observation-based distance sought in this paper. A substantial problem is that the Hachisu \& Kato model requires the extinction during eruption to be near zero yet also to be $E_{B-V}=1.10$ mag in quiescence. They resolve this large difference in extinction by one sentence ``we may suggest that the intrinsic absorber of V394 CrA is blown off during the outburst". Apparently, the idea is that the eruption radiation destroys nearby dust (leading to low extinction during outburst) and the ejected material later forms new dust of its own (leading to high extinction during quiescence). But this idea won't work because the observations at quiescence are taken at such a late time that the dust will have dispersed to invisibility. And my detailed calculations of dust formation shows no detectable dust formation. And the theoretical model has the critical inconsistency that the extinction during the eruption (or anytime) cannot be smaller than $E_{B-V}\sim 0.25$ mag due to the intervening ISM in the first kiloparsec. In all, I do not see a consistent theoretical scenario. So we are in an unsatisfactory situation where there is no acceptable theoretical model and the observational situation only has broad limits. I will adopt the unsatisfactory observational results that the distance is $10,000^{+15,000}_{-3,000}$ pc.
For T CrB, the presence of the red giant companion star can be used to get a distance. Harrison, Johnson, \& Spyromilio (1993) used the infrared colors and the K-band magnitude to estimate $E_{B-V}=0.10\pm0.02$ mag and a distance of 1020 pc. Patterson (1984) gives a distance to T CrB of 1200 pc based on measures of the secondary star. Bailey (1981) used an empirical calibration of the K-band surface brightness as a function of V-K color to derive a distance of 1180 pc. Krautter et al. (1981) used the spectral type of the red giant to get an absolute magnitude and a distance of 1300 pc. Belczy\'nski \& Miko{\l}ajewska (1998) derived a distance based on the red giant (assuming it fills its Roche lobe) to be $960\pm150$ pc, which is similar to my value by the same method in Table 27. I adopt the distance from the average of the three methods in Table 27 (and the mean 23\% uncertainty) of $900\pm200$ pc.
For RS Oph, distance estimates in the previous literature has been summarized by Barry et al. (2009), while a critical review is given by Schaefer (2009). Historically, RS Oph distance has been estimated to be over a rather wide range ($\lesssim$ 540 to 3180-3290 pc), however the most cited value has been 1600 pc from Hjellming (1986) as supported by a limit of $\lesssim$2000 by Cassatella et al. (1985). However, Schaefer (2009) has corrected errors and added the conservative effects of systematic uncertainties to find that the literature distances are 1050-4200 pc (for the MMRD relations), no limit (Monnier et al. 2006), 1200-4900 pc (Rupern, Mioduszewski, \& Sokoloski 2008), $>$1000 pc (Hjellming et al. 1986), no limit (Cassatella et al. 1985), and `3200?' pc (Livio, Truran, \& Webbink 1986). These independent distance evidences have not decided whether the RS Oph is sufficiently far away so as to require that the red giant fill its Roche lobe (with a distance of 4200$\pm$900 pc or $\gtrsim$ 3200 pc) or not fill its Roche lobe (with distances $<$3200 pc). Schaefer (2009) demonstrates that the wind accretion model for RS Oph cannot account for the observed rate of mass accumulation on the white dwarf by a factor of 100,000$\times$. Thus, the only way to get a large enough accretion rate is for the red giant to fill its Roche lobe. With this, the best distance measures are based on the companion star (Table 27), for which I average the three methods and adopt a 23\% error. The best estimate distance is 4200$\pm$900 pc.
For V745 Sco, the presence of the red giant companion star can be used to get the distance. My MMRD analysis gives a distance of 14,100 pc. Such a large distance is relatively unlikely as the RN would then be well outside the galactic bulge (with this being by far the highest density of stars along the line of sight). Harrison, Johnson, \& Spyromilio (1993) used the assumption that a M4 III star would have a K-band absolute magnitude of -5.5 to derive a distance of 4600 pc, although they point out that such a canonical luminosity could well be far off for any particular star (there is much vertical scatter amongst red giants in the H-R diagram), especially one that is in an interacting binary. With my orbital period for V745 Sco, we can now get a reliable size for the red giant and derive the distance to the companion star as the average of the three methods in Table 27. With this, we have a distance of $7800\pm1800$ pc, and V745 Sco lies in the middle of the galactic bulge.
For V3890 Sgr, we have many measures of the extinction (cf. Table 26), but few measures of the distance. Harrison, Johnson, \& Spyromilio (1993) estimate $d\approx5200$ pc as based on the assumption that the interacting red giant star has the same absolute magnitude as a canonical isolated red giant, with this assumption possibly being far wrong as we know that the red giants have a lot of vertical scatter in the H-R diagram. My MMRD analysis (Table 27) gives a distance of 7,600 pc. Fortunately, again the discovery of the orbital period allows for a reliable size of the red giant and a distance to the companion star (Table 28). With this, the distance to V3890 Sgr is $7000\pm1600$ pc and it resides in the bulge of our Milky Way galaxy.
All the distance and extinction estimates from this section have been collected into the top of Table 29.
\subsection{Is the RS Oph Red Giant Filling Its Roche Lobe?}
The general presumption is that the companion stars in all nova systems are filling their Roche lobes. This strong result arises mainly from the need to get matter falling into the accretion disk at a high rate. This imperative is particularly strong for RNe, where the accretion rate onto the white dwarf must be something like $10^{-7}$ M$_{\odot}$ yr$^{-1}$. However, this imperative might not be required in the case where the companion star is a red giant. Red giants have substantial stellar winds, some fraction of which will accrete onto the white dwarf, so the Roche lobe can possibly be underfilled and yet the system can still have significant accretion.
Of the four RNe with red giant companions, a case for an underfilled Roche lobe has only been made for RS Oph. In this case, the accretion onto the white dwarf would have to be driven by a stellar wind perhaps mediated through a ring around the companion star. Garcia (1986) interpreted profile changes in the Fe II line as being like those seen in a Be star, which would imply that the red giant cannot fill its Roche lobe. But this model hardly seems unique and it is based only on an analogy, so the underfilling of the Roche lobe was just a suggestion. The primary reason given by Garcia for RS Oph not filling its Roche lobe comes from a simple comparison of the calculated Roche lobe radius (for presumed masses) with the canonical radius of an M0 giant. But this argument was critically based on the reckless presumption that the star in a weird binary just happens to have the same radius as some isolated star with the same temperature. Garcia quoted the Roche lobe radius as 74 $R_{\odot}$ while his canonical M0 giant has a radius of 50 $R_{\odot}$ and hence concluded that the Roche lobe is not filled. But I see no reason why the real companion star can't be just a little bit higher up in an H-R diagram and totally filling its Roche lobe. That is, red giants of the same temperature (all along a vertical line in an H-R diagram) come in a wide range of sizes (spreading up and down along that vertical line), so there is no real constraint on the size. As such, the original argument against Roche lobe filling is far too weak for serious conclusions.
A more sophisticated version of the same argument comes from detailed modeling of the fluxes and temperatures of the companion star (e.g., Hachisu \& Kato 2001). The derived size of the companion star critically depends on the adopted distance to the star. Hachisu \& Kato adopted a distance of 600 pc and concluded that the red giant only fills 25\% of the Roche lobe (by radius), while Dobrzycka et al. (1996) adopted a distance of 1500 pc and concluded that the red giant only fills 60\% of its Roche lobe. If the distance to RS Oph is truly as close as these old assumptions, then everyone would agree that the Roche lobe is underfilled. So the primary question really comes down to the distance. If RS Oph is $\gtrsim$3000 pc away, then it is filling its Roche lobe, while if RS Oph is nearer then it will not be filling its Roche lobe.
Historically, the RS Oph distance question went horribly awry. The first (apparently) reasonable distance estimates to RS Oph were by Casatella et al. (1985) and Hjellming et al. (1986), and so these are always the papers that later works cited. No one in the later literature spotted the simple and killing error that completely invalidated these distance estimates. (In particular, for a galactic latitude of 19.5$\degr$, the line of sight leaves the gas and dust in the disk within a kiloparsec of Earth, so there would be no absorption from the Carina Arm in any case and the naive linear relation between distance and absorption is far wrong.) The Hjellming et al. paper and its claimed distance of 1,600 pc has been cited 63 times, always as a primary reference for the distance to RS Oph. And then, no accurate or reliable method for distance determination came along (see previous section) as a substitute for the invalid 1,600 pc distance estimate. So our community has been left with the original (invalid) distance of 1,600 pc plus a number of inaccurate and unreliable measures that convinced no one. By virtue of repetition and nothing better, our community has taken the distance of 1,600 pc as the default answer. At the conference on cataclysmic variables in Tucson in March 2009, I systematically asked all four speaker who talked about RS Oph as to what they thought the distance was, and they all answered a hesitant ``1,600 pc". This shows the legacy of the old error. And the error is reinforced by the circular argument of modelers adopting a short distance scale, finding that the red giant does not fill its Roche lobe, developing models with wind accretion, and then later researchers look back on these models as justification for the short distance scale. So, historically, the current default distance of 1,600 pc (and the conclusion that the red giant does not fill its Roche lobe) has arisen largely from a bandwagon effect (cf. Schaefer 2008) that was started as a simple error in an old paper.
The stellar wind accretion hypothesis (required if the red giant underfills its Roche lobe) can be tested. We know from various independent methods what the mass loss rate of the red giant is and what fraction will accrete onto the white dwarf between eruptions. We also know from multiple independent methods how much accreted material is required to trigger the eruption. We can then compare the two, with the wind accretion hypothesis being plausible only if the mass accreted onto the white dwarf is comparable or larger than required for the eruption. Schaefer (2009) has performed this analysis, finding that the wind from the red giant in the RS Oph system has a mass loss rate of $\sim 3.7\times10^{-8}$ M$_{\odot}$ yr$^{-1}$, with $\sim 5.4\times10^{-10}$ M$_{\odot}$ falling onto the white dwarf in the average inter-eruption interval of 14.7 years, while roughly $5.9\times10^{-5}$ M$_{\odot}$ is required as the trigger mass. The wind accretion model fails by a factor of $100,000\times$. Thus, the only way to feed the white dwarf fast enough is with Roche lobe overflow. So I confidently conclude that the red giant fills its Roche lobe.
\section{Galactic Distribution}
What is the galactic distribution for the RNe? We will not be able to make a high precision measure of the distribution, because we only have ten stars and because of the usual interstellar medium masking. To quantify the positions of the known RNe, I have collected the distances (from the previous section) and the galactic coordinates ($\ell$, $b$, and the angle from the galactic center $\theta_{GC}$) into Table 29. From this, I calculate the height above the galactic plane for the best estimate distances, $Z_{best}$. I have also tabulated the heights above the galactic plane for the minimum acceptable distances, $Z_{min}$. A top view and a side view of our Milky Way, along with the projected positions of the RNe (and the uncertainty associated with the distances) are displayed in Figure 69.
The RMS scatter in the $Z_{best}$ values is 1600 pc, while the RMS scatter in the $Z_{min}$ values is 1300 pc. This is very large compared to the observed value of 190 pc made from a large demographic study of cataclysmic variables, while the intrinsic value (after correction for interstellar absorption) is more like 150 pc (Patterson 1984). Patterson's sample is composed mainly of relatively nearby systems, so this would be representative of the scale height of the disk population. As such, we see that the RNe are not a simple disk population. Shafter (2002) and others have pointed out that the classical novae are a combination of a disk and a bulge population. With the large scale height for three RNe and the concentration on the sky towards the galactic center for eight RNe, we see that the RNe must have a large contribution from a bulge population. But RNe cannot be entirely a bulge population, because no more than 50\% can be within 3 kpc of the galactic center and because the remaining RNe are similar to a disk population with its RMS scatter (400 pc) much closer to that of general CVs. Another measure is that we see 4-out-of-10 RNe $>30\degr$ from the galactic center, while close to half of all CNe are $>30\degr$ from the galactic center, which implies that the bulge-to-disk ratio is comparable for RNe and CNe. While estimates are necessarily crude due to the low numbers, it appears that RNe are roughly evenly divided between thick disk and bulge populations, in a situation similar to that amongst CNe.
\section{Absolute Magnitudes}
In the preceding sections, I have been reporting on the best values for the peak magnitudes, the quiescent magnitudes, the distances, and the extinctions. With this, we can now calculate the absolute magnitudes for the RNe. We can use the standard relation that the absolute magnitude is $M=m-5\log(d)+5-A$, where $m$ is the observed apparent magnitude, $d$ is the distance in parsecs, and $A$ is the absorption from the interstellar medium in magnitudes. The absorption is a function of the observed color excess ($E_{B-V}=A_B-A_V$), with $A_V=3.1~E_{B-V}$ for the V-band, and with $A_B=4.1~E_{B-V}$ for the B-band. The intrinsic color ($(B-V)_{0}$) will equal the observed color corrected for the color excess, as $(B-V)_0=(B-V)-E_{B-V}$. As an intermediate step, I have tabulated the observed distance moduli ($\mu=m-M=5\log(d)-5+A$) for both the B-band and the V-band (see Table 29).
In Table 29, I collect the various magnitudes for the ten RNe. These include the peak B-band magnitude ($B_{peak}$), the peak V-band magnitude ($V_{peak}$), the average B-band magnitude in quiescence ($\langle B_q \rangle$), the average V-band magnitude in quiescence ($\langle V_q \rangle$), and the range of $V_q$. I have converted all these apparent magnitudes to absolute magnitude with the observed distance modulus, and these are also placed into Table 29. I have also added the intrinsic colors at peak and at quiescence.
The peak absolute magnitudes in the V-band have an average value of -8.0 mag, which is close to the value for CNe (Shafter 2002). The intrinsic colors at peak have an average value of $(B-V)_0=-0.1$ mag, with this being close to the value for CNe. Thus, I am seeing no difference between RNe and CNe at the peak. This is surprising to me, as I would expect that the greatly different mass and velocity of the ejected envelope (between RNe and CNe) would have made for systematically different peak conditions.
The average quiescent absolute magnitudes vary substantially, with $-4.1\leq M_V \leq 3.2$ mag. As expected, the most luminous quiescent systems are those with red giant companions. The canonical $M_V$ for a M0-M5 red giant is around -0.3 mag. Both V3890 Sgr and RS Oph are substantially brighter than this, suggestive that there might be a substantial contribution from the accretion disk, and this is confirmed by the relatively blue $B-V_0$ values at quiescence. Both V745 Sco and T CrB are less luminous than the canonical red giant, but this is not a cause for concern because we have no reason to think that the companions are canonical. (Again, red giants show a lot of vertical scatter in the H-R diagram.) The intrinsic colors for these two stars suggests that the contributions from the accretion disks are relatively small. For the six remaining systems that do not have a red giant companion, the range of $M_V$ in quiescence is 0.0 to 3.2 mag with a median of 2.2 mag. For comparison with the CNe with reliable distance measures (Shafter 1997), the range of $M_V$ in quiescence is 1.1 to 7.0 mag with a median of 4.6 mag. Thus, in quiescence, RNe are substantially and significantly more luminous than CNe, by almost an order of magnitude. This is easy to understand, because theory requires that RNe have a very high accretion rate, and many of the RNe have red giant companion stars, with both of these effects making for brighter systems.
\section{Spectral Energy Distributions in Quiescence}
What is the spectral energy distribution (SED) for the galactic RNe in quiescence? That is, how does the spectral flux (say, in units of jansky) change across the spectrum. For most of the RNe, I have measured magnitudes in the UBVRIJHK bands, and with the extinction estimates, I can calculate the SED. For the short period systems like T Pyx and IM Nor we will see the accretion disk contribution alone, while for the long period systems we will see the accretion disk plus the red giant companion star. By getting the flux across a wide range, we can separate out the disk and companion components. An integral under the accretion disk component will give the accretion flux, which can then be converted to the accretion luminosity with the distance estimates given in Section 14. From the accretion luminosity and the mass of the white dwarf, we can derive the accretion rate ($\dot{M}$). The RN accretion rates are important for measuring whether $M_{ejecta}>\tau_{rec}~\dot{M}$ and for determining the lifetime (and hence the death rate) as $\sim 0.2 M_{\odot} / \dot{M}$. With both of these questions, the determination of the RN accretion rate is critical for the most important questions.
This path to $\dot{M}$ has two substantial uncertainties. The first big uncertainty is that much of the accretion luminosity comes out in the far ultraviolet and the extreme ultraviolet. The SEDs in this paper will only cover from around 0.36-2.2 microns of wavelength, which does not cover where most of the energy comes out. Nevertheless, the observed SED will rise and fall with the accretion rate, so model fits can still produce reasonable values. Also, in some cases, far ultraviolet observations might be available to substantially extend the SED so as to greatly increase the accuracy of model fits. In particular, I have GALEX observations of three RNe in quiescence, for which I have simultaneous BVRIJHK data from the SMARTS 1.3-m telescope. The derivation of the needed optical SEDs (for joining with the GALEX SEDs) requires good comparison stars, simultaneous photometry in quiescence, and the best possible distances and extinctions, and has been a primary motivation for much of the work reported in this paper. The second big uncertainty is that the derived accretion rate depends on the square of the distance and the distance is often poorly determined. It is for this reason that I have done so much preparation leading up to the results in Section 14.
In this section, I will only derive the SED, with the further task of deriving the $\dot{M}$ reserved for a later paper. The first part of this task is simply to collect magnitudes and colors for the ten RNe. These should all come from nearly simultaneous observations. For this, I have selected SMARTS observations with UBVRI or BVRIJHK observations of nine of the RNe (V2487 Oph was not known as a RN when these series of observations were made) that were all taken within a 15-minute time interval. (The GALEX observations were also taken at the same times.) The U-V colors were taken at other times, yet were nevertheless taken to be representative of the U-V color at the quoted times. This is a reasonable equality as we can see from Table 25 that the colors do not change greatly. For some stars, I also had to take the V-H and V-K colors from some other date, and again this is reasonable because the colors do not change greatly. My collected magnitudes and colors are given in Table 30. The magnitudes do change on the usual flickering time scales, so even my 15-minute interval of observation will have some error due to non-simultaneity, and I will estimate that the one-sigma uncertainty is typically 0.05 mag for all the RNe. With this, the errors associated with variability will usually dominate over measurement errors. For the few cases where the measurement errors are significant, I will correctly propagate the uncertainties into the next table.
The second part of the task (constructing the SEDs) is to correct for the interstellar extinction so as to get the extinction-corrected magnitudes. For this, we have to use the estimates of the $E_{B-V}$ values from Table 26. From these, we can calculate the extinction-corrected magnitudes by subtracting the extinction for each band, $A(\lambda)$ in magnitudes, with $A(\lambda)=[A(\lambda)/A(V)]\times 3.1 E_{B-V}$. Here I have taken the canonical numerical constant of 3.1 applicable to the ordinary extinction in our Milky Way. The quantity $A(\lambda)/A(V)$ depends on the bandpass, and these are tabulated in Table 31 along with other standard properties of each band (Mathis 2000). With this, we can convert all the colors in Table 30 into extinction-corrected magnitudes (denoted with the subscript `0'), along with their one-sigma error bars, and I have tabulated these in Table 32. The error bars are being dominated by the uncertainties in the extinction. So the errors for individual points are correlated.
The third part of the task is to convert the extinction-corrected magnitudes into physical units. I have chosen to convert them into flux densities ($f_{\nu}$) with units of jansky (Jy), with this flux unit equal to $10^{-26}$ watt per square meter per hertz or $10^{-23}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$. For this conversion, I have tabulated the flux in jansky for a zero-magnitude star in each bandpass (see Table 31, as taken from Bessel 1979; Campins, Reike, \& Lebovsky 1985). A simple scaling from the zero-magnitude flux to the magnitude reported in Table 32 will give the flux density, with these values reported in Table 33. Both T CrB and RS Oph have been detected with the $\it IRAS$ satellite, with T CrB having a 12 $\micron$ flux of 0.70 Jy ($6\%$ error bars) and a 25 $\micron$ flux of 0.3 Jy ($14\%$ error bars), and with RS Oph having a 12 $\micron$ flux of 0.43 Jy ($10\%$ error bars) (Schaefer 1986). These flux densities as a function of frequency constitute the SED for each RN. I have plotted these SEDs in Figure 70.
The fourth part of this task is to correct for the distances, that is, to convert from fluxes to luminosities. For this, we have $L_{\nu}=4\pi d^2 f_{\nu}$, where the distances are taken from Table 29. I have taken $\log(L_{\nu})$ in units of watt per hertz and presented these in Figure 71. I have not propagated the uncertainty in the distances, which can be substantial for V394 CrA and V2487 Oph. With this figure, we can finally see the RNe relative to each other with the distances taken out.
The SEDs show the expected features of the red giant companions and the accretion disks. The four systems with red giant companions are all bright in the near infrared and have peaks $\sim$14.3 in $\log (\nu)$. With the equivalent of Wein's displacement law for the peak of the flux distribution corresponding to temperatures of around $3,400\degr$, as appropriate for M giant stars. RS Oph has another peak in its SED at around $\log (\nu)=14.7$ (in the V-band), likely due to continuum from the cF-type source (associated with the accretion stream) and nebular emission lines (Anupama \& Miko{\l}ajewska 1999; Brandi et al. 2009). The CI Aql, T Pyx, IM Nor, and U Sco systems all display a power law rise with the classic signature of an accretion disk ($f_{\nu} \propto \nu^{1/3}$). I see no evidence of the companion star. However, the presence of a moderate secondary eclipse on U Sco in the I-band and the ellipsoidal effects on CI Aql demonstrates that there are indeed contributions to the flux from the secondary stars. All of the RNe have their SEDs falling from the B-band to the U-band, and this cannot be due to measurement errors because everyone sees these same colors and it cannot be due to under-correcting the extinction because the B-band fluxes would rise correspondingly and the size of the extinction error bars are too small to lead to a rise towards the ultraviolet. I think that the systematic fall towards the U-band is real, and I do not understand it.
All RNe in quiescence vary on all time scales (minutes-hours-days-weeks-months-years-decades) usually with amplitudes of over one magnitude (see Sections 10-12 and Figures 42-67). So how can one SED and its derived $\dot{M}$ represent the accretion rate over a whole eruption cycle. One adequate answer is that the single derived $\dot{M}$ will provide a typical value that is approximately correct, and this is a lot better than the alternatives. But we can do better. With the colors not changing much (outside of eclipses and eruptions), the SED can be scaled by the V-band magnitude. Or rather, the SED scaling should go as the V-band flux ($\propto 10^{-0.4V}$). With many magnitude estimates throughout an inter-eruption interval, we can take each V-band flux, scale the reported SED to each V-band flux, calculate the $\dot{M}$ values for each scaled SED, average all the $\dot{M}$ values to get the average accretion rate over the inter-eruption time interval, and then multiply by the time of that interval to get the mass accreted between eruptions. This plan is straight forward, practical, and follows the complexity of the RNe variability.
\section{Open Questions for Observation and Theory}
A variety of questions have been raised in this work, and I will itemize these questions here. I will not separate out the various higher level questions. Such questions as ``What are the average $\dot{M}$ values?", ``Are the white dwarfs gaining mass?", and ``What is the RN death rate in our Local Group?" will be addressed in other papers. Instead, I will select questions with some relevance for photometric issues, be it concerning observational facts or theoretical interpretation:
\subsection{Observational Questions}
(O1) What is going on with the sudden fading around day 33 in the tail of the U Sco light curve? Is the fading temporary, only to have the nova rebrighten and return to its prior fading decline (as would arise from the formation and dispersal of dust in the ejecta shell), or is the fading monotonic (as would arise from a turn off in the nuclear burning)? And how does this fading behavior depend on color? Does the x-ray brightness turn off at that time? All of the extant data for prior events have already been examined, so the only answer will come from detailed photometry of the upcoming eruption.
(O2) What is the orbital period for V2487 Oph? The single most important datum for any cataclysmic variable is its orbital period, and this is especially true for RNe due to their very wide range of $P_{orb}$. V2487 Oph likely has a period of near one day, but this is guessed only on a perceived similarity with the RNe like U Sco.
(O3) What is the frequency of plateaus in the light curves of classical novae? If they are common, then maybe their presence in RN light curves does not mean much. If plateaus are in only a fraction of classical novae, then this might provide an indication of the fraction of classical novae that are really RNe. Also, plateaus amongst the classical novae light curves could point to good RN candidates, for example, for use in archival plate searches for the earlier eruptions.
(O4) Another observational imperative is to create a long series of accurate minima times for both T Pyx and IM Nor. This task has three science returns: First, with an accurate orbital period, we can await the next eruption and then measure the post-eruption orbital period so as to measure the change in its orbital period. This is an important task that wil require a large and sustained observing program. Unfortunately, both RNe are expected to have their next eruptions only many decades or longer from now (see Table 21). Second, we can measure whether the T Pyx period is increasing linearly. That is, does the O-C curve rise quadratically as predicted if the period change were caused by conservative mass transfer? If so, then this will force models to take the high derived $\dot{M}$ at face value, despite theoretical difficulties in understanding. Third, we can measure the period for IM Nor using only minimum times during quiescence, and this will provide a good ephemeris to see the deviations in eclipse times during the tail of the 2002 eruption. For U Sco and CI Aql, I see large deviations in eclipse times made during the eruption with these being caused by the shift in the center of light in the fading nova. With the excellent data of Monard for the 2002 eruption, we can trace the deviations throughout the eruption, for direct comparison with theoretical models.
(O5) Someone really has to make a modern radial velocity curve for the white dwarf in T CrB. This is to replace the 1950's radial velocity curve of Kraft (1958) for which the amplitude is based on only two plates. The study of Kenyon \& Garcia (1986) had high spectral resolution and good time coverage, but did not cover any emission lines. As the velocities are small, high spectral resolution is required, the spectral range should be carefully selected to include emission lines, and the observations should be well-spaced over a several year time span.
(O6) I am predicting that five of the RNe (V394 CrA, V2487 Oph, U Sco, V3890 Sgr, and V745 Sco) will go off in the next decade. With the expected uncertainties in such predictions, any one of these could go off any night now. This creates an observational imperative to keep regular daily monitoring of all five so as to promptly catch their next eruptions.
(O7) How do the long-term variations in the quiescent light curve compare between CNe and RNe? That is, do the CNe have long-term variations of high amplitude as shown by most of the RNe? Do the CNe show any systematic rise or fall after nova events?
(O8) How do the very-low-frequency PDSs connect with the higher-frequency PDSs? That is, for at least RS Oph and T CrB, we see $1/f$ noise for time scales longer than a week and shorter than a few hours, but we don't know how they connect. In principle, with correct normalization of the PDSs, we can see whether the high and low frequency behaviors are all part of one continuous power law. Alternatively, we could construct a PDS from a time series with nearly uniform coverage from time scales of hours to months, with the possibility of breaks between. Within the ideas mentioned in this paper, the short time scale variations should only extend to something like the orbital period, while the convection cells on the red giant will provide little power on short time scales.
(O9) Is IM Nor a supersoft source and does it have a shell from an earlier ordinary nova event? Like T Pyx, IM Nor has a very high accretion rate for which ordinary mechanisms cannot account, hence making it likely that the system is also driven by a supersoft source after an ordinary nova eruption. Deep observations with {\it Chandra} or {\it XMM-Newton} might be able to detect a residual supersoft source, while the {\t Hubble Space Telescope} might be required to find a shell. Problems for this task are that any IM Nor supersoft source is turning off and the source is near the galactic plane (latitude $+2.5\degr$) with moderate extinction.
\subsection{Theoretical Questions}
(T1) Why did T CrB suffer a distinct, significant, and unique fading in the year {\it before} its 1946 eruption? And why would this fading behavior be different in the B and V bands? The fading is by around one magnitude below the usual level of the system, with this going to two magnitudes below the usual level in the B-band at a time 29 days before the eruption. My first thought is that the accretion turned off (for unknown reason) hence making the system lose the light from the accretion disk, but maybe the depth of the drop will require the red giant companion to be dimmed somehow. And what is the physical connection between this fading and the subsequent nova eruption? That is, how can the turning off of accretion {\it anticipate} or {\it trigger} the nova event?
(T2) Why did T CrB have a secondary maximum? This event carries a large amount of energy. The maximum started after the system had returned to quiescence for fifty days, so why the delay? Are there any other novae with secondary maxima?
(T3) Why do the quiescent light curves of half the RNe (CI Aql, U Sco in the red, V394 CrA, V745 Sco, and V3890 Sgr) have an asymmetry where the maximum just after the primary minimum is higher than the maximum preceding the primary minimum? I do not know of any precedent for this amongst the classical novae, while other types of cataclysmic variables that have an asymmetry have the brightest phase just {\it before} the primary eclipse\footnote{http://cbastro.org/cataclysmics/atlas/}. I suggest that the extra light arises from the hot spot, and if so, then we need an explanation for why it appears brightest around phase 0.15.
(T4) Why do most of the RNe display large amplitude variations on long time scales (years to decades)?
(T5) How does the eclipse amplitude and phase change throughout the eruption. This will depend in detail on the optical depths of the envelope as a function of time. We already have amplitudes and phases for three RNe (IM Nor, CI Aql, and U Sco), and we can anticipate wonderful measures from the upcoming eruption of U Sco. In principle, the observations can be used to test theoretical models by providing an observed optical depth all the way to the center of the nova shell throughout the entire eruption. Also, I hope that this optical depth information can be combined with spectral data so as to derive a value for the total mass ejected during the eruption.
\section{Conclusions}
This paper is addressing a fairly restricted topic (observational issues arising from photometry of one small class of stars with only ten known members). Nevertheless, these data provide a fundamental basis for front-line questions of broad importance (hibernation and the Type Ia progenitors). To address these big questions, I repeatedly realized that I needed comprehensive and correct photometric measures of the few known RNe. But I found that the published data were widely scattered and not-modern, while the majority of the data have not been published (not even counting the huge AAVSO data sets). So this paper represents my attempt to collect all RNe photometry and put it together in a consistent and modern way. I will be using the results from this paper as the fundamental observational basis for a variety of future papers, while I expect that a wide variety of researchers will also be able to pull out the best numbers on RNe from this paper.
Here I will summarize the new points. Many of the new results are also tabulated in Table 34. Additionally, I will itemize the new points for each of the ten RNe, and then summarize the broader conclusions:
(T Pyx) T Pyx is unique for having a flagrant plateau in its eruption light curve that starts after a precipitous drop (2.0 mag in 20 days). T Pyx is unique in having a secular decline from 1890 to present across many eruptions.
(IM Nor) The light curve shows a sharp drop after a plateau. For the first time, the depth of eclipse in a nova system has been measured throughout the entire eruption all the way to quiescence. The amplitude increased from 0.01 mag at a time 110 days after peak (3 magnitudes below the maximum), to 0.25 mag in the late tail, to 0.4 mag in quiescence.
(CI Aql) The 1941 eruption of CI Aql was discovered with an exhaustive search through the Harvard plate collection. This changes the recurrence time scale from 83 years to 24 and 59/N years (for N=1, 2, or 3). With the quiescent magnitude changing relatively little over the last century, the best idea is that the accretion rate and recurrence time scale has been roughly constant, suggesting N=2 or N=3, with this being consistent with a discovery efficiency of 86\% over the time between 1941 and 2000. The light curve of CI Aql has a mysterious asymmetry in that it is brighter in the time interval just after the primary eclipse than in the time interval just after the secondary eclipse. I have measured 80 eclipse times for CI Aql, with the best period being $0.61836090\pm0.0000005$ days. Previous claims to having measured a period change for CI Aql are certainly wrong because they were using eclipse times from during the eruption and these times are systematically shifted earlier by 0.006 days, with this producing the entire claimed effect.
(V2487 Oph) As part of our work, we predicted that the CN was actually a RN, and we tested this prediction at Harvard and Sonneberg. The prediction proved to be correct when we discovered the eruption in 1900 (Pagnotta, Schaefer, \& Xiao 2008; Pagnotta et al. 2009), and this provides confidence that our indicators (high excitation lines and high expansion velocities during eruption) are valid pointers to RNe. Only $30\%$ of the eruptions are detected, so the real recurrence time scale cannot be the naive 1998-1900=98 years, but instead must be more like 18 years. To keep the distance to V2487 Oph reasonable (i.e., in our galaxy and preferably not far from the bulge), the peak magnitude might have to be several magnitudes brighter than observed, perhaps as bright as seventh mag.
(U Sco) I discovered two previously-unrecognized eruptions on the Harvard plates (in 1917 and 1945), while another eruption in 1969 was found in the archives of the RASNZ. With nine known eruptions, we see that U Sco erupts every $10\pm2$ years with missed eruptions around 1927 and 1957. U Sco rises from quiescence to peak in roughly half a day. With $t_3=2.6$ days, U Sco is the fastest known nova of any type. In the new 1945 light curve from the Harvard plates, I find a sudden and steep drop in the B-band magnitudes starting 33 days after the peak and dropping by over two magnitudes within the next several days. Other than this 1945 eruption, only 4 magnitudes have been recorded after day 33 (these on days 44-57 for the 1979 and 1999 eruptions) and these V-band data do {\it not} show evidence for the sudden drop in blue brightness on day 33. U Sco is predicted to have its next eruption in the spring of 2009 (within a year), and this provides a wonderful opportunity to prepare an observing campaign from x-ray to infrared. U Sco has a classic eclipsing light curve, with a total eclipse at the primary minimum and ellipsoidal effects visible in red light. I have 47 eclipse times for U Sco from 1989 to present. Prior claims to have found a period change are certainly wrong as a key role was given to eclipse times during the eruption for which systematic shifts of up to 0.015 days earlier occur. The light curve for the uneclipsed system is relatively flat in the B-band, but it displays ellipsoidal modulations plus a secondary minimum (with amplitude 0.3 mag) in the I-band. The light curve during eclipse has relatively little flickering and apparently has a flat bottom with duration of $0.0253\pm0.0025$ in phase. Using the visual magnitude of the companion star alone (during the total eclipse), the companion star temperature (from the observed spectral type), and the companion star radius (from the Roche lobe diameter), I derive a distance of $12,000\pm2000$ pc.
(V394 CrA) V394 CrA has its average brightness level changing up and down by roughly one magnitude. Prominent sinusoidal oscillations appear when the system is faint, and they get lost in the flickering when the system is bright. That is, the slowly changing brightness (presumably from variable accretion) is superposed on top of unchanging periodic modulations (presumably from ellipsoidal, reflection, and eclipse effects). The accurate orbital period is $1.515682\pm0.000008$ days. The folded light curve shows a primary minima that appears to be an eclipse, while a shallower secondary minimum is also visible. Outside of eclipse, the system is systematically brighter soon after the primary minimum.
(T CrB) An old question is whether the mass accreting star in the system is a white dwarf, but by now many very strong arguments and proofs have been presented that T CrB has a white dwarf and the nova is caused by the usual thermonuclear runaway on its surface. In the year before the eruption, the brightness dipped with large color swings by 1-2 magnitude to unprecedentedly faint levels (with its all-time faintest level being 29 days before the eruption). This is too close a coincidence to be by chance, so there is likely some causal connection by which the fading is physically connected with the eruption. During quiescence, the claimed periodicity of 9840 days (Leibowitz, Ofek, and Mattei 1997) is certainly wrong. In quiescence, the power density spectrum is rising as a power law from seconds to hours and from days to decades. The observed power law index (-0.8) is somewhat shallower than that of variations on a red giant (with index -1.37 to -1.54) or variations on cataclysmic variables (with indices typically -1.5 to -2.0).
(RS Oph) Two additional eruptions were identified in 1907 (by myself) and in 1945 (Oppenheimer \& Mattei 1993) on the basis of seeing the characteristic post-eruption dip as the star came out from behind the Sun. These post-eruption dips last for 100-500 days after the peak, with the exact levels likely determined by the brightness of the red giant companion at the time. During quiescence, the system brightness behaves chaotically, with fast and slow flares, with secular rises and declines, and with decades of relative constancy. The power density spectrum shows a $1/f$ noise power law rise (with index -1.8) down to extremely low frequencies corresponding to many years. This PDS shape is closely similar to those of many other red giants, which suggests that the long-term variations are arising in part from the red giant companion. But a scattered correlation between V-band magnitude and the Balmer line flux shows that at least part of the variations are associated with the accretion. So apparently, the long-term variability and the power law PDS are caused by both mechanisms, convection cells on the red giant and accretion instabilities, with comparable contributions. The suggested periodicity of 2016 days (Oppenheimer \& Mattei 1993) is certainly wrong. Prior claims that the red giant gets as `hot' as G5 (Adams, Humason, \& Joy 1927) are mistakenly based on a retracted measurement (Humason 1938), so the entire observed range is from K5 to M4 (or maybe just K5 to M0). Over this range, the spectral type varies linearly with the V-band brightness from 10.7 to 12.2 mag.
(V745 Sco) V745 Sco has sinusoidal oscillations with an amplitude of 0.4 mag and period of 255 days. These variations have alternating high and low maxima plus deep and shallow minima, so the real orbital period is $510\pm 20$ days.
(V3890 Sgr) V3890 Sgr has been claimed to have pre-eruption dwarf nova outbursts, but this claim has been proven to be wrong. (In addition, the putative pre-eruption dwarf nova events for V446 Her and PU Vul are shown to not be dwarf nova events.) I have discovered two photometric periodicities; 103.8 days likely from pulsations in the red giant star, and $519.7\pm0.3$ days which equals the orbital period. The system was apparently fainter on average (by roughly one magnitude) in the first half of the last century as compared to the last half-century.
(1) I propose that the RNe can be usefully divided into those with short orbital periods (T Pyx and IM Nor) and long orbital periods (the other RNe). This division separates the RNe by the cause of the high mass accretion rate (a continuing supersoft source heating the companion versus evolutionary expansion of the companion star) and by whether the system will become a Type Ia supernova (no and yes, respectively).
(2) The RNe have distinctly different average properties from the CNe, even though their distributions have substantial overlap. The RNe have a median $t_3$ of 11 days (with a full range of 2.6-80 days, including the all-time fastest nova U Sco), while the CNe have a median $t_3$ of 44 days (with a full range of 3.2-900 days). The RNe have a median amplitude of 8.5 mag (with a full range of 6.2-11.2), while the CNe have a median amplitude of around 12.5 mag (with a full range from around 7-16 mag). The RNe have a median orbital period of 1.4 days (with a full range of 0.076-519.7 days), while the CNe have a median period of 0.17 days (with a full range of 0.059-5.8 days). The RNe have a median absolute magnitude in quiescence of 0.8 mag (with a full range of 3.2 to -4.1 mag), while the CNe have a median $M_V$ in quiescence of 4.6 mag (with a full range of around 7.0 to 1.1 mag). However, both RNe and CNe have an essentially identical absolute magnitude at peak ($-8.0$ mag) and color at peak ($\sim 0.0$ mag). Thus, RNe are generally faster, brighter in quiescence, and with longer orbital periods when compared to the CNe.
(3) All the eruption light curves from a single RN are consistent with a single invariant template. That is, the light curves are always the same from eruption-to-eruption. This tells us that the eruption light curve depends on system parameters (like the white dwarf mass and the composition) that do not vary from eruption-to-eruption (like $\dot{M}$).
(4) The RNe light curves are similar in shape to those of many classical novae, except that RNe lack the diversity of CN light curves (e.g., with no DQ-Her-like dust dips and no long and chaotic symbiotic nova episodes). The eruption light curve shapes scaled by $t_3$ for the ten RNe are approximately the same. That is, other than an overall time scale, the declines from peak are similar for all the systems. There are definite differences between RNe (such as the oscillations for T Pyx, the sharp drop for U Sco, and the secondary maximum of T CrB), but largely the light curves lie on top of each other in a log-log plot with the magnitude scaled to some common epoch. In particular, the early declines have power law indices of around -1.75 while the late declines have indices of around -3.0, in reasonable agreement with the theoretical prediction of a 'universal decline law' by Hachisu \& Kato. The break times for all the RNe (except for IM Nor and RS Oph) is very fast (as short as 6 days) which points to the RNe having white dwarf masses close to the Chandrasekkhar mass.
(5) The presence of plateaus in RN light curves is mixed, with six showing definite plateaus (T Pyx, IM Nor, CI Aql, V2487 Oph, U Sco, and RS Oph), one definitely showing no plateau (T CrB), and the remaining three having possible-but-inconclusive plateaus.
(6) The discovery efficiency for RN events is horrifyingly low. Undirected searches have an efficiency of from roughly 0.6\% to 19\% with a median of 4\%. This implies that most galactic RNe with peaks brighter than tenth magnitude never even have one event discovered, that a small fraction have only one event discovered (and the system is cataloged as a classical nova), and a much smaller fraction have two-or-more eruptions discovered (and labeled as RNe). The ratio of RNe-mislabeled-as-CNe to RNe is roughly a factor of 6-10, which implies that 60-100 `CNe' already in the nova lists have really had multiple eruptions within the last century.
(7) Directed searches have average efficiencies (from 1890 to 2008) ranging from 30\% to 100\% with a median of 60\%. This implies that the majority of RNe likely have additional nova eruptions missed over the last century and that their real recurrence time scale is substantially shorter than the naive calculation based only on the discovered eruption dates.
(8) Five RNe (V394 CrA, V2487 Oph, U Sco, V3890 Sgr, and V745 Sco) have predicted dates of the next eruptions within the next decade. Other than for the case of U Sco (which should reliably go off in the next year or so), the uncertainty on these predictions is large, so the other four could go off any night now or not for two more decades.
(9) Five RNe (CI Aql, U Sco, V394 CrA, V745 Sco, and V3890 Sgr) have an identical pattern in their folded light curve that the maximum following the primary (deeper) minimum is significantly higher than the maximum following the secondary minimum.
(10) All RNe display the usual flickering on time scales from minutes-to-hours and with amplitudes up to half a magnitude.
(11) All RNe display surprisingly large amplitude variations on time scales of months-years-decades-century. This behavior has many types of morphology. Seven-out-of-nine RNe (with long-term light curves) have variations of greater than one magnitude on time scales of from 10-100 years. Four of these RNe (those with red giant companions) have a power density spectrum for these long time scales that might arise from the normal variations on red giants. T Pyx has a large-amplitude secular decline over the 119 years likely caused by a feedback mechanism where the supersoft luminosity of the white dwarf is fading. And I have no idea as to why U Sco and V394 CrA have variations on the time scale of decades.
(12) For fourteen eruptions with good pre-eruption light curves, zero showed a pre-eruption rise while one even showed a significant pre-eruption dip.
(13) The RNe do not show any consistent secular trends after the eruptions. Instead, some rise, some fall, some rise and fall, one stays constant, and several vary chaotically. This demonstrates that some other mechanism or mechanisms (in addition to the possible hibernation slow turn-off) are operating and are dominating the behavior in quiescence. With the existence of these apparently chaotic competing effects (which dominate over any trends from hibernation), it will be difficult to test the hibernation model by seeking slow post-eruption declines.
(14) The RNe are apparently divided with about half as a thick disk population (with a scale height of 380 pc) and half as a bulge population (within 3 kpc of the galactic center). This is the same as for CNe.
(15) The spectral energy distributions of the RNe can be divided into two groups. Four RNe have a prominent peak in the IR caused by their red giant companion star. Six of the RNe have the classic $f_{\nu} \propto \nu ^{1/3}$ as expected for accretion disks.
(16) The RNe in quiescence have a wide range of luminosities with $-4.1\leq M_V \leq 3.2$ mag. The four brightest are those with red giant companions. Of the remainder, all are dominated by the accretion disk (as demonstrated by the $\nu ^{1/3}$ spectral energy distribution), and these have a median absolute magnitude of 2.2 mag (and a full range of 0.0 to 3.2 mag). This is greatly brighter than CNe (with median absolute magnitude of 4.6 mag), likely because RNe must have a high accretion rate.
\section{Acknowledgments}
In any such paper as this, I have had help and used observations beyond those with citations in the text. The largest source of data is that from the dedicated people with the AAVSO. The many observers are quality astronomers producing a wonderfully useful data set. Indeed, the AAVSO data are so voluminous and useful that it now provides about 98\% of all RNe light curve measures. Without the heartfelt work of the amateurs, the whole topic of recurrent novae would be barren, both for the lack of known eruptions (only 11 known eruptions from 3 RNe would be now known without amateur work) and for the lack of data on those eruptions. I thank the many hundreds of amateur astronomers over the last century who have sacrificed their sleep with no immediate reward. The AAVSO organization provided many items of great use for this paper. In particular, I have made frequent and extensive use of the various finder charts, and the comparison star sequences have taken many magnitudes from AAVSO sequences, most measured by the current AAVSO Director Arne Henden. The second source of data and discoveries is the Harvard plate collection. Harvard plates provide the only data on 11 eruptions and the primary data on 4 more. Without Harvard data, the discovery of the RN phenomenon would have been delayed until 1946 and the fourth member of the class would have been found in 1987. The historical plate archives are still producing front-line science, yet they are under a variety of threats, and this is not acceptable. I would like to thank the many observers and curators who contributed to the Harvard collection over the last 120 years. Without the amateur nova searchers and the Harvard plate collection, we would even today not know about {\it any} RN. For my own observations, I have used a vast amount of `small' telescope time over the last 21 years, and this requires a wide variety of support. So I would like to thank the observers, allocation committees, night assistants, and managers of the SMARTS consortium, Cerro Tololo Inter-American Observatory, McDonald Observatory, and the ROTSE collaboration. In particular, Suzanne Tourtellotte, Charles Bailyn, Michelle Buxton, and Rebeccah Winnick provided valuable long-term close support. A large article like this one needs to have knowledgeable readers to help improve the text, and so I thank Arlo Landolt, Ron Webbink, Joe Patterson, and the anonymous referee for spending longer-than-normal time at checking the manuscript. The National Science Foundation and the National Aeronautics and Space Administration provided funds under grants AST-0708079 and 05-GALEX05-23.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,638 |
## THE CURVE
A NOVEL
## THE CURVE
A NOVEL
## JEREMY BLACHMAN
CAMERON STRACHER
JB: For Nina and Micah, hug-hug.
CS: For Adam and Erica, partners in crime.
## Contents
1 The Clock Strikes One
2 Love Me Tenure
3 The Priapic Professor
4 Lost in Translation
5 Where Jobs Go to Die
6 House of Cards
7 Point. Game. Match
8 Love and Other Drugs
9 A Kick in the Kidney Pie
10 Money for Nothing; Drinks for Free
11 If It Quacks Like a Duck
12 Cat Scratch Fever
13 Invasion of the Body Snatchers
14 The Bernie Madoff Bypass
15 A Dump with a View
16 Solitaire
17 Grades, Gold, and Steele
18 Dropping the F Bomb
19 All Men Are Socrates
20 Bologna Dreams
21 Does Never Work for You?
22 Saving Mrs. Palsgraf
23 Hail Mary
24 Down and Out in Gowanus
25 A Light Shines in Brooklyn
26 Snap, Crackle, and Pop
Acknowledgments
##
## 1
## The Clock Strikes One
AS THE CROW FLIES, IT'S less than three miles from Wall Street to the banks of the Gowanus Canal in Brooklyn. The last crow who tried it, however, got sucked into the jet engine of a 727 landing at LaGuardia Airport, and then spat in pieces into the murky depths of the water. Ever since, the crows have steered clear of Gowanus. Navigating flight paths is difficult enough; it's the indignity of being discarded into a Superfund site that really gets them.
But not everyone is as smart as a crow.
The gray brick building that teetered at the Eastern edge of the canal, near Twelfth Street, was filled with human beings who should have known better, but didn't. Its entrance marred by pigeon droppings, the building had been designed by an architectural student with one eye and no ruler. None of the walls met at right angles, and the ceilings drooped in the middle like pancakes weighed down with syrup. If the students who sat in those crooked rooms were more alert, they might have noticed the metaphor.
Instead, they were playing solitaire.
Not just solitaire. Minesweeper. Hearts. Online poker. Checking their fantasy baseball teams, watching movie trailers, browsing through profiles on OKCupid, Tinder, Hinge. One guy in the corner, sweatshirt, baggy jeans, downloading a pornographic movie so raunchy it was illegal in eight states. Not that he knew (or cared) about the statutory violation—even though that was the law he was supposed to be studying.
A girl in the back row was buying and selling stocks in real time, her screen filled with market tickers. A middle-aged man, a stripe of pale skin still visible where his wedding ring used to be, was on a Skype call, watching his daughter crawl for the first time, in a house he was still paying for, three time zones away—evidence of a former life, a dead end he thought law school would give him the tools to escape from. But all it was doing was adding $50,000 a year to his debt load. His baby girl knew just about as much law as he did—and, sadly, was a heck of a lot more employable.
None of this was a mystery to their professor, Adam Wright. He had been in their seats not much more than a decade earlier—albeit nicer seats, in a newer building, at a school with an endowment, and with a rational reason to believe his law degree wouldn't be worthless. The Internet connection was a little slower back then, the earphones a little more conspicuous, but he knew all his students' tricks. He could tell by the books on their desks whose pages never turned, the smiles that didn't fit the words he was saying, the way their eyes darted back and forth, back and forth. He did not have their attention. He merely had their attendance, hyperlinked from the absent recesses of their brains.
He sighed, and shifted his focus to the seating chart he had made the students fill out on the first day, in the hope that the threat of being called on would force them to do the reading and pay the slightest bit of attention. He thought it had worked—until he read some of the names on the chart. He was pretty sure the joker selling drugs via Craigslist in the third row wasn't going to answer to Donald Duck. And if the woman in the fourth row on the aisle, buying the drugs, was actually Lady Gaga, she'd clearly borrowed this morning's outfit from a boring, impoverished friend.
And then the sound of a cell phone, as if the entire second row playing first-person shooting games on their laptops wasn't insulting enough. Adam made a point to remind everyone to turn off their phones at the start of each class—but there had yet to be a class where he got through fifty minutes uninterrupted. He recognized the ring tone. What was that song? It reminded him of a younger, hipper period in his own life. But then he heard the lyrics and cringed: "Fuck tha police!" This was what a law student had chosen for a ring tone?
Adam watched as a young woman in the second row—olive skin, gleaming black hair—fumbled for her phone... and answered the call.
"Yo 'sup," she whispered. "Torts."
At least they respected him enough to whisper, he thought.
"Yeah, it's a total joke."
He tried to focus on the lecture notes in front of him.
"You know, the new guy with the Labradoodle hair."
He could stand it no longer. He searched his seating chart, praying she had written down a real name. "Ms... ah... Ms. Fortunato," he said.
"Fuck," she said, as all eyes turned to her. "Gotta go. Sext me later." She swiveled her phone back into its case, bejeweled with cubic zirconium. It fake-sparkled like the smile she gave him.
"Hey, Prof," she said.
"Could you please tell us the facts of Palsgraf v. Long Island Railroad?"
There was a slight gasp among the students, which may have been related to the question but, more likely, meant there was a blip in the Wi-Fi. But Adam pretended the gasp was for him, and that the students were actually, suddenly, afraid.
Although the practice of calling upon students was still fairly common in law schools, Adam hadn't set out to be one of those old-fashioned, rigid interrogators. He started out as part of a kinder, gentler breed of faculty, eschewing the Socratic method for something resembling a real, authentic dialogue about the law. He believed students learned very little from having their arguments shot down and being made to feel foolish. Instead he believed in coaxing the answers from them, and making them feel like partners in the search for fundamental principles like Duty, Care, and Causation.
But he was coming to realize no one ever did the reading. Leaving him no choice but to bully them all into submission.
"Uh, what page is that on?" Ms. Fortunato asked. "The red book, right?"
Staci Fortunato, said the seating chart. A marketing major at Syracuse. Adam imagined she'd skated by on her looks and charm. Probably never met an assignment she couldn't seduce a teacher to let her avoid. Unfair assumption, but Adam was suddenly cranky.
"Page one-thirteen," Adam said. "And, yes, the red book. The one with 'Torts' on the cover. Because this is Torts. Or at least it's supposed to be." Adam was surprised Staci had even acknowledged his question. Most students, if called on, would simply ignore their names. Yes, the professors had the chart in front of them—but would they really challenge a student who pretended not to be there? In fact, if "being" meant "present," they might not have even been pretending.
"I think I skipped that case. Maybe I have the wrong casebook? Mine's blue, and I don't see anything about a railroad."
Adam stared at her with what he hoped was a menacing look. "I'm sure it's there. Can you skim the case quickly now?"
"Like, here?" she said.
"Yes."
"It may take me a few minutes. I'm a slow reader."
A slow reader? Then don't waste our time. Call your mother and ask her to pick you up. Tell her you'll be living at home while you look for a different career. This is what Adam's own Torts professor might have said to him when he was a 1L at Harvard. But, of course, Adam didn't say it. Although he wanted to be one of those professors who struck fear in their students, who made them feel that if they didn't do the reading, they might as well not show up at all—the time and place had changed. When Adam was in school, professors still gave C's. Now, a C was the new F, and an A− the new B. At Manhattan Law School—which wasn't in Manhattan at all, but on the banks of the Gowanus Canal in Brooklyn—even the curve was curved. If a student got a C, it meant either he didn't show up for the final exam, or he'd been caught in a compromising position with the professor's golden retriever.
Adam looked at Staci, her wandering eyes and her lost soul. Her petite frame squeezed into a low-cut blouse, her hair and makeup perfect. Why was she here? What cultural forces had driven her to waste hundreds of thousands of dollars of her parents' money and sit in class as if she were waiting for a manicure at her favorite salon? Did she realize she wasn't a kid anymore, and was staring down the barrel of the rest of her life?
The truth was he felt bad for her. It wasn't her fault her parents had expectations. And it wasn't her fault the system let her slide for all these years. She tried her best on the LSAT, but the logic games were a mystery. Why would the girl in the puzzle have to sit next to Jacob, but avoid Kevin, Linus, and Mark? Why couldn't she just invite them all to the table for bottle service?
Adam held her gaze, and Staci's eyes flickered, then turned away. In that brief instant, however, Adam saw something: call it shame. It was just a spark, but he saw it before it was gone and it gave him hope. After all, wasn't that why he left his big corporate law firm and took a teaching job at this third-tier school? The glimmer of intellectual engagement; the possibility of making a difference in a student's life; the challenge of educating a new generation to succeed in a complex world. He didn't become a professor to use his pulpit to embarrass the weak and ill-informed. Instead, God help him, he had hoped to inspire and provoke them.
He looked up at the class and asked for a volunteer.
Daniel Hamburger's hand shot up in the front row. It had become a familiar, and rare, hand. He rattled off the facts and twisted procedural history of Palsgraf with confidence, speaking in complete paragraphs and even concluding with a critique of the majority holding. "Query," he said, "whether proximate cause is so attenuated as to preclude recovery for a set of events—although implausible—not impossible to foresee."
It was a command performance, earning Hamburger the envious stares of several classmates, and the rolled eyes of the rest of the class who, although relieved Adam wouldn't have to call on anyone else, wouldn't hesitate to push Daniel into the Gowanus Canal should the opportunity arise.
"Excellent answer," Adam said, as Hamburger beamed, the grin on his face just as bright as the guy in the back who was checking his fantasy baseball scores. Adam couldn't help but feel a tiny drop of pride in Hamburger's performance—a student who cared enough to do the reading, even if he was the kind of guy Adam had hated at Harvard. At least Daniel Hamburger gave Adam the feeling that all was not lost. Coupled with his sympathy for Staci, he thought that maybe—just maybe—he could turn around these students' academic futures.
Just then, the woman next to Hamburger leaned too far backwards and her chair slipped out from beneath her. Her arms windmilled, and she went down, taking her coffee, laptop, and a stack of papers with her. It was as if a small brown shrapnel bomb exploded as students on both sides jumped out of the way. In the commotion, a soft-covered book tumbled from Hamburger's desk to the floor, landing not far from Adam's feet.
It was a teacher's guide. The same one Adam used to prepare for class. The one he'd use to write the exam questions. The one that the casebook publisher assured him was never made available to students. Hamburger saw him looking and shrugged. The shrug seemed to say at least I tried.
Adam felt sick. This was not what he expected when he gave up the town cars, Manhattan co-op, and Hamptons share for the idealized life of an academic. But no one warned him teaching at a place like Manhattan Law School would be like this—disappointment, frustration, and a feeling of pointlessness nearly equal to his work at the law firm. In fact, when he was hired, no one had said anything about the teaching at all. Instead, the faculty committee asked about his scholarly intentions, reviewed his meager writings, and called a couple of the partners who'd supervised his work. No one bothered to see what he would be like standing in front of a classroom or told him what to expect. It turned out he was a prop. A babysitter. A dartboard on which the students flung their disdain.
But he wasn't prepared to fold. Not after only three weeks. Not yet. "Let's review the case as a class," he said, when the student picked herself up from the floor and the laughter died down. Adam grabbed a piece of chalk and turned to the old-fashioned blackboard. He rarely used the board—his penmanship showed it—but things couldn't possibly get any worse. He scribbled as quickly as he could, the chalk making a reassuring clacking sound and kicking up puffs of dust as the words took form. Even as the bell rang he powered on. "I'm going to ask you to stay in your seats for another minute or two," he said to the board. "I want to get the facts up here and give you a few questions to think about for when we pick it up tomorrow."
He raced to get it all written down, a concise diagram reducing the case to its simplest parts—the train in the station, the passenger rushing to catch it, the dropped package of fireworks, an explosion, pandemonium, and poor Mrs. Palsgraf struck by the falling clock. He looked at his creation for a moment, pleased that he could turn a disaster into a true teaching opportunity. He turned around, ready to call on anyone who could read. No advance preparation necessary.
But the room was completely empty.
## 2
## Love Me Tenure
THE METAL VAT IN THE faculty dining room steamed with "Golden Delicious Chicken," or so claimed the placard on the rickety folding table. In truth, its color was closer to lead than gold. As for its taste, no chickens had been injured in its preparation. But at least it was pasteurized. The faculty insisted on this after an outbreak of giardiasis traced back to the "homemade" muffins served in the student cafeteria. Now, although the food was mostly tasteless, it wouldn't kill them. Not today, anyway.
Laura Stapleton lifted the serving spoon and stifled a grimace. She had seen legal briefs that were more appetizing. If she had a choice, she wouldn't be eating here. Not that there were a lot of other options in the wasteland that bordered the land of waste on which the campus sat. But anything would have been better than the tasteless food and colorless company. Face time, however, was everything—especially when most faculty rarely showed their faces outside of scheduled classes, hiding from students in their locked offices or home in their apartments. If Laura wanted to advance up the academic ladder—better committee appointments, fewer teaching responsibilities, more money for research assistants—she would have to eat and greet in the dining room.
As the only woman of color on the faculty, Laura was both constantly celebrated and consistently underestimated. She was trotted out for every candidate interview, accreditation meeting, and panel discussion involving anything linked to anyone of ethnicity. At alumni events she was positioned front and center. Her picture was photoshopped into most of the brochures released by the marketing department, and placed on virtually every page of the law school's website. On the other hand, her scholarship was viewed with suspicion and, occasionally, disdain. Her subject matter—civil procedure—was mainstream, doctrinal, and, to the disappointment of many on the faculty, entirely unrelated to her race. She had yet to propose a class on Racism and the Law, the Civil Rights movement, or the Voting Rights Act, and had not written anything titled, "The Founding Fathers of Bigotry: A Race-Based Interpretation of Marbury v. Madison." This was a huge frustration to the Promotions and Tenure Committee, who would have loved a good reason to reject her appointment when she came up for a vote. Instead, she wrote articles on perfectly normal topics, cited by an appropriate number of journals and judges, and her book on the ascendancy of the federal judiciary was published by Yale University Press and nominated for a National Book Award. So the Old Guard had no choice but to grant her tenure.
Laura took her tray and looked for a safe place to eat. Old Man Porter sat at his usual table in the corner, gumming his mashed potatoes, having lost his teeth somewhere between his home in Queens and the International Tax elective where he taught three students about new regulations in East Germany, Czechoslovakia, and the Ottoman Empire. Rochelle Tucker sat over by the window, reapplying her lipstick and adjusting her brassiere. She was gunning for tenure any way she could, and she didn't care who knew it. Rumor was she had even paid Old Man Porter a call—during visiting hours, of course—but no one wanted to think too hard about that image, let alone investigate the truth. At the large table in the middle of the room—the only one with a floral centerpiece—sat the school's Associate Dean and two of his faculty henchmen, along with the Chief Marketing Officer, who had managed to lose the rights to the law school's domain name to a porn site. Now the new website sat, uninvitingly, at seemanhattanlawschool.com. (It had been effectively parodied in the student newspaper as "Semen Hat and Law School." Sadly, this was now how most students and faculty remembered the address.)
Steering wide of the Dean, Laura sat at the table near the (broken) fire exit with some of the other junior faculty—second-class citizens in the professorial pecking order. They, at least, were tenured, or tenure-track. The non-tenure track faculty—contract employees and adjuncts—didn't even get a table, or chairs; they took their lunch to go in recycled Chinese food containers that still bore a faint trace of moo goo gai pan, and ate in the stairwells. As the conversation swirled around her, Laura poked at the shreds of chicken on her plate. After a couple of bites, however, she discreetly spat the unchewable parts into a napkin and gave up. She sipped instead from her bottle of Perrier, which, untouched by the food service subcontractors, at least had some effervescence.
"Mind if I join you?" a voice asked.
She looked up to see one of the new faculty hires hovering with a plate stacked high with the crusty ends of the chicken. He was tall and lanky, with floppy brown hair that he really should have had cut. Laura cleared her purse from the seat to her left.
"All yours," she said, and resumed fake chewing.
"Adam Wright," he said, extending a hand.
"I know who you are."
"You're Laura Stapleton."
"I know that, too."
If Wright was offended, he didn't show it. He smiled and sat beside her, then asked her to pass the hot sauce.
"Just finished teaching?" she asked, softening her tone. She regretted being sharp with him. He seemed harmless and inoffensive. One of a steady stream of young faculty who would last a few years, then be passed over for tenure. Barely anyone made it, and the number had only dwindled in recent years. The tenure system protected the perks and privileges of an entrenched oligarchy, and kept a younger generation from challenging the status quo. Laura was an exception to a jealously guarded rule.
"I guess you could call it teaching," said Wright. "Pleading is more like it." He picked up the bottle of hot sauce, noticed the expiration date, and put it back on the table. "Thanks for letting me join you."
Laura nodded. "Why not? We're on the same team."
Before Wright could respond, they were distracted by a shout coming from the center table. "I'm telling you," Prof. David Wheeler had said, loud enough for the whole room to hear, "it is not even a debate. The Emancipation Proclamation was a complete bust." Laura felt Wright looking at her, while a silence fell over the room. It was, Laura knew, exactly the reaction Wheeler hoped to provoke.
Wheeler taught Constitutional Law and had been named one of New York magazine's 100 Most Influential Lawyers in 1983, an honor he still had taped to his office door. He also ran the school's mostly ignored program in "distance learning," Web-enabled course offerings that recycled videotaped lectures for the Internet, packaging them with the promise of an "online J.D.," an unaccredited degree that was worthless yet cost $1,500 per credit hour. He was a big man whose hair grew entirely on the lower half of his head. Now, he called over to Laura. "What do you think, Professor Stapleton? Did the slaves get a bum deal?"
Laura refused to take the bait. Like all bullies, Wheeler was best ignored. He took a perverse glee in tormenting her to compensate for his extremely small manhood—not that Laura had personal knowledge, but she knew the type.
"Emancipation is overrated, don't you agree?" Wheeler's voice could crush glass.
"Laura, come join us," said Associate Dean Jasper Jeffries. Although Erwin Clopp was officially the law school's dean, Jeffries was the true power behind the throne. He was everything Clopp wasn't, physically, mentally, and otherwise. Muscular, bald, and clean-shaven, Jeffries could pass for a mascot on a bottle of cleaning fluid. Clopp, on the other hand, with his white hair, white beard, and mellifluous voice, looked and sounded like Burl Ives in Rudolph the Red-Nosed Reindeer. Jeffries made all the decisions while Clopp was rolled out for faculty and board meetings.
Laura glanced at Wright. She didn't want to be rude, but she couldn't ignore a direct invitation from Jeffries. The other junior faculty members at the table had resumed their conversation, studiously ignoring Laura and Jeffries for fear of being singled out.
Jeffries saved her from the Hobson's choice by adding, "Bring the doofus."
"I think he means you," said Laura.
"Charming," said Wright.
"It's a term of endearment," Laura assured him. "You should hear some of his insults."
She stood and waited for Wright, then took her tray and moved to the table in the middle of the room. Wright followed, squeezing into a chair between her and Rick Rodriguez, faculty advisor to the Law Review. A former Supreme Court clerk, Rodriguez was a rabid Republican and staunch immigration foe—and extremely, almost preposterously, gay. He was trailed constantly by a coterie of young, handsome students he was advising, although there was talk that some of these young men weren't even enrolled at the school. But no one bothered to check. As long as they didn't cost the school any money, no one cared who Rodriguez kept in his closet, or what they did there.
Laura pretended to listen to Wheeler pontificate. She snapped to attention, however, when Jeffries laid a thick finger on her forearm. She turned just in time to see the woman from marketing—who may have had a name, but Laura couldn't recall it—glaring at her. Was she jealous? Impossible. And yet that would explain a lot of things, like why Laura's name was misspelled in the alumni magazine and why her photo on the website was a hideous shot taken without her knowledge when she was sick with the flu.
"Laura, my dear," said Jeffries. "How is the book coming?"
Laura was working on a follow-up to her book on the federal judiciary. Although it earned her the enmity of her colleagues, her first book made her the most-recognized scholar at the law school. It helped that publication coincided with a bruising Supreme Court nomination battle, which made Laura a minor celebrity on NPR. It also gave her a certain amount of clout, which she was loath to exercise, but quietly wielded during the tenure process when Jeffries questioned her "administrative responsibilities."
"Truthfully, I've barely started."
"That's a shame," said Jeffries, as he fingered her arm. "A brilliant and beautiful mind like yours should be focusing on her scholarship."
"It's been so busy with the new semester."
He lowered his voice. "You would have plenty of time for your writing if you had accepted our proposal."
"I know," said Laura, being careful with her words. "Unfortunately, my answer is the same."
"I thought recent events might have changed your feelings."
"What events?"
"Your living situation."
"Excuse me?"
"That lovely building you live in. It's going co-op. I assumed you'd heard. It would be a shame if you couldn't afford to buy your apartment. All that time and effort you've put into fixing it up."
For the last ten years, Laura had rented a beautiful apartment in Park Slope that she could afford because the landlord loved her and was too lazy to raise the rent. But if it were true that it was on the market, even at an insider price it would be beyond her means. She hoped Jeffries was wrong, although she suspected he probably wasn't.
"Then I better start saving my nickels."
"You'll need more than nickels."
"I appreciate the concern, Jasper, really. But I think I'll manage."
"It's your decision. Although I must say I remain mystified."
"It's not that mysterious. I'm just old-fashioned that way."
"I've always admired that about you." Jeffries's finger inched along her arm like an oily caterpillar. "But remember, we're in this together. Sink or swim. One big happy family."
The three men were not her family, Laura thought, and she would be damned to sink with them. But she didn't respond. She caught Adam observing her, his brown eyes like question marks, dark and uncertain. A spasm of guilt racked her insides—or maybe it was the food. She clenched her jaw until the feeling passed, but it left her hollow and shaken, and the taste in her mouth was like something had gone there to die.
## 3
## The Priapic Professor
A WEEK HAD PASSED SINCE ADAM sat next to Laura in the faculty dining room. A week during which Adam spent his time plotting, scheming, and hoping to accidentally run into her again, without success. With their offices on separate floors, it was a challenge to engineer a random encounter, and apparently one meal per week was the limit of Laura's appetite for culinary danger. No matter how long he lingered at the lunch table he didn't see her. He was supposed to be working on an article that would keep him employed, but instead he spent his time trying to map out her most likely route to class. He even ventured into the faculty reading room where he found Old Man Porter asleep in a puddle of drool. But no Laura.
At most schools, he could have merely waited in the lobby for her to pass. Indeed, during the cash-flush 1990s, Manhattan Law School had spent $7 million constructing a magnificent main entrance atrium—cathedral ceilings, imported wood, great legal quotations from history etched into the walls (some without errors)—but the ventilation ducts had been accidentally reversed and blew exhaust from the street into the atrium. As a result, even though it was the only entrance to the school, it was impossible to wait there for more than a few minutes without being overcome by carbon monoxide fumes and the ever-present stink from the canal. The security guards wore masks and worked in short shifts, while everyone else held his nose and walked through briskly.
Adam tried to sit in the atrium each morning and casually read the newspaper at one of the old Apple IIe computer terminals. But his vision blurred before the modem finished dialing up the server, and he found himself taking regular walks around the building, where the particulate concentration was not as severe. On his fifth day of circling, when he had almost given up on seeing her again, he nearly ran into Laura as she raced around the corner, a cup of coffee in her hand. Adam's hands were stuffed in his jacket pockets, and his head was down as he gulped fresh air like a drowning victim.
"Adam?" she asked, as he narrowly missed colliding with her. "Are you okay? You seem like you're... lost."
"Oh, no, I just come out here for the fresh air sometimes," he replied, flustered. "My office gets a little claustrophobic."
"I can imagine," Laura said. "I was in a closet my first two years. People kept coming in looking for paper clips."
Adam laughed. "Last week the janitor asked me for cleaning supplies."
"Well, don't stay outside too long. You might come back to find him sleeping in your chair."
"I was just heading in. You?"
"Teaching at ten-thirty."
"We should grab lunch sometime," Adam blurted before he lost his nerve. "We never finished our conversation."
"That's true, we didn't," said Laura. "Give my assistant a call and we can set something up."
And that was how he and Laura ended up in a Carroll Gardens coffee shop, the kind of place with exposed brick, bearded hipsters, and overpriced milky beverages. Laura looked radiant as always, in a yellow blouse and black trousers that cinched high at the waist, while he felt slovenly and underdressed. It was warm outside, and his pants itched uncomfortably. He scratched at his legs, then rubbed one foot against his other ankle like a praying mantis. His hair felt suddenly too big for his head, and his skin felt sticky and loose.
Laura didn't seem to notice his awkwardness. She stirred three spoons of sugar into her coffee and shrugged when she saw him watching. "I have a sweet tooth," she explained.
"I'm not judging."
"No, it's a serious character flaw. Goes right to my hips."
"You can't possibly tell."
"Oh, I can tell."
"Well, we all need a good character flaw. It's like a well-placed mole. Accentuates the positive."
"I have others."
"Moles?"
"Character flaws."
"Let me guess," said Adam. "You hate children."
"I love children! As long as they're not mine."
"Dogs, then."
"I love dogs. I've got a retriever mutt."
"You put empty ice cream cartons back into the freezer."
"I hate people who do that."
"I'm stumped, then. What else could be wrong with you?"
Laura's eyes flicked toward the fire exit. She took a healthy sip of her coffee. Then she asked, "Do you like it here?"
"In this café?" Adam replied, a little thrown.
"No, I mean at MLS. It's not exactly Harvard."
When Adam first announced his career change, friends told him how much they envied him. After all, everyone in the rough-and-tumble corporate world was stretched too thin and too far, overworked, underpaid, unappreciated, and generally miserable. The work, at best, was amoral; at its worst, it was positively corrupting. The academic life seemed like the idyllic antidote, despite the pay cut. Only his brother, Sam, understood the complicated motives and emotions, how the choices their father made years ago cast a shadow over nearly every aspect of Adam's life.
"The students are a mixed bunch," Adam said carefully.
"What was your first clue?" Laura cocked an eyebrow, a neat party trick.
He allowed himself a smile. "I had a student take a phone call in Torts the other day. Then she emailed me to ask if I could send her the reading. I couldn't even come up with a response that wouldn't get me fired."
"That's nothing. Two students were suspended last year for having sex in class."
"Sex?"
"S-E-X."
"Like actual intercourse?"
"Well, no, I don't think so. We didn't get into the details. There was digital manipulation, shall we say. And they had to replace the chair."
"Wow. At least back in the day we snuck behind the library stacks to make out."
"Professor! I'm shocked."
"Same old, I suppose. Every generation complains about the next. Soon I'll be saying, 'That's not music; it's noise.'"
"I already do."
They shared a chuckle, then Adam said, "But students really are different today."
"They wear less, for one thing."
"True. And all their devices—you can't get their attention. And if you do, there's this sense of entitlement, like what have you done for me lately?"
"It's the customer service version of higher education."
"The consumer is always right."
"Exactly."
"For two hundred thousand dollars in debt, they want something for their money. Even if it makes us feel like short-order cooks."
"I suppose I don't blame them, but someone should teach them not to email their professors and ask for the reading material. What will they do when they have jobs?"
"They don't get jobs."
"Is it really that bad?"
Laura nodded. "It's bad. No one wants to tell our students the truth."
"Well, it's not as if their eyes aren't open."
"True, but we still don't level with them."
Adam had seen the glossy brochure and marketing video produced for the school at great expense by some NoHo PR firm. Reviewing the materials, you would think Manhattan Law School students had perfect hair, skin, and teeth, and liked nothing better than to read The Federalist Papers while paddle-boarding in the Central Park reservoir. Adam knew the reservoir was really a sound stage in Queens, and the "students" were unemployed actors, but he wondered if the real students—or, more importantly, their parents—did.
"You think it's different at other schools?"
"I don't know," said Laura. "I'll report back to you soon. I've been talking to the Dean at Berkeley."
"UC Berkeley—in San Francisco?"
"There's another Berkeley?"
"No, I mean—that's a serious school. Top ten. Congratulations." Adam tried to keep a straight face, but he felt like he had been sucker punched. They just met, and now she was leaving? Whoa, slow down, he told himself. No one is going anywhere today. Drink your coffee. Breathe. Stalking a woman he barely knew was surely not the best way to date her.
"The thing is, I'm not sure I want to move."
"Why would you stay if you could teach at Berkeley?"
"I'm a New Yorker. Born and raised here, and lived here all my life. Besides, I'm comfortable at MLS. Sure, there are problems, but at least I know where the bodies are buried."
"There are bodies?"
"You know what I mean. No institution is perfect. In the end, you live with the imperfections you can handle."
Adam understood, better than anyone, a life with imperfections—banging on the windows to get inside. But sometimes you couldn't keep them out; sometimes, they crawled up underneath and then it was them, or you.
Laura glanced at her watch, which Adam took as a bad sign. "Are we late?" he asked.
"I'm sorry," she said. "Directed Legal Studies. I should probably be on time, even though my students aren't."
"How'd you end up teaching that class anyway?"
"It's a long story. The short answer is I didn't get to choose. Call it a form of penance."
"Did they lighten your teaching load?"
"Are you kidding? I don't even get paid for it."
The school's administration was notoriously cheap, which Adam knew from his negotiations over salary with Dean Clopp, who offered him a new lamp and a box of staples instead of the additional $5,000 he requested. Yet law school was also a big business, a money-making machine that produced steady returns. At sixty grand a pop, and 400 students per class, MLS took in $72 million in annual tuition revenue alone. Even with the drum-beat of bad news about jobs and the legal profession generally, applications still exceeded available places by a margin of three to one. Given its low-budget location and lack of amenities, the school was like a corporate raider's plum waiting to be picked.
Laura cleared their cups and the empty packets of sugar. "Should we walk back to school together?" she asked.
"If you think you can be seen with me," said Adam.
"I'll take the chance."
It was unusually warm for the end of September. Laura carried her jacket over her shoulder and Adam was in his shirt sleeves. The streets of Carroll Gardens were tree-lined and flowered, and bustled with delis, markets, restaurants, and bars. Adam felt a sense of peace, a serenity that comes in the presence of a kindred spirit. Maybe he was seeing the world through Laura-colored lenses, but it had been a long time since he met anyone who stirred his soul, and even longer since his last real relationship. Jane Van Dyke had been his law school girlfriend, and they continued to date for a time after graduation. But it was the same old story: she lived uptown; he lived downtown; and then she cheated on him with her boss. After that, he hunkered down and focused his excess testosterone on the law.
Was it too much to hope that Laura shared his feelings? He was relatively attractive, clean, and had all his teeth. Why could it not work out between them? He knew he was getting ahead of himself, but he couldn't help it. He felt like a schoolboy with a crush on the prettiest girl in town. Here she was walking next to him, lithe and loose-limbed, while the breeze rippled the leaves on a sun-dappled afternoon and blew his cares into the far reaches of Queens.
Laura grabbed his elbow as they stepped around a particularly treacherous pothole and held it one second longer than absolutely necessary. Her touch rippled down his arm into the tips of his fingers. He stole a quick glance and saw her smile, a flash of white between red lips, and that was enough to give him hope.
They turned the corner, talking about nothing consequential, and ran into a student Laura knew. He was thin and furtive, and wore a wool cap, although it was sixty-five degrees. Laura introduced him, but Adam didn't catch his name. The boy looked anxious and preoccupied, and Laura soon released him.
"He's going to be late for my tutorial," she said.
"Where's he going?"
"I don't know—to get a coffee?"
"Doesn't look like he needs any."
She shook her head. "Some of these kids are lost souls. I don't know what they're doing in law school in the first place."
"Where else would they go?"
"Prison?"
Adam looked back after the boy, but he was gone.
A knot of students smoking cigarettes lingered outside the school's entrance. Adam envied their careless indifference to health and breath. It was illegal to smoke near a school, but the students puffed away, defiant and proud. Once there were protests against apartheid; now there was smoking.
Adam and Laura made their way through the smokers and entered the atrium. It was foul-smelling and hot, but Laura did not seem anxious to leave. Students streamed past, backpacks like heavy weaponry. A herd of young men in baseball caps were shouting about last night's football game. A group of young women with belly rings and tattoos laughed uproariously at something one of them said. Everyone was texting, talking, walking, posting, snapping, chatting. A stranger couldn't have guessed this was an academic institution, where the next generation of lawmakers and judges were being trained.
"I'd like to ask you a favor," said Laura.
Adam stopped, then started. "Okay," he said.
Laura looked away. A small bird had flown into the atrium and was fluttering around in confusion. It banged against a window as if it expected it to open, then flew back up to a ledge where it rested before making another attempt.
"Don't say anything about Berkeley," said Laura.
"Of course not!"
"And if you know anyone there, they don't need to know what goes on here, either."
"What goes on here?" Adam asked, confused. He scrutinized her face for a clue, but her brown eyes reflected no light, just the flatness of someone who doesn't want to be asked more questions.
"Exactly." She pursed her lips and nodded, then held out her hand to shake goodbye. It felt awkward to shake hands as if they were strangers or business acquaintances, but the hand was waiting there and Adam took it. Laura shook briskly, her palm warm but not inviting. Her smile polite but demure. Then she turned, disappeared into a throng of students, and was gone. Adam slowly made his way in the opposite direction, betwixt, befuddled, and not a little bit priapic.
## 4
## Lost in Translation
GARY DERANGER DID NOT SET out to be a stalker.
In the beginning, he was simply looking for signs of regret. A frown, a faraway glance, a quivering lip. If he'd learned one thing in law school—and he hadn't learned much more—it was that intent was everything. There was no crime without a guilty mind. But when her ISP started spitting back his emails, and she blocked him on Twitter and Gchat, unfriended him on Facebook, and changed her cell phone number, he really had no other choice. She had broken up with him in April, right before exams, claiming she needed to concentrate on studying for finals. Then she disappeared from all the places he used to find her: the student lounge, the Starbucks they frequented, the women's bathroom next to the shuttered library. He spent the rest of the spring and summer in a mounting frenzy of longing, searching for her like a hypercaffeinated paparazzo (Starbucks made him buy something if he was going to sit in there all day). He camped out in front of her building until some local hoodlums tried to set him on fire. Then he spent thirty-two hours straight at LaGuardia Airport when he heard a rumor she was flying home to visit her parents. His repeated encounters with the Port Authority police landed him on the "No Fly" list, even though he hadn't flown since his first panic attack eight years ago. The Administrative Law judge who heard his appeal told him he had crossed the line, and warned him to leave Ann Marie alone.
By the time school began again in the fall, she had dropped the classes they had in common and seemed to know exactly where he'd be waiting in the halls. On the rare occasion he was able to get close, her friends intervened with a well-timed shove or a shot of pepper spray. She traveled in a pack like the Queen of England or a major party presidential candidate. There was always a giant lug beside her—a thick-skulled future prosecutor—willing to take a bullet. Gary's only option had been to follow her to class and peer through the shatterproof windows while she unpacked her books and laptop and took her seat in the front of the room. He was always an outsider, looking in.
On this day, as the students streamed into the classroom, jostling him as he stood by the door, he watched her chatting with the girl on her left, a brunette in a bustier, the kind of girl who might normally catch his attention if she had not been sitting next to Ann Marie. As it was, he could no longer even remember the names of most of his classmates. They morphed into a giant faceless blur, indistinguishable in their tank tops and cargo pants, tattoos and hair gel, bleached teeth, and depilated follicles.
That Gary was still in school was something of a small miracle. He had passed his second semester classes by the skin of his teeth, with two C's, one C−, and a D. As a result, he was forcibly enrolled in the law school's remedial program, something euphemistically called "Directed Legal Studies," or "DLS," but more commonly known as "Law School for Dummies" or "LSD." This required him to attend special workshops focused on test-taking and essay writing, and a course devoted entirely to basic English grammar. The school's ostensible purpose was to help him raise his grades, but really it was to keep him paying tuition. Every student lost was a $60,000 check down the drain.
Not only did the DLS program keep them in, but the school structured it so that a DLS student couldn't fulfill all of his requirements in just three years. Sure, Gary could quit. But quitting meant his already-spent tuition would have gone to waste, something he vaguely remembered was called "sunk cost." Even more important, quitting meant losing Ann Marie forever. And this, more than anything, was the reason he continued to read and outline for his classes—barely enough to follow his professors' lectures, but sufficient, he hoped, to stay in DLS.
If Ann Marie saw him peering through the window, she didn't acknowledge it. Not that it was easy to see through the glass, caked with years of law student spittle. The girl in the bustier continued to talk to Ann Marie as she pulled her books out of her bag. The class had filled, and nearly half the seats were occupied. Students had their computers open in front of them, already Tindering for potential dates or browsing the day's deals on Scoutmob. Two students were furiously finishing a game of Words With Friends. Gary stood on his tiptoes, mentally willing Ann Marie to look his way, until he felt a meaty hand on his shoulder, uncomfortably heavy, flesh swollen and loose, attached to a large, ursine creature.
"Are you in this class?" Professor Wheeler asked, his breath like a dead squirrel.
"No, I... uh... I," said Gary.
"And the reason you are blocking the egress is...?"
"Uh, no reason."
"Then please find no reason somewhere else." Wheeler stepped past him with a sneer, leaving Gary wobbling in the door frame. The class stared out at him, and for one instant he caught Ann Marie's eyes before they dropped to her keyboard.
But that was enough. She had seen him! Never mind that he had been humiliated; it was worth the price of embarrassment for just one look directly into those velvet blue eyes, so dark they could be black. Gary remembered how they used to stare up at him as they made love, slowly closing as Ann Marie's excitement mounted, then fluttering awake in surprise.
Wheeler had already begun his lecture. A student in the front row squirmed nervously as Wheeler peppered him with questions about the day's case, a monster Con Law decision that had radically changed the world just before the casebook was written in 1942. Even through the clouded glass, Gary could see the sickly expression on the poor student's face and the beads of sweat that stood out on his forehead. Constitutional Law was a required course for second year students, even though it was the least likely subject any of them would encounter in their careers (those who were lucky enough to have careers). It was like teaching plumbers about the molecular structure of H2O. Interesting work if you could get it, but worthless when there was a leak.
Gary considered waiting for the class to end in the hope of catching Ann Marie again. But he was already late for his tutorial, and had been warned that two absences would mean an additional "administrative fee" in order to continue participating. He was in enough debt that it would take three hundred fifty-four years of minimum payments to pay it off. He took one last lingering squint through the glass, then turned and headed for the elevator. The police tape across the doors reminded him that they were out of service. Panicked, he raced the other way to the stairs. If he wanted to be on time, he would have to sprint.
Although the law school spanned ten floors in one large building, the elevators were temperamental, prone to mechanical failure, and, prior to the tape, had caused no less than a dozen students to miss final exams, class-dropping deadlines, or days of sleep. Last week a librarian had been trapped for eight hours until school security noticed the alarm had been pulled and quickly finished lunch in order to go and rescue her. They found her chewing pages from Powell's treatise on Property, half-crazed from glue and the Rule Against Perpetuities. Now everyone except the fully tenured took the stairs. (Their elevator, the school's lone functioning one, was no luxury ride either, not since a queen hornet flew in and built her nest there.) So the stairwells were overwhelmingly crowded with students, staff, and the members of the faculty unfortunate enough to lack both job security and health insurance, all trudging up and down, most too out of shape to be in a stairwell in the first place, moving slowly, panting, resting, taking a break for a snack.
Gary followed a young woman with a tattoo of a fox (or possibly a squirrel) just below her waistline. As he leaned in to examine it closely, he nearly tripped on the feet of a man he assumed was one of the school's "nontraditional" students: old people who had lost their jobs and believed that saddling themselves with additional debt would save them. Gary yelped, frightening the tattooed woman, who escaped quickly into the herd.
"Sorry," said the man.
"Fuck!" said Gary.
"Excuse me?" The man seemed offended, as if it were Gary who had tripped him, and not the other way around. But now that Gary's eyes refocused, he realized he had seen the man before—it was the new Torts guy, Professor Wright, who had been outside walking with Professor Stapleton.
"Tourette's," Gary explained, faking a spasm and a series of eye tics. Then, spotting a pair of fire doors above them, he quickly pushed past Professor Wright, hoping he wouldn't be followed. When he got out in the hallway, he ducked into a bathroom stall where he waited a good ten minutes before safely (and uneventfully) resuming his trudge upstairs.
Professor Stapleton's office was on the eighth floor, near the faculty dining room. The hallway smelled of sauerkraut, rotten eggs, mold, and wet dog. It was a faculty smell, familiar to Gary from too many closed door meetings with associate deans after his first semester when the law school started talking to him about the Directed Legal Studies program, issuing him a glossy brochure he could show his parents in case they wondered where his tuition money was going and why it seemed like he would never actually earn his degree. Prof. Stapleton's tutorial met in a small conference room two doors down from her office. When Gary arrived, the other four students in the tutorial were already there. They avoided looking at him, as men condemned to death often avoid looking at their fellow inmates in the hope of eluding their fate.
"Welcome, Mr. Deranger," said Professor Stapleton. "Glad you could join us."
"Sorry I'm late," said Gary. "I, uh, I was talking to Professor Wheeler, and then Professor Wright."
Professor Stapleton regarded him quizzically, but if she thought Gary was lying she didn't call him on it.
The class was conducted in workshop format. The students wrote short assignments each week, and then, led by Professor Stapleton, reviewed each other's work. Gary imagined the class might have been helpful if he could actually concentrate. He liked Professor Stapleton. Besides dressing better than most professors—always wearing both a top and a bottom—she seemed genuinely interested in helping him. Too bad Gary spent most of the tutorial scribbling pictures of Ann Marie in the margins of the other students' papers.
It was not as if he hadn't had other girlfriends. And it was not as if Ann Marie was the first woman to have broken his heart. She wasn't even the first to call the cops—that honor went to his high school prom date, who flagged down a policeman when Gary refused to stop dancing. On top of her car. While she was driving. On the highway. But Ann Marie had been his anchor during a difficult first year, when his self-esteem and confidence were attacked by professors bent on making him feel stupid and impotent. Each day they chipped away at his façade, and each day his ego cracked a little more. His hold on reality, never strong to begin with, began to fracture. His very self began to split. He went from being shaky but secured to unmoored and dangerously unbalanced.
Now he forced himself to focus on the faces of the students talking. He saw lips moving, eyes raised. There were three other men and one young woman, the woman vaguely attractive in a Staten Island kind of way: wavy black hair, olive skin, plump arms exposed in a halter top. The words she was speaking sounded to Gary like "blurbedy, blurbedy, blook." He nodded and smiled encouragingly, then scribbled some more pictures of Ann Marie.
The class might have lasted an hour, or maybe it was only twenty minutes. Gary knew in theory it occupied a seventy-five minute block on his schedule, but time was just another subjective measurement like space or sanity. It seemed like mere seconds before Professor Stapleton was giving them their assignments for the following week, although Gary saw his notebook was thick with scribblings and blunt pen marks like fence posts.
As the students were packing their belongings, Professor Stapleton asked Gary to stay. He felt his neck burn while his classmates picked up their books and rushed off to their next courses. He knew the news wouldn't be good, and they did, too.
When the room emptied, Professor Stapleton sat in the chair opposite him, so close he could see a chip in her white incisor and smell her lilac-scented shampoo.
"I'm worried about you, Gary," she began.
"Me?" Gary asked.
"Your writing assignments are marginal, at best. You barely even completed last week's drafting exercise. And I know those aren't notes you're scribbling on your classmates' papers."
Gary flushed again, then looked down at his fingernails. Boy, were they dirty. When was the last time he had washed his hands? When was the last time he showered? He was suddenly conscious of a bitter smell rising up from his armpits, and was embarrassed to think Professor Stapleton probably smelled it, too. He squeezed his arms close to his body in the hope of trapping the odor before it escaped, or at least muting it enough so it could merge with the cafeteria odors.
"Is something going on?"
"Going on?" Gary repeated.
"Something at home? I mean, with your friends, or a girlfriend, or something?"
Gary looked up at Professor Stapleton. She was old—at least thirty—but except for some crow's feet around her eyes, her skin was smooth and unlined. In seventeen years of schooling, Gary never had an African American professor before, and he found his thinking muddled by his parents' prejudices. Although they were semi-educated people, his father believed the Nation of Islam blew up the World Trade Center, while his mother insisted it was the Jews.
Now, as Professor Stapleton waited for an answer, Gary wondered what his father would think if he ever dated a black woman, and then his mind wandered to what it would be like to date Professor Stapleton, or any professor. To someone who hadn't been touched by another soul in months, even Professor Wheeler's firm grip earlier had been strangely appealing.
"Everything's fine," he said. "How's everything with you?"
"Me?" Professor Stapleton arched an eyebrow.
"With teaching and everything." Gary shrugged. "I mean being the only black professor?"
Professor Stapleton raised the other eyebrow, and for a moment Gary wondered whether he had gone too far. Then she said, "I appreciate your concern, Gary. But we're not here to talk about my problems. We're here to talk about yours."
"Well," Gary took a breath, and something slipped inside of him. "I have been a little distracted. My girlfriend and I broke up."
Professor Stapleton nodded. Gary continued. "But everything's fine now. We're friends and all. It's just hard sometimes."
"Have you talked to anybody about it?"
"Talked? Like to a therapist?"
"Or to the school counselor? He's in his office most days until noon."
"No. I've just talked to my friends." In fact, Professor Stapleton was the first person to whom he had admitted that Ann Marie and he had broken up. Although he suspected his parents sensed something was amiss, every time they asked about Ann Marie he invented some story about a party they had been to or a cute south Brooklyn restaurant they discovered—authentic Vietnamese food served by gun-toting three-year-olds, for instance. It played into everything his parents imagined Brooklyn life was like. As for friends, he had none.
"If you ever need someone else to talk to," said Professor Stapleton, "I'm always here."
For a minute, Gary felt like crying. He wanted to lay his head on Professor Stapleton's lap and let himself go. He was so tired. At night his sleep was punctuated by staccato dreams that left him disoriented and exhausted. During the day he was assaulted in class by a teaching style that resembled machine gun fire. But he still had enough of a grasp on reality to know that a student should not lay his head on the lap of a professor, not if he wanted to remain in school and out of a mental institution. His instinct for self-preservation won out against his fundamental craziness, and he bit his lip to stop from crying.
"I appreciate that," said Gary. He stood to leave.
"Gary?" She stopped him.
"Yes?"
"Your work. It has got to get better."
"I know."
Professor Stapleton held his eyes, and he grew mesmerized by their complicated shades of brown speckled with green. If he squinted, they appeared to be slowly rotating like a kaleidoscope. The feeling was not unlike the first time he had smoked marijuana in college: everything spinning and slightly beyond his reach. Back then, he had laughed with his roommates at how easily a beer bottle evaded their hands, how difficult it was to navigate the doorway. Now he grasped the side of his chair to steady himself.
"Then buckle down, do the reading, and put some effort into your writing—at least use spell-check. Otherwise, Gary, you're wasting both of our time."
Gary nodded and stood still, until he realized Professor Stapleton was holding open the door and it was time to leave. Then his feet carried him out of the office and down the hallway where the smell of wet dog wrapped him in a shroud. Most days, Gary felt like a wet dog himself. And looked like one, too.
## 5
## Where Jobs Go to Die
THE STUDENT CAFÉ WAS NO more than a dirty corner on the law school's second floor, with a coffee urn and some week-old pastries wrapped in plastic. "Made in Brooklyn," claimed the lettering on the wrapping, which gave everyone the warm fuzzies—and later, the cold sweats. The half dozen tables overflowed with case books and study guides, and students crowded around them like bums at a trash fire.
Adam sat at one of the tables with four of his five "mentees," first-years in his Torts class, whom he was supposed to advise, guide, and steer clear of the pastries. His fifth mentee was just now making her way past a giant mimeograph machine the school finally retired in 2003, stopping every few feet to say hello to a classmate, plant a kiss on a cheek, and receive a high five or a pat on the ass. Her dark hair fell perfectly to the top of her shoulders, and her teeth shone like a predator's. Staci Fortunato was every boy's dream: lush, dark, and dangerous. She lured men in, then bit off their heads.
When Adam arranged this meeting, he had emailed all the students. Only Staci replied. Considering their first encounter in class, Staci was surprisingly cheerful and friendly, as if she'd been his star pupil rather than a first-class slacker. Of course, most of her communications were entirely inappropriate, but at least she responded. The other four students he had to track down by calling their emergency contact numbers, one of which turned out to be the cell phone for the student's drug dealer.
Now that student, greasy-haired with muttonchops, waved to Staci, who slalomed toward their table. Adam sat up straight, and kept his hands clear of the filthy surface.
"'Sup, Prof?" Staci asked as Muttonchops lifted his size-14 feet off the seat of the chair he had been saving for her.
"Hello," said Adam, not exactly sure how he should respond, and trying to maintain his dignity amid the squalor. "I guess that's all of us. I'm sorry we had to meet here. I didn't realize it was so... limited. Next time we can go out for something to eat."
"I have a friend whose advisor took them for sushi," said Muttonchops, whose name, Adam recalled, was Gregg.
"We could do that," said Adam.
"I don't like sushi," said Gregg.
"No biggie," said Staci. "Most people don't even meet their faculty mentors."
"I heard there are some professors who don't even come to school," added Samantha, an unnaturally tanned brunette who sat on the aisle in Torts, and seemed to make far too many bathroom trips for a seventy-five-minute class.
The other two students at the table were also in Adam's class, although one of them—Peter—had yet to attend, and the other—Derwin—appeared not to know it. During class he sat in the second row flipping through his Contracts case book as if he might find his Torts cases there. Right now he appeared to be asleep.
Technically, Adam was supposed to guide these students, answering questions they might have about law and law school, and otherwise making them feel as if the administration cared about the education they received. In reality, he had not been given any instruction on how to accomplish these things, and no money to facilitate them. He tried to strike the right balance between friendliness and formality; hence, the student café. But he realized he had made a mistake the moment he saw a squirrel scurry out of the top of the coffee urn from which he had just filled his cup.
"So tell me a little bit about yourselves—something I wouldn't know from Facebook." This was a question Laura had suggested. An icebreaker that would get them talking. Instead, it spawned complete silence.
"Anything," Adam added helpfully.
"I'm bi," said Samantha.
"Everyone knows that, Sam," said Staci.
Samantha shrugged. "He said anything."
"He meant anything that nobody knows."
"Then why didn't he say that?"
"I used to be a gangsta," said Gregg.
"A gangsta from Scarsdale," said Staci.
Gregg looked deflated. "I've been to Compton."
"In a tour bus."
"Facebook is lame anyway," said Samantha.
Adam tried a different tack.
"How about questions you haven't been able to ask your other professors? Maybe questions about grades? Or study groups? When I was in law school everyone said to join one, but I didn't think it was very helpful."
"How do we get a job?" asked Peter, pulling out a notebook and pen.
"Good question. Cutting right to the chase."
"My Dad says that," said Samantha.
Adam wasn't quite old enough to be Samantha's father—although he realized, biologically speaking, it wouldn't be an impossibility. Maybe that's what the students needed: a hip professor who spoke their own language and knew which social media sites were still cool. But even if Adam knew the difference between a tweet and a tag, which he didn't, that wasn't him. Instead, he decided the best tack was honesty, integrity, and a good dose of realism.
"It's a difficult market right now," said Adam. "You're competing against a lot of students from some very good schools."
"Your old firm is hiring," said Staci. She waved a flyer she had picked up from the career counseling office, also known as The House Where Jobs Go To Die. "How can we get an offer there?"
Adam was impressed that Staci had bothered to read his faculty bio on the law school's website—and that she managed to find it. But she had no chance of getting hired, especially as a first-year. "You should probably think about an internship or volunteering at a nonprofit this summer," he told her.
"What firm are we even talking about?" asked Gregg.
"Cranberry, Boggs & Pickel. The one that did the big oil spill litigation. Remember, it turned all those seagulls that yucky brown color? In Louisiana?"
"I thought that was Katrina."
"There was a hurricane, too. The firm proved the seagulls were brown to begin with, and got the case dismissed."
"Those firms pay like a hundred seventy-five grand," said Peter.
"Yeah, but you've got to defend cigarette companies and Nazi war criminals," said Samantha.
"You don't have to defend cigarette companies," said Staci.
In fact, as Adam recalled, the firm's litigators had a lucrative practice defending some of the leading tobacco companies. But he didn't mention that.
"If they pay so much, why are they interviewing here?" Gregg had been in law school for less than two months, yet the pecking order of law firm recruiting was painfully obvious to everyone. The top firms treated Manhattan Law School like a leper colony, avoiding its grounds and shunning its inmates.
"Because you worked there, right?" Staci asked Adam.
Adam explained that one of the firm's top partners, Howell Goldreckt, was an MLS alum. He threw his weight around—literally—to hire at least one summer associate from the school. It wasn't a pretty sight, but it was effective.
"You have to be on Law Review to get an interview," said Peter. "With straight A's."
"Not everyone on Law Review has straight A's," said Staci. "I know a guy, he told me he had a B– average first year. He got on through the writing competition."
"Everyone except Associate Editors get on through the writing competition."
"I know, but he had B's and C's and he still got on."
Adam had not been on Law Review at Harvard, but he was familiar with the grueling competition. Over a three-day weekend, students had to weed through several thousand pages of source material and edit a one hundred page article. Many of the successful candidates stayed awake for seventy-two hours straight. It was rumored that President Obama himself went to the emergency room with heart palpitations and a ruptured bladder after he turned in his packet.
"Law Review is a great credential," said Adam. "But there are other places to work besides a big firm."
"Like where?" asked Peter.
"Professor Copeland is hiring this summer."
"Yeah, a nanny," said Staci.
"Isn't she too old to have a nanny?" asked Samantha.
"Not her, Sam," said Staci. "Are you brain damaged?"
"Sorry we can't all be Rambda Lambda Ding Dong."
"Rho Lambda."
"Isn't that for lesbians?" asked Gregg.
"No, jerkoff. It's a leadership organization. You have to be elected."
"I don't know. Sounds hot." Gregg leered at her.
"Ignore them," she said to Adam. "Would you write me a letter of recommendation?"
"For the nanny job?"
"For the law firm."
Staci gave him her sweetest, most seductive smile. Truthfully, it was difficult to resist. But Adam did because, for one thing, it was a violation of school policy to accept sexual favors from anyone except third-year students. In any event, whatever pull he had with Howell Goldreckt wouldn't work for a first-year who didn't do the reading—despite her good looks. Goldreckt might be a letch, but he was also a savant, and he expected his associates to keep up.
"Like I said, you should focus your attention on other possibilities this year," said Adam. "Getting good grades should be a priority. The jobs will follow."
"Next year, then?"
"Sure. We'll see."
"Thanks, Prof!" Staci gave him another enticing smile, and Adam felt himself blush. By the time he regained his composure, the conversation had morphed into a debate over the best places to work in the summer (Nantucket was a top choice, followed by Jackson Hole, and then Taos), and from there into a discussion about whether global warming would make it harder to get a good summer share in the Hamptons. On the one hand, Gregg argued, rising sea levels would create lots of new beaches. On the other hand, said Samantha, hotter temperatures would make more people want to take vacations. Peter suggested they get Al Gore to speak on the topic at the law school. The students then asked Adam if he knew Al Gore. Adam said he didn't, but commended the students on their ambition.
This started another round of conversation about whether it was better to marry someone who was rich or someone who was famous. At some point in the conversation, Derwin woke up. He blinked and looked around like a hermit crab coming out of its shell.
"Yo, dog, what up?" asked Gregg.
"Huh?" said Derwin.
This provoked a round of laughter at Derwin's expense, and even Adam couldn't help smiling despite his best efforts to keep a straight face. It was comforting, he knew, to find someone more guileless than oneself. In the cutthroat world in which these students fought for jobs, a classmate like Derwin was a blessing. He lowered the curve and improved the odds for the rest of them. For a moment the students basked in the communal glow, warmed by their gratitude for the dimwitted and daft.
But they didn't read the fine print. None of them did. As the students gathered their books, and Adam exhorted them to stay on top of their classwork, only Derwin lingered behind. English was not his best language—though it was his only one—and many printed words were beyond his grasp. Even before his collision with the UPS truck, he had never been the sharpest tack in the box. Once, he lost his way in his own apartment and ended up living in a closet until his roommates found him rambling about liberating General Tso's chicken. Another time he mistook a flower pot for a hat and knocked himself unconscious. Yet in a pinch he could read, and even write, although not at the same time.
He scanned the flyer Staci left behind. The law firm was seeking summer associates—that much was true. Top dollars paid for summers spent dining in midtown, frequenting Broadway shows and Yankee Stadium, yachting up the Hudson. But there, at the bottom of the page, was the contact information for a different career counseling center, one located on the other side of the river, at another law school in a faraway borough. In her enthusiasm, Staci had picked up the wrong flyer, delivered by a postal clerk who didn't know the difference between Gowanus and Greenwich Village.
Derwin made a move to tell her, but then stopped, distracted, lost in space. By the time he regained his bearings he had already forgotten, and Staci was a faint blur in the distance, a smudge of red and black. Whatever it was, it wasn't important. He shrugged on his backpack and put his best foot forward. Then he followed with his other one, too.
## 6
## House of Cards
THE LAW REVIEW DIDN'T HAVE a proper office. It had, instead, a mildewed windowless room in the sub-basement, between the boiler and the overflow cafeteria storage facility, filled with boxes of potatoes and freeze-dried meat that had been around since the Eisenhower administration. The room was previously used for mothballed equipment by the Director of Maintenance. He died while snorting bleach in an incident buried by the school and never reported to the police. As lawyers, they knew better than to involve the justice system. To punish the maintenance department for causing trouble, however, the room was converted into an office to house the law school's student elite.
Asher Herman was the Law Review's Editor in Chief. He was not the smartest student on Law Review, although he was the most enthusiastic. On the short side, with unruly brown hair and perpetually wet lips, his one outstanding feature was a prominent nose that kept watch over the rest of his face like a hawk. Now he and his nose sat with his fellow editors in a corner of the room they had converted into an editorial suite by erecting partial walls constructed from old exam answers. There was the Managing Editor in Chief, the Articles Editor in Chief, the Editor in Chief of Notes & Comments and the Editor in Chief of Comments & Notes, as well as the Online Editor in Chief, the three Supervising Editors in Chief, and the two Editors in Chief at Large. In the outer room, about twenty Associate Editors in Chief were hard at work at the copy/scan machines, shouting to be heard over the noise from all the whirring, clicking, jamming, and cursing.
"Vermouth! We need more vermouth," said Charlie Spires, the Managing Editor in Chief. He fancied himself a future Hollywood mogul and never lost an opportunity to sound like one.
"Duly noted," said John Tarantula, Supervising Editor in Chief #1, whose job consisted of ordering liquor for Charlie, drinking liquor with Charlie, and maintaining restroom supplies.
"The condom dispenser in the women's bathroom is jammed," said Binky Paratha, the Editor in Chief in charge of the publication's nonexistent website.
"I'll email the Dean," said John.
"And the overhead lights don't work."
"That's because they're video cameras."
"Isn't that illegal?" Binky had worked on Capitol Hill prior to law school and she vaguely recalled a scandal involving a Congressman, a bathroom stall, and a videotape.
"Not in a federal prison."
"But this is a law school."
"Yeah, and I'm Santa Claus."
"Not funny, John. If someone's taping me trying to buy condoms, we have a serious problem."
"No one's taping you, Binky," said Asher.
"Yeah, and you shouldn't be using those condoms anyway," said John. "I thought you had a latex allergy."
"That was a yeast infection."
"You win. I move to buy a box of condoms for the women's bathroom."
"Seconded," said Charlie.
"Any objections?" asked Asher. "Seeing none, the motion is approved."
With the important agenda items out of the way, Asher turned to the unpleasant task of publishing legal scholarship. "What's up next in the pipeline?"
"Federalism," said Willow Summer, the Articles Editor in Chief, who was anything but willowy.
Nods all around the table. Federalism was a solid topic. It had constitutional notes with a bouquet of states' rights and a whiff of musty British history. Not everyone was sure what it meant, but it sounded smart, and that was all that mattered. None of the editors had done very well in their classes, at least not by traditional measures, but they enjoyed the patina of their faux meritocracy.
Willow distributed a stack of printed articles from the pile in front of her. Asher took a quick look at their first pages. They certainly seemed impressive. Each page crammed with footnotes, headings and subheadings galore, lots of Roman numerals and Latin words, a complete lack of anything resembling spoken English. They were exactly the sort of gaseous claptrap that bestowed accolades and tenure on their authors.
"Looks perfect," said Asher. "You'll assign the Assistant EICs?" The question was directed at Supervising Editor in Chief #2, a young woman named Raya Kurdle who chewed on her nails, although she had no nails left to chew.
"Is this volume three?" asked Raya.
"Four," Asher reminded her. "Three is the Symposium on Pets' Rights."
"I thought that was two."
"No. Two is Queer Theory and the Law."
"Got it." Raya made a note in a marbled black book she used to schedule important quarterly events, like her next meal.
"And make sure they use the good scanners. Last time everything came out blurry."
"Good scanners. Check."
Asher squinted at the agenda, but he couldn't read the print because the page was soggy with Red Bull and doughnut crumbs. "Is there any new business?"
"Jobs," said Charlie.
"What about them?" asked Asher.
"When are we going to get them?" Charlie had received straight C's in his first year, but now that he was on Law Review he expected better. "I mean, what the fuck are we paying for?"
"Dean Jeffries says they can only give us the grades," said Asher. "It's up to us to get the jobs."
"How can we get the jobs if all they give us are grades?" complained Charlie. "Can't they call some firms? For all the money we pay, you'd think they could punch a few numbers."
"I heard Naomi is working for Trapp & Dore," said John Tarantula. Naomi Stein was an Associate Editor in Chief who was also President of the Jewish Jurists Society.
"Yech," said Willow. "You couldn't pay me to work for that firm."
"No one is," said John.
Willow glared at him, but stayed silent. The two had a history, dating back to first year when John drunk-dialed her and professed his undying lust, only to realize he had the wrong number. Since then, they pretended the call had never occurred, and circled each other warily like welterweights.
"We are the best and brightest," said Charlie. "If we can't get jobs, what hope is there for everyone else?"
"Very little," said John.
"Fuck that," said Charlie. "If the school wants my money, they better get me a job."
"You had a job," Asher reminded him. Charlie's father owned a waste management service in New Jersey and pulled more than a few strings to get his son hired last summer. "No one forced you to throw up in the back of that partner's car."
"It wasn't my fault," said Charlie. "The guy was a terrible driver."
"The car was parked."
Charlie shrugged and winked at Raya. He loved playing the drunk Casanova. In truth, he was just a drunk.
Raya smiled back. Charlie wasn't attractive, but there was something appealing about his oversized belly and large bank account. Raya felt most secure beneath a man who outweighed her by at least one hundred pounds and could write a check that didn't bounce like a Super Ball.
"Maybe we should set up a committee," she said.
"Good idea," said Asher. "Raya, Binky, Charlie, and John. You'll be responsible for keeping track of the hiring situation. If things don't improve, we'll create a PowerPoint for Dean Jeffries."
"That'll show him," said John.
Asher ignored John's sarcasm. He already had a headache from the stress of running this meeting, and severe gas pains from all the Red Bull he had been drinking. He wanted a job as badly as his classmates, but believed his law review credential would eventually get him hired somewhere. That's what the deans had promised him, and why he joined law review in the first place. Yet so far only four law review members had received permanent job offers, and three of them were lowly Associate Editors (no Chiefs among them), selected for law review based entirely on their grades. It had begun to make him wonder whether law review—and law school, in general—was one giant scam.
The irony was not lost on him.
He rationalized his cheating by telling himself that students at other schools had their own unfair advantages. No one in this world ever got ahead without the help of someone else. Whether it was a well-placed uncle, a college roommate, or simply a good word between the sheets, an unfair advantage was the American way. The cards were marked at birth and dealt to the highest bidder.
"Is there any other new business?" Asher asked. He peered at the agenda, and then back up at John, but both were inscrutable and needed a good cleansing. "Seeing none, the meeting is adjourned."
The editors pushed back their chairs and made their way out of the suite. Willow snuck another doughnut when she thought no one was watching, while Charlie looked at his watch and declared it was never too early to start drinking. Raya said she would join him, but Asher begged off. What he needed was a couple of aspirin and a gallon of water. He knew if he went out with Charlie things would end badly, as they often did. He couldn't afford another night in front of a toilet or in a police precinct.
At that moment the door opened, and a tall blonde woman entered. A current of nervousness rippled through the editors, and there was an imperceptible quickening of movements. The Associate EICs finished up their jobs at the copy/scan machines and tucked their papers safely into folders. Willow flipped the articles she was carrying so that only their blank white faces showed. Even Charlie covered his mouth when he burped.
Everyone in the room knew Ann Marie Kowalski. She was one of those pesky Associate Editors with grades too high to be ignored. Exempted from the write-on competition and the special orientation for write-on editors, the five Associate Editors were given the most demanding tasks that required travel to the far reaches of the Bronx and Staten Island. Last week one of them was sent to the special Polish War Veterans museum in Pittsburgh, and he had yet to return.
Asher smiled at Ann Marie in a way he hoped was welcoming but felt stiff and fake. He didn't enjoy keeping the Associate Editors in the dark about the inner workings of the Law Review, yet he accepted it as one of the hazards of the job. The risks were obvious, and both Dean Jeffries and Professor Rodriguez, the Law Review's faculty advisor, drummed it into him on the day they approved him as Editor in Chief. Since then he felt like Lot's wife every time he was around Ann Marie: paralyzed by fright yet unable to look away.
"Sorry," she said, when she saw him staring at her. "I just had to get my new assignment."
"No!" said Asher, with too much vigor. "I mean, no worries. Mi casa es su casa." He gestured around the room, feeling thirty pairs of eyes surreptitiously observing him.
"Thanks." She brushed a wayward strand of hair from her lips. "It's a lot to get used to so early in the semester. But I'm really enjoying the work."
"It's very rewarding."
"Yes... well. Guess I should get to it. Nice talking with you."
He watched as she made her way to a computer terminal, admiring the curve of her hips and the outline of her butt in her jeans. Lost in a reverie, he almost allowed her to sit down right in front of a screen on which another editor had uploaded a scan of a recent article published in the NYU Law Review.
"Asher!" shouted John from across the room.
Asher snapped awake and bolted toward her. "Ann Marie!"
Startled, she turned around. "What? Did I do something wrong?"
"It's nothing," said Asher. He was panting like a puppy. "It's just we have a different assignment for you, that's all." Asher's mind raced through possible tasks he could give Ann Marie that would get her out of the room and away from the monitor.
"What kind of assignment?" Ann Marie asked warily.
"An important one," said Asher. "You're not allergic to latex, are you?"
## 7
## Point. Game. Match.
OUT!"
"Out?"
"Out!"
Adam knew his shot was a good six inches inside the line, inside the squash court, inside an exercise club so exclusive he wasn't even allowed to know its name, despite having been invited to play there once a month by Howell Goldreckt, the attorney who had made possible his escape into academia. If Adam cared at all about squash, he would have argued the call, but instead he cared about getting off the court alive, with no damage to his reproductive organs or his relationship with a man who could, with one phone call, make Adam's job go away. And so he gave up the point and the game went on, fourteen points for Adam and three points for Goldreckt. After all, cheating can only get a man so far when he weighs nearly three hundred pounds and it takes a full four minutes to find the ball when it's sitting right at his feet, because he can't see his feet, and the ball is practically invisible to him because it isn't food.
Two points later Adam finally put away the winning shot, which glanced off Goldreckt's fleshy face and caused the law firm partner to let go of his racket and fall to the ground. Supine, he looked like something to climb with a couple of Sherpas and an oxygen tent. In fact, Adam thought he might need the tent until Goldreckt groaned and sat up, the imprint of the ball visible as a white circle on his red skin.
"Fucking beginner's luck," he said.
Adam and Goldreckt had been playing squash for the last ten years, ever since Goldreckt introduced him to the sport when Adam was a summer associate at Cranberry, Boggs & Pickel. Goldreckt was fifty pounds lighter back then, though still enormous, and he was drunk. Very drunk. It was almost certainly an accident of fate that Adam was the summer associate to whom Goldreckt started unburdening his soul. But there Adam was, in Goldreckt's private bathroom as the partner threw up a half dozen times, and in between each hurl he regaled the young law student with the story of how he'd sacrificed to make it to the top. Months spent sleeping in his office, wearing the same clothes, eating cold food—and not because work kept him, but simply because he had no better place to go.
Yes, the firm was one of New York's biggest, with offices in twelve cities across the world, specializing in the kind of high-stakes litigation and corporate deals that justified billable rates above one thousand dollars an hour for top partners. And, yes, Goldreckt's specialty was the most lucrative of all: leveraged buyouts and institutional mergers and acquisitions. Big hospitals buying each other; big corporations buying small countries; big pension funds dumping their obligations and leaving senior citizens begging their children for help paying their bills. But all that money still didn't buy the man love.
It ended up being one of the only times Adam saw the inside of Goldreckt's bathroom, unlocked with a special key and off limits to associates, reserved for Howell alone so he could defecate in private. Even the cleaning woman, Adam later discovered, could only enter after signing a nondisclosure agreement. But that night, Adam was thrust into Goldreckt's inner circle, forced to listen to the man go on about why he had become a lawyer. In high school he was a champion player of Dungeons and Dragons and Strat-O-Matic Baseball ("on a board, not a video screen," he insisted). In college, the fraternities paid him not to rush. He went to law school because it was a way to maintain his baseball card collection without worrying about getting a job. But in law school, he told Adam, something finally clicked. The coolness of legal argument while bodies lay bleeding on the ground. The pure transactional value of human enterprise and behavior. He graduated at the top of his class, won every prize his law school could offer, and despite his lowly pedigree and social awkwardness, was hired by one of the best firms in the city at a time it needed more bodies in a booming market. He was so good at what he did, and so trusted by clients, that before long the partners had no choice but to ask him to join them.
When Goldreckt sobered up, Adam was still there, washing his tie that had been splattered in the line of fire. He helped the partner back to his office and onto his sofa, then found a blanket shoved in a closet filled with clothing and unpaired shoes. He covered as much of the man as he could, then tiptoed out of the room under cover of darkness and the earth-rumbling sound of Goldreckt's snores.
From then on, Adam was the man's confidant. Goldreckt would call him late at night while Adam was studying at Harvard and regale him with tales of personal woe and professional triumph. His need to be loved was so obvious and childlike that Adam couldn't help feeling sorry for him. Sorrow became pity, and pity became—not friendship, exactly, but an understanding, a symbiotic relationship where Goldreckt got a willing listener and Adam got an education. So when it came time to weigh the firm's job offer, Adam reasoned he could do worse than working at one of the most prestigious law firms in New York City for a partner who trusted and relied upon him.
Until the day Adam betrayed him by walking into his office and asking to quit. "This life's not for me," Adam explained.
"It's not for anyone," Goldreckt replied.
But Adam's decision was about more than just the typical complaints of a law firm associate: the long, unpredictable hours, the mind-numbing work, and the lack of the tiniest bit of control over one's life. It was the sudden realization, while he listened to a colleague as she walked a client through the proper procedure for minimizing its obligations to fund a health plan for its employees, that he was flying too close to the sun. The line between what the law permitted and what lawyers advocated for was sometimes so faint that it was easily blurred. By staying at the firm he risked becoming part of the machinery that encouraged line-blurring. Already he could see how the older attorneys ignored the lines entirely, their only boundaries whatever was necessary to achieve results. Advocacy became expediency, and expediency became the new norm. It was as close to his own father's life as he ever wanted to get.
So he decided to leave. He always had an intellectual passion for the academic life, and as a law professor he could advocate against the kind of expediency that turned lawyers into enablers. To get there, however, he needed Goldreckt's help.
The only job more difficult to land than a job teaching law was a job teaching law in New York City. Lawyers were leaving (or getting booted from) their firms in droves, and law school seemed a kinder, gentler place to land. Teaching salaries were decent; summers were free; and the students never aged. After three long hours of wheedling, pleading, and eating (because Goldreckt had to), the partner finally agreed to put in a good word with Dean Clopp at Goldreckt's alma mater, Manhattan Law School. The school owed Goldreckt—many millions, in fact—and his word carried the weight of his bank account. Adam knew it was a third-tier school, but he was also realistic: without a PhD or any publications to his name, it was slim pickings in the job market. And, he reasoned, once he had a foothold, he could write his way up the academic food chain.
In return, Goldreckt demanded an additional five hundred billable hours over Adam's assigned target of three thousand, and a promise to help him with recruiting from MLS.
"Tall," he insisted. "Slim hips. Breasts like melons."
"I don't think that's legal," said Adam.
"It's illegal now to have big breasts?"
"Wouldn't you rather hire someone who's competent?"
"Competent I've got." Goldreckt made a shooing motion with his hands. "Clopp sends me their grades. But the dean wouldn't know a hottie if she sat on his face. That's where you'll come in."
Goldreckt wanted facts and—most important—figures. His partners were giving him a harder and harder time leaving a slot open for an MLS student. In the good old days—back when Adam was hired—Goldreckt would never have needed to make a case to extend someone an offer. The firm hired as many as ninety summer associates a year, threw them lavish parties at the partners' country clubs (the ones that allowed minorities), and took them on cruises up the Hudson. But the work had soured and so had the lawyers. These days the parties were gone, replaced by discount vouchers to Broadway shows and free admission to the Transit Museum. Financial constraints also limited the number of summer associates the firm could hire. Now, fifty was a reach, thirty-five even better, and those who did not earn their keep wouldn't be invited back.
It was a battlefield out there, and Goldreckt was fighting a rearguard action against the other members of the hiring committee: Steve Spivey, a specialist in foreign mining leases, who helped multinational corporations strip the mineral resources from third world countries; Mike Weirhammer, who practiced admiralty law, a subspecialty so obscure its practitioners still wore wigs; Ursula Font, the only female partner at the firm, although technically she had failed the firm's mandatory chromosome test; and Cepeda Wilkins, who, two years earlier, tried to stab Goldreckt with a ballpoint pen but it didn't penetrate the polyester blend of his suit. All of these men, and the one woman, attended first-tier schools, and they would stop at nothing to keep the firm's bloodlines pure. At their hourly rate, as Steve Spivey put it, clients wanted more than "Schmuck, Putz, and Dreck."
Even with the proper pedigree, the partners rejected anyone who was too needy, clingy, or closely resembled an ex-spouse, ex-girlfriend, or, in Ursula Font's case, two ex-children she had fathered and then legally disowned in a precedent-setting legal dispute. They kicked to the curb anyone they thought might be pregnant, unvaccinated, or need medical care that could increase health insurance premiums or require paid leave under federal or state law. There was no such thing as a perfect summer associate, but the partners held fast to a Platonic ideal. Each championed his own favorite; each refused to see the virtues in another's top choice. Mostly they fought over perceived miniscule differences between their alma maters, with Weirhammer, from NYU, maintaining that Columbia grads were spineless and weak-willed, and Spivey, from Harvard, arguing that Yale grads were godless Communists. Font and Wilkins, who were in the same class at Stanford, claimed the elite Northeast schools were just part of a liberal conspiracy designed to hide their intellectual failings.
Goldreckt reminded Adam of his promise after they had showered and were dressing in the locker room. Adam did his best to avoid looking at Goldreckt's backside, but he couldn't help marveling at the size of the man's underwear, which was an engineering miracle on the scale of the Suez Canal. For a large man, however, he was incredibly dainty in his grooming habits, and took a good twenty minutes to apply various unguents to regions of his body that Adam wished he didn't know existed. Then, when Goldreckt had finished, he applied enough cologne to set off the fire alarm, which brought several staff members running in a futile attempt to silence the din.
Adam lowered his head and followed Goldreckt into the club canteen where the partner ordered himself a calorie-busting "Coffee No-No," which, as far as Adam could see, consisted of several drops of coffee, a heap of sugar, and a thirty-two ounce dollop of whipped cream. Adam ordered a bottle of vitamin juice, which had neither vitamins nor juice, but was a syrupy concoction that tasted like green chalk. They sat at a table overlooking one of the squash courts where a young woman was thrashing a man who looked old enough to be her father. With each shot he returned, she drove the ball into the opposite corner and sent him flying.
"So, can you do it?" Goldreckt asked. "Can you do this for me?"
Adam realized that the alarm had drowned out Goldreckt's question. Now he was waiting for an answer, and looking like a schoolboy about to receive his grades.
"I'm not sure," said Adam, stalling for time.
"Not sure?!" roared Goldreckt. "After everything I've done for you?" The big man's face reddened and Adam worried he might burst a blood vessel. Then his lower lip quivered, and he said in a voice so quiet Adam could only make it out by reading his lips. "Please?"
On the squash court the woman jammed an elbow into the man's gut as he lunged for the ball. It could have been accident, but it looked purposeful to Adam. The man stumbled, staggered, then crashed head first into the wall while the woman pumped her fist triumphantly.
"Okay," said Adam, not sure what he had just agreed to.
"Thank you," said Goldreckt. He looked, at that moment, like the little boy he must have once been. Bright-eyed and rosy-cheeked. Stubby little fingers and teeth. Milk breath. "Her name," he said, "is Ann Marie."
Adam didn't recognize the name. But whomever she was, the student clearly had the partner's number. His eyelashes fluttered as if he might start crying, and Adam had to turn away to save both of them the embarrassment. When he looked back down at the squash court the man was on the ground again while the woman stepped calmly around him and ricocheted the ball into his head. Point. Game. Match.
## 8
## Love and Other Drugs
ADAM UNSCREWED THE TOP FROM the container of hair product and examined it closely. It had been so long since he last used it that the gel had hardened into a solid mass. He sniffed it, checking if it had gone rancid (could hair gel go rancid?), then ran the container under hot water to loosen the goop. He caught a palmful as it streamed out, and massaged it into his hair. It wasn't until he put on deodorant that he realized he had confused the two products, and now had spiky underarm hair and his scalp smelled like "Fresh Breeze." He climbed back into the shower and rinsed himself clean, then started all over again.
It had taken him a few days after their last meeting to get the courage to ask Laura out again. Even so, he did it over email, afraid of face-to-face rejection. He spent more time crafting the note than he had preparing for that day's class, but it was worth it. She said yes. And now, here he was, buttoning his only pair of not-jeans-but-not-business-casual pants, tucking in the brand new Banana Republic shirt his brother, Sam—always better at luring the ladies, although not necessarily the right ones—picked out for him online, and giving himself one last look in the mirror before heading out to meet Laura at Oink, a month-old, no-reservations, farm-to-table bistro in the East Village that he had read about in New York magazine. "A surprisingly romantic vibe," the review concluded, "perfect for first dates that you hope might lead to more. And save room for the bacon-shell cannoli for dessert."
The line was already out the door when Adam arrived, twenty minutes early. He meant to get there sooner but, as usual, the MTA had forgotten people lived in Brooklyn. The F train was under construction and replaced by a shuttle bus, so he walked to Atlantic Avenue and caught an N train. The N was making local R stops, until it stalled at Canal Street and the conductor told everyone to get off. Adam, quickly becoming drenched in sweat, walked through the station to catch a 6 train, took that to Astor Place and, finally, arrived at the restaurant. Next time he swore not to be such a cheapskate and splurge on a cab.
By the time Laura arrived, the estimated wait time had fallen to just over an hour. Adam surreptitiously popped a mint when he saw Laura approach. She was wearing a simple but sexy black dress, a book in her hand. She greeted him with a hug that felt too brotherly, but then gave him a quick peck on the lips as if to make up for it.
"I put our name down, but they claim it's still an hour," he said. "Apparently three different improvisational comedy teams sat down to eat at the same time, and so they're really backed up."
"It's okay," said Laura. "It's a beautiful night. I don't mind the wait. Besides, they have a bar."
Before long they had squeezed their way between a young couple arguing over who was more "emotionally available" and two bankers complaining about the high cost of doggy daycare. "That doesn't even include the Christmas bonus!" barked one. The bartender, a young woman who looked about sixteen years old and six feet tall, brought them their drinks—a margarita for Laura and a Scotch for Adam. Laura raised her glass and proposed a toast.
"To Manhattan Law School."
"Home of the brave."
"But not the free," quipped Laura.
Adam admired the relaxed way Laura held her glass and took a healthy sip of her drink. It was a welcome difference from the pinched imbibing of Jane Van Dyke, who drank modestly and never lost control.
"So what's a nice boy like you doing in a place like Gowanus?" she asked.
"Got on the F and just kept going."
"Funny guy, I'm being serious here. How'd you end up at Manhattan Law School?"
Adam quickly recited the story he'd told countless times before—to family, to friends, to colleagues who thought he'd lost his mind—although he skipped the part about avoiding his father's fate. Work at the law firm was making him crazy. The hours were too much to bear. He was good at it, but didn't care about the clients. Helping big corporations get bigger had no meaning, and since that was what he did all day (and night), it felt as if his life had no meaning. Looking around the firm, he could see his future: the older, divorced, burned-out partners whose kids ignored them and ex-spouses who hated them. Adam knew it was not for him.
"And now here you are," Laura concluded.
"Yes, I am," said Adam, with a twinge of sadness.
"It's not that bad," said Laura, misinterpreting Adam's melancholy for regret about the school where they both taught. "I have friends who've ended up in all sorts of places. Nebraska, Montana, a law and economics class at a community college in South Dakota."
"I hear the rents are cheap in South Dakota."
"There's a reason for that," said Laura.
"Listen to you—such a New York snob."
"If I were a New York snob I wouldn't be teaching in Gowanus."
"Why are you teaching here, anyway? You must have had lots of other options."
Before Laura could respond, a table opened up, and Adam's name was called. They followed the hostess to a cozy corner where, once seated, each ordered another drink. Adam felt the alcohol relax him, and listened as Laura shared her own carefully edited biography.
"They made me an offer I couldn't refuse," she said. "I was finishing a two-year clerkship and it seemed like a good way to stall. I didn't want to go to a big firm, but I didn't want to go to Iowa, either." She shrugged. "Okay, guilty. I am a New York snob."
"So where does Berkeley fit into the picture?"
"Maybe I'm done stalling."
"That would be a pretty dramatic move. All the way across the country. Isn't Columbia hiring? NYU? Or even Harvard?"
"Even Harvard! Listen to you. Let's just settle for Harvard."
Adam blushed. In his zeal over Laura's job opportunities, he felt like he had recklessly proclaimed his undying love.
Laura, however, appeared not to notice. "You know those jobs don't come calling," she continued. "It's more competitive than it has ever been. Too many lawyers; too few opportunities; and everyone who's working is looking for an exit strategy."
"But you're a rock star," Adam insisted. "You practically won the National Book Award."
"Please. That was four years ago." She crooked her neck and did her best imitation: "What have you done for me, lately?"
"Janet Jackson, circa 1986."
"Very good."
"Thank you."
"Anyway, sometimes a fresh start is needed. Someplace new. Without the baggage."
"You've got baggage?"
"We've all got baggage," Laura answered a little more seriously than Adam expected.
Adam's own baggage, he reflected, was neatly packed away at his father's death, although his mother periodically hauled it out for special family occasions.
The waiter arrived to inform them about the specials: a Hamptons-raised rabbit braised in a shiitake and red wine reduction, Long Beach oysters stirred in a hurricane brine, and an authentic Lower East Side poached squirrel with an acorn purée, garnished with pigeon feathers and a "homeless foam." They both ordered something more mundane—salmon for Laura, the trout for Adam—and after the waiter left the conversation turned to lighter subjects. High school sweethearts, college roommates, favorite movies and cartoon characters, books they'd read, worst New York City subway adventures, boxers or briefs.
After a few more drinks, Laura repeatedly touched Adam's hand for emphasis while she was telling a story, and somewhere in the middle of the tale Adam's fingers entwined with hers and held the hand there. Their legs brushed beneath the table and, having found each other, stayed. Soon, Adam was feeling a steady pressure on his thigh that radiated warmth through his pelvis. When Laura began to stroke his arm with her free hand, Adam quickly flagged down the waiter for the check. He paid with cash, leaving an outrageous tip because he was too impatient to wait for change, and they stumbled outside onto the street. Soon they were kissing in the back seat of a fifteen-minute, thirty-dollar taxi ride to Laura's apartment.
Adam had just swiped his credit card through the fare meter when he noticed Laura staring at a young vagrant sitting on the front stoop of her brownstone. In the harsh glare of the street-lights he looked awful. His face shone with several days of accumulated sweat. His hair was matted and mashed. And his clothing appeared to have been rolled in grease, dusted with soot, and wiped clean with a pizza slice.
As they got out of the cab the kid rose and motioned toward them.
"Careful," said Adam to Laura. In college one of his roommates had been attacked by a homeless man brandishing a set of keys, and in the scuffle he received a slash across the face that required thirty stitches to close. The only weapon Adam had was his red Pilot G2 pen, which he raised now in self-defense.
"It's okay," said Laura. "I know him."
"You know him?" Adam was about to lecture her on the dangers of befriending psychotic homeless people.
"What are you doing here, Gary?" she asked.
"You said I should call if I needed help."
"Yes," Laura said tentatively.
"But I only had your office number."
"I see."
"I found your address on the Internet. It was easy. You really should do something about that."
"Gary, this is my home. It's Saturday night."
"But I need to talk."
"Now?" She looked to Adam for help.
"What's going on?" Adam interjected. He recognized the kid now as the student they passed on the street after their lunch date, and who nearly tripped him in the stairwell. He looked as if he'd lost twenty pounds—and half his hair.
"I'm going through a bad time," said Gary. His hands shook from four espressos and some pills he'd found in a law school wastebasket.
"Maybe you should talk to someone," said Adam. "Else," he added.
"What is it, Gary?" asked Laura.
"I'm sorry, Professor. I didn't mean to ruin a special night." He looked from Adam to Laura as if it had just dawned on him that Laura wasn't alone.
"It's okay. What do you need to talk about?"
"Can I come inside and use the bathroom?"
"No, Gary. That's a line you can't cross."
Gary fidgeted on the stairs. The street was relatively quiet, but the noises of city life surrounded them: cars honking, music drifting from a nearby club, laughter, shouts, a bell. Normal people having fun on a normal weekend night.
Gary unzipped his fly and began to urinate on the sidewalk.
"Gary!" said Laura.
"It's bigger when I'm sober."
Adam made a move to stop him but then thought better of it. Instead, the two of them stood there awkwardly while Gary finished his business.
"Sorry," he said. "When you gotta go, you gotta go."
"Speaking of going," said Adam.
But Gary sat back down on the stoop. Then he put his head in his hands and moaned. After a couple of minutes, when he hadn't stopped, Laura looked over at Adam with a worried expression on her face. Neither of them needed a suicidal student on their hands—especially on a full stomach. She went to Gary and sat down next to him. "Talk to me," she said.
So Gary did. He told her his story about Ann Marie—at least the expurgated version—and Laura listened politely. He was not the first young man she knew to be distracted by heartache and loss. But he was the craziest.
"Listen to me, Gary," Laura said when he was done, as calmly as she could. "Everyone has his heart broken. This is a normal part of life. People treat each other terribly, and they hurt each other. Some days we feel like we can't go on. But life goes on. We go on." She regarded him intently. He seemed to be listening, but he could have been hearing the radio transmitter in his head. "Do you understand what I'm saying?"
"I think so." Gary looked up at Adam. "Do you?"
"Me?" said Adam.
"Not him," Laura interrupted. "I'm not talking about him. He's a good man. I wouldn't be here with him if he wasn't."
"But you said—"
"Go home, Gary. Think about what we just discussed. This, too, shall pass."
Gary sat there for a minute, considering. Then he heaved himself to his feet. "What's the reading for tomorrow?"
"Tomorrow is Sunday."
"Oh, yeah." A semblance of a smile cracked his lips, which looked chapped and painful. "Okay. Well, thanks for talking to me. I really appreciate it." He held out his hand and Laura shook it, then he came around and offered it to Adam, as well. Adam hesitated, but over Gary's shoulder he saw Laura silently urging him to do it. So he took Gary's hand and gripped it. The hand was surprisingly firm, rigid and warm, not at all what Adam expected. It was the handshake of a CEO or a law firm partner.
"Have a good night," Gary said like a normal person. Then he turned, waved once, and walked off into the night.
Adam and Laura watched him go until his back was clearly out of sight.
"Wow. What was that?" asked Adam.
"Poor kid. I feel sorry for him," said Laura.
"I feel sorry for you."
"I'll be okay," said Laura. "Nothing some good sex won't cure."
Adam gulped. "I'll try my best."
"No heartbreak, right?"
"No."
Then she took his hand and led him upstairs.
## 9
## A Kick in the Kidney Pie
ADAM IGNORED THE BLINKING RED light on his living room phone when he returned to his apartment. He was probably the last person in New York City with a landline, and no one of any importance ever left him a message anyway. The phone was included in the cable and Internet bundle he was forced to purchase even though he didn't need one hundred eighty television channels or another telephone. But if he wanted the high-speed Internet connection and ESPN, there was no choice but to buy it all. Like the reading assignments he gave his students, the phone was a superfluous irrelevancy.
He puttered around his one-bedroom for half an hour, blowing dust bunnies into the corners, shifting dirty dishes in the sink, and figuring out whether he had enough clean socks to get through the week without doing laundry. He swapped an old photo of his parents with one of him whitewater rafting in Colorado with Sam. His brother no longer had the time, and Adam no longer had the money, but the trip brought back happy memories.
Finally, out of mindless things with which to distract himself from thinking about Laura, he picked up the phone and listened to the message.
At first, he didn't recognize the name. The voice, however, was unforgettable. High-pitched, reedy, a British accent by way of Rochester, New York. He said his name was Campbell Wiley, and it took Adam several minutes to realize this was the same Donald C. Wiley who sat next to him in first-year Contracts and Torts. They weren't exactly friends, but shared a bond common to POWs, reformed alcoholics, and former cult members.
Campbell, f/k/a Donald, had not survived to graduation. He abandoned Harvard after their first year and went back to journalism, his undergraduate major. Over the years, Adam had spotted his byline in a steady climb up the newspaper ladder: The Rochester Daily, The Great Lakes Gazette, The Huffington Post, Newsday, and, finally, The Gray Lady herself, The New York Times. There was a scandal in his background, Adam recalled, a reason he had abandoned journalism and gone to law school in the first place. Something about burning a source while on his college paper at Dartmouth, a member of the administration whom Donald outed after promising him confidentiality. The details were fuzzy, but Adam recalled Donald gleefully repeating Janet Malcolm's famous line over a couple of beers after their last final exam: "Every journalist... is a kind of confidence man, preying on people's vanity, ignorance, or loneliness, gaining their trust, and betraying them without remorse."
It was curiosity rather than a death wish that motivated Adam to return Donald's call, but it would have the same effect. Donald represented the road not taken, and Adam wondered where it had led. Although they ended up in different places, perhaps they hadn't traveled as far as he once expected.
"Hey, old chum," said Donald when Adam reached him.
"How are you, Donald? Long time, no speak."
"It's Campbell now. I'm called Campbell."
"I heard that on your message. Didn't know who you were at first."
"Same bloke I've always been."
"So what do I owe the pleasure?"
"I thought we could meet for a pint."
"A pint?"
"There's a terrific pub not far from you. Used to be a crack den."
"How do you know where I live?"
"I'm a journalist, Adam," Donald laughed. "You free in an hour?"
They made arrangements to meet at the Spotted Lout, overlooking the Brooklyn-Queens Expressway. The Lout, as it was known among the locals, served warm beer and kidney pies, and had a loyal following among the British expat crowd. It was rumored Prince Harry got naked in a back booth once, although older patrons swore it was Hugh Grant.
When Adam walked in, the jukebox was playing the Sex Pistols' God Save the Queen. He ordered a Guinness from a comely bartender and watched her pour while he waited for Donald. The bar looked to be about two miles long and carved from an ancient oak. The wall behind it was filled with amber liquids, most of which Adam assumed were whiskey, but could easily have been gasoline. Two men about his age were having a vigorous argument about Manchester United while next to them a young mother was breast-feeding her baby and drinking a beer.
D. Campbell Wiley was about five-foot seven, stocky, with a thatch of black hair. He was wearing a blue pea coat and tan boots that made him three inches taller when Adam spotted him at the door. He waved, then came over to the bar.
"Hello old boy," he said and nodded approvingly at Adam's Guinness. "I'll have one as well," he instructed the bartender with a wink.
"Good to see you Don... uh, Campbell," said Adam.
"You too, old chap."
As Adam recalled, Donald had not been happy at law school almost from the day he arrived. He missed the "rough and tumble" of the real world, as he had called it, and disliked the "pinhead professors" whose heads were in the clouds.
"Congratulations," said Adam. "I've been reading your articles in the Times."
"Thanks, mate. Congratulations to you, too. Got out of the rat race."
"Out of the rat race and into the rat business," said Adam.
"Ha! Bloody good! Must be challenging."
"Definitely. Especially at a place like Manhattan Law."
"No kidding. Not like when we were in law school."
"You went to law school?"
"Funny, mate." Donald's beer arrived, and he lifted his glass in a mock salute. "We can't all sell our souls; otherwise, there would be nobody left to sin."
"Touché. I'll remember that."
Donald signaled the bartender for two more beers, although he had barely sipped the first. "So what are your students like?" he asked.
"Oh, they're a mixed lot," said Adam cautiously.
"How so?"
"The dumb, and the dumber."
Donald slapped the bar. "Toots! You're hilarious!"
Adam suddenly regretted belittling his students, even as a joke, so he added, "Actually, there are some very bright ones."
"Really?"
"Sure. I've been impressed."
"So you're not giving out A's just to satisfy the curve?"
"No, of course not."
Donald quickly jerked his head, distracted by something to his right.
"Blimey, take a look at that," he said.
The mother had finished nursing her baby and now one of her breasts, full and firm, was partially exposed. Adam turned, then looked quickly away.
"Mama! Give me some milk," said Donald.
"It's nice that she feels comfortable nursing in public."
"I'm all for it," said Donald. "Let's pass a law."
"Actually, there is a law."
"Even better!"
The two men arguing about Manchester United stopped talking for a moment and looked their way. It occurred to Adam that the nursing mother might be married to one of them. He broke out into a nervous sweat, and steered the conversation onto safer ground. "So what's new in your life? Married? Kids?"
"Nah. I was married, but it only lasted about six months," Donald said, as he took the final gulp of his beer.
"What happened?"
"Thought I was in love. Turns out it was the whiskey." He laughed. "How about you?"
"Still single. Haven't met the right person yet."
"What about that Jane girl? From law school?"
"Van Dyke. Didn't work out."
"Sorry to hear that. Terrific knockers on her."
As Donald prattled, Adam questioned why the hell he had agreed to meet him in the first place—and, more importantly, what Donald wanted from him. How long would he have to stay before it was polite to leave? He was not-so-subtly checking his watch when Donald said, "I'm writing an article about law schools."
Adam's ears perked up. "Something I should read?"
"You know, how they're just a giant Ponzi scheme."
"Well, not all of them."
"Not all, true. But ones like Manhattan Law." He held Adam's gaze.
"It depends," Adam said carefully. "For some students they can be a good deal. We offer a lot of financial aid."
"I'm sure," said Donald.
"And our best students get good jobs," Adam added, although he had seen no proof of it yet.
"That's not what I'm hearing."
"What are you hearing?"
"There are rumors about Manhattan Law."
"Like what?"
Donald shrugged as if he didn't pay much attention to rumors. "Rumors about students buying grades."
"That's ridiculous," said Adam.
"And still not getting jobs, to boot!"
"I don't believe it."
Donald's tiny eyes drilled into Adam like black lasers, and Adam squirmed under his gaze. In the back of his mind he recalled Laura's comment about not telling anyone what really went on. Was this what she meant? Impossible. He refused to believe that about Manhattan Law School, or about her.
"You've got some angry, ripped-off students on your hands," Donald said.
Somehow Adam had finished his second beer, although he didn't remember drinking it. He signaled the bartender for a third.
"You haven't heard anything?" Donald pressed.
Adam recalled once, at his law firm's summer outing, an associate stripped down to a bikini and went for a swim in the pool. Two days later a popular legal website reported she had been skinny dipping with a partner. It took an entire year to dispel the rumor, and there were still people who referred to the firm's Trusts & Estates practice as the T & A practice. The point was that stories quickly took on a life of their own, morphing into far more interesting and sordid tales, especially among lawyers who loved nothing like a good scandal. Adam told himself it must be like that.
"Where did you get these reports?" he asked.
"You know a journalist can't reveal his sources." Donald gave him a sweet smile that made Adam feel sick.
"Well, they're not true, and I wouldn't make such a serious accusation if I were you."
"Is that some kind of threat?"
Adam hadn't intended to sound menacing, but he felt defensive and protective of Laura, if no one else. He was also a little bit tipsy—three beers on an empty stomach had gone right to his head. "Not a threat. Just a caution. I'd hate to see you ruin someone's career on the say-so of a couple of frustrated students."
"That's why I'm talking to you, old friend."
"Like I've said, I've never heard of it."
"Pity. You'll keep your eyes and ears open though? For your old Torts mate?"
"Sure." Adam would be damned before he told Donald anything, even if there were any truth to the rumors, which he hoped there wasn't. All he wanted was to get the hell out of the pub and away from the faux-Brit as quickly as possible. He signaled to the bartender and asked her for the check.
"Now there's a piece of work," said Donald, after the bartender delivered their bill.
Adam pulled out a twenty to pay for his share.
"We should leave her a little more, don't you think?" said Donald.
Adam looked at the bill, looked at the twenty, then looked at Donald. He realized the guy expected him to pay for their drinks.
"I thought we would split it," he said.
"Come on, mate. You're the one with all the money." He laughed.
"Not anymore," Adam said weakly. But Donald had left him in an impossible situation. Either he was a tight-fisted jerk or a gullible rube. He pulled another twenty out of his pocket and laid it on the bar. Then he stood.
"You taking a cab back?" Donald asked.
"I think I'll walk," Adam lied. He didn't want to get stuck in a cab with Donald. "How about you?"
Donald stared wolfishly at the bartender who was just picking up Adam's money. "Nah. I think I'll stay for a few more rounds."
The bartender counted out the change and laid it back on the bar.
"Keep it," Donald said to her.
She smiled and dipped down to pick up Adam's change, the white crescents of her cleavage shining like twinned moons. "Thank you," she said to Donald in a lilting accent.
"You English?" asked Donald.
"Irish," she said.
"You've got beautiful eyes."
When Adam walked out the door Donald was leaning in to the bar, his head practically nestled between the bartender's breasts. The bartender was laughing as if he were the funniest man in the world instead of a cheap, slime-ball, low-life journalist.
The door slammed shut and a foul gust of diesel fumes blew in from the expressway. Adam zipped up his jacket and walked quickly toward home.
## 10
## Money for Nothing; Drinks for Free
STACI FORTUNATO LIVED IN THE one high-rise apartment building in Gowanus that wasn't infested with rats, lice, bedbugs, or all three. A building that was far too expensive for a student, but located in far too unappealing a location for anyone else. It had been designed by a famous Italian architect who thought Gowanus was a charming up-and-coming neighborhood in southern Manhattan rather than a trash heap on a toxic canal. As a result, the building stood out like a raised middle finger among sore thumbs, calling attention to itself and telling its neighbors to go screw themselves. The developer went bankrupt in the middle of construction when the scope of the architect's plans were revealed, and a Saudi bank bought the rights at a fire sale. It converted the top two floors to penthouse suites, reserved for a Prince who never set foot inside, but whose relatives often used the lobby as a soccer field or a place to tether their goats.
Staci's apartment had sweeping views of lower Manhattan, blurred only by the fumes that wafted from the crematorium across the street. Her father bought the apartment when Staci started law school. It was the perfect residence for his beloved daughter, and an occasional refuge when his wife locked him out of their house. He wanted to buy something in Manhattan, but Staci was both logistically and philosophically opposed to the public transportation system, and licensed cabs refused to travel to Gowanus. Normally, she would never have deigned to live in a borough without a Prada boutique, but she visited the apartment on a day when the setting sun cast a golden glow off the building's soft marble and the green-tinted windows shimmered iridescently in the fading light like newly printed hundred dollar bills. Also, that day the goats were at the slaughterhouse.
The doormen loved Staci—as older men always did—and greeted her effusively each time she entered the lobby. After all, she was one of the few residents who didn't eat trash and shed all over the carpet. They held the door for her, and gave her boyfriends the evil eye. Once they even banished an ardent suitor who tried to carry a very drunk Staci into the elevator. The building's superintendent escorted her to her apartment instead, and made sure she had brushed her teeth before shutting the lights and double locking the door behind him. She had that effect on people: solicitude, followed by heavy security precautions.
Now, as she stepped out of the shower and dropped her towel, she padded across the cherry floors without a care as to who could see her through the floor to ceiling windows. She threw open the double doors of her walk-in closet and scrutinized the shelves for the perfect outfit. On the left were tops and jeans—stacked from the darkest black to the most faded blue—while the right wall was entirely shoes. Directly in front were four drawers of lingerie and leggings, and one drawer devoted entirely to contraceptive devices. She settled on a conservative thong and matching bra, along with black tights and a top that barely covered her bottom and couldn't possibly keep her warm. A pair of black boots with a stiletto heel added three inches to her petite frame. Hair extensions added another six to her locks. In just ninety minutes, she was dressed.
It took another hour to pluck, tweeze, polish, and apply her makeup. She was fastidious about giving her skin exactly the right shine: too much and she looked like Joan Rivers, too little and she wouldn't glow in the dark. Finding the right perfume—her own blend of Marc Jacobs Daisy and a glass of Kendall-Jackson Chardonnay—and carefully misting herself took another thirty minutes. Finally, she spent another quarter of an hour scrunching her face in front of the mirror to be certain no lines showed and her foundation didn't crack. All in all, it was 10 PM by the time she was ready to leave.
In the lobby, her shoes clacked over the granite tile, distracting the men (and the goats). She smiled at all and none of them, and ran a hand along the curve of her thigh to smooth her body-hugging leggings. One of the men—a Saudi diplomat wanted for tax evasion—lifted a hand to call her over, but she ignored him. Ever since watching Lord of the Rings, she had something against people from the Middle East.
Fernando, the doorman on duty, greeted her with a kiss. He felt about Staci the way a man would feel about his own adopted stepdaughter—if that man were in prison for incest. He scolded her for not being properly dressed for the elements, and insisted she wait inside while he hailed a cab. When he returned, his face was flushed and a jet black Lincoln Continental idled by the curb.
Her destination was less than a half mile from her apartment, but Staci did not believe in walking. Especially not in heels that carried a warning label from the American Society of Podiatrists. The limo driver was not pleased by the short fare, but he brightened when Staci adjusted her top and gave him a better glimpse of her breasts. She tipped him generously, but only because she was bad at math.
Bar Bar was the watering hole for the law school's serious drinkers, those who couldn't afford a more expensive prescription drug addiction or whose dealers were out of town. Founded in 1867 to provide sustenance for workers dredging the Gowanus Canal, its rise and fall followed the history of that ill-fated waterway. By World War One it was one of the most popular bars in the region, and men spilled onto the street through its saloon-style doors both day and night. But with the building of the nearby expressway and construction of the F subway line, the bar sank into disuse. Although the canal usually smelled worse, there were nights when the back of the bar rivaled it. Different proprietors tried their hand at everything from darts to karaoke to cockfighting, but the bar didn't turn a profit until the latest owner renamed it and constructed a Wall of Shame on which he invited students from the neighboring law school to paper their law firm rejection letters. The Wall now had over thirty thousand papers, on which patrons scrawled graffiti. It had recently been featured in the Styles section of the Sunday New York Times ("Dear [Your Name Here]/Local bar puts a face to the rejection").
Staci greeted the bouncer with a double air kiss. She had been in law school only three months, but already she felt like a regular. Nearly everyone at the bar was either a student or a recent unemployed alum with nowhere else to go. Each year a new class claimed Bar Bar for itself, the way a group of tourists glom onto a tour guide or a John falls for a prostitute, confusing business for affection. Bar Bar cared for the students about as much as Manhattan Law School did—it just offered more for the money.
Staci's friends were gathered at a table in the rear. The boys were sharing a pitcher and Samantha was drinking a Manhattan on the rocks. Sam was dressed like a retro chic punk rocker with clip-on safety pins and temporary tattoos. She was scared of needles, and had only one permanent tattoo: a jet coming in for a landing on her pubis. She whooped when she saw Staci, and gave her a wet kiss on the lips.
"Girlfriend!" she shrieked.
"You're drunk," said Staci.
"And beautiful!"
Staci took a sip of Samantha's Manhattan, but the bourbon twisted through her gut like a trapped opossum. She was a wine drinker, normally. She refused to drink beer because it reminded her of frat basements and keg parties, wasted hours she preferred to forget. She looked around for someone to buy her a real cocktail, but the boys were engaged in a loud argument about the Jets and she couldn't get their attention.
Frustrated, and slightly annoyed by the drunken Samantha, she stomped off to the bar to buy herself a vodka and Red Bull. She carried no cash, but was well-armed with her father's credit cards, some of which he knew she had. As she pushed her way to the front, she ignored the young men who stared at her hungrily as if she were a baby seal waiting to be clubbed.
The bartender, a giant with a shaved head, attended to her immediately, ignoring her classmates who clamored for another round at the far end.
"Here you go, sweetheart," he said as he set down her glass of amber and ice. "That'll be twelve dollars."
Staci withdrew the black AmEx from her ostrich skin wallet.
"No credit cards," said the giant, pointing to a huge sign behind the bar that said Cash, Grass, or Ass—Nobody Drinks for Free.
Staci blinked uncomprehendingly. "Is that new?"
"No."
"But.... then how do I pay for it?"
"You know those green pieces of paper? With the picture of Alexander Hamilton?"
"I don't have any cash," said Staci, pronouncing "cash" as if it were "herpes."
The bartender withdrew the drink from the counter. "Sorry. No tickey, no laundry." Although he could have made an exception and given it to Staci on the house, he hated her. Truly, he hated everyone in the place: the spoiled, unjustly entitled Manhattan Law School students who rarely tipped, hardly paid, and looked mostly awful (Staci being the exception). His real passion was the drums, but his thrash metal band was stuck in litigation with their manager who had impounded their equipment. The law was killing him—all day and all night.
"Nooooo!" Staci wailed.
As he lifted the drink to spill it into the sink, the undersized man standing next to Staci slapped a twenty on the bar.
"Drink's on me," said Asher Herman.
Staci turned just as the Jets scored a touchdown. A roar from the crowd, and the blue light from the television screen cut the hollows of Asher's face like the sun glinting off a glacier. He had never looked more rugged or attractive, and never would again. Staci cozied up against him. "I'll have two, then," she said.
"Make it three," said Asher, feeling expansive as he slapped down another twenty.
The bartender frowned, but mixed up two more cocktails.
Staci took her vodkas and raised a glass to Asher. "I love a man with cash."
"I'll drink to that," said Asher. He clinked her glass and grinned like a dervish.
At that moment the Jets converted the extra point, and John Tarantula, Asher's friend and fellow editor, shouted "Score!"
"Keep it in your pants," said Staci, who thought he was talking about Asher. It took more than two drinks to get to her end zone.
Asher quickly explained John was talking about the Jets. "He gets excited."
"I can't help it," said John, wiping away tears.
"We don't let him out of the Law Review office except on game day."
Staci stopped. "You're on the Law Review?"
"We both are," said Asher. "I'm the Editor in Chief. The most important one."
The Law Review writing competition, Staci knew, was judged by students. It was supposedly anonymous, but already she'd heard rumors of favoritism and preferential treatment. Like any student-run organization, it was easily corrupted.
Staci smiled flirtatiously at Asher, and shook her hair loose. "I heard you guys get the best jobs."
"No one gets better," said Asher truthfully.
"And there's a writing competition where you pick the new editors."
Asher hesitated. "Uh, that's one way we do it."
"Is there another?" Staci leaned in close so that her breasts were practically rubbing against Asher's shirt.
Asher looked to John, but John's attention had returned to the Jets who were now being beaten back on defense. The Law Review's rules were quite strict, and Asher knew them. But he had never stood next to someone as attractive and good-smelling as Staci, and the pheromones played havoc with his neurons.
"Can I trust you?" asked Asher.
"Of course you can trust me. Don't I look trustable?" She gave him her best trustworthy look: eyes lowered, lips pouting, thumb hooked inside her waistband and fingers splayed beneath.
He lowered his voice. "There's Pay for an A."
"Pay for an A?"
Asher could have sucked Staci's ear, he was that close. As he whispered the details, his lips brushed the whorled ridges that guarded entry to the deeper parts of Staci's brain. It sent a chill down her back, not because of the physical contact, but because his words were so naughty, delicious, and utterly corrupting.
"So that's how it works," he concluded.
"And all you have to do is invite me?"
"I'm the Editor in Chief." He sounded offended.
"Then I accept your invitation."
"I think there's a vote, too," Asher stammered. "At the end of your first year."
"I thought you said you were the Editor in Chief?"
"I am."
"Then you just voted." She gave him a quick peck near his lips, but it was enough to send his circulatory system into overdrive, shutting down his brain and shunting his blood to his nethers.
Asher ordered another round, and three drinks became five which soon became nine. The jukebox spun through a selection of tunes about love, loss, and late nights wasted on quixotic pursuits. In a dark corner, someone lit his torts casebook on fire, and the smell of burning paper mingled with stale beer and expensive perfume. Conversations rose and fell, the noise punctuated by a shout or a burst of laughter. At midnight, Staci's friends left without her. Samantha murmured a cautionary word into her ear. "Loser."
But Staci ignored her. She knew what she wanted; it was her friends who would go begging for jobs. Hard work and diligence were nothing if you couldn't find the ways around them. Those were the lessons her life had taught her. Law school perplexed her at first with its emphasis on actual reading and writing, but she knew she would figure it out. She always did.
The warmth of the alcohol suffused her, making her extremities tingle and her adrenaline flow. She wanted to jump Asher's bones, dig in her spurs, ride him until he barked. By now she was sitting in his lap and stroking his four-day-old stubble while he struggled to remain upright on the bar stool.
"Take me home," she whispered.
"I have roommates," said Asher.
"I don't."
That was how Asher found himself playing twenty questions with Fernando, who eyed him with suspicion and scorn.
"It's fine, Fernando," Staci reassured him. "It's only for the night."
Fernando hated to see another man with Staci—especially one as young and sexually inadequate as Asher—but unless she was blind drunk or passed out there was little he could do. Instead, he shrugged and then, when Staci wasn't looking, caught Asher's eye and drew a finger across his throat.
"Nice guy," said Asher when they were safely in the elevator.
"Ah, he's a sweetheart. He just doesn't like men, that's all."
Staci pulled Asher down the hall and kicked open her door with the heel of her boot. They fell into the room and onto the couch in a tangle of clothing and undergarments, their body parts bumping and knocking into and against each other. Asher was neither well-endowed nor particularly sober, but he managed to do his part. Staci didn't see supernovas, but the occasional comet that streaked across her vision kept her from falling asleep beneath him.
In the morning, Staci called her father to tell him the good news. It took a little arm-twisting to get him to cough up the money, but he capitulated when she threatened to date a Mexican.
"I thought I already paid your goddamned tuition," he complained.
"You did, Daddy. This is an extra activities fee for Law Review."
"That's a hell of an activities fee."
"Hold on, Daddy. Manuel is calling me on the other line."
After that, things went smoothly, and when she met Asher again that night at Bar Bar she was a member of the club for which she had paid to be a member. There were drinks all around, and much good cheer, and even the bartender didn't mind when Staci rubbed his bald head for luck. No genie appeared, but fortune shined on the few. Of which she now was one.
## 11
## If It Quacks Like a Duck
THE STEAKHOUSE WAS WHERE MEN died and went to heaven (literally—there was a defibrillator mounted by the bathroom): red brocade and draperies, waitresses whose décolletage left little to the imagination, hushed conversation punctuated by uproarious laughter, a humidor and cellar devoted to red wine, single malts, and Cubans.
Howell Goldreckt was already seated when Ann Marie arrived. She saw him waving down the waiter and ordering another cocktail even as he drained the one he had in his hands. He was an enormous man, nearly as wide as he was tall, and she could hear his booming martini order across the restaurant. When the maitre d' asked if she was waiting for someone, she just pointed at Goldreckt, and he nodded sagely.
"Ah, yes. My sympathies."
Ann Marie smiled politely. Not counting her callback interview lunch with a group of lifeless associates at Goldreckt's firm, her only two expensive restaurant meals were in her hometown of Buffalo, New York. One followed her high school graduation (with her father and sister), and the other coincided with the loss of her virginity (with her first serious boyfriend, who, unlike her father, let her order dessert). Although Goldreckt frightened her, the maitre d' frightened her even more. She dreaded saying something that would indicate she was an outsider in this world of privilege and money, or betray her disbelief she belonged here. Silence always served her well in these situations. It was better to keep your mouth shut and be thought a fool, her father was fond of saying, than open it and prove everyone correct. Ann Marie was not a fool.
The maitre d' guided her through the narrow aisles to her table. The hubbub and harrumphing ceased momentarily as she followed him, and a reverential hush fell over the restaurant. Ann Marie was unaware of the effect that her physical presence in a simple navy dress and black shawl had on the male population, but Goldreckt looked up from his drink and nearly choked as he watched her approach.
"Mr. Goldreckt," she said, and extended her hand politely.
"Howie, please," said Goldreckt as he wiped his palm against his trousers before grasping Ann Marie's cool, delicate, and dry fingers.
Ann Marie nodded demurely, but she couldn't possibly bring herself to call Goldreckt by his first name. Not here; not ever. She waited while the maitre d' pulled her chair out, then ushered her into the seat. The waiter hurried over to ask if she wanted something to drink, and she flitted over the wine menu for several minutes before settling on a glass of reasonably priced California Chardonnay. Goldreckt ordered another martini.
The waiter scooted off to the bar, and an awkward silence settled upon the table.
"Well," said Ann Marie. "How are you?"
"Excellent."
"That's excellent," Ann Marie responded.
"Never been more excellent," Goldreckt confirmed.
Ann Marie hadn't known what to expect when she got Goldreckt's voicemail. The rumor was the firm had begun to extend summer associate offers, but when she returned his call he didn't say anything about it. Would he invite her to a fancy restaurant just to tell her there was no job for her? A simple phone call or letter would be cheaper, and a lot less discomfiting. Yet here they were: Ann Marie waiting for the axe; Goldreckt waiting for his drink.
She knew the odds. Despite her GPA, her letters of recommendation, her good looks, and her work habits, Goldreckt's firm had a pattern of taking just one summer associate from Manhattan Law School each year. Although Ann Marie was confident in her own skills, she knew there were others who had outperformed her, students with more exalted Law Review titles, or cozier relationships with some of the professors. She expected the worst, and was prepared for it. Unlike most of her classmates, she did not believe the world owed her something. She worked for what she achieved, and knew that there would always be those who worked harder.
Ann Marie gulped her wine when it arrived. It tasted acrid, like grapes soaked in gasoline. Goldreckt's martini, meanwhile, appeared to have been pure ethanol: it fueled his conversation and sent him sputtering on about his latest deal—the acquisition of a health-care plan by the largest private operator of prisons in the United States, who proposed to replace nonessential medical personnel with low-level felons. Ann Marie nodded politely, and tried to follow the thread of the conversation, but she would have been confused by the intricate legal details even if she weren't distracted and already buzzed as a result of an empty stomach and generally low tolerance for alcohol. Goldreckt's deals tended toward the arcane and barely legal, skirting administrative regulations and venturing into a statutory no-man's land.
"So when the EPA rolled around we told them it was Chinese fireworks—nothing a little topsoil removal wouldn't fix—and the dopes believed us!" Goldreckt exploded with laughter, and Ann Marie followed as best she could.
More war stories followed, the theme of which was Goldreckt as a cunning and cutthroat deal-maker. Pity the fool who mistook the blood on his tie for ketchup. Goldreckt ground him up for hamburger.
The waiter returned to take their order. Goldreckt ordered the biggest steak on the menu with a side of creamed spinach and a baked potato. To maintain his body at fighting weight, he required a minimum of four thousand calories a day. Ann Marie chose the red snapper. Goldreckt insisted they split a dozen oysters and also ordered a Caesar salad as a concession to healthy eating. "Hold the lettuce," he added.
It took the waiter a moment to realize Goldreckt was joking. Then he forced a chuckle, and gave Ann Marie a painful smile.
"My ex-wife thought I was going to have a heart attack before I turned forty," said Goldreckt. "Joke's on her for taking the lump sum!"
Ann Marie felt a pang of compassion for the aforementioned Mrs. Goldreckt, whom she imagined folding towels at a tanning salon where she worked part-time to supplement her meager remaining funds. Goldreckt couldn't have been an easy husband, and he certainly wouldn't be a generous ex. When Ann Marie married, she intended to be able to support herself. It was not the only reason she went to law school, but having a career that would allow her not to worry about next month's rent payment was certainly a priority—and if it meant suffering through dinner with Howell Goldreckt, she would endure it.
Meanwhile, the partner prattled on. He had plans to renovate his home in Bedford, and it was important that Ann Marie know the details. The architect had feminine notions of style against which Goldreckt struggled. Not that he knew anything about style; but he knew what he didn't like: cupolas. The contractors and craftsmen were in cahoots with the supply houses, and everyone was in thrall to the local zoning board. Goldreckt fought them all relentlessly, and would beat them back like barbarians at the gate.
There was a slight lull in the conversation as Goldreckt drained the last of his martini, but then the waiter returned with the oysters, and Goldreckt spent the next ten minutes trying to cajole Ann Marie into eating one. He finally succeeded, although Ann Marie struggled for air as it went down.
"Oysters are an aphrodisiac," Goldreckt noted.
"Maybe if you're an octopus," said Ann Marie, as the color gradually returned to her face.
"I've never met an octopus who liked oysters," said Goldreckt.
"Have you met a lot of octopuses?"
"Octopi," Goldreckt corrected her.
"Pizza pie," Ann Marie giggled. She was, she realized, a little drunk. It was a pleasant feeling, like wrapping her brain in a fuzzy blanket. Most of the time she was so damned serious, reading and studying as if her life depended on it, which it probably did. She was determined to pay back her father's miserly contribution to her education, and even do some good in the world—although exactly what that good would be she still did not know. Getting drunk was not part of her plan.
The waiter refilled her wine, which she didn't recall ordering. Goldreckt had stopped talking and, to her surprise, she was telling a story about the dog her sorority adopted. A little mutt they named Chet, after one of the girl's ex-boyfriends. The college threatened to fine them if they kept him, so the women took turns hiding Chet in their room. They couldn't bear to let the dog go, even though he was never housetrained, chewed their boots, and howled when they tried to sleep late. One day, after receiving their third warning letter from the college provost, one of the sisters "accidentally" let Chet outside, and he promptly ran away, never to be heard from again. Although they were heartbroken at first, the girls quickly realized they were better off without the dog. His absence was a blessing, and they soon forgot all about him.
Ann Marie had told the story before, but this time as she was talking she realized it was a metaphor for her relationship with Gary. The mewling, weeping, and soiling of sheets. The threats and provocations. The open window through which Gary jumped that required a night at Methodist Hospital but no lasting injury. Unlike the dog, however, she couldn't just remove Gary's collar and hope he would go away. He was permanent, like a bad tattoo.
Yet somehow she had managed to survive her first year of law school, and even to ace her final exams (it helped that she broke up with Gary right before they began). She made Law Review, got an internship with a local judge in Buffalo for the summer, and was one of a handful of students to land an interview with the few firms that hired from MLS. By the time of Goldreckt's phone call, she had managed to put the craziness of that year behind her.
Now the future speared an untouched oyster from her plate. "Are you going to eat that?" asked Goldreckt. It was down his gullet before she could respond.
"Please, help yourself," said Ann Marie. The wine had emboldened her, but Goldreckt missed the sarcasm.
"Some weekend I'll take you to the shore. Best oysters in the world."
That should have been her first warning, but Ann Marie was so inured to men making passes and inappropriate remarks that she had developed a survival mechanism known as Selective Defensive Deafness. It enabled her to put up with men like Goldreckt. Like a trauma victim, certain synapses simply refused to fire, leaving her in blissful ignorance of boorish behavior. It was how Ann Marie ended up with Gary. She listened to his tales of woe until the wee hours of the morning, then slept with him because it seemed like the kindest thing to do. Fixing the mistake took most of the ensuing year and all the antipathy she could muster.
Goldreckt, on the other hand, was so used to women batting, slapping, and spitting at him, that he immediately assumed the absence of hostility was an invitation. Ann Marie owed him one, after all, even if she didn't know it. Thanks to his help she had jumped to the front of the line. Now he would extract his pound of flesh.
"We went there last summer after my pool party," said Goldreckt.
Ann Marie nodded smartly, as if Goldreckt's social life were the most interesting thing in the world to her.
"For the associates," he added.
Her ears buzzed. Was he making her an offer? Had she heard correctly? Although she was bad with innuendo, she was pretty good at logical reasoning and had scored in the ninety-third percentile on that section of the LSAT. She came to law school on one of MLS's few merit scholarships, and passed up a place at Fordham to secure it. Since then, Ann Marie had lived up to her promise as a bright light of her class. She was a pleaser, and drawn to men who needed saving—from themselves, their wives, their lives. They, in turn, were mesmerized by her preternatural calm, her sweetness, and her hourglass figure.
Now Goldreckt sat before her, panting like a puppy. There was lust, but abject need, as well. For all his wealth and bluster, Goldreckt was a lonely, lonely man. In his secret heart he was still the fat boy with no friends. He had bought his ex-wives' affection (in one case, literally), and regularly paid for companionship, yet Ann Marie treated him with genuine kindness, which he had never experienced. She couldn't help it—the emotion was hardwired in her personality—but for Goldreckt it was a revelation. Until then his work and personal life had been an endless series of transactions. For Ann Marie, however, he would have sacrificed anything—if only she would say the word.
Ann Marie felt that power, and it was better than oysters.
"Why Mr. Goldreckt," she said coyly, "is that an offer on the table?"
"Howie, please."
Ann Marie's mouth formed the syllable, but the best she could manage was, "How can I ever thank you?"
"So, you accept?"
"Yes!" If Goldreckt had tried to kiss her at that moment, Ann Marie might have complied. Fortunately, for her, he did not. All she could think about was calling her father to tell him the good news. He had encouraged her to go to Fordham, even if it meant she had to mortgage her future. He was cheap, but not stupid. He had read up on Manhattan Law School, and claimed it was a racket, a pyramid scheme that took students' money and left them floundering. She might get a degree, but she wouldn't have a job. But she couldn't bear the thought of all of those loans, debt that would take decades to pay off. And now she had a job—a good one for the summer—that, if she played it right and worked hard, would fill her pockets, sharpen her mind, and brighten her future.
"This calls for a celebration," Goldreckt insisted. He called the waiter over and ordered a bottle of expensive champagne. He touched Ann Marie's hand when he raised his glass and she didn't flinch. The champagne was excellent, dry with a hint of citrus. Goldreckt gulped his down in one swig.
It was a night to remember, although Ann Marie would have trouble recalling specifics the next day. At one point in the evening she could have sworn Goldreckt started quacking like a duck, but she chalked it up to a hallucination. For the most part, he was a gentleman, and Ann Marie was disarmed and enchanted. They left the restaurant together, and he hailed her a cab, making sure her seatbelt was fastened before he shut the door. When the cab sped up the avenue she turned around to see the partner standing in the middle of the street without his shoes or socks, dancing what looked like a waltz with an imaginary partner as the maître d' and several waiters tried to usher him back inside.
## 12
## Cat Scratch Fever
THE GRADES ARRIVED WITHOUT WARNING, stuck between bar exam review brochures ("never too early!"), loan refinancing solicitations ("never too late!"), and stale Christmas fruitcake courtesy of the Manhattan Law School Christian Fellowship. The school had accepted that email was here to stay, but grades were still delivered in hard copy, inside transparent white envelopes that turned the purest members of the student body into criminals, unable to resist sneaking a peek inside the boxes of their friends, or, more often, their enemies.
Despite the liberal curve and the scourge of grade inflation, there were still more C's than A's, more dissatisfied customers than happy ones. The grades seemed random and unjustified, an unfair and inaccurate representation of a student's worth, as disconnected a reflection on the ability to practice law as a law school degree itself. Each year the Student Governance Counsel proposed switching to an honors/pass/fail system ("just like Yale!"), but there was too much resistance from the faculty and members of the Law Review who, after all, had a lot invested in letter grades.
Staci carefully stepped over a student who was sleeping in the mail room and searched for her box. She had only been to this corner of the basement once before, when a friend left her a dozen pills of Adderall hidden inside the handle of a plastic gavel. She found her mailbox stuffed with credit card offers and invitations to open houses from student legal organizations, intramural sports teams, and obscure religious sects. Stuck to the inside of the box, taped closed—not even sealed—she found the envelope and quickly opened it. Straight A's, just as Asher had promised. But then she blinked and rubbed her eyes, and one of the letters fattened and turned on her like a high school frenemy. She shook the page, but the grade was still there. She threw it to the ground, stomped on it, then picked it up once more. But there it was, albeit smudged and splotched with brown mashed cockroach. Professor Wright had given her a C– in Torts.
She kicked the sleeping student. He rolled over and blinked at her. "Mom?" he asked.
"Shut up, jerkoff," she said.
"Name's Gary," he said, then went back to bed.
Staci ignored him. How could Professor Wright do this? She paid good money for those grades. It was more than unfair; it was practically criminal. Her father would be furious, although she had no intention of telling him. He had already mortgaged a good part of her sisters' inheritance to send her to law school, and though she felt some guilt for it, she had no desire to pay them back.
It was 2:45 on a Friday afternoon. The new semester had barely begun. The registrar had surely left already. The administrative offices were probably locked for the weekend. And who could she complain to anyway? Who would hear her plea for equality in dishonesty, justice for the corrupt? No, she would have to do this herself. She withdrew her rhinestone-studded compact from her purse and touched up her makeup, then tightened her shit-kicking boots. With one last look at her grade for inspiration, she went right for the man himself, straight to Professor Wright's office.
It took some time, however, to find it. For security reasons, there was no directory that listed office numbers. Staci had also thrown away the class syllabus on which Professor Wright detailed his office hours and location. Plus, the office was in an obscure part of the building that could only be accessed by going to the fifth floor and then walking down three flights, which Staci discovered when she asked directions from a janitor who was looking for cleaning supplies suitable for huffing.
Professor Wright was packing a leather satchel with the Torts casebook and a thick stack of articles when Staci banged on his partially open door, then stomped into the office.
"Can I come in?" It wasn't a question.
"Make yourself at home," he said, although sarcasm was lost on her.
Now that she was actually inside Professor Wright's office, her resolve wavered. She had never been in a professor's office before—at least not for an academic reason—and was surprised by how spartan and shabby it looked. Professor Wright's desk was crammed into one corner, between a heating duct and a large rat trap. With no bookshelves, his papers and books were piled on the floor. A single light bulb swung from a wire in the ceiling. All in all, it resembled a cell on Rikers Island—at least from what Staci had seen on Law & Order.
"How can I help you, Ms. Fortunato?" Professor Wright asked.
"I just got your grade. My grade," she corrected herself.
Professor Wright sighed, as if he had hoped to avoid this conversation. He seemed to be judging the distance to the door to see if he had enough room to escape.
"I'm sorry. But there's a strict curve, and I don't set the numbers. Your performance on the exam—"
"Yes, but we had a deal."
"A deal?"
"You were supposed to give me an A."
Wright cocked his head. "I don't remember making that deal. Like I said, your performance on the exam—"
"With the Law Review. Pay for an A, you know?"
Staci could have sworn she heard Professor Wright gasp, but she decided it must have been a burp, which was just plain rude.
"Are you saying you expected to pay for a higher grade?"
"Duh. I paid for an A. And it wasn't cheap," said Staci.
"Is this common?"
Staci hesitated. Asher assured her all the professors knew about the system, and everyone had signed on. But Professor Wright sounded like he had never heard of it, and he did look a little confused. He was new, Staci reminded herself. Perhaps he just didn't understand the details.
"First-years don't participate. Usually," Staci explained.
"But everyone else does?"
"Well, most everyone. I don't know. I assume so. I mean, I assumed you knew."
Professor Wright shook his head. "I had heard some rumors, but...." his voice trailed off.
"Oh shit. I'm sorry I said anything. Forget about it." Staci took one step backward toward the door. "Fuck. Sorry about the 's' word," she added.
"Wait," said Professor Wright. "Who told you about it?"
"My, uh, boyfriend." Soon-to-be-ex-boyfriend, she thought. "He's on Law Review."
"And the professors. You said everyone participated?"
"I don't know. Don't you have faculty meetings about this kind of stuff?" Now Staci was nervous, and her palms and neck began to sweat. She hated her own perspiration, almost as much as she hated body hair, and now she was grossing herself out.
"Did everyone else give you an A?"
Staci nodded.
"And you studied the same for their exams?"
"Of course."
The tiny room was making Staci claustrophobic, and Professor Wright was staring at her as if she had forgotten to tweeze her eyebrows.
"Anyway, Prof, I gotta run," she said. "Sorry about everything. Really."
She bolted out of his office without looking back. By the time she reached Asher's building, three blocks away, she was red-faced and winded. Her breath fogged in the winter air, or maybe it was the smoke from her nostrils. The building was squat and wide, and resembled a fortress. It had been built for low-income housing, but was repossessed by the city for housing code and drug violations, then sold to the law school for a box of pencil erasers and an assumption of debt. The school's housing clinic quickly got the violations dismissed, then locked out the low-income tenants and dumped their possessions along the bank of the canal. The building had no lobby or doorman, and the front door was made of reinforced steel and bulletproof glass. It was yawning open on two broken hinges when Staci shoved through it with the heel of her palm. Inside it smelled like stuffed-crust pizza, old condoms, and socks. Asher's apartment was on the top floor, a five-flight haul, and Staci took the stairs two at a time, her little legs churning like an Olympic hurdler. One floor below the landing she encountered Asher's roommate, a Mormon who had gotten off track on the way to his mission and remained in Gowanus.
"Hi, Staci," he said brightly.
"Fuck you, Hal," she said as she sprinted past him.
Asher was sitting on his Craigslist couch watching SportsCenter and eating Lucky Charms straight from the box. His feet were propped up on the empty Con Ed cable spool they used as a table, next to the carton of milk and surrounded by burning candles that smelled like cheap aftershave. On his iPod, a rapper spat out angry rhymes about privileged white kids. He waved broadly when he saw Staci and motioned her inside.
"Heyyyyyy babe." He removed his headphones and handed her the box of cereal.
"I got a fucking C minus!" she said.
"Wha?" said Asher.
"C minus. In Professor Fucking Wright's Fucking Torts class."
"Wow. That sucks."
She advanced on him like a matador. "You said it was Pay for an A. I paid. I want my A!"
"But you got a C minus."
"No shit, Sherlock. And I paid for an A."
Asher's brain worked slowly, even in the best of circumstances. Now he gradually put it together like a Contracts hypothetical: Student S pays four professors to give her an A; three accept, but the fourth declines. Does she have a cause of action? "It's a unilateral contract," he explained. "It's not enforceable because there's no meeting of the minds."
"What the fuck are you talking about?"
"Wright is new. He probably doesn't know about the system yet."
"He doesn't!"
"See. I told you." Satisfied, Asher pulled a jug of milk from beside the couch and took a gulp.
"But what about my money?"
Asher shrugged. "You still got three A's, right? Have some milk." He offered her the container.
"I paid for four A's."
"You're lucky you got three."
"What the fuck does that mean?"
Later, Asher would regret his choice of words. Perhaps, if he'd been more politic, things would have turned out differently. But he suffered from the illusion that being handed something was the same thing as earning it.
"Come on Staci. You're not the brightest bulb in the socket. You know that."
"Who's talking about bulbs?"
"In a normal semester you'd be lucky to get four C's," he continued. "We gave you a gift."
"I paid you!"
"We don't usually let first-years buy in. We did it as a favor to me." He winked.
Staci looked around for something to throw at him. The TV was far too big; the box of cereal too small. Then she saw the cat, scratching the plastic legs of the TV stand. She grabbed it with both hands and flung it before Asher had time to react. The cat, however, was quicker. It pirouetted in mid-flight and extended its claws, landing on Asher's face and digging in before it fell to the table, knocked over the container of milk, then landed feet first on the carpet where it began to lap up the milk.
"Ow! Shit! The fucking cat!" yelled Asher.
"Serves you right," said Staci, but a little less sharply. She loved cats even less than dogs, and only slightly above goats, but she hadn't meant to scare it.
"I'm bleeding," said Asher. Three ragged lines ran down his face like war paint.
"Oh, baby, I'm sorry. Let me see." Staci tucked in next to Asher on the couch.
"It hurts," Asher whined.
"Poor baby." Staci examined the wound, then said, "You should get that cat checked for rabies."
"Really?"
"Yeah. This cut looks a little infected."
Asher bolted upright on the couch. "You think so?"
"No, you idiot," said Staci. "You can't get rabies from a fucking house cat."
"But, you said...."
"And you believed me. Which makes you the dumb bulb."
Asher nodded knowingly. "I get it. You're just mad because I said that thing about the C's."
"And you're so smart? You've been buying your grades for the past two years."
"I'm the Editor in Chief of the Law Review."
"Really? Who'd you blow to get that job?"
"I'm sorry, okay? I'll talk to Professor Wright about the grade."
"Like that's going to change anything."
"It's not right."
"No. Fuck the grade. And fuck you and your pricks on Law Review. Strutting around like you earned it. Well, you know what? You didn't earn shit. You may have paid for it, but you don't deserve it."
"What's the difference? An A is an A. No one cares how you got it."
"Maybe that's why you don't have a job. Because people know you don't deserve it."
"I don't have a job because our school sucks."
"No, you suck."
"Whatever." Asher touched the wound on his face. He missed and poked himself in the eye instead. "Dammit," he said.
"Loser," said Staci.
But if he was a loser, what did that make her? A loser's girlfriend who couldn't even buy an A. It would be wrong to call the moment an epiphany, but something shifted, and a slight crack of self-awareness shined a light through the opacity of her narcissism. In the shadows, she could see her better half.
"Do you smell something?" she asked. "Toast?"
"Like someone burned it."
"Must be Hal."
"No. Mormons can't eat toast."
"I'll check."
Asher rose from the couch, but it was too late. The smoke was beneath them and then all around them, billowing from the carpet where the milk container had knocked over the candles. A lick of flame rose from the sparkling fibers of flammable polyester. The cat clawed at the front door trying to escape. Staci saw her life pass before her eyes, the regrets and missed opportunities, the lies, and unexamined possibilities. But then she realized it wasn't her life, or it didn't need to be. So she pushed past Asher, opened the door, and left that life behind.
## 13
## Invasion of the Body Snatchers
ADAM AWOKE IN A RAY of light. It bathed his face in a golden glow and flitted across his eyelids like a beckoning promise. The blankets around his body were warm and the pillow beneath his head was soft. In the air was the scent of lavender, and outside there were real birds chirping near the open window. He rolled to his side, and reached for the woman lying next to him.
Laura.
He had been spending nights at her apartment, even though his was closer to school and classes were back in session after Christmas break. Hers, however, was better decorated, cleaner, and didn't need mouse traps. It also had Laura.
He trailed his fingers over the smooth skin of her back, a cocoa color that faded to pale tan below her waist. Her hair fanned across one shoulder, a wavy brown-black that kinked and curled as it fell. She sighed in her sleep but didn't move. He rested his hand at the base of her neck, the narrow points of her shoulder blades climbing on either side. He could feel the rise and fall of her breathing, the gentle ridges of her spine. Laura was a championship sleeper, he had learned; it took two alarm clocks and her dog, Monk—a German shepherd/retriever mix—to get her awake.
His previous girlfriend, the aforementioned Jane Van Dyke, was an insomniac and a closet bulimic. Usually both at once. He would awake in the middle of the night to find the bed empty and the lights on in the bathroom, Jane kneeling on the ground, alternately vomiting and typing on her laptop, which was perched on the edge of the sink. When she was in full neurotic mode she subsisted on Cheetos, Twizzlers, Halls cough drops, and Diet Barq's Root Beer, and could go for days without a real meal. In fact, most of the women he dated were hyperactive overachievers who thought they could outsmart the digestive process. Somehow Laura managed to get the hyperachievement without the overactivity, a skill she attributed to yoga, psychotherapy, and a daily multivitamin, but which Adam believed was heredity, self-confidence, and sufficient rest. She was born with the gene for quality sleep.
Monk greeted him as he slipped out from the blankets and tiptoed to the bath. The dog waited patiently while Adam did his business, then followed him into the tiny galley kitchen.
"Okay," said Adam, knowing what Monk wanted. "But coffee, first."
Laura was a bit of a coffee snob, and her espresso machine and French press sat next to a ceramic jar of beans and a dual-burr grinder. The setup intimidated him, and he had yet to make himself a fresh cup in the morning. Instead, he found the thermos with yesterday's brew and reheated a cup in the microwave. Then, cup in hand, he grabbed a coat from the closet and leashed up Monk for the walk outside.
The coat was ridiculously short—Laura's Burberry rain poncho—but it covered his ratty, old Harvard Law T-shirt and hid the fact he was still in his boxers. Although he looked like a flasher, at least he wouldn't be arrested—as long as he kept the coat closed.
Adam would have preferred a walk along the promenade that went beneath the Brooklyn Bridge, but Monk pulled him in the opposite direction toward his favorite spot, a tiny patch of grass the city insisted on calling a "park." Monk circled around until he found a spot to his liking, then squatted on his haunches. What did dogs smell, Adam wondered, that made one location preferable to any other? Dogs were so particular about where they made their mark, and yet it all looked the same to Adam, one bit of dirty city indistinguishable from another. It was, perhaps, the same thing dogs wondered about humans.
Monk finished, and Adam fished in his pockets for something to pick up his droppings. He had forgotten a plastic baggie, and his sense of civic duty compelled him to clean up the waste—that, or fear of being arrested. He didn't want to be a snarky "Law Professor Above the Law" headline on Gawker. He looked around for something to cover his hands, and spotted several newspapers lying atop a garbage can. He grabbed the top one and tore off the first page. There, below the fold, was an article by D. Campbell Wiley about the implosion of yet another Wall Street law firm. Adam wrapped the article around the dog feces and tossed them both in the trash.
When he got back to the apartment, Laura was still sleeping. He reheated another cup of coffee and sat down at the kitchen table. The birds chirped, but now he could hear the grinding gears of a garbage truck backing into the alley behind Laura's apartment. It beeped loudly in an octave designed to attract raccoons and drive humans crazy. As far as he could tell, city employees worked from 4 in the morning until daybreak, unless they were working on the street outside his classroom window, in which case they worked on Mondays, Wednesdays, and Fridays, from 10 until 10:50.
Adam got up and slammed the window shut.
He wished that student—Staci Fortunato—had never said a word. Why couldn't she go home, pout, drink it off, lie to her parents? She could rant on a message board, tweet his name in infamy; he wouldn't have cared. Instead, she had to spill the beans.
In truth, the clues had been there all along: the subcommittee meetings behind closed doors, the jokes about "expense accounts" and "donation dinners," Jasper Jeffries's new BMW 535i. If students were paying for grades, it would explain a lot of things.
Yet he had no proof. An untrustworthy pseudo-journalist and an angry student did not constitute guilt beyond a reasonable doubt. They weren't even a preponderance of the evidence. He had no documents, no photographs, no surreptitious recordings. Instead, he had suspicions. Even in the court of public opinion that wouldn't do.
He peered over at the bedroom where Laura rested peacefully. How deep did the lies go? He couldn't believe she was involved, but he couldn't believe she wasn't. Her warning from months earlier rang with an eerie clarity: They don't need to know what goes on here. Nights they had lain together, Laura's head resting on his chest, telling each other everything. Or, now it seemed, nothing.
Monk nuzzled his foot. He bent down and petted the dog absentmindedly while gazing out over the living room. Laura's apartment was small but elegant, with expensive kitchen appliances and designer furniture. That espresso machine was no joke. Her bathroom was tiled in Italian ceramics, and the paint in her bedroom had been hand-detailed by a local artisan. Now he wondered how she could afford these luxuries on a professor's salary.
He rose and walked to her bookshelves. There were the usual rows of shamans and sophists—Hobbes, Mill, Marx, Locke, Rousseau, Descartes, Kant, Hume, Gramsci, Foucault, Derrida. Beneath them, the serious nonfiction, biographies, and histories. Then the novels and short-story collections, the books of poetry and art history, and finally the paperbacks and popular works of fiction. He poked around in them, removing one from the shelf and rifling its pages, then returning it and pulling out its neighbor. He wasn't even sure what he was looking for. An account ledger? A black book with names, dates, and receipts? A wire-transfer record or a stack of bills stashed in a hollowed-out novel?
"What are you doing?"
He jumped as if burned by a hot poker. "Me?" he asked, as if she might be talking to Monk.
"Are you looking for pictures of old boyfriends? There are probably a few of them in there." Laura was in a nightshirt and boxers, her hair still gathered at one shoulder, her bare legs muscular and brown.
"No. I was just trying to find something to read."
"Then why do you look so guilty?"
"I don't."
"You can't see your own face."
She was a tough prosecutor. And he was an even worse liar.
"Laura." He hesitated. "A student came to me."
"Okay. But her photo's definitely not in my bookcase."
"I know."
Laura shook out her hair. "Should I be jealous?"
"She told me something."
"Something about what?"
"The system. Pay for an A. She thought she had bought a better grade."
He expected Laura to deny it. Or laugh. Or plead ignorance. Instead, she sighed and said, "Well, now you know."
"It's not true, is it?"
"Of course it's true. I warned you."
"But how could they...? How could you...?"
"Sell out?"
"Yes. And take such a huge risk."
"There are all kinds of risks."
"Laura. It's crazy."
"No, what's crazy is the system that rips these kids off in the first place. I didn't lie to them about law school or their future. I'm just their professor. I teach, do my research, and keep my head down."
"Even if you forget the ethical issues," said Adam, "you could get caught. Disbarred."
"I think you're exaggerating."
"Students are paying for their grades. That's bribery. Fraud. Larceny."
"This is a private institution." She pulled her shirt tight; her arms wrapped around like a straitjacket. "The government can't regulate a strictly private transaction."
"Prostitution is a private transaction, but it's illegal. So is murder for hire."
"Now you're comparing it to murder?"
"If anyone learned about it, the whole school would blow up," Adam warned. "It would be a huge scandal."
"Maybe. But no one's going to find out."
"How can you be so sure?"
"Because it's been going on for years and no one has found out yet. And, really, is it so different from what happens elsewhere? You hear all the time about professors grading exams by tossing them down the stairs or giving them to their kids to read."
"Those are just rumors. Urban myths."
"There's a random element to grading even in the best of circumstances. You know that."
"Randomness is not the same thing as fraud. You're screwing with our students' lives. They work hard for those grades."
Laura raised one eyebrow at him. "How are they harmed? The ones who deserve good grades still get good grades. It's just that there aren't very many of them."
"You screw with the grading curve and hurt the students who don't get A's. Not all of them are useless or corrupt. The ones who aren't deserve better."
"It's too late to change things." Laura bent down to pet Monk. "There's nothing we can do. We're here, and this is how it works."
"No."
"Come on, it's not worth getting upset over."
Adam wheeled on her. "Not worth it? This is how people's lives are ruined. Trust me, I know what I'm talking about. You can't sit around and close your eyes to it."
"What do you want me to do? File a grievance? Write an op-ed? Call the fire department? Adam, my hands aren't clean. And I don't intend to ruin my career by exposing myself."
"It doesn't have to be that way."
"Maybe not. But it will."
"Laura. Listen."
Laura straightened. "You listen, Adam. Do you really want to lose your teaching job? For what? Some students who treat you like hired help and couldn't get jobs anyway? You're a nice guy, and I like you very much. You're cute, and you're not bad in bed. But don't be naïve. This isn't some abstract debate about Aristotelian ethics. It's the nasty rough and tumble of the real world."
He couldn't believe what he was hearing. Beautiful, clever Laura. With whom he thought things might work out. "What you're giving up is more than just your career."
"I'll be the judge of that, thank you."
"Well, I'm not going to have anything to do with it."
"If you want to keep teaching here, you'll have to."
"Never. And I couldn't be involved with anyone who is."
Laura's eyes did not waver. "You know what you're saying?"
"I do," said Adam.
He felt like he had taken a punch and fallen down a flight of stairs. His teeth hurt, and he was bruised and aching. But he was still standing, and not afraid where his next step would lead.
"So you're saying we're done?"
"Is there anything else to say?"
The words hung between them like a guillotine. Then Laura turned and shut the bedroom door behind her.
He packed his few things and gathered his books. He unplugged his phone and shut off his laptop. And all the way down the hall he could hear Monk barking in vain.
## 14
## The Bernie Madoff Bypass
THE EDITORS WERE IN A panic. They'd gotten a Snapchat from Asher telling them to meet at the office, along with a photo of his bloodied head and the cryptic command, "BURN IT ALL!" Although Asher was prone to hyperbole, the scratches on his face looked real, and the fear made his eyes—already lopsided—skew like a Dali nightmare.
Willow Summer was already in the office, hard at work. Not on articles, of course. She'd been nominated to handle the most important job—recruiting. Pay for an A wasn't as simple as it sounded. The faculty couldn't just advertise that grades were for sale. Instead, each current Law Review member was permitted to invite one new student onto Law Review at the end of every year. Willow was tasked with scanning through the financial aid statements of the first-year class to weed out students whose families were too poor to afford the added cost of Law Review membership. From the list that remained, she would strike troublemakers, hotheads, and do-gooders. Then, the editors would all go into the field and conduct in-depth interviews, probing for problem spots ("do you have trouble keeping a secret?") and running credit checks. On the last Saturday in April, the entire Law Review (except for the Associate Editors) gathered to approve the selections with the assistance of Professor Rodriguez. The writing competition that followed was a complete sham, as fake as the school's hiring statistics, but necessary to preserve the illusion of merit.
"What's going on?" asked Raya Kurdle, who arrived breathless in black Lycra. She had been interrupted in the middle of her jog along the canal just before she would have passed out from the fumes.
"I don't know," said Willow. "He looked freaked."
"That's the way he always looks," said John Tarantula, who followed Raya into the office and nearly decapitated himself on a book shelf because his gaze was fixed on Raya's ass.
"Yeah, but there was blood."
"He cut himself shaving," said John.
"He doesn't shave his forehead," said Raya.
"How do you know?"
Charlie Spires arrived, martini shaker in hand. "Anyone have vermouth?" he asked.
"This is serious, Charlie!" said Willow.
"That's why I'm drinking."
Meanwhile, the object of their concern was only several hundred feet away, filling out paperwork in the basement of the law school. Asher had never been to the student health center before. As far as he knew, no one had. Rumor was that a few years earlier Prof. Ogden Templeton, born 1906, had died there, and the nurse, afraid she would be blamed, simply threw him in a file cabinet and pretended it never happened. At least that would explain the smell of formaldehyde that pervaded the place. But Asher's health plan didn't cover doctor's visits or emergency rooms, and Asher—face scratched, fingers singed—was desperate, so into the bowels of Manhattan Law School he descended.
In the cold, empty waiting room, he was convinced he was dying. His life unspooled before his eyes like the damaged print of an old movie. The regrets, the lost chances, the phone number of that runway model—or perhaps she'd said "runaway model"—he'd accidentally left in his pants before he washed them. In high school he had wanted to be an actor. Then his voice changed (late puberty, but that was another story) and he decided to become a radio deejay. After the university shut down his college station for broadcasting the answers to a chemistry exam, he turned to modern interpretive dance. Yet somehow he ended up here, on the banks of the Gowanus. How did it all go so wrong?
Asher's existential musings were interrupted by Nurse Brinda —or so read her nametag—who called him into the examining room.
"How are we today?" she asked. "Quite a nasty boo-boo we have there."
"I think I might have rabies," said Asher.
"Where did Professor Hotchkins bite us?"
"He didn't bite me."
"Are we a student at the law school?"
"Is there a doctor I could see instead?"
"Open your mouth so we can draw some blood."
"Actually, I think I'm feeling a little better."
He tried to get up from the exam table. Brinda pushed him back down.
"You stay here. We'll be right back with some gauze for your wounds. We're not allergic to gauze, are we?"
"I'm not. Are you?"
Brinda laughed. As soon as she left the room, Asher grabbed his backpack and ran for the door.
"Come back here!" she yelled. "We haven't tested for a hernia yet!"
Asher kept running.
Although the Law Review office was also in the basement, it was in a different section than the nurse's office. To get there required going to the fifth floor, crossing through the Bernie Madoff bypass, then ducking through a small ventilation tunnel before heading back down another stairwell. By the time Asher arrived he was panting, out of breath, and still bleeding from the cat scratches. His hair was matted and clumped, and he was sweating like a pipe in August.
"He's not wearing deodorant!" Willow shrieked.
"What the fuck happened to you, dude?" asked John.
Asher collapsed in a chair. "Water," he managed.
Charlie brought him a bottle of water from a stash he had stolen off a Poland Spring truck earlier in the week. Asher gulped it down, and then burped loudly.
"Staci," he said. "She screwed us."
"Who's Staci?" asked Raya.
"The chick he's been screwing," said Charlie.
"That's gross."
Charlie shrugged. "I don't know; I think she's kind of hot."
"What happened?" asked John.
Asher told them about Staci's grade and her confrontation with Professor Wright.
"Professor Wright scratched you?" asked Willow.
"Not him. The cat."
"Your cat scratched you? Why? And why do you look all... burned?"
"Aren't you listening?" Asher was getting agitated. "Staci told Professor Wright everything."
"So what?" said Charlie. "What can Wright do?"
"He could tell Dean Clopp."
"The Dean knows," replied Charlie. Everyone knows."
"No. I don't think he does. Professor Jeffries said he didn't."
"How is that possible? He's the Dean."
"Have you seen him lately? At 1L orientation they had to pretape his remarks so he could lip-sync them."
"I heard he pooped on the floor of the library," said Raya.
"That wasn't him," Charlie admitted.
John pulled up a chair and perched on the back of it, resting his feet on the cushion. "You've got to go talk to Wright," he said to Asher.
"Me? Why me?"
"Because you fucked this up."
"But it wasn't me! It was Staci."
All four editors gave him a long silent look. "Enough said," said Raya.
Asher hung his head. A drop of sweat rolled off the end of his nose. His scratches stung and his hair itched. He regretted the primal urges that made him want to impress Staci. He should have let her chart her own course through the law school's capricious grading system. But now, of course, it was too late.
The school kept professors' home addresses in the same hidden, password-protected directory as it stored information about the school's legal settlements, health code violations, and data concerning the cancer cluster centered around the student cafeteria. Within a few minutes, Willow had cracked the code ("password123") and printed Professor Wright's address on a slip of paper. "It's a twenty-minute walk," she said.
"I'll expense a cab."
"Not a chance," said Raya. "You're not using the slush fund because you're too lazy to walk."
So he walked.
The neighborhood around the law school was barren and bereft. But as he got closer to Professor Wright's neighborhood, the streets came alive with people: young mothers pushing strollers, kids running home from school, hipsters carrying messenger bags weighed down with Kerouac and David Foster Wallace. Asher gaped openly. It had been so long since he had seen a real person that he forgot they existed. Kids, for example—who knew they were so small? And babies—they couldn't even walk! In his bewildered state he nearly collided with a stroller.
"Watch your fucking self!" the mother yelled at him.
"Fucker!" said the father, glaring over his Warby Parker frames.
"Sorry! Sorry," said Asher. He backed away and banged right into a scrawny guy carrying a guitar and an espresso.
"Fucking dude!" said the guy.
"My bad," said Asher.
"You don't look too good, dude."
"I know. A cat scratched me."
"You should have that checked out. Dude could have rabies."
"Very funny."
"I'm not kidding. Buddy of mine got scratched by a feral hamster and got hepatitis C. I'm telling you, that shit's serious."
Asher scrambled away as fast as he could.
"It's your life, dude!" the guy yelled after him.
Asher was pale and shaken when he arrived at the front stoop of Professor Wright's brownstone in Cobble Hill. The day had started badly and was about to get worse—he had a feeling. He wasn't a moral person, but the wrongness of what he was doing struck him at that moment. It was bad to buy grades, and worse to threaten faculty about it. Yet he was swept up in a system that led him slowly, inexorably, to this point of no return. He blamed society.
He rang Professor Wright's doorbell. No answer. He waited, then rang again, a little longer this time. Still no answer. He rang a third time for a good minute until the ringer sputtered and died.
Relieved, he was about to turn and go home when he heard the clomp of footsteps behind the door. He considered running, but the street was long and wide open and Professor Wright would surely see him before he got very far. Instead, he stood frozen on the stoop like a statue to stupidity.
"Yes?" Professor Wright opened the door. He was dressed in jeans and a white T-shirt. His brown hair curled down around his ears, and he had two days of stubble.
"Professor Wright, I'm Asher Herman."
"Am I supposed to know you?"
"Asher Herman. Editor in Chief of the Law Review."
"You realize you're bleeding?"
"Yeah, a cat scratched me."
"I think you broke my doorbell."
"Sorry. I didn't know you were home."
"I was writing. I don't like to be interrupted."
"Yes. Well. I'm here on a very important matter."
"If it's got to do with the Law Review, I haven't written anything for it. And if I did—if you need to talk to me—I have office hours."
"I'm sure you do," Asher interjected. "But this is a matter that's outside regular office hours."
"Like I said, I'm trying to do some writing. Do you think you could send me an email?"
"It's about Staci Fortunato."
"What about Staci?"
"It's about her grade."
Professor Wright's face darkened. "I've already talked to Staci about her grade."
"I know. And I don't think you understand the system."
"Oh, I understand it quite well."
Asher peered over Professor Wright's shoulder. He could see through to the kitchen: white and black tiled walls, a loaf of bread on the counter, a jar of pickles beside it. "Do you mind if I come in?" He felt exposed standing on the stoop, and he suddenly had a craving for pickles.
Wright opened his door wider, and motioned for Asher to come inside.
"Thank you for inviting me in," said Asher.
"I didn't invite you. You invited yourself."
"Well, thanks for opening the door, at least."
Wright nodded. Standing inside, Asher noticed that the apartment could use a coat of paint. There was a large crack running along one wall and a chunk of plaster was missing from the ceiling.
"I know you're new and all," Asher began. "But there's a system in place; it's been around forever. It works a certain way and everyone knows how it works. They're supposed to tell you—the other professors? There was like an orientation or something."
"There wasn't an orientation. But I know how it works."
"Then you should know that Staci was supposed to get an A."
"Staci got the grade she deserved based on her performance on the exam."
"But she paid for an A."
Professor Wright stared at him. Asher had never been this close to a professor before. He could see the hair follicles on Wright's unshaven face and a small scar above his left eye. There was a loose thread on the neck of his T-shirt where it rubbed against his throat. When Wright exhaled Asher could smell the pickles on his breath.
"No one pays for an A," Professor Wright said.
"You don't understand. She did."
"No, you don't understand. No one in my class pays for an A, a B, or otherwise. Paying for grades is wrong. Didn't your parents teach you that?"
Asher shifted uncomfortably. "My parents had... flexible morals. Besides, everyone does it. And it looks like you could use the money."
"Of course I could use the money, but if all I wanted was money, I'd be a partner at a law firm. Believe it or not, I became a professor because I wanted to teach students, not become rich. Although when I meet students like you I wonder why I'm in this business at all."
"I'm sorry we couldn't all go to Harvard."
"Even at Harvard students work for their grades."
"That's because they're at Harvard. We have to buy our grades because our degrees are worthless."
Professor Wright grimaced. "I think it's time for you to leave." He opened the door and gestured outside.
Asher stood his ground. He decided to try a different tactic. "Come on, Staci's exam couldn't have been that bad. Maybe you can reconsider?"
"If she'd written an A exam, I would have given her an A."
"We both know she couldn't do that."
Wright shook his head sadly. "You realize you're just hurting the students who least deserve to be hurt, don't you? The ones who actually deserve their A's. By perpetuating this scam you're debasing everyone's grades and telling the world you can't hack it as a law student. That's something, I think, you don't want the world to find out."
"Why would the world care?"
"The people who matter will care."
"Who's going to tell them?"
"The law may not always tell us how to act. But that doesn't mean you should sit idly and watch a crime happen. Maybe it's too late for you to learn this—although I hope not." Then he took Asher by the elbow. "Now please get out of here."
Asher stumbled as Professor Wright dragged him from the apartment. The guy was stronger than he looked. Asher caught himself from falling on the outside railing, then turned just in time to have the heavy wooden door slam shut in his face.
"I was leaving anyway!" he shouted.
He descended the stairs and found a patch of sidewalk that looked relatively free of dog feces. Then he sank to the ground. He sat there with his head in his hands trying to remember what the world was like before he decided to go to law school. When he looked up, it had started to rain.
## 15
## A Dump with a View
IT WASN'T FLEEING, EXACTLY. NO one had chased her or driven her out. And it wasn't retreating because that implied being beaten and running away. Yes, they had fought. Yes, it was over. But as Laura stepped off the plane at San Francisco International Airport, she allowed herself a faint glimmer of optimism about the future. The clear blue skies, the lift in the breeze, the moist Pacific air. In the distance the hills sparkled and shined, while wildflowers bloomed in the sunlit valleys below. It was winter in New York, but in San Francisco the season glowed.
She had not been looking for another job although, in truth, it was never far from her mind. But she was always too busy to update her resume, attend the job fairs, or gather her thoughts for a faculty presentation. When a former classmate and friend was appointed Dean at Berkeley in August, however, the second person she called was Laura. (The first was her agent at the Speakers Bureau to tell him to increase her fee.) Rosalind Minsky had been the student graduation speaker at Yale, a Supreme Court clerk, and the youngest tenured professor at Boalt Hall. Fiercely intelligent, opinionated, and sober, she was everything Dean Clopp was not. Since then, she and Laura had traded messages, emails, and updates about their lives, until Rosie finally prevailed upon Laura to pay a visit.
Now, as her cab swooped down the steep inclines of the city, Laura felt the vertiginous thrill of new beginnings. Maybe she really could start again, leave the failed promises behind her. Manhattan Law School had trapped her—not just the system that corrupted and changed her, but New York City and all it signified: home, family, and, sadly, Adam.
She hadn't told him about the interview, or even the trip West. Since their fight in her apartment they had not spoken. He avoided her in the cafeteria, and she never saw him in the halls or on the street. One time she called him after blocking her number, but she hung up after the first ring. She had too much pride to ask for forgiveness, and every time she almost did she got angry at Adam for even making her consider it. It was easy to play the scold without a stake in the game; but once vested, what did he expect her to do? And at what cost?
Her arrival at the hotel forced her mind in other directions. The doorman helped her from the cab, then insisted on carrying her single small bag. In the lobby he transferred it to a bell boy who escorted her to the reception desk. An Asian woman with cheekbones like razor blades checked her in, and informed her she had been upgraded to a deluxe suite and all charges were prepaid—even the minibar. It was a far cry from her interview at MLS where the finance department refused to reimburse her for the cab fare downtown and the cafeteria manager sent her a bill for lunch.
In her room there were flowers and a gift basket with Ghirardelli chocolate and two bottles of Napa Valley Pinot, along with a welcoming note from Rosie. Outside she could see the bay from her window and, in the distance, the Golden Gate bridge. She ordered room service, then took a shower, liberally applying all the gels and creams the hotel helpfully provided. The food arrived, and she curled up in the king-sized bed with a good book and a glass of the Pinot. Twenty-one minutes later she was asleep.
The next morning she violated a city ordinance by taking another shower and using two towels to dry off. Then she ran the blow-dryer and the coffee maker at the same time. Emboldened by her lawlessness, she flushed the reusable tissues down the toilet. She laughed to herself thinking how Adam might view her environmental mischief, then felt a keen urge for his presence. But she pushed it aside as she hustled out the door.
Her two student escorts were waiting for her in the lobby, more Ghirardelli chocolate in hand. Both were blond and wind-tanned from outdoor activities Laura could only imagine. The boy took her bag while the girl took her elbow and guided her to their car—a wheatgrass green Prius. She laughed when Laura asked if they were a couple, the toss of her head indicating she was too post-millennial to be "coupled" with anyone for more than a weekend.
"Did you have a good trip?" asked the boy, whose first name was Kennedy ("after the President, not the Justice").
"Very nice," said Laura.
"What did you do with your carbon offsets?" asked the girl.
"I, uh, gave them to Greenpeace," Laura lied.
"They do good work," said Kennedy.
"Puh-leez," said the girl.
"Remember In re Trans-Atlantic Cable Litigation?"
"Hello? Rainbow Warrior?"
While they bickered, Laura marveled at their quickness and breadth of their knowledge. Most of all, she marveled at their desire to argue about anything except Mets versus Yankees, Kanye versus Jay-Z, vodka versus tequila. She had nearly forgotten law students cared about things other than getting a job, getting stoned, and getting laid, not necessarily in that order. Of course, the students at Berkeley had real prospects, which freed them to debate the kind of topics they would never encounter in the real world. Her students did not.
The law school sat inland, near the entrance to the main university campus, surrounded by low-slung buildings with red Spanish tile roofs draped by vines and bordered by grasses. The sun burned through the remnants of morning fog and tendrils of light shot through the mist. To Laura, it felt as if she were in another country, one where grime and rudeness were banned by treaty. When she stepped from the car it was like stepping onto the yellow brick road—if Dorothy were braless and the Tin Man sported muttonchops.
Rosie met her at the entrance. She smiled widely as Laura approached.
"Laura!" she cooed. "So good to see you."
They hugged and exchanged air kisses. Rosie was not wearing deodorant, Laura noticed. Her brown hair was also unruly and longer than in law school, cascading over her shoulders in a snarl of curls. She had the weather-beaten look that seemed to be a status symbol in the Bay Area—a sign of second homes, summer vacations, and leisure activities—and her bare arms were sinewy and muscular like a rower's. She also had something in her hair—not quite a flower—an herb, perhaps. Cilantro? Parsley? Organic, for sure.
"Come, come," Rosie gestured after they unclenched. The students followed but she shooed them away. Instead, they stood at a respectful distance and said polite goodbyes to Laura, then waved as she walked down the hallway. "Students," sniffed Rosie. "Who needs them?" She grinned, which could have meant she was joking, although Laura suspected she wasn't.
As Laura recalled, in law school Rosie had been a bit of a snob. She was two years older than most of their classmates (a congressional internship), and acted as if that made her the smartest girl in the room. She had edited Laura's law review Note, and was obsessive over the use of the "em" and "en" dashes. She took personal offense at improper italicization, and nothing made her angrier than incorrect use of the semicolon. Several times Laura's draft was returned with a red circle around the offending mark and the word "punctuation!" written beside it.
These things came back to Laura as they walked past the sparkling student cafeteria. It actually smelled good—unlike the one at MLS that reeked of backed-up sewage and dead cats. Here Laura could smell fresh bread, the tang of citrus, and hints of the field where the grass fed cows had lived good lives before being turned into cracked pepper bresaola for the evening meal. An espresso machine hissed noisily while a woman wearing a long cardigan and a beret expertly foamed milk. Two students waited patiently for their coffees, and they were neither underdressed nor visibly tattooed. No one was dressed like a hooker, stripper, or minor league baseball player.
Laura and Rosie whooshed upstairs in a clean silver elevator that didn't shudder or suddenly drop half a floor, and the doors opened without the need of the Fire Department or the jaws of life. Stepping out, Laura admired the modern artwork in the hallway—pastel landscapes in angular patterns that reminded her of Richard Diebenkorn—carefully hung without duct tape or thumb tacks. There were no holes in the wall where the Building Department had drilled to determine whether the structure should be condemned, and no bait stations in the corners for rats.
As they walked, Rosie explained their schedule. First stop was a casual coffee with a handful of younger colleagues. Informal, nonevaluative, get the blood flowing. Then a series of meetings with tenured faculty. Three to four at a time, polite, but more rigorous. They would want to know about her work, and would ask pointed questions, probing both her intellect and her politics. "Hint," said Rosie. "Karl Marx is too conservative for these people." Then it was on to her faculty presentation, where she would be expected to present a paper in progress to the room and entertain questions for forty-five minutes. Finally, a glass of wine back in Rosie's office, where they could debrief, decompress, and compare notes for the day.
"Easy enough?" asked Rosie.
"I think I'll go back and take a nap now."
"You're a superstar!" said Rosie. "Nothing to worry about. Anyone gives you trouble just remind them you got the Wootfried prize for Civil Procedure at Yale."
"That gets 'em every time."
Rosie looked at her seriously for a moment. "Your first book practically won the National Book Award. You have written more articles than anyone else I know our age. And you're gorgeous. Who can resist you?"
Laura smiled, but suddenly she felt like a fraud. What would the editors at the Yale University Press say if they knew about Pay for an A? Her academic reputation would disappear down the toilet as quickly as her recycled tissues earlier that morning. That was why she needed to give this job interview her best shot. She would put three thousand miles between herself and New York, and would worry later about her mother, her apartment, Monk, Adam.
Rosie left her in a cozy conference room with a bright-eyed group of dweebs. There were seven—four men and three women—and she quickly forgot everyone's name. In fact, two were so androgynous she forgot their gender. All of them munched happily on sugared beignets with rhubarb dipping sauce and chatted eagerly about eco-feminism and trans-race theory. Laura could do post-Modernism with the best of them, and had even read Foucault in the original French (after which she had a headache that lasted through the winter), so the hour passed quickly, albeit nonlinearly.
Then it was on to meet the more established faculty. They were an intimidating group that included one rejected Supreme Court nominee (too liberal), two retired federal appellate judges (too old), and a former Junior Deputy Assistant Attorney General (too many titles). All had their own views on Laura's appointment to the faculty, which were directly proportional to Laura's admiration of them. So they probed and prodded for compliments, which Laura dutifully delivered, leaving them flattered and impressed with her acumen.
At lunch she sat with the two genderless professors she had met that morning and learned that they were, in fact, transgendered. Having tired of gender studies, the legal academy had recently turned to transgender studies, spawning articles like "Penis 2.0: Civil Rights, Transgenderism, and the Law" and "Transgender Theory: What We Talk About When We Talk About Hermaphrodites." They were joined at the table by a coterie of associate deans and even a couple of nonacademic staff, one of whom was either wearing some kind of indigenous outfit with a headdress or was on her way to a costume party. No one gave the woman a second look; she was just another colorful light in a vibrant palette that wasn't white-washed and gray-haired. It would be a relief, Laura thought, not to be the poster girl for diversity.
For her presentation to the entire faculty, Laura had chosen carefully: a modest work in progress with impeccable support. No risks here. She crafted a solid thesis that built off established foundations rather than tearing them down. The point was not to blow anyone away with her cogitations, but to present herself as a solid and reliable colleague. Like so many faculty candidates, once she had the job she could go off the reservation. But she was still nervous. She hadn't done a job talk since she was first hired by MLS, and there she had won them over by breathing.
She needn't have worried. No one was really listening; they were just waiting to demonstrate their superior intelligence. As soon as Laura finished, the questions began—which weren't really questions but soliloquies with inflections tacked to the end. "The Fourteenth Amendment transmogrified a loose coalition of diverse political philosophies into a monolithic uniformity, don't you agree?" Or "The jurisdictional limits of the federal courts instantiate the Constitution's driving ethos toward restraint and limitation, haven't you found?" Laura just had to nod and say, "That's a good question," before the speaker went off on his own frolic and detour that left her glassy-eyed and dazed. By the time the presentation concluded, her feet had gone numb from standing at the podium and her face felt like it might crack from the smile frozen on her lips.
Rosie saved her. She pulled her away from the yammering throng who continued to pontificate, each on his own soap box, and pushed her unapologetically out of the room. "Otherwise, they'll never shut up," she explained.
Laura protested feebly that she was enjoying the monologues—everyone was so smart and articulate—but she did have a plane to catch and all that freshly roasted coffee had left her desperately in need of a restroom.
"No excuses needed," said Rosie, as she led Laura to her office. "You know how we love to talk."
Laura smiled, but a spasm of guilt cut through her. She told herself she wasn't betraying Manhattan Law School by leaving; she was saving herself. Yet she felt like a traitor, nonetheless. The ship was sinking, and she was leaving the rats to drown.
Rosie's office was a modern suite, with three desks and a small conference table in an outer reception area, and then an inner sanctum that glowed with a greenish light. Everything was stainless steel and redwood, burnished and polished to a dull sheen. It reminded Laura of a partner's office at a corporate law firm. The two secretaries who snapped to attention when they entered made her feel underdressed. One handed Rosie a stack of message slips while the other handed her a bound report. Rosie dismissed them and ushered Laura into her office.
"You realize this is all a formality," she said when they were seated.
"Your office?"
"Your visit. The presentation. If you want the job, it's yours. Say the word."
"You can do that?"
Rosie laughed. Deep and husky like a smoker. "Honey, I do what I want."
Laura didn't doubt it. Despite the unruly hair and peasant skirt, there was a regality to Rosie that she carried wherever she went, a liberal entitlement to fine Merlot, unpasteurized cheese, and gluten-free biscuits. The whole school stank of it. These people weren't going somewhere; they had already arrived, and the school merely stamped its approval on their foreheads.
"So, what do you say?" asked Rosie.
"It's a big move," said Laura.
"It's the right move. Now. For your career. You can't keep teaching at that toxic dump—no offense."
Laura shook her head. "Manhattan Law School is a dump," she agreed. "But it's my dump."
"Maybe it's time to find a new dump."
Maybe it was, thought Laura. A dump with a view.
Rosie smiled at her. Her Cheshire cat grin expanded and Laura felt herself being drawn inside. It was dark in there, and confusing. Laura reached out to steady herself but she was falling, falling, falling. Where would she land?
## 16
## Solitaire
ADAM WAS STARING ABSENTLY AT his computer screen when he heard a knock on his office door. It was a quiet afternoon, and he was distracted. School had resumed from its short February break, but teaching was the furthest thing from his mind. For weeks he had been trying to forget Laura, hoping desperately that the Pay for an A scandal was something far less awful than it seemed. He had even started running, a sport he hated, nearly killing himself on the frozen Cobble Hill sidewalks. For the hour of diversion, however, it was worth the risk to life and limb. But the more he tried to put it out of his mind, the heavier it weighed upon him. No amount of effort could loosen the burden.
Maybe he'd misunderstood. Maybe it was limited to a few students and a handful of professors who were making some extra tutoring money on the side. Maybe Asher and the rest of the Law Review were creating something out of nothing. It wouldn't be the first time lawyers had overreacted. Theirs was a profession of exaggerators and dissemblers.
So maybe it was like that, Adam thought. And then came the knock. And his hopes evaporated like the milk in the faculty cafeteria.
"Hello?"
The door creaked open, and the unmistakable bald head of Professor David Wheeler poked through the opening.
"May I come in." It wasn't a question.
"Sure, David. Sit down."
The last time a professor had knocked on Adam's door, it was Professor Harding looking for the men's room. Adam was hoping Wheeler wouldn't repeat Harding's mistake and use the corner of his office as a toilet.
Wheeler swung open the door to reveal Rick Rodriguez and Jasper Jeffries standing right behind him. Adam knew right away this wasn't going to be fun.
"I'm not sure I have enough seats for all of you," he joked.
"Maybe you should yield your chair to tenured faculty," answered Jeffries, dead serious. Without thinking, Adam stood. Jeffries clomped behind the desk and collapsed in the seat, testing the chair's structural integrity. Adam tried to recall the last thing he had been working on, and hoped he remembered to save it. From the way his forearms rested on Adam's keyboard, it appeared Jeffries was intent on destroying it.
"We can see you're busy," said Wheeler, flicking some leftover lunch from his shirt onto the carpeted floor, "so we'll get right to the point."
"Please do," said Adam, bracing himself.
"We've heard that you've been deviating from the grading curve. Not your fault, really. You weren't properly informed of the way grading works here. We take full responsibility for the oversight, of course. We just want to clear the problem up, and make sure something like this doesn't happen again."
"I'm not sure I know what you're talking about, David," replied Adam, trying not to say more than necessary.
"Exactly. That's good. Very good. You see there's a special curve here at MLS that I don't think you were effectively briefed on. A lot of professors pick it up on their own, but sometimes we have to spell it out a little more clearly." He coughed and scratched a suspicious black mole on the side of his head. "We have... let's call it an arrangement. We're able to supplement our earnings, so to speak, in exchange for a willingness to be a little more flexible with our grading than we might otherwise be inclined. Nothing crazy. Just trying to do our students a service as they go out into the world. It's a shame when talented young lawyers don't get jobs. We make it a little bit easier for them. They make it a little bit easier for us. You understand what I'm saying?"
"I understand what I think you might be saying."
"Good. Very good. It's not something you really have to worry much about until you start teaching upper-class electives—that's why you didn't know about it—but there is one small problem that has arisen this year."
"And what's that?"
"Staci Fortunato."
"What about Ms. Fortunato?"
"My colleagues and I took a look at her midterm exam," interjected Jeffries, as he leaned forward on Adam's chair, making disturbing cracking sounds as he did. "We think you were too harsh. You didn't take the curve into consideration. We believe it was an A paper, nothing less."
"I believe you're mistaken," answered Adam.
"No," said Jeffries. "We are not."
"It's completely understandable given the circumstances," added Wheeler. "You're not used to the caliber of work here. Maybe you're holding them to a Harvard standard."
"Some changes in perception will have to be made," said Jeffries.
"For the institution."
Adam felt a rush of anger like a tropical storm descending. It swirled around him, violent purples and reds blackening his vision. He swam up through the darkness, taking several deep breaths before he spoke again.
"Can we cut the crap?" he asked.
The three professors stared him down. Or tried to. Like many academics, each suffered from a mild form of autism, so the staring consisted of their eyes focusing intently on Adam's ears and chin.
"I know all about what's going on here," Adam continued. "The grade-buying, the pay-offs, the violations of every rule in the faculty handbook. It's unethical and surely illegal. And it's not something I will have anything to do with."
Now it was Rodriguez who spoke, in his high-pitched faux Latino-inflected English that he saved for true histrionics. "You want to discuss ethics?" asked Rodriguez. "Let's discuss ethics. How ethical is a system that teaches students to do a job that doesn't exist?"
"I see," said Adam. "It's the system's fault."
"Yes," said Rodriguez. "It is. When students are not getting what the system promises, that is the system's fault."
"And the remedy is cheating?"
"Not cheating. Modifying. Redirecting. Repairing."
He let the words hang in the air for a moment as if they were snowflakes.
"We don't promise jobs," he continued, "We promise grades. And that's what we deliver."
"You're veritable saints."
"We like to think so."
"What about the others?"
"What others?"
"The ones who don't pay for good grades."
Rodriguez shrugged, but Wheeler answered for him. "They're no worse off than they would otherwise be. We don't lie to them."
"Not telling the truth is lying."
"Not telling the truth is an omission. There is no affirmative misrepresentation; no mens rea."
Adam shook his head. "I don't buy it. If the system doesn't work, you fix it. You don't corrupt it further. You're running a criminal organization here."
"And you're part of it."
"Never."
Rodriguez clucked disapprovingly. "Don't be naïve, Wright. Either you're with us, or you're nothing."
"Or, here's another option: I turn you in."
"You wouldn't," said Jeffries. "It would put your own career at risk. Can you imagine what your association with the law school would mean to your future? No other school would touch you, and good luck getting your law firm job back." He reclined in Adam's chair and wiped the sweat from his hands on the arm-rests. "No. We're all in this together," he said contentedly. "One happy family."
"Not me."
Jeffries pushed himself forward. "Don't be a troublemaker," he said. "The new ones often are, but they come around."
"And if I don't?"
"The Promotions and Tenure Committee doesn't look favorably on troublemakers."
"You see, we want to help you," said Wheeler. "But first you have to help yourself."
Adam wished he was taping their conversation. His phone, however, was on his desk, next to Jeffries' elbow, out of reach. It would be, instead, his word against the rest of the faculty. One crazy conspiracy theorist against the sober hordes. And even if he made himself heard, then what? What would be the point? Jeffries was right: speaking up would ruin his career. He would be damaged goods, tainted by scandal, and carrying the poisonous whiff of impropriety. Even that might have been acceptable at one time, his cross to bear, if not for Laura. They hadn't spoken since he walked out of her apartment, but she was always present. His brutal runs were testament to that, if nothing else, and still easier than missing her.
"Shall we take another look at Ms. Fortunato's exam?" asked Wheeler.
Adam sighed. "I'll take another look, but I'm not promising anything."
"Of course. We understand."
"And I'm not accepting money from anyone."
"That's your prerogative."
Maybe in the light of a new day, with another read, Staci Fortunato's unfortunate exam would look a little better. Maybe he had read it too quickly, didn't notice the subtleties, the incisive analysis buried between the ramblings, misstatements, and simplistic, third-grade analysis. But he knew that exams, like cheap wine, rarely improved with age. Instead, imperfections became more glaring, flaws less fixable, and the whole construct exposed as shaky and unsustainable.
"Now, if you don't mind," said Adam, motioning toward his computer. "I'd like to get back to my work."
"Suit yourself." Jeffries stood, and motioned to Rodriguez and Wheeler. "Come, gentlemen. Our business here is done."
"Remember," said Wheeler, as he followed Jeffries. "We're all on the same side."
"Are we?" asked Adam.
"Of course. You went to Harvard, didn't you?"
Then the three of them turned and left his office.
Adam sat back down at his desk and regarded his computer screen. Jeffries had destroyed his work, as he feared. The neat rows of reds and blacks were mangled, and the ordered numbers in disarray. Adam tried moving the cards back to their proper positions, but it was too late. The damage was done; his game of solitaire ruined. The Jack of Spades sat atop the King of Diamonds, his one-eyed smirk welcoming cheaters, liars, crooks, and con men.
Adam leaned back in his chair, and it creaked ominously, like a dying man's last wheeze. He straightened, and placed both hands back on his desk. But it took several minutes before the chair steadied, and the feeling of uneasiness stayed with him the rest of the day.
## 17
## Grades, Gold, and Steele
DEAN CLOPP RETURNED TO HIS office holding his usual post-lunch beverage, a glass of whiskey mixed with vanilla Ensure. He took several healthy sips before he spied the brown-skinned woman sitting in his favorite wing chair by the window. She was young and attractive, but that was no excuse for slacking off on the job, and he told her so. There was still dust visible on the mantel.
"I am not the cleaning woman," Laura said. "And I have an appointment. Your secretary told me to sit here."
"Wait a minute," said Clopp, a vague recollection coming back to him. "Shouldn't you be in class?"
"I don't have a class right now."
"When I went to law school, class was sixty hours a week, the reading took another sixty, and, if you were lucky, you had a few hours left for drinking."
"I'm not a student. I'm a professor."
"A law professor? Here?"
"That's correct."
"I highly doubt that."
"I've been teaching here for nearly a decade."
Clopp shook his head. Sometimes the pace of social change was dizzying. First, the right to vote. Now, this. Heaven knew what the liberals would concoct next.
"Well, then I don't want to keep you." Clopp looked at his watch. "Besides, I have another appointment." He motioned for Laura to go. She remained seated.
"I came to tell you I'm leaving."
"Please do."
"I got an offer from Berkeley."
"Surely you can find work closer to home."
"Dean Clopp, do you understand what I'm saying? I am a tenured member of your faculty, and I'm here to tell you I've accepted a job offer from Boalt Hall."
The dean took another sip of his whiskey Ensure and fixed Laura with his most inquisitorial glare, although with his burly belly and bushy white beard he didn't intimidate anyone except children who were frightened of Santa Claus.
"My dear, you've given yourself away. First, you said Berkeley, but now it's Boalt Hall."
"Boalt is the law school at Berkeley."
"So you say."
Laura grimaced. "I'll be leaving at the end of the semester, as soon as I turn in my grades. We both know what we're doing here can't go on indefinitely. Eventually the truth will come out, and when that happens, I don't want to be around."
"Then off with you." He shooed Laura out of the chair with his foot, spilling half of his drink in the process.
"Dean Clopp!" Laura stood. "You'll put an end to this business, if you know what's good for you."
"I have no idea what you're talking about. Business has never been better. Applications are down, but we keep dropping our standards so matriculations are up. Jeffries explained it to me."
"It won't end well."
"For you, perhaps," said Clopp as he nudged her to the door.
At the threshold, Laura made one last grab at the jamb. "Please," she said, "think about what I said. There are good people being hurt by a bad system. You can change things here." Her eyes held the dean's, and Clopp noticed the dark shadows below her lids. He made a mental note to speak to Health Services about the drugs they were dispensing to students. Then, he kicked her in the heels and closed the door behind her.
Satisfied that he was finally alone, Clopp went back to his desk and plunked himself in his swivel chair. He spun around to stare out the window at his unobstructed view of the canal while he finished his drink in peace.
Unlike everyone else at the school, Clopp happened to enjoy being situated on a toxic tort site. It gave him a sense of adventure, a bit of the Wild West. When he was teaching at Fordham several centuries ago, the worst thing that had ever happened was someone stole the bust of William Rehnquist he kept in his office. Here, his skin could flake off and blood might seep out of his pores from some waterborne plague that thrived in the sewage-infested canal. It was almost as exciting as a talking picture—in full Technicolor.
He swiveled back to his desk and lifted the thick dossier his assistant had prepared for his next appointment. It was the first real work Clopp had given her in six months, and she tackled it with enthusiasm. Henry Steele had made his fortune from the failures of the educational system, and his net worth was now at over five hundred million dollars. He had started with a small tutoring business that charged parents by the hour to help their underperforming children get B's. This grew into a larger tutoring service that helped the formerly-satisfied-with-B students get A's. It was an easy move from tutoring into test-taking services, for high school and college at first, and then subsidiaries focusing on medicine, law, business, and podiatry.
The passage of charter school laws gave Steele an opportunity to move into elementary education. Soon, Steele Educational Enterprises, Inc., was running eight public schools in New York State, and fifty others across the country. With the Internet came "distance learning," and the addition of two online colleges and a graduate program in finance. But Steele had a bigger plan: he wanted to transform higher education and run it like a for-profit corporation, answerable not to students and their parents but to shareholders and investment banks. He wanted, in short, to ruin it.
He quickly found that the college market was saturated, with little room for a new player. Medical schools required too much capital investment, while business schools required too little. Steele needed barriers to entry, students held captive by a regulated monopoly and lured by false promises of wealth and status. What he needed was a law school.
The dean ignored most of these details, focusing instead on the ones that really mattered, like net profits. When he was first introduced to Steele by Associate Dean Jeffries at a Bar Association seminar entitled "Getting Out While the Getting Is Good," Clopp was surprised to learn how profitable even a failing law school could be. Since then, he had come to see the inadequate services, shabby facilities maintenance, and rote course materials as pure financial wizardry on his part. He wasn't a neglectful and absent dean with a penchant for drinking himself into a stupor, but the leader of hidden economic gem about to be polished for the world to see.
After a brief nap, Clopp was awakened by his assistant just in time to welcome the educational visionary into the office. Henry Steele greeted him with a smile whitened to one shade shy of blinding and a handshake firmed up with the help of a personal hand trainer. He wore a crisp gray suit with a lavender tie, and his salt-and-pepper hair bristled like a porcupine. He inquired after Clopp's wife, the son who worked for the SEC, the married daughter with the two girls in Chappaqua, and the Filipino cleaning lady whom Clopp had deported when she asked for a raise. Clopp loved a good conversation about himself, and regaled Steele with tales of his darling granddaughters using names that were almost the ones that belonged to them.
"Sit, sit," said Clopp, after he had displayed one photo of a toothless grandchild playing the recorder, and another of a horse which Clopp confused for the child's nanny.
Clopp's office furniture was banged and dinged, and looked as if it hadn't been dusted since the Pleistocene era. His shelves were disarrayed and disorganized, with mountains of documents, piles of books, and the occasional empty liquor bottle that tumbled to the floor. In short, he didn't inspire confidence so much as the need for a good vacuuming.
Steele sat in the chair recently vacated by Professor Stapleton. He folded his manicured hands in front of him and waited. Time was on his side. Clopp and men like him were a dying breed. Guardians of a more genteel era, where no one sought to question the value of the services they provided. Now they needed to get out before they were trampled—or throttled.
"You've developed a number of impressive programs," said Clopp. "But tell me: Why law?"
Forty million dollars in yearly tuition revenue, Steele might have said, amortized and securitized into AAA bonds yielding a steady six percent. Instead he said, "A keen desire to shape and improve the future of our legal system."
"Excellent, excellent," said Clopp, who could smell money from twenty blocks away, although he could no longer keep the denominations straight.
"You have a fine institution here, and a valuable investment asset."
"We do fairly well for ourselves," agreed Clopp.
"But you could do better."
"A man can always do better." Clopp tried to wink, but his right eye got stuck, so he squinted at Steele like a pirate instead.
Steele explained that the law school was an attractive opportunity for investors looking for a steady and reliable stream of cash. But the spigot couldn't run forever. The school had leveraged its Superfund credits to pay for professors, chairs, and free popcorn at every other faculty meeting. Already, applications had dropped precipitously. Eventually, the banks would get wise. And then what? What would happen to the older faculty past the age of retirement, burning through their TIAA-CREF accounts? What would happen to Clopp, if pneumonia didn't strike him first?
That was where Steele came in.
Clopp no longer had a full deck, but he understood the need for an exit strategy. There were only so many papers Jasper Jeffries could sign for him, only so many speeches he could lip sync. It was only a matter of time before the law school's Board of Trustees removed him. He needed to make sure he didn't lose his long-term care plan at the Scottsdale Home for the Doddering, and he wanted his wife to have enough to last the rest of her days. So he listened to Steele's plan with growing interest. The law school couldn't survive its staggering debt load, constant tuition increases, and pathetic career opportunities without a bailout. The consultants had made that clear. At some point students—or their more sensible parents—would scream uncle. Clopp needed to go while the going was good. One step ahead of the angry and vindictive masses.
That cleaning woman had the right idea, Clopp concluded. Why change things when you could leave them behind? Start afresh in another country without an extradition treaty.
As for the students, who needed them? The ones who weren't criminals or certifiably psychotic were brain dead, oblivious, or overindulged. Their manners were tragedies, and their clothing wasn't much better. Clopp hated them. If only Steele could design a program that eliminated students altogether, that deposited their parents' checks directly and cut out the middle man. He had no love for his fellow faculty members, either. Although he had checked out long ago from the true administrative responsibilities of running a law school, and no longer taught any classes, he was still required to preside over faculty meetings and hobnob at social functions with his colleagues. They bored him to tears when he understood them, and put him to sleep otherwise. The law had not been an academic pursuit when he started teaching, and he never felt comfortable with the long German words and fancy French theorists.
All was not well in higher education, and at Manhattan Law School in particular. Clopp knew this, although he couldn't quite finger the reason. Perhaps it was the extra money that appeared in his bank account every few months, or the inability of Law Review editors to spell "Manhattan" correctly on their resumes, or the angry messages (dutifully transcribed by his assistant) from parents. In any event, ignorance was bliss—until it became a RICO violation.
Clopp took Steele's hand—at least what he thought was his hand but turned out to be his elbow—and shook hard. "Here's to doing better," he said.
"I'll drink to that," said Steele.
"It's five o'clock somewhere."
Clopp refilled his glass and poured another whiskey for Steele. Then the two men toasted each other and poured another. Several hours later, Steele slipped quietly out of Clopp's office, leaving the dean asleep on the floor, curled up like a baby with his thumb in his mouth.
## 18
## Dropping the F Bomb
WHEN THE CALL CAME FROM The New York Times, Asher Herman was taking a nap. This was not an unusual occurrence for the Law Review Editor in Chief. Having spent much of the last few months interviewing unsuccessfully for law firm jobs, as well as trying to get back into Staci's pants, he was tired. With a few dozen resumes out there, and a handful of interviews still on the calendar, he wasn't finished, but he was losing hope.
"Hello?" he slurred into the phone.
"I'm sorry, is this Asher Herman?" said a vaguely British-sounding voice at the other end of the line.
"Are you from Allen & Avery?" said Asher, trying to remember the name of the best British law firm he could. He hoped that even if it wasn't them, whoever was on the line would be impressed that it could have been, and bump his application to a higher priority.
"Allen & Overy," said the caller.
"You are?" asked a shocked Asher, now shooting up from his couch and wiping ash from face, as if the caller could see him through the headset. The apartment was soot-stained and smelled of burned plastic since the fire, but the couch was mostly intact though charred and blackened.
"No. I was just correcting you. You said Allen & Avery. The firm is Allen & Overy. But, no, I'm not from there. I'm from The New York Times. D. Campbell Wiley. Friends call me Wills."
"Wills?"
"Yes. Nickname. Like it?"
"Okay. Wills."
"I'm writing an article about law schools and grading, and I'm wondering if you would have a few minutes to talk with me."
"Uh..."
"Specifically, it's about the grading at Manhattan Law School. The Law Review, the write-on competition, the fact that students are allegedly buying their grades, and, as Editor in Chief of the Law Review, you're the one managing the entire scheme."
Asher turned a ghostly shade of white and, operating entirely on instinct, hurled his phone out the open window of his fifth-floor walkup. After a beat, he realized that was probably not the smartest thing to do. He found it about ten minutes later in the gutter, screen cracked, battery already stolen by someone who clearly knew more than he did about how to jailbreak an iPhone. Asher swallowed hard, put the remnants of the phone in his pocket, and took off for campus.
He found Professor Rodriguez in his office, his shirt untucked and his door half open. Rodriguez regarded him with alarm, then his eyes quickly dropped back to his computer monitor.
"Professor—" Asher interrupted.
Rodriguez sighed. "If you're going to show up at my door unannounced, at least be someone else."
"Excuse me?"
"Never mind. I assume you're here because of the Times?"
"Yes. Did they call you?"
"D. Campbell Wiley. I've done some research."
Asher could see now that Rodriguez's computer monitor showed Wiley's Facebook page, and the image on the screen was a shirtless vacation picture of the pasty white reporter. Rodriguez saw him looking.
"It's important to know who we're dealing with. I've already called David, and we're meeting in ten minutes."
Rodriguez was talking about Professor Wheeler, whom he had reached on his cell phone in the middle of recording a lecture for his online course about Star Wars and long-arm jurisdiction. It was Wheeler's thesis that by transporting Luke Skywalker from Tatooine, Han Solo should have reasonably expected to be hauled into court on that planet.
Asher followed Rodriguez through the back hallways of the law school. The two professors had arranged to meet away from the prying ears of the school's IT department, which had inadvertently connected the school's phone system to the loudspeakers in the lobby—leading to a host of embarrassing public revelations whenever a professor accidentally hit the pound sign on his handset.
Instead, they met at the law school bookstore, although it took Wheeler twenty minutes to find it. While he waited, Rodriguez regaled Asher with the last time someone had inquired about the grading system: several years earlier a reality television crew had spent a week with a Law Review editor filming a program where six young nymphomaniacs shared a beach house with six Orthodox Jews. The show—called "That's Not Kosher!"—never made it past the pilot, but the crew had filmed the student skipping his final exams, which led to phone calls from network fact checkers, followed by letters from the law school's attorneys, followed by the excision of the offending segment (and the student). It was then that the professors came up with their emergency plan.
Wheeler finally arrived, carrying a light saber.
Rodriguez grabbed the toy from Wheeler's hand. "Where were you?"
"Sorry. I got lost. Was the bookstore always here?"
"Since the '60s, David."
"Who buys books anymore anyway?" Then he looked at Asher. "Who are you?"
Asher extended his hand. "Asher Herman. Editor in Chief of the Law Review."
Wheeler grabbed his light saber back from Rodriguez. "It's not what it seems," he said to Asher.
"It seems like a toy from Star Wars," Asher said.
"Well, it's not."
Asher didn't think much of Wheeler's Star Wars–themed teaching—none of the students did—and his distance learning program was known to be a serious gut. But now the entire future value of his considerable law school investment rested in the portly professor's hands. "What are we going to do?" he asked.
"Ed Con Five," said Wheeler.
Rodriguez nodded. "Let's inform Jasper."
Associate Dean Jeffries' office was on the law school's top floor. Although it was designated the "penthouse," it was better known as the "flophouse." The roof leaked regularly and the windows were prime targets for pigeon droppings. To save money, the ventilation system and the elevator stopped one floor below, and the only access was a set of metal stairs that had never passed inspection. Nevertheless, because the floor was situated furthest from the classrooms—and, therefore, students—it was prime real estate, and Jeffries occupied the corner office.
"I've never liked the Times," said Jeffries, when Rodriguez had finished talking. "Although that Maureen Dowd isn't bad looking."
"She speaks like a mouse on helium," said Wheeler.
"You don't have to speak with her," said Jeffries.
Asher couldn't believe professors would talk about a Pulitzer Prize-winning journalist the same way he and his buddies talked about women they met at a bar, but it was liberating to know that men were jerks no matter their age or education. It was simply a matter of hardwiring in the DNA. He couldn't wait to explain it to Staci.
"Never mind that," said Rodriguez. "We've trained for this eventuality, and it's time to put our training into action."
What the professors had planned would require work—lots of it—the old-fashioned kind. Late nights burning the candle in the library, stacks of books piled high before them (although they'd have to find some first). Greasy chicken, M&Ms, and burned deli coffee. But there was no alternative. They could hide, but they couldn't run—at least not forever. The professors knew this moment would come, sooner or later, and the fact that they had covered up their scheme for so long was a testament to the rigor of their fraud and the willing stupidity of those around them. Law school had always been a shell game, with the nut moving quicker than the eye. As long as everyone was a winner, no one complained. But lately the list of losers had grown too large to hide and the pot of money too big to ignore. It was time to face the music.
The job, however, was too much for a few brave hearts to handle. It wouldn't take long for a nosy journalist to discover that Law Review editors spent their time copying and scanning articles already in print, rather than researching and writing them, and that the profits flowed to the faculty. They had evaded detection so far because the Law Review was not available in the legal databases and its website was coded so that it was unsearchable. But now, because Wiley might actually read an article (the first outsider to do so since a librarian at Pace with chronic insomnia used it to help her fall asleep in the mid-1980s), they had to cover their tracks. And that meant work.
The three professors divided the Law Review editors into six teams, each led by two faculty members who outlined one Article, two Notes, and four Case Comments. Regular classes were suspended for two weeks on the pretext of an Asian flu pandemic, and the school was shuttered to everyone except those in the know. Extra security guards were hired who stood sentry outside the law school's main entrance like SS officers. Food, beds, and portable showers were trucked into the basement where trusted administrative staff set up a FEMA-funded shelter. The toilets backed up from overuse and discarded pizza crusts, while a foul stench rose through the hallways like purulent socks. The teams worked round the clock in eight-hour shifts to create an entire year's paper trail of Law Review articles that no one—except D. Campbell Wiley—would ever see.
It was all hands on deck, steel balls to the floor, do or die, now or never, backs against the wall. Everyone was expected to give one hundred and ten percent, take no prisoners, and leave no stone unturned. There was no time for regrets, no "I" in team, and no crying in baseball. In this crucible, friendships were forged and old enmities forgotten. Swords became ploughshares, and ploughshares blossomed into footnotes.
That was how it came to be that Staci Fortunato found herself working on Asher's team, writing a section of an article Professor Rodriguez had outlined, entitled "The Conundrum of the Penumbra: Repealing the Ninth Amendment." She was the only first-year on the team—indeed, in the whole building—but like the others she had a vested interest in protecting the system from exposure. She hadn't exactly volunteered—in fact, Asher threatened to give her name to the Times if she didn't help—but she found she had a knack for legal argument, especially as outlined by Professor Rodriguez. Ever since her quasi-epiphany in Asher's burning apartment, Staci had been doing her reading for class and trying her best to keep up. Sometimes the Latin words overwhelmed her, and other times the sheer nonsense of disputes over widgets and peppercorns bored her to tears, but her efforts began to pay off. Professor Rodriguez praised her efforts when she turned in her section ten hours early, spell-checked, properly formatted, and reasonably coherent. Then he set her loose to write her own Case Comment which, once completed, he arranged to be published immediately in the bound volume the school had contracted for at great expense with an overnight printing company in Tulsa, Oklahoma. The first (and only) original work of scholarship ever published by students at Manhattan Law School.
D. Campbell Wiley was duly impressed. In fact, he spent most of his visit trying to get Staci's phone number. He practically ignored the cartons of newly minted Law Review issues, the manuscripts ready for cite-checking, the source documents piled high on the shelves. He interviewed Asher and the other top editors with one eye on Staci's quite visible décolletage. He blew off his lunch with his former law school classmate, Adam Wright, to brown-bag it with the students (which conveniently included Staci). When it was time to leave, he thanked each student personally, and slipped his card into Staci's outstretched hand. You're so hot you should be illegal, he wrote. Call me. Staci, who knew a dork when she saw one, tossed the card in the shredder after he left the building.
That night, to celebrate deceiving The New York Times, Professors Jeffries, Wheeler, and Rodriguez took the students to a popular Tapas bar for food and drink. Staci arrived in thigh-high black boots, black bolero, and an electric green brassiere, which nearly gave Asher another heart attack. Emboldened by their accomplishment, he cornered her at the bar.
"Buy you a drink?" he asked, although Staci had one and drinks were free, courtesy of the Law Review's slush fund.
"I don't think so."
"That was good work you did," he said.
"It was more fun than I thought."
"I may have been wrong about you."
Staci smiled, just a bit. The bar was crowded with law students and Gowanus hipsters who had gotten lost on their way to Williamsburg. Wedged into a seat next to her was a classmate who looked familiar, although she couldn't place him. He wore a moth-eaten sweater and nursed a beer through a straw. Not everyone had his life served up to him on a silver platter, she thought. There was sadness, and madness, and people who were just plain lazy. But not her. Not anymore.
She turned back to Asher and stomped on his big toe with the pointed heel of her boot. Asher cried out in pain, a noise that was lost in the electronic throbbing from the restaurant's sound system, and grabbed at his injured foot. As he bent over, Staci kneed him in the groin. The lights flickered, and Asher went down for the count.
The boy on the bar stool, who had seen the whole thing, rocketed out of his seat.
"Stay," said Staci, pushing him back down. "Buy me a drink."
"But drinks are free tonight," said the boy.
"What's your name?"
"Gary Deranger." He was trembling slightly.
"Gary, I hope you're not a big loser."
"I'm not."
"Good. Because it's your lucky day."
And it was.
## 19
## All Men Are Socrates
FOR TWO WEEKS ADAM HID in his office and his apartment, venturing outside only when absolutely necessary for food or air. He left the law school after dark, and avoided human beings on the street. He made no phone calls, and returned only the most urgent emails and messages. It wasn't that Jeffries intimidated him, but he didn't trust himself not to do something foolish. He had grown up amid foolishness, and saw what it did to people, and swore he wouldn't make that same mistake.
One of the emails he returned was to Laura, after she left a half-dozen messages on his phone, culminating with the one that informed him she had accepted a job offer at Berkeley. ("I wish I didn't have to do this on voicemail," her recorded voice said, "but it's not like you're giving me other options"). He replied, simply: "Got your vm. Happy for you."
Of course he wasn't happy—anything but. When he read the email he felt the heart-crushing sickness of the high school valedictorian who watches the girl he has pined for waltz off with the football captain. All his diligence and moral uprightness ignored in favor of big pecs and good hair. True, it was his decision, but what choice did he have? Hadn't she forced him to choose between the right thing and her? Walking out her apartment door felt more like being pushed off a cliff than jumping into a pool. His bones still ached at the thought of landing.
Thus, it was with a sense of doom that he came to the realization he couldn't keep teaching at Manhattan Law School. Not after what he knew. Quitting was for cowards, but it was also the only way to survive. Jeffries was right: if he exposed Pay for an A, it would taint everyone and take them all down, Laura included. So, despite his noble pronouncements, he would have to slink out quietly with his head low, mouth shut, and fingers holding his nose. He had no choice. He cared too much for Laura to hurt her.
With nowhere else to turn, tail between his legs, he called Howell Goldreckt.
"If you're calling about Ann Marie," said the lawyer after his secretary left Adam on hold for fifteen minutes, "I nailed her."
"I'm very happy for both of you," said Adam. "I'm sure you'll make a lovely couple."
"No thanks to you."
Despite promising to feed inside information about the law student to Goldreckt, Adam had conveniently forgotten. Apparently, Goldreckt had not.
"Let me apologize at lunch."
"I don't have time for squash this week. I'm training a service dog."
"A what?"
"For my PTSD."
"Since when do you have posttraumatic stress disorder?"
"Since I learned they board first on airplanes."
"Bring the dog," Adam said. "I'll buy the martinis—for both of you."
If there was one thing Goldreckt couldn't resist, Adam knew, it was free alcohol. They made a date, and Adam promised to reserve Goldreckt's favorite table.
After Adam hung up his hands were clammy and his feet itched. There were other law firms where he might land, he knew, but getting a job at one of them could take months. Meanwhile, every day at Manhattan Law School he slipped deeper into the moral morass. If he was going to do nothing about Pay for an A, he needed to do nothing quickly.
For his lunch with Goldreckt, Adam dressed in a suit. Goldreckt hadn't seen him in one since he left the firm, and Adam didn't want to take any chances. Cranberry, Boggs & Pickel adhered to a strict dress code; no one dressed "business casual"—not on Fridays, not during the dot-com boom, not on Halloween or at Goldreckt's annual pool party in the Hamptons, never. The only exceptions were the firm's Scottish partner who wore his kilt on Hogmanay and the one Orthodox Jewish partner who wore his yarmulke to meetings until a client accused him of making fun of the Pope.
Adam was early, and the maître d' escorted him to the table with a look of pity. It was a look with which Adam was familiar, seen on the faces of messengers, secretaries, and older associates. Been there, taken that, worse for the wear. No one who worked for Goldreckt for more than six months survived without the scars. Adam was a veteran, and proud of his wounds, even if they disfigured him. At least with Goldreckt you knew what you were getting, he reasoned, and there was no doubting the man's talent or energy.
To fortify himself against the latter, Adam ordered a Scotch. The drink was harsh and burned the back of his throat, but it warmed his insides and filled him with a relaxed glow. Thusly prepared, he gently eased into the soft cushion of his chair.
A commotion at the door distracted him. He looked up and saw Goldreckt trying to shove a small poodle back into his briefcase while the maître d' and several waiters flitted around him offering assistance. The dog didn't want to go, but after much snarling and snapping—and the dog wasn't too happy either—Goldreckt succeeded in shoving the pooch inside. He flipped the latches shut, leaving just a tuft of hair sticking out from the side.
"Goddamn mutt," said Goldreckt when the maître d' sat him at the table.
Adam had never seen Goldreckt with a domesticated animal—at least not a living one. The partner explained that the firm wouldn't let him leave the dog alone in his office. It had already chewed up three pairs of his secretary's shoes and defecated twice in a conference room. Not unlike Goldreckt himself.
As if to protest the comparison, the dog began barking, his howls muffled slightly by the leather. Goldreckt kicked the bag, and the dog yelped, but then stopped. All around them people were shaking their heads and muttering, and Adam expected the arrival of PETA imminently.
"Is he getting enough air in there?" Adam asked.
"He'll be fine. The leather breathes for him."
The waiter arrived to take his cocktail order, and Goldreckt ordered the "usual," which was a gin martini in a glass big enough for a milkshake. "Doctor's orders," he said, indicating the glass. He took a healthy swallow and then placed both arms on the table. "So, let's get down to business. I know you're not wearing a suit to take me dancing, at least not that one."
The last time Adam had lunch with Goldreckt it was interrupted when the partner's third wife served him with divorce papers during appetizers. The ensuing fist fight made Page Six of the Post. Adam reasoned it couldn't be worse. He plunged ahead.
"I'd like to come back to the firm."
Goldreckt laughed. "I knew you'd come crawling back eventually. The money doesn't smell so bad once you've left it behind."
"It's not the money. I don't expect partnership track."
"The Socratic Method not as much fun as you had hoped?"
"It isn't that either."
"'All men are fools; Socrates is a man; all men are Socrates,'" Goldreckt recited.
"Not exactly," said Adam.
"Annoying bastard. No wonder they killed him." He signaled for another martini. "Make it a double," he instructed.
The waiter balked, but took the order and walked away muttering about lawyers and liver disease. Goldreckt continued. "You should know the law isn't a bunch of philosophers drinking bark. It's plumbers kicking each other in the balls when the ref isn't looking."
When Goldreckt got going like this it was best to let him roll. He thrived on offending people, and to take the bait was only to encourage him to be more offensive. "I'm not defending the Socratic Method," Adam said, "but there's a place for making students question their assumptions."
"That's what wrong with you professors. You only care about the questions. Clients don't want questions. They want someone to ram their adversary up the ass. At least Socrates got that part right."
Now it was Adam's turn to order a second drink, which he did as soon as the waiter brought Goldreckt's martini in what appeared to be a flower vase. Perhaps he would become an alcoholic, Adam thought. If he was going to work for Goldreckt again, it might be for the best. A steady dulling of the senses leading to brain cell pickling and an early death.
"Anyway, it's all academic," said Adam, smiling slightly at his pun. "I'm done with teaching. I want to come back to the firm."
"That's not going to happen," said Goldreckt.
"Excuse me?"
"No refunds, returns, or exchanges."
Adam smiled wanly. Surely, Goldreckt was joking. It was all part of giving him a hard time and making him grovel. He deserved it. But in the end the economics worked in Goldreckt's favor: a senior attorney who didn't draw a partner's salary was billable gold for any law firm. Or so Adam thought.
"You don't need to decide right now," he said. "Take it back to the partnership. Bring it up for a vote."
"I've already voted for everyone."
"I understand you were angry when I left, but I'm willing to admit I made a mistake. You know I can do the work, and you know it will be done right."
"I had a little chat with Associate Dean Jeffries after you called me. Just in case you had something like this in mind. It seems you've gotten your panties in a twist over the school's grading curve. Not a good idea to cause trouble your first year in a new job. Jeffries and I discussed it, and we decided the best thing for everyone was to let you get a little more experience before you made rash decisions."
"You know about Pay for an A?"
"Of course I do. You think I want to hire idiots?" His laugh was loud enough to leave a stunned silence throughout the restaurant in its wake. "Each recruiting season Jeffries gives me a list of the Law Review editors who didn't pay for their A's. Then I hire the prettiest ones."
Adam tried to process what he had just been told. It sounded like the school's proudest alumni was saying he knew the school sold grades like black market pharmaceuticals. But that couldn't be right. For all his flaws, Goldreckt was a meritocrat. He believed in giving the just what they deserved, and despised the elites who rigged the game. At least that's what he claimed. But here he was acting like any other insider, stealing from the poor to keep everything for the rich.
"If you know about Pay for an A, how can you let it continue?" Adam realized he was raising his voice when a waiter approached their table and asked if everything was okay. Goldreckt waved him away. When he left, Adam lowered his voice and hissed, "Don't you care the school is misrepresenting students' qualifications based on their grades?"
"I've never met a law student whose grades weren't a misrepresentation."
"That's not true. Grades aren't perfect, I know. But there are students on Law Review who can't even do basic legal research. How do you think employers feel about Manhattan Law School when they hire one of them?"
"Who cares?" Goldreckt belched.
"You should. Isn't that why you give so much money to the school?"
"I give money so I can get the good students." He cackled.
"But there won't be any good students left if this continues."
"There will always be at least one. Jeffries promised me."
It was a cynical and mercenary attitude, but Adam didn't doubt it was true. The law school could limp along for years, relying on financial aid or a poor sense of direction to lure a few top candidates to Gowanus. And, as long as Goldreckt kept the charade alive by hiring a lucky (and comely) few, the rest would follow like lemmings.
But there was one thing that didn't make sense to Adam. "If you don't care about the school's reputation," he asked, "then why do you care if I leave?"
"Can't trust you. You're a goddamn pussy. If word leaks out, then there really won't be any good students left. The damn accreditors will shut the school down."
Adam shook his head sadly. "I'm not going to cause trouble. I've already sold out. I just want to leave quietly and get my old job back."
"Sorry. I promised Jeffries."
"Promised him you wouldn't hire me?"
"Promised him no one would hire you."
"That's ridiculous. You can't stop me from finding another job somewhere else."
"But I can, Adam. And I will."
Goldreckt put the flower vase to his mouth and gulped. His gullet bobbed like a snake's, and a driblet of gin ran down his tie. "But we're still on for squash next week," he added, then drank the remainder.
Adam watched in horror, realizing that he was just a small mammal trapped in that snake's mouth, struggling to breathe as the life was slowly crushed out of him. His cries for mercy swallowed whole.
## 20
## Bologna Dreams
THE TWO-WEEK SHUTDOWN TO FAKE the Law Review's archives had finally given Dean Clopp a chance to finish the scrapbook of his trip to Italy nearly a decade and a half earlier. Over time he'd stuffed all the photos, menus, and magazine clippings between the pages, and sorted everything into a rough chronological order, but it wasn't until he was able to carve out an uninterrupted stretch that he could make real progress on pasting his final selections into place. "Risotto or ricotta?" he asked himself, debating the merits of a meal they'd eaten in Bologna with Giuseppe Borgese, the Dean of the Faculty of Law, and his wife. He was just about to glue a photo of the couple into the book when the phone rang.
"Pick it up, Millicent. It's interrupting my work," he yelled, using the name of his long-deceased first wife. When the phone rang for a second and then a third time, Clopp threw the photo to the floor where it adhered permanently to the carpet and shuffled over to the receiver himself.
"This better be important," he barked.
"It's Henry Steele," said the voice on the other end of the line. "My lawyers have done their due diligence, and we're ready to make an offer."
"An offer for what? Who is this, and what are you trying to sell me? I'm on that do-not-call list, you know. I'm aware of my rights."
"It's Henry Steele," the voice said, this time slower, and louder. "We met a few weeks ago, in your office. I was the one looking to buy Manhattan Law School."
"Ah! Why didn't you say so? No need to be so secretive about it. We've been expecting to hear from you." Clopp put his hand on the mouthpiece, and shouted to the empty room, "Millicent, it's him!"
"Yes, well, I just wanted to touch base," Steele continued, "and let you know I'll be sending some papers over for your review. You can have your lawyers take a look at them."
"I am a lawyer, Mr. Steele."
"Yes, of course, but I meant the legal department. At the school. The people who read the contracts."
"Those are our students. Send the papers over by courier."
"I can have them emailed, if that would be easier."
"This is the twenty-first century, Mr. Steele. Mail will take days. Courier, please, at your expense."
After he hung up the phone, Clopp marveled at the archaic ways of communication among the younger generation. Instead of using a modern invention like the telephone, they wrote illiterate messages on tiny typewriters with their thumbs. Instead of scrapbooks, they took "Snapchats" that barely lasted for more than ten seconds. The students on Law Review didn't even know how to use the mimeograph machine that the law school had purchased in a bake sale from the church next door. He wondered what would become of them. Fortunately, however, that would no longer be his problem.
The courier arrived with the papers in forty-five minutes. Once Clopp found his glasses in the refrigerator and was able to read the numbers, they were staggering. He recalled that they had discussed how the deal would enable Clopp to retire, but Clopp figured that meant a hundred, maybe two hundred thousand dollars. Combined with his TIAA-CREF account, he thought it would be enough to buy a small apartment and live in Bologna a few months a year as long as he could wrangle a part-time teaching position and his wife kept her job at Nordstrom's. But if he was reading Steele's papers right—and there was a small chance he was—Clopp realized his share alone would be more than a million dollars. With that much money, he wouldn't have to teach, and his wife could find a job at any Italian department store she wanted.
Clopp immediately picked up the phone to call Jasper Jeffries. On the third try, he got the number right, and Jeffries picked up.
"What is it, Clopp?"
"How'd you know it was me?"
"For the millionth time, phones tell you who's calling now."
"Seems pointless. I could tell you just as easily myself."
"But I'd have to answer first."
"Why wouldn't you answer?"
"Because.... Never mind. Why did you call?"
"I've got some wonderful news. That man laid it on the table. And it's a big one."
"Excuse me?"
"You know, the man I told you about. The investor."
"Oh, thank God. I thought we had a really serious problem on our hands."
"Of course not. I told you everything would be fine. And now he's going to make us all rich. Well, most of us, anyway."
"Can you email me a copy of the offer?"
"I'm sure I can't."
"Can you at least read me the terms?" Jeffries asked.
After Clopp finished reading the one-page term sheet, the phone went silent. "Hello?" he asked. "Well, that was rude," he said to the phone. "He hung up on me."
Clopp had never liked Jeffries, but he tolerated him because he realized he could never manage the law school without him. In fact, there were days he couldn't find the faculty dining hall without Jeffries. But soon he would be rid of the man and his cronies—the homosexuals, libertarians, and federalists. In Italy, he would associate only with the pure of heart and simple of mind, and leave politics for the opera.
A light was blinking on his telephone, but when Clopp pressed the button, there was no one there. He made a mental note to get Millicent to have the phone replaced with something that functioned properly, without all the bleeps and squawks that distracted his callers when he tried to type a simple message using the keypad.
He returned to his scrapbook, and had just glued a piece of rigatoni from his favorite restaurant onto the page when he was interrupted by a banging at his door.
"Millicent!" he shouted. But the banging continued. The woman really had grown deaf and batty in her old age, he thought. In fact, the other day she insisted her name was not Millicent. He would have to talk to someone about her, if he could only remember whom.
He shuffled to the door and opened it to find Jeffries standing there, looking exasperated.
"Jasper! You hung up on me. That was very unprofessional."
"I didn't hang up on you. You hung up on me!"
"Now why would I do that?"
Jeffries rolled his eyes, but he took the purchase and sale agreement from Clopp's hands. As he read it, his eyes widened. When he was done, his bald head was sweating.
"We're rich!" he announced.
"Let's not park the car before the house."
"Horse. Cart before the horse."
"It's still subject to a faculty vote."
"It will pass unanimously."
"No hiding behind anonymity. We will vote in the open."
"Clopp...." Jeffries shook his head.
"Yes?"
"Never mind." Jeffries went to Clopp's desk and opened the bottom drawer. He pulled out two glasses and a bottle of whiskey. "This calls for a celebration."
"That's where Millicent put the Scotch!"
Jeffries poured two glasses. "Here's to wealth and happiness," he said, raising his glass. "Ours."
Clopp was already imagining the faculty meeting where he would present Steele's proposal. He could picture it all in his head. Applause—a standing ovation, perhaps—and a whole spread of food. Yes, for this meeting they would have real food. Clopp decided they should go all out: sandwiches. Maybe even some of those wraps the kids were all talking about. Sure, there might be some questions about ethics, and the future of the school, but they'd crossed that line long ago. Who could argue with selling out to an educational robber baron when they'd already been selling out their students for years?
"I can't believe all Steele wants us to do is sell the school," said Jeffries, still flipping through the agreement. "I would have thought for this much money, I'd have to blow someone."
"Blow someone what?" asked Clopp.
"Blow someone's horn," Jeffries improvised.
"Toot, toot," said Clopp, which surprised both men, though for different reasons.
The only thing that made either man pause was a set of terms and conditions at the back—contingencies that needed to be met before the deal would go through. The faculty vote and approval of the Board of Trustees—but those would be mere formalities. An audit of the books, but Steele already knew the salient details. And a handful of other terms Clopp expected would cause no significant problems—an inspection of the building by an outside engineering firm (paid off to avoid water and soil sampling), a review of all employees' immigration status (which would cost them some legal writing instructors—but none who couldn't be replaced), and a morals clause that would void the deal if something occurred that would bring the school into "disrepute" before the end of the academic year—unlikely, given its reputation.
Clopp scheduled the faculty vote for Wednesday, the third day back from the Asian flu scare. The professors were not told in advance why they were required to gather in the Summer Associate on Ally McBeal moot court/gymnasium at noon sharp, but the Evite invitation insisted it was mandatory—for all but Professors Chang, Chen, Huang, and Lee who would be joining via video-conference, a final precaution against potential infection (Prof. Lee wasn't even Asian—he was eighth-generation American, from Charleston, South Carolina—a direct descendant of the famous Civil War general, but the school was taking no chances). Those who hadn't RSVP'd by Tuesday received a follow-up email threatening to block their access to Pinterest unless they confirmed attendance at the meeting. It was an empty threat—no one on the faculty used the website—but it sounded frightening, and few were technologically sophisticated enough to risk it.
The court room/gymnasium was one of the few well-maintained gathering places in the building, which is why Clopp chose it for his announcement. The Summer Associate was a law school alum, and although her acting career went nowhere, she inherited a multimillion dollar fortune from her father, who had invented plastic, and donated a portion of it to fund a performance/workout space at Manhattan Law School. One side of the room contained three sets of free weights, four Nautilus machines, and a half-dozen tread-mills, while on the other side the chairs were arranged in a wide semicircle beneath paintings that depicted two landmark Supreme Court cases, Dred Scott, and Bowers v. Hardwick, both of which had since been overruled, to the dismay of some of the law school's more conservative alumni.
"Thank you for coming," boomed Dean Clopp, when everyone was seated. "You may wonder why we've gathered you here. We have some exciting news."
"We're finally shutting our doors?" yelled an elderly Professor Emeritus from the back.
"Actually, in a manner of speaking, we are." A few audible gasps were heard. "No reason to get alarmed. It's good news. We're entertaining a sale of the law school to a company called Steele Educational Enterprises."
"A nonprofit known for its charter school initiatives," added Jeffries, leaning in toward the microphone from his place at Clopp's side.
Rick Rodriguez stood up from his seat in the front row, pulled an index card from his pocket, and turned to his colleagues. "They did quite a profitable turnaround on a number of elementary schools in Connecticut, did they not." It was supposed to be a question—he was supposed to have rehearsed it—but it came out sounding like a statement.
David Wheeler followed, and pulled his card from his pocket as well. "And I hear the teachers made out like band-aids."
Jeffries cleared his throat. "You mean bandits."
Wheeler squinted, looked closer at his card. "Yes, I guess I do."
"Excellent points, gentlemen," said Clopp. "Steele has come in with a very generous offer. Tenured faculty will retain their positions, subject to certain revisions in their employment status, and receive bonuses of as much as one hundred thousand per year of service, up to a maximum of two million dollars."
A louder gasp this time, followed by a wave of chatter and then a crescendo of greed. Clopp's promises were vague, but that didn't prevent faculty members—who weren't very good at math anyway—from trying to run some quick calculations on their phones.
"I'll also be retiring," added Clopp. "Lucinda and I have long wanted to travel and spend more time in our beloved Bologna." Lucinda was the correct name of Clopp's secretary, and this caused some more gasps. Mostly, though, there was admiration—at least from the male faculty members—who elbowed each other enviously at their dean's good fortune. "You dog!" shouted a Tax professor.
"Thank you," said Clopp. "I'm sure a suitable replacement will be hired, and it should come as no surprise that I've recommended Jasper for the position."
"Let's all give Dean Clopp a round of applause for his years of service," said Jeffries, lifting his glass of triple-filtered water. There was no response from the crowd.
"It's been an honor and a privilege," Clopp said, over the silence. "And now, unless there are questions, it's time for tenured faculty to vote on the sale."
Adam Wright raised his hand from the back. As a nontenured member of the faculty, he would have little to gain from the deal, and in fact wouldn't even have a vote. Clopp reluctantly gestured at him, and he stood.
"Will Steele be making any substantive changes to the school's policies, such as our grading curve?" he asked.
"Sit down, jackass!" shouted Rick Rodriguez from across the room.
A few others added their own hoots and hisses, but Clopp raised his hand to silence them. "I'm sure there will be policy changes, but it would be premature to speculate."
"Will the faculty be involved in those changes?"
Clopp looked to Jeffries for help, who looked to David Wheeler. Wheeler raised his hand. "I move to close debate."
"Seconded," said Rodriguez.
"Objection," said a female voice from the left center of the room.
Heads swiveled as Laura Stapleton rose to her feet. She was dressed all in black, except for a red sash that held back her hair, and it made her a look like a samurai or a ninja. Her voice, however, was soft and sad.
"I think we all know the truth about our grading curve," she said. "Cutting off debate won't make the problem disappear."
"There's a motion on the floor," said Rodriguez.
"I'm embarrassed we've allowed it for so long," Laura continued. "No, embarrassed is the wrong word. Shamed. Humiliated. Horrified."
"Please, spare us the ethics lecture."
"But it's not too late to change things. Maybe we're being given a chance here to start again. Maybe this deal will give us the chance to put our house in order. But how will we know if we don't ask the hard questions? The ones we should have started asking years ago?"
"Point of order," Wheeler said loudly.
"Yes?" asked Clopp.
"As I recall Professor Stapleton has accepted a job offer from the University of California at Berkeley. Therefore, under Faculty Rule two point three dash 'a' small 'i,' she has divested herself of eligibility to vote or participate in debate."
"That's true," said Clopp.
"Her remarks, therefore, are a nullity, and should be stricken from the record."
"So struck."
"My motion is pending."
"All in favor?"
A rousing chorus of ayes shut down further discussion, and Clopp brought the deal to a vote. It passed, with only a handful of objections, and then the partying began, turning the courtroom/gymnasium into a spectacle worthy of an HBO series. Over sandwich wraps and boxes of wine, the faculty partied well into the afternoon, and by the time the celebration was over, Dean Clopp was asleep in a Front Lat Pulldown machine with a half-eaten turkey wrap in one hand, David Wheeler's underpants in the other, and his head filled with dreams of Italy. He had thought he was going to die in his office at Manhattan Law School, but Henry Steele had given him a way out. Now he could die like everybody else, in a nursing home, alone. He couldn't wait.
## 21
## Does Never Work for You?
ANN MARIE LISTENED INTENTLY, ARMS folded across her breasts in a defensive posture, while Henry Steele held court at a makeshift lectern in the Law Review offices. Even in normal circumstances the office was cramped and poorly ventilated. Now, with the entire Law Review present, the room resembled a broken-down subway car during summer rush hour—sweaty bodies crammed together and one guy in the corner with his possessions in a trash bag.
Steele, however, appeared cool, calm, and collected. His porcupine hair was brushed back neatly from his forehead, and his cheeks shined with the vigor of the newly shaven and recently exercised. His teeth were white and even, and clicked precisely when he spoke. He wore a dark blue suit which was creased so sharply it could cut paper, along with his trademark lavender tie. The only imperfect thing about him were his eyes: they darted from point to point like a hummingbird, or a man looking to make a quick escape.
Steele's skittishness was understandable. He had come to tell the editors about his purchase of the law school and to recruit them as poster children in his plans for scholastic domination. He didn't care about their education, but he needed them, and worried that his desire was unrequited. After all, most of them had paid double the normal cost of law school tuition and now were facing a lifetime of unemployment and penury. They could rightly blame the lightly regulated educational market, in which Steele thrived. He was like a politician preaching the values of fiscal conservatism to an audience of auto workers while cutting import taxes on Maseratis.
But he needn't have worried. They loved him. His attention lavished upon them the illusion that their lives had meaning and their careers promise. If such a well-coiffed and dentally hygienic businessman sought to curry their favor, then they must have something worth currying. They beamed in his reflected glow, and gazed at him like lambs before the slaughter.
Ann Marie, however, felt herself growing cold, then hot, and then finally steaming. She couldn't believe what she was hearing, and yet everything suddenly made a sick kind of sense. The furtive glances, the closed doors, the whispered voices. It was like awakening from a dream to discover the world in which she lived was just a computer simulation.
Asher Herman stood, and thanked Mr. Steele for taking the time to talk to the Law Review. "What you're doing here is very important," he said. "I know I speak for all of us when I say we're proud to be part of your vision."
"Thank you, Asher. I believe in meritocracy. It's what's made this country great. If you've paid for something, you deserve to get your money's worth."
"Agreed, sir. There are too many people with their hands out."
"Precisely. Given the expense of higher education today, we owe it to this generation of students—and parents, I might add—to eliminate the guesswork from the evaluative process. If we can quantify the inputs, we should be able to quantify the outputs. It's simple math. A plus A equals twenty thousand dollars." He chuckled, then continued. "What's made the system here work so well is that students know the exact dollar value of their education. I want to implement that system across the country, beginning with law schools but moving eventually into graduate education everywhere. With your help, Manhattan Law School will lead the way, shining its light down a rocky yet fruitful path."
There was much applause, followed by whistling and stomping, and even a few yodels.
Finally, Ann Marie raised one trembling hand. Steele's eyes lit like a shaman's as he called on her.
"Are you saying we've been living a lie?" she asked.
The room quieted. "Define lie," Steele responded.
"A fraud; a sham; a mockery. A false representation of reality."
"Most of you have, yes."
"But that's... that's awful. Monstrous."
Steele shrugged. "It's what you paid for."
Asher interrupted. "Ann Marie is one of our grade-ons," he explained.
Steele looked at her with a mixture of sadness and pity. "You seem like a sensitive young woman," he said. "Look at it this way: You've been lucky enough to win the lottery. But what about your peers who haven't been as fortunate?"
"I haven't won a lottery," she said. "I worked for my grades."
"You mean you scribbled some sentences into a blue book which a professor tossed down the stairs and then picked up and scrawled a letter onto it?"
"No. I mean I studied my ass off while these jerks were out drinking."
Hoots and jeers, followed by catcalls and farting noises.
Steele raised one hand to quiet the angry throng. "Now, now. That's not fair. Drinking games are part of the educational experience. Perhaps you should have gotten out more."
"Why? So I could waste my parents' money getting wasted? At least I have a job."
"Ann Marie accepted an offer from Cranberry, Boggs & Pickel," Asher explained.
"Congratulations." Steele's smile was all incisors, pointed and deadly like a shark's. "That's Howell Goldreckt's firm."
Ann Marie nodded weakly. "Yes."
"Howell, of course, is one of the program's biggest contributors, if you know what I mean."
"I'm not sure I do." What was Steele saying? Suddenly, Ann Marie felt queasy. When she was hired by Goldreckt, she became a minor celebrity at the law school—walking proof that all was not lost. For days she moved as if on air, gliding effortlessly through her classes like the patron saint of legal reasoning. Even an accidental Gary sighting did nothing to dampen her mood. She assumed Goldreckt hired her because she was smart and industrious, a Law Review editor, and top-ranked in her class. But now the doubts crept in.
"I could never have made this acquisition without his blessing," explained Steele. "His commitment to augmenting the resumes of students who have paid for the privilege is crucial to the success of the program as we move forward."
"He knows?"
"Yes, he knows."
Ann Marie's stomach knotted and for a moment she feared she might have to dash to the bathroom. She clenched her legs until the feeling passed, but it left her hollowed-out and wretched. Men had always misjudged her—mistaking her good looks and eagerness to please for stupidity. She spent her college years fighting off advances from professors and frat boys who took a polite smile as an invitation. Gary was the first man who valued her brains, not her body. It explained a lot about why she stuck with him even after her friends (and his social worker) told her to flee. Now, after finally succeeding on her own terms—or so she thought—it turned out her job offer was just another excuse for some cretin to get into her pants.
She stood. She was furious. Her hands shook and her lips had gone white. But when she spoke her voice was clear and strong.
"You should be ashamed of yourself. Peddling your lies. This isn't Wall Street. It's a school; a place of learning. You can't change the truth by buying it."
More boos, louder this time, along with shouts to "shut up!" and "sit down!" and "take off your shirt!"
Ann Marie turned and faced her peers. "You should all be ashamed. You think you're getting something for your money, but you're not. You're getting screwed. How much did you pay for your A's? Ten thousand? A hundred thousand? For what? How many of you have jobs? How many of you will be living with your parents next year? This man is talking like you should be grateful for his plans, but when you're gone he'll be laughing behind your backs. The joke's on you—on all of us!"
There were a few grumbles, and several students even nodded their heads. But someone else tossed a half-eaten wad of sandwich that narrowly missed her, and a group of boys in the back started chanting "blow job."
"You're being unfair to Mr. Steele," said Asher. "He's investing a lot of money in the law school."
"Henry, please," said Steele.
"Henry," said Asher, smiling shyly at the older man, who towered over him like a sequoia.
"Why don't the two of you get a room?" said Ann Marie.
This time the other half of the sandwich hit her in the side of the head. It was chicken salad on soggy white bread and it wasn't thrown very hard, but it felt like getting hit by a rock. Her cheek stung and her eyes filled with water, but she refused to cry. Instead, she shook one fist at her fellow editors and stormed out of the room.
Safely in the hallway, however, she burst into tears. Even at her lowest point with Gary—Methodist Hospital at 4 AM—she had never felt so humiliated. At least then she was needed and respected, her value measured by more than her waist size.
Now what was her value? Another dumb blonde for Howell Goldreckt to grope and fondle? He didn't respect her. He didn't care if she was intelligent, or even if she could speak English. She had come this far, only to have a man treat her as a doormat—again.
But she wouldn't stand for it. Not anymore.
She climbed out of the basement and emerged through a fire exit on a side street. She ignored the alarms and hailed a black livery cab that was idling at a stop light. The driver wore mirrored glasses and a plastic gun dangled from his rear view mirror (at least she hoped it was plastic), but she bargained the fare down to twenty dollars. The ride to Manhattan was potholed and broken, and the cab's shock absorbers not much better. Each bump made her teeth ache. On a hard curve the passenger-side door popped open, and the driver leaned dangerously across the seat to close it. He kept up a steady patois with a colleague on his cell phone, and nearly sideswiped a half-dozen cars on the FDR Drive. By the time they were crossing west on 53rd Street, Ann Marie's organs were jumbled and her brain hurt.
Finally, the driver pulled to a stop at the corner of 54th and Sixth, although the cab continued bouncing like a toddler on a ball. The driver swung around in his seat and demanded forty dollars.
"We said twenty," she said.
He grinned at her. "Okay, I say for you, pretty lady, thirty."
She stared at him for a moment, then slowly, without moving her eyes, she withdrew a twenty from her wallet.
"It's illegal for a black car to pick up a street fare anywhere in the five boroughs," she said. "Section nineteen dash five-zero-six, NYC Administrative Code." She handed the driver the twenty. "Have a nice day."
He stared at her wide-eyed as she skipped out of the cab and made her way into the glass and steel skyscraper.
Through the revolving doors, into the chill of the atrium, across the marble lobby. The guards at the security desk ignored her while they texted their girlfriends and watched highlights from last year's Puppy Bowl. Ann Marie strode past them purposefully and didn't look back.
Up, up, up, she flew in the elevator. The metal box was a silent transporter between two worlds. In one, she was an impoverished law student; in another, she would become a well-paid attorney. The decision was hers. She had worked hard for the choice, and believed she deserved it. But the just in justice had gone sour, leaving the desserts with a funny smell.
The doors opened. Harried looking women in heels wobbled past and glanced at her sympathetically, while men in shirts starched stiffly as corpses leered. Across the floor the receptionist perched like an exotic bird.
"May I help you?" she asked in a frosty tone that suggested she had no interest in doing so.
"I'm here to see Howell Goldreckt."
"Mr. Goldreckt is in a meeting," said the receptionist without missing a beat.
"How do you know?"
"Sweetheart. That's my job."
Ann Marie frowned. "Do you know when he'll be available?"
"No, I'm sorry; I can't help you."
"Can't or won't?" The question slipped out before she could stop it.
"Is there a difference?"
Ann Marie was about to respond when she heard a familiar voice booming down the hallway.
"A five iron in a sand trap! I told you the woman's an idiot."
A round of guffaws, and then four men came into view around the corner.
"Well, here's a surprise," said Goldreckt, stopping in his tracks. "Hello. Isn't your start date next month?"
Behind Goldreckt's back the men winked at each other knowingly.
"I need to talk with you," said Ann Marie, ignoring the free-floating testosterone and extra chromosomes.
"What about?"
"About that start date."
Goldreckt looked at his watch. "Give me twenty minutes."
"Now." She held Goldreckt's eyes until he blinked first. Then he turned to his partners and told them he would catch up. He led Ann Marie down the hallway to his office. Although it had commanding views North and East, the office looked and smelled like a locker room. In addition to the sofa which Goldreckt used as a bed, his squash sneakers were on a shelf next to a fistful of socks. The sneakers were cracked and hardened like rosin, and there was dried blood on one toe.
"Is there a problem?" the partner asked once he had safely closed his door and she was seated.
"Is it true?" she asked.
"True? Is what true?"
"That you're behind Pay for an A?"
"Ah," said Goldreckt, fingering his chin.
Ann Marie stood up. "I should have known."
"Wait! I've got nothing to do with it. I know about it, sure. But everyone knows about it."
She turned and took a step toward the door.
"Don't go!" For a large man, Goldreckt moved quickly. He was down on one knee in front of Ann Marie before she could stop him.
"I need you," he said.
"Yech! Gross." She could barely contain the bile. "Get up."
"I'm serious. I think about you all the time. I can't think of anything else."
"You're making me nauseous."
"Those are butterflies."
"No, it's vomit."
"I count the days waiting for you to work for me."
"I would never work for someone who hired me when he thought I bought my grades."
Goldreckt gazed up at her. His face, in the weak afternoon sunlight, seemed like a small child's. There was even a misshapen halo around his head, as if he were an angel who had collided with a wall.
"I don't think you bought your grades," he said.
"You said you did."
"No, I didn't. I said I knew how the Law Review worked. But I never thought you were part of it."
"You're so full of it."
"I'm telling the truth! Why would I hire someone who bought her grades?"
Ann Marie glanced down at her breasts, then flicked her eyes back to Goldreckt. "I'm not an idiot," she said. "I know what I look like to you."
"You're beautiful." Goldreckt's voice caught in his throat. "But I'm a businessman. I let my competitors hire the idiots."
Ann Marie placed one foot on Goldreckt's forehead, the point of her heel inches from his right eye. She felt powerful, as if she could kick a hole right through his skull. Perhaps she would.
"If there's one thing I've learned from law school," she said, "it's that a fact without support is just someone else's lie." Then she pushed hard against Goldreckt's forehead, and he toppled over.
She left him there—on the floor—and walked briskly to the elevator, a spring in her step, a song on her lips. The tune she was whistling sounded like freedom.
## 22
## Saving Mrs. Palsgraf
THE PLACE WHERE ADAM GREW up couldn't be reached by subway, bus, or Pony Express. Nor by airplane, Amtrak, or Greyhound. The town had its own post office and small bank, but no town hall, municipal courthouse, or even a public library. When Adam graduated there were forty-eight students in his class, a third of whom hadn't shown up to school in months, and another third who were either pregnant or responsible for it. On a typical Saturday night there were bar fights and drag races, and on Sunday mornings Adam and his brother would find beer bottles, skid marks, underwear, and broken teeth on the streets near their house. Once they found a finger, neatly severed with the nail polish unchipped. Adam wanted to bag it and bring it to the police station (which, conveniently, shared space with the town liquor store), but his brother convinced him to leave it where it lay so the owner could come retrieve it (and, indeed, a few hours later, she did).
When Adam went to college, he swore he wouldn't return, and though he was forced over the years to make the occasional exception, since his father's funeral he hadn't been back.
Until the morning his cell phone buzzed as he waited for Laura in the hallway outside the faculty meeting. He had bolted as soon as the vote was taken, and assumed she would, too. While he waited, he prepared a little forgiveness speech: Yes, she had taken the cowardly way out by accepting the job at Berkeley, but at least she stood up for her principles in the end. Yes, she could buy him a drink. Yes, if she insisted, he would go home with her and they could have make-up sex. But she ended up trapped in the doorway by David Wheeler, who held a sandwich wrap in each hand like a billy club. Adam hesitated, then her eyes caught his, and he pushed his way against the crowd just as his phone began buzzing. He glanced down and saw Sam's number flashing.
"Sam," he said.
"It's Mom."
"Is she okay?"
"She's alive."
Adam moved to an alcove and cupped his hand over the earpiece to block out the noise of the faculty party. His brother explained there had been an accident, and their mother was unbowed but not unbroken. Then there was a polite but intense argument about which brother was too busy to rent a car and drive two hundred miles, which Sam won by playing the doctor card. Although Adam's students were suffering, none were on life support. More important, none would sue him if they died.
There was an Avis near his apartment, and it only took about three hours and an FBI background check to rent a car. He meant to call Laura during the ordeal, but it required all his patience and concentration to avoid a fistfight with the clerk and then with the other customers when a solitary car rolled into the lot. Finally, safely on the road, he was too agitated and distracted to speak with anyone, and drove in silence—without even the radio—at eighty miles an hour.
He found his mother at the regional hospital where she was propped up in bed, giving the nurses a hard time. Gauze wrapped her hands where they had been burned by the airbags, and bruises darkened the skin below both her eyes. Although he called her faithfully every Sunday, he had not seen her for two years—since her last "theater weekend" in New York City with two of her neighbors. The bruises shocked him, but worse was the way her hair had grayed and thinned.
"Adam!" she cried. "Could you please tell these people that I am fine? Give me some band-aids and pain killers, and let me go home, for Chrissakes."
"As soon as the doctor reviews your chart you can go home," said one of the nurses, a heavy-set woman in her mid-forties. "We've already explained that to her," she said apologetically to Adam.
"I'm sure it will be soon," said Adam.
"Hah! I was married to a doctor, in case you forgot."
"No, Mom. I didn't forget."
"With your father it was always late, and later. Medical time, we called it."
Adam didn't have the heart to remind her that his father hadn't practiced medicine for the last twenty years of his life; that, in fact, his sons barely recalled the life they lived in Manhattan and were forced to leave when Adam was only five years old. There were enough new wounds without raising old ones. Instead, he held her hand and listened to her talk until the doctor finally arrived. He was a young man who looked about sixteen, with thin wisps of hair growing from his chin. He took Adam's mother's pulse and listened to her chest with a stethoscope.
"There's nothing the matter with my heart," she said.
"Mom," Adam cautioned.
"What? I've been in a fender bender. I didn't have a heart attack."
"It wasn't exactly a fender bender."
"A few coats of paint and the car will be fine."
In fact, the car had been totaled, and his mother was driving with an expired license. She had forgotten to renew it in time, and then failed her driving test twice. But she neglected to share this information with anyone—including her insurance company—until she was hospitalized and facing criminal charges.
His mother was not yet seventy, but she had bad eyesight, and terrible coordination. Adam and his brother long feared the phone call that would tell them she broke a hip, fractured an elbow, cracked her skull. Instead, they got calls about traffic accidents. Small ones, and then big ones, although never her fault. This one was driving too quickly; that one stopped too short; the last one didn't see her signaling.
"Mom, you turned left on red," said Adam.
"But he plowed right into me."
"He couldn't stop in time."
"Exactly."
The doctor poked and prodded while she talked, eliciting various howls of protest, until he finally declared her free to leave. Adam signed a dozen forms, pledging his 401(k) and a pint of blood as collateral should her health plan choose not to pay the bill. A nurse brought a wheelchair, which his mother adamantly refused. Instead, Adam helped her limp to the elevators and then to the front entrance where she waited on a bench while he got his rental car.
It was touch and go getting her into the passenger seat, but finally after reclining it nearly flat she was able to slide inside. On the ride home she talked about the new car she would buy with the insurance money. An electric one, of course, "with a dimmer switch to save energy."
Adam didn't tell her she was probably uninsurable, even if she could get her license back, which he doubted. She would need a driver, and probably rehab. His father had died suddenly of a heart attack, with no time to prepare. Now, it felt like his mother was at the beginning of a long, slow decline, and he was overwhelmed by it. Equal parts grief, sadness, and practical concern. For a moment he wanted to stop the car and get out, run for the woods, be anywhere but the claustrophobic front seat where he was belted beside his mother and stuck behind the wheel.
But then the familiar landscape opened in front of him: craggy hills and rolling meadows, family farms with their crooked silos and dark earth, one-block townlets with their filling stations and grocery stores. Soon, he was listening to his mother tell him about her plans for opening a yogurt bar/bookstore—"because who isn't buying books these days?"—and he couldn't help smiling. His mother's energy never flagged—even in the darkest times. It wasn't that she overcame obstacles; she simply ran right through them.
They drove up the gravel road to the weather-beaten Cape Cod hidden behind ash and elder trees. A rusted bicycle leaned against the garage, and several broken folding chairs lay scattered in the side yard. The house was about ten years overdue for a good paint job and, from the look of it, new gutters. Adam had been living in apartments for the last two decades, but the green-brown streaks down the sides of the house were unmistakable and ominous.
He parked as close as he could to the front door, then got out to help his mother. It was tough going on the slippery wooden steps, and he could see that would be another logistical problem that needed a quick resolution.
Once they were safely inside, and his mother seated at the kitchen table with a cup of tea and a plate of her favorite biscuits, he called his brother to report her condition. He took the phone into his old bedroom for privacy, but when he emerged the first thing his mother said was, "I am not an invalid. I do not need a nurse, or a wheelchair, or a cemetery plot."
"Glad to see you've got no problem with your hearing."
"I'm fine. I told you there was nothing wrong with me."
"Except for the walking part. And the driving."
"What's wrong with my driving?"
"For one thing, you don't have a license. Or a car."
"That's two things."
"How are you going to get around?"
"You'll help me get my license back. You're the lawyer."
"Mom, I've got a job. I've got to get back to the law school."
"I thought you were on break."
"I canceled my classes for the week."
"That's plenty of time. We can go down tomorrow and I'll take the driving test. It's like riding a bike."
His mother was in no condition to take a driving test—or ride a bike. Even if she could walk, she obviously couldn't drive. She could barely see. When he pointed that out, she said, "We'll go to the ophthalmologist first. I've been meaning to check my prescription anyway."
"When was the last time you went?"
She thought for a moment, then shrugged. "Twenty years ago? Twenty-two?"
"Are you serious?"
"I never needed to go."
"No wonder you keep getting into accidents."
"No." She shook her head. "It's not my eyes. I'm a bad driver."
"I can't believe you're admitting it."
"I have terrible spatial perception. That's why your father always drove."
"You used to fight about it all the time. You were a miserable backseat driver."
"I couldn't judge distances. Still can't. Of course, I never told him."
"I'm sure he knew."
"Yes."
They were silent, but Adam knew they were both thinking the same thing. His father was the elephant in the room, the skeleton in the closet, the Emperor's new clothes. They never talked about the past, but it hung over their home like a shroud, a rarely told tale that darkened their memories. Years ago his father had shared a medical practice with a classmate in lower Manhattan. This was before the days of hospital mergers, when a couple of general practitioners could make a decent living treating sick people. They weren't rich, but they did well enough that they could afford not to accept insurance. At least that's what Adam's father thought. Until one day he discovered his partner was billing Medicare for procedures that patients had already paid for out of pocket. He reported the fraud immediately, and for his diligence both men lost their licenses. The partner, after a messy divorce, decamped for Florida and made millions selling snake oil to the elderly. Adam's father didn't lose his family, but he went from being a doctor in Manhattan to a pharmacist in Nowhere, Pennsylvania, hoping everyone would forget the press coverage.
It didn't seem right. His father had done nothing wrong; he was guilty only of naïvete. So why did he accept his punishment? Why didn't he fight back? Surely, he could have at least kept his license. How different would their lives had been then? Adam could have grown up in an apartment with Central Park views, instead of spending his weekends working in the back of the pharmacy, wearing a smock and a name tag, fending off classmates who'd beg him to sneak them narcotics.
His father refused to discuss it. "What's done is done," he would say, leaving Adam to fumble for answers in the press clippings, but finding only unjust character attacks and mean-spirited labels.
"There isn't a day I don't miss him," his mother said into the silence. "He was very proud of you. Both of you boys."
"I thought he hated lawyers."
"No. He didn't hate anyone." His mother drained the rest of her tea. "Your father wasn't a complicated man. He didn't agonize over his choices. He made them, and moved on."
"That still doesn't explain why he let them take away his license. He could have hired a lawyer and sued. He probably would've won."
"Adam, you know that wasn't your father."
"I don't know anything."
"He didn't like confrontation. You remember he used to leave the house when we fought. But that wasn't the only reason. He knew the toll fighting would take, and he couldn't go through it."
"Because he was afraid?"
"Because he loved us. He didn't want to drag his family through years of hearings and court cases, and let his problems take over our lives. He wanted to spare you and Sam by making a clean escape."
"But he gave up everything. Look at where we lived."
"Was there something so wrong with it? He wanted you to grow up in a place where you could make your own lives and not be saddled with his. He believed the best thing for the two of you—for the three of us—was a fresh start someplace new. The only way he could get through his pain was by knowing he'd spared his family."
Adam considered this. It wasn't that he doubted his father's love—even though it was rarely expressed—but the idea that his father might have sacrificed himself to save his family greater pain was something new. It made a certain sense, though, and Adam could understand how saving yourself at the cost of hurting people you cared for was actually more painful than accepting the consequences, however unfair. "He didn't want to drag us through the mud," Adam said.
"And, really, what good would it have done? Maybe he'd still have his job, but at what price? Years of public battling, details about his business practices, all to satisfy his own ego? I didn't care if he was a doctor or a hobo. You don't do that to someone you love."
Adam nodded. Slowly, then all at once, he told her about Pay for an A. The floodgates opened, and, somehow it all gushed out, relief at being able to finally tell the truth to someone. Her eyes widened and her jaw dropped, but she didn't say a word until he was finished, which was probably a personal record.
"Bastards! It's always the rich and entitled who think they aren't rich and entitled enough."
"What should I do?" Adam asked, surprised he was asking his mother for advice, and yet more surprised that it felt so normal to ask.
"You have to turn them in."
"And then what? Become like Dad?"
"You may find this hard to believe, but your father was happy. We were happy. We raised our children and made a life here. It was our home."
"But I'll lose my job."
"You'll find another one. Life isn't about what you do, it's about who you are. You don't love your father any less because he was only a pharmacist. If he made one mistake, it was that he truly thought he wouldn't be blamed or lose his job, until he did. But once it happened, he made the best decision he could. You can find a new job, be safe no matter the fallout. And, you know, you can always move back home with me."
"Mom."
"I make some mean pancakes."
"Mean" was not the word he would use, but he didn't say anything. He also didn't tell his mother that Goldreckt promised to make finding a new job impossible. And even if he could, there was still Laura to consider. No cause is without its effect—wasn't that what he taught his students in Torts? A man trips carrying fireworks, and hundreds of feet down the platform a woman is knocked unconscious. The law holds him blameless, but is he? What about Mrs. Palsgraf, her head bloodied and ringing? Who will care for her? As she staggers to her feet she curses fate, the railroad, and the ignorance of her fellow passengers who think they can play with explosives without getting hurt.
Outside it was growing dark. Adam was tired and hungry. His last meal had been a plastic-bagged sandwich from the hospital cafeteria. It made the food served in the law school faculty dining room seem like filet mignon. He scavenged his mother's pantry and found some pasta and a barely-expired can of tomato puree. In the refrigerator she had a head of iceberg lettuce and some radishes. There was an ancient jar of dried basil sticking out of the spice rack, and an old tin of anchovies in the back of the cupboard. She was not a cook, but his father had been, and Adam inherited a bit of the cooking gene from him. In half an hour he managed to make dinner for both of them, and his mother's praise for the meal was entirely genuine. Afterwards, he cleaned up while his mother returned phone calls from various well-wishers. Then he helped her get upstairs (another problem to be remedied), washed, and into bed. He brought his own reading, and looked forward to a couple of quiet hours catching up on the academic journals, but he was asleep before his head hit the armrest on the comfortable couch downstairs.
Two days later he took his mother to the optometrist where they discovered her vision was 20-200, and her prescription strength had tripled. She bought stylish new glasses (clunky black frames; all the rage in Williamsburg, he assured her), and the next day she passed her vision and written tests at the DMV. Although it was still difficult for her to get in and out of the car, they practiced driving for the next three days, and on the day Adam was supposed to leave, scheduled a driving test with the help of a former patient of Sam's who pulled some strings at the DMV to bump his mother to the head of line.
The singular focus on his mother's needs was a pleasant distraction from the criminal enterprise that was Manhattan Law School. In addition to driving, he went food shopping for her (new basil, better tomato sauce), cleaned up the yard, and got several estimates to have the house painted in the spring and the gutters reattached. His brother arranged for a caregiver to come for four hours a day, and although his mother moaned and groaned about it, Adam could tell she appreciated the help. It also gave Adam a break, which he spent taking long walks in the woods where there wasn't even cell phone service. The silence and isolation was a tonic of relief, and it reminded him that there was a world outside Gowanus—a place where things could still be pure, simple, and clean. And he knew then what he needed to do.
The day of the driving test Adam's mother drove them to the DMV office. She ran a stop sign, and nearly sideswiped a parked car. Bad omens for sure. Her usual confidence and bluster were gone, replaced by something he was surprised to recognize as nervousness.
"You'll be fine, Mom," he said, with as much false confidence as he could muster.
"Of course I will," she snapped.
And she was. The DMV examiner agreed, even though she failed to parallel park correctly ("Who parallel parks anymore?" she asked). They left the office with her shiny new license, and the safety of other drivers in the hands of the State of Pennsylvania and Geico Insurance. He kissed her goodbye and promised to visit more often, which, just for the moment, he truly meant.
As he drove home across the George Washington Bridge, with the sun setting red over the Hudson River, a plan had begun to assemble in his mind. The pieces clicking into place like a children's puzzle. But first there was something he needed to do, a person he needed to see. Without her, there would be nothing.
## 23
## Hail Mary
LAURA WAS NOT IN HER office when Adam returned to Manhattan Law School. He had texted her about his mother's accident, but otherwise they hadn't communicated since the faculty meeting. Twice he picked up the phone to call—late at night after several, or perhaps more than several, sorrowful beers—but then lost his nerve and had to use the bathroom, anyway. Besides, he wasn't quite sure how to begin. I'm sorry. You're sorry. What are you wearing? Thinking about her only made sleeping in his old twin bed as painfully uncomfortable as his senior year in high school when every girl wracked his hormonally addled dreams.
He wandered the law school, searching for her. It was too late for lunch. The library was recently closed for fumigation. The streets deserted. Eventually, her faculty assistant took pity and called up Laura's teaching schedule. She was in the small conference room preparing for her DLS seminar. He found her leaning over a casebook, highlighter in hand, lips pursed in concentration. Her other hand played with a curl that corkscrewed over one ear.
"Hello," he said.
She looked up, eyes still focused on legal citations, and then she saw him.
"Adam," she said softly. "You're back."
"I'm back."
"How's your Mom?"
"She's fine. It's the other drivers I'm worried about."
She smiled politely at his joke, then tugged again at her curl.
"Anyway," he said, "I wanted to thank you for standing up at the faculty meeting."
"You don't have to thank me. It was the right thing to do."
"Too bad the sale was approved."
Laura shrugged. "Maybe not. With Clopp gone things might change."
"Not as long as Jeffries stays. Besides, it shouldn't matter to you. You're leaving." He didn't intend it, but some of the bitterness crept back into his voice.
"Is that why you're here?"
"No. I came to say thank you."
"You're welcome."
"I don't want to fight with you anymore, Laura."
"We're not fighting."
"You know what I mean."
"We both said some things we probably regret. I forgive you."
"Water under the bridge."
"Exactly."
"It's kind of convenient, though, don't you think?"
"What does that mean?"
Adam shrugged. "You go off to some cushy professorship miles away where you never have to deal with what's gone on back here."
"You think I should stay?"
"No. But I think you should do something."
"Do what?"
"Take responsibility. Acknowledge it was wrong." The conversation was going in a direction Adam hadn't intended, but he couldn't stop. "You were wrong."
"If it makes you feel better, I admit it. I was wrong. I got wrapped up in the nonsense of this place. But at least I'm getting out. What are you doing?"
"I'm not taking money to give out fake grades."
"Neither am I."
"But you did."
"Never! What do you think I am?"
"You're tenured."
"I earned my tenure," she said. "The old-fashioned way. I wrote a book, in case you didn't know—a pretty good one—and a bunch of articles. But I never did Pay for an A. Never. I refused to take the money, and I graded my own exams. Jeffries told me I wouldn't get tenure, but then he changed his mind. He made me a deal and I took it, even though I knew it was wrong. Yes, I'm guilty for keeping quiet. But I never sold my grades, and I never would. I can't believe you would even think that."
"What did you expect me to think?"
"I expected you to know me better than that."
"Well, if you really thought Pay for an A was wrong, you wouldn't be ignoring it—you'd be stopping Jeffries from using the law school as his personal ATM."
"Maybe I will!"
"You should!"
"Okay!"
And then they kissed. For ten minutes. They kissed, and didn't say another word. Their lips locked, and the world spun, and gravity loosened its grip. Adam felt as if he were floating, high above the earth, the moon beside him like a fat old friend. They kissed until Adam was dizzy, and he imagined a student's voice calling out to him through the haze.
"Professor Wright?" asked the voice.
Laura nudged him, and Adam peeked out from behind her neck. The young woman he imagined standing in the doorway was tall and blond, and looked very uncomfortable. He wanted to reassure her that as a figment of his imagination she had nothing to worry about, but the longer she stood there awkwardly, the more his brain resumed its normal activity, and he realized that she was a very real student and he was really embarrassed.
"I'm sorry to interrupt," she said. "The website says you have office hours, but you weren't in your office. Professor Stapleton's assistant said I could find you here."
"She did?"
"Yes."
"Well, Professor Stapleton and I were just discussing a few things." Adam straightened his hair and did his best to look professional, while Laura tried to suppress the giggles. "How can I help you?"
"I'm Ann Marie Kowalski. I'm not one of your students, but I'm on the Law Review, and Staci Fortunato mentioned—"
Upon hearing Staci's name, Adam flinched. "Whatever Staci told you is incorrect. And I have to warn you that campus security can be here in ten seconds." It was an empty threat, of course. The phones hadn't worked in the security office since Hurricane Sandy flooded the telecommunications equipment with sewage.
"This is awkward," she began, and glanced at Laura. "Is there somewhere else we can talk?"
"If it's a personal matter," said Adam, "it might be better to speak with a professional. A licensed professional," he added.
Ann Marie hesitated. She seemed pained, and Adam waited. Finally, she said, carefully, "Staci mentioned you didn't give her the grade she expected in your class."
"I hope she isn't still trying to get me to change it. It's a little late for that."
Ann Marie's laugh was soft and nervous, like a bird landing on a pillow. "No, no, that's not why I'm here. I was just interested in talking to you about the grading system." Her eyes flickered again to Laura.
Suddenly, Adam realized he knew this Ann Marie. She was the young woman Goldreckt had hired. Was it just a coincidence that she was here? Now? He eyed her warily.
"What about the system?" he asked. "I heard it worked out for you."
"I turned that job down."
"You did? Why?"
"I had my reasons."
Laura inched closer, and Adam could feel the fabric of her shirt brushing his arm.
"I don't think our students can afford to be turning down jobs," he said.
"Maybe I should go." Ann Marie made a move for the door.
"Wait," said Laura. She was still holding her highlighter. Her lipstick, Adam noticed, was smeared and several strands of her hair had come loose. She looked as if she had just returned from a night of drinking. Soon, he knew, she had to teach. That meant more students coming in to gawk at the two professors making out like high school kids. Laura, however, didn't seem to care. She waited until she clearly had Ann Marie's attention, then spoke very deliberately.
"If you know something—something about the grading system—it could be very important. Professor Wright and I care about the school, and we could help you."
Ann Marie took a deep breath, and then plunged ahead.
"Staci told me Professor Wright wouldn't change her grade—even though she paid for it."
"That's true," said Adam.
"Then you've got to stop it."
"Stop what?" He was being cautious.
She gave him a pitying look. "Pay for an A, of course. You can't let it go on."
"You know about Pay for an A?"
"Everyone knows about it! That's why the Law Review editors were so busy at the copy machines. I thought it was for cite-checking, but I should have known something was rotten. I feel like an idiot! They were always sending me out of the office—up to Columbia, and Pace! Have you ever been to Pace? They don't even have books. And now someone's buying the school who's going to make things ten times worse."
"Henry Steele."
"Yes." Ann Marie's eyes unexpectedly started to well up with tears. "That's why I came to talk to you. Mr. Steele spoke to us on the Law Review. He said Pay for an A was his model. He wants to spread it across the country, to all the schools he owns. He says it's the 'free market,' and the future of education. Can you believe it? He actually thinks schools are cheating students who pay big tuition bills and don't get good grades. He thinks he's a revolutionary!"
Laura withdrew some tissues from her handbag and passed them to Ann Marie. The young woman blew her nose, and continued.
"It makes me angry, really mad. I worked hard to get to law school and get good grades here. It's not fair that people who don't deserve it can buy the same thing."
"No, it's not," Adam said quietly.
"What happened to fairness, equality, justice? Isn't that what you teach? That's why I went to law school. Because I believed in those things." Ann Marie's face had reddened and her eyes were a vivid ultramarine. "Instead, we don't learn anything on Law Review. We can't get jobs because firms know our grades are a joke. And if we do get hired, it's only because some partner wants to sleep with us."
Goldreckt was a letch, but as far as Adam knew, he sinned with his eyes, not his hands. "I'm sure you're more than qualified for the job Howie offered," he said.
"I don't care. I'm not working for that firm."
"He'll be upset to hear that."
"He won't stop calling me."
Adam was familiar with Goldreckt's obsessive qualities. It's what made him a relentless dealmaker as well as one of the world's leading collectors of Jerry Garcia ties. Once he had his eye on something—or someone—good taste did not stand in his way. "He can be hard to ignore," Adam agreed.
"It's wrong. The whole system is wrong. You can't stand here and let it go on."
"She's right, Adam," said Laura. "We have to do something."
"I know." In fact, he had known for some time, but he hadn't wanted to hurt Laura, or end up like his father. That life—hang your head, bear your sorrows, slouch off to the far reaches of another state—was not for him. Yes, his father was a good man, but Adam was not a martyr. He wouldn't go quietly into the night. The student standing in front of him was reason enough to do something, if more reasons were needed.
But the plan he had concocted was marginal, at best. A Hail Mary pass with three seconds remaining. He had never been a quarterback, or much of an athlete in general, and he preferred his sports as metaphors rather than the dirty, painful reality. Yet Laura and Ann Marie were looking at him as if they expected him to lead the drive downfield. He met their eyes, and realized that maybe he did indeed have a winning play. It would take good timing, and great coordination, but with the three of them they might be able to pull it off. Or so he hoped.
"Listen," he said, and then he explained.
He finished just as the students trickled in to Laura's seminar. They glanced at him briefly, but most were so downtrodden they could barely lift their heads. They probably assumed he was another Associate Dean who had come to berate or flunk them. Laura gathered her notes and cleared a space at the head of the seminar table, transitioning seamlessly into her professorial role. Ann Marie, on the other hand, stood like a deer in the headlights, her face visibly paled.
He was about to reassure her—her part in the plan was without risk—when he noticed the clean-shaven, recently showered, young man in a suit at whom Ann Marie stared.
"Hello Professor Wright," he said jauntily. "Good to see you again."
"Gary?" asked Laura.
"I know I've missed the past few classes," he said to her. "I was in rehab."
"Rehab?"
"My girlfriend made me go. You wouldn't believe the kind of people you meet there—I have so many future clients lined up." He extended his hand to Ann Marie. "Hi, Ann Marie. How are you?"
Adam cast a glance at the young woman to see if she was okay, or if he needed to throw his body between the two of them and absorb the blows.
"Girlfriend?" she asked.
He nodded. "I'm sorry to spring that on you. It just seemed like it was time to move on, you know?"
"It's okay," said Ann Marie, a bit flustered. "You look really good," she added.
"Thanks. You look great, too. I hope it's okay for me to say that."
"Thank you, Gary."
The entire class was watching now, spellbound, including Adam and Laura. Even those who didn't know the specific details of their relationship had heard the tale—now an urban legend—of the stalker and the first-year. It had made the rounds of the legal blogs—and the local police blotter—and was even a case study at NYU's psychopharmacology program.
"I understand now how my obsession over you was really a struggle with my own insecurity during my first year of law school and my problematic relationship with my parents."
"That's okay, Gary."
"Really? Because in my group we talked a lot about the collateral damage of substance abuse. Hurting people you love, damaging heavy machinery, things like that."
"I'm fine, Gary. And your parents' insurance paid to repair the bulldozer."
"Good. It's important to make amends. My girlfriend says we shouldn't fixate on the past, but I think we shouldn't ignore it, either."
"It was an interesting year," agreed Ann Marie. "We learned a lot, and grew up a lot. I know I did."
"There's still so much more to learn, though. Don't you think? We won't be in school forever. And then what? What will we do? What kind of people will we become?"
"That's a good question."
"I'm excited to see what happens next. Aren't you?"
"I am."
"Um, Gary," said Laura, breaking the spell. "Class has begun?"
"Oh, I'm sorry Professor Stapleton!" He laughed almost normally. "I'll take my seat. It was nice talking to you again, Ann Marie."
"Nice talking to you."
Adam watched him take little goat-steps to an open chair on the opposite side of the table. Then he withdrew a shiny laptop from his backpack, a casebook, a legal pad, two pens, and a highlighter. He arranged them in perfect square angles, then sat up straight and gave Laura his full attention. His face was beaming.
In that moment, Adam saw the future, and knew it was theirs to lose.
## 24
## Down and Out in Gowanus
THE LAW SCHOOL LOOKED AS if a bomb had exploded in the main entrance. Papers were scattered everywhere; chairs were toppled; and the guards had abandoned their posts. When Adam pushed through the front doors, a glass pane broke with a sharp crack that sounded like a gunshot and left a spider web across its surface. The security turnstiles were stuck, and his ID didn't work, so he scrambled over them awkwardly, tearing a hole in his trousers in the process.
He walked to the stairs without seeing a soul. The only sound was a low growl, like someone praying, dying, or worse. But when he opened the stairwell doors, there was no one inside. Instead, it reeked of stale sweat and marijuana. Someone had taped the article from The New York Times across the fire alarm and scrawled "Manhattan Law Skool Sux!" in a black marker above it. At least there were still students who read the printed paper, although the bad handwriting and spelling suggested it could be a professor.
The faculty, of course, had plenty of reasons to be angry. Two hundred and fifty million reasons, to be exact. The article single-handedly killed the sale of the school to Steele Educational Enterprises, Inc. In light of the bad publicity, the estimated value of the institution fell by eighty percent. Unlike other schools with valuable real estate, or at least a name with valuable goodwill, the law school had few assets besides the cracked and stained furniture it could sell on Overstock.com and the expired snacks in its vending machines. The day the article was published, Steele exercised his termination rights under the contract's "morals clause." He had promised the tenured faculty $100,000 for each year of service to eliminate tenure, end the school's 401(k) contributions, and accept at-will employment status. The old guard would become millionaires, while new hires would become wage slaves, and everyone could be fired for publishing the type of claptrap that gave the eight-point law journal font a bad name. But now the older faculty's fantasies of riches—sold on their younger colleagues' backs—vanished in a flash.
Everyone's dreams had died. The students, who had few jobs to begin with, found their offers revoked. The faculty, both innocent and guilty, were immediately tarred by Pay for an A and rendered unemployable. The school instantly became the subject of a joint investigation by the Department of Education and the District Attorney's office, and an accreditation review by the Association of American Law Schools. Even the faculty dining hall was shuttered when the school's outside food vendor (a company dealing with its own legal crisis, as dozens of office workers in buildings it served had come down with a rare strain of feline hepatitis) demanded payment in cash up front.
"My God, what have you done?" Laura asked Adam when the article appeared.
"We," he said. "Remember?"
"I do, but now I'm wondering what we've gotten into."
"It's not too late for you to jump ship."
"Of course I won't. I just hope you know what you're doing."
"I knew he couldn't keep my name out of it. A guy like Donald can change his accent, but he can never change his stripes."
In violation of his promise of confidentiality, D. Campbell Wiley had revealed the source for his article. Since then, Adam's phone had not stopped ringing with other reporters, angry alumni, and enraged colleagues. Former colleagues, actually, because one of the callers had been Jasper Jeffries, who informed him he had been fired. "Just as soon as you make up some grades for the idiots in your class," he added.
Now, as he made his way to his office, Adam was no longer as confident in his plan. It was a Tuesday morning, but the halls were deserted. He passed a seminar room where there was normally a document drafting workshop and saw a student sitting alone sullenly smoking a cigarette, which of course was prohibited indoors. The student spotted him, then slowly and deliberately flipped him the middle finger. Adam was, he realized, the scapegoat for all that had gone wrong.
Had he done the right thing? Certainly the student didn't think so. Whatever teaching prospects Adam had were gone, and the law school's name was like a scarlet letter burned into his chest. But the students didn't have jobs to begin with, and they were laboring under an illusion that hurt all of them, even the ones who paid for their A's. Unless there was real change, they were all on a sinking ship where even the rats would drown.
Adam was halfway down the hall when the door to the bathroom opened and Professor Rodriguez emerged. He looked wild-eyed and unshaven, and his normally lacquered black hair was tangled and dulled. Adam nodded politely and tried to slide by, but Rodriguez put out a hand and stopped him.
"You've really fucked us, haven't you, Wright?" he said.
"Not me," said Adam.
"Not me," Rodriguez mocked him in a whiny, sing-songy voice. "You've fucked us, Wright. Yes, you have. Right up the ass. The ass. Where people get fucked. But you'll get yours. I promise you that."
"Thanks for the warning."
"It's not a warning. You won't see it coming. We'll slam you so hard your shit will come out of your ears."
"Is that so?"
"Yes, it's so."
Adam had never liked Rodriguez. Like Wheeler and Jeffries, teaching was an excuse to pursue his own esoteric interests while his students floundered and sank. The truth was most of his colleagues were in the teaching business for all the wrong reasons, concerned more with their obscure political agendas and solipsistic scholarship than imparting anything worth learning. Adam felt an enormous freedom knowing he wouldn't miss them. Which explains what he said next.
"Let me ask you something, Rick. Were you born a pompous prick or did you just become one?"
He saw the punch coming from a mile away. Rodriguez was not quick, and he wasn't athletic. He punched like a kangaroo, all hopped up on his toes. Adam sidestepped, and Rodriguez's fist went right into the wall—the thin coating of plaster no match for a human hand. Unfortunately, the plaster was just a veneer for the discolored, but still solid, brick beneath. The crack of bones was unmistakable, and both men stood looking at Rodriguez's hand as if it were a foreign object for two silent seconds before Rodriguez let out a howl.
"You moved!" he cried.
Adam shrugged. "You missed."
"You broke my hand!" He put his knuckles in his mouth, sucking on them like a baby.
"Sorry." Adam left him there, doubled over in pain.
In the safety of his office, Adam shut the door and collapsed into the bean bag chair Facilities Maintenance had given him to replace his creaky desk chair. Although the "beans" gave off a plastic chemical smell, it masked the moldy bacterial smell of the carpet. In fact, the guy from Facilities informed him that the chemicals in the chair would kill the mold in the carpet, and reassured Adam he would be fine as long as he didn't spend more than two hours with the door closed.
And yet he would miss the place. There was a certain satisfaction to being down and out in Gowanus. Like the Bad News Bears or the '63 Mets, the school's haplessness was its draw. The promise of better things glowed faintly in the distance if you squinted your eyes hard enough. A fan just had to believe.
He took another minute to compose himself, then reached into the back drawer of his desk and found the digital recorder, which he pocketed in his pants. Then he turned and left his office.
He had no difficulty hailing a livery cab on the street where they circled like flies hungry for a fare. Business was bad, and the driver spent the first half of the ride complaining about Uber and the second half complaining about lawyers. "Jackals and thieves," he spat. "The devil take them." Then he quoted Adam a fare that was double the price they had agreed on.
Adam didn't have time to argue, and the driver appeared slightly unhinged. He gave the man his money.
As he approached the building, he could see the tall blonde woman standing alone and aloof at the far corner of the security desk. In heels, she was taller than many of the men who leered at her while pretending not to. Regal and statuesque, she was like a fairy tale ice princess from a Northern land.
Yet Ann Marie was nothing like that, as Adam had learned. She was, instead, the type of student he had gone into teaching to teach: committed, curious, law-abiding, and impassioned. It wasn't her fault she attracted men like Goldreckt, men who preyed on her natural empathy and her inability to kick a man to the curb.
But that was about to change.
She saw Adam and waved. "Professor Wright!"
He smiled and walked over to her, ignoring the stares like daggers from his fellow members of the male species. Men always acted true to character. At least that's what he was counting on.
"Ready?" he asked.
Ann Marie made a face. "I guess so."
"It will be fine," Adam promised her. "He'll behave."
"That's not what I'm worried about."
"He told you the truth. If there's one thing I know about Howie, it's that charity is not in his nature."
Ann Marie sighed. "What if I still don't want to work for him?"
"If the plan works, you won't have to."
"And if it doesn't?"
"We can both get jobs at Starbucks."
When she smiled, her whole face shone, and Adam felt woozy from the reflected glow. He steadied himself with one hand against the wall until the blue spots quit dancing in his peripheral vision. Then he straightened his spine.
"Let's go make a deal."
They took the elevator to the thirty-fifth floor. They rode together in silence, contemplating the LCD screens that flashed news updates and, once, the headline, "Law School Grading Scandal." Adam winced, but was glad to see that Ann Marie had missed it. They stopped at thirty-two to pick up a harried-looking young man who ignored them because he was too busy chewing his nails. When they got off, he was unlacing his shoes.
The receptionist on thirty-five nodded vaguely when she saw them, and Adam realized she had been his suite-mate's secretary many years ago. But she gave no hint of recognition, and simply called Goldreckt's office to announce their arrival. They waited a few minutes in the richly appointed reception area, with its nautical themed prints, musty leather-bound books, and astringent smell of floor cleaner. An associate passed in the hallway, then stopped quickly when he saw Adam.
"Dude!" he said. "Returning to the scene of the crime?"
Adam introduced the associate to Ann Marie, whom the associate already knew from interview season. "How could I forget?" he asked. "They're already scalping tickets for the pool party."
Ann Marie smiled politely, and Adam steered the conversation in a different direction.
Finally, another secretary came to fetch them. The poor woman looked like a trauma victim, with bald patches on her scalp and red patches on her face. She spoke extremely quickly and quietly, as if words hurt her tongue. When she walked, her feet appeared to be running away from each other. But she calmed down when Ann Marie lightly touched her forearm, and managed to correctly steer them in the direction of Goldreckt's office.
When they entered, the first thing Adam thought was that he had never seen Goldreckt without his usual bluster and spittle. But there he was, sitting at his desk, hands steepled, thumbs twiddling, hair partially combed. If Adam didn't know better, he'd say the man looked nervous. He had even cleaned his office—or at least stacked things into assorted piles that teetered dangerously on the verge of collapse. He took one look at Ann Marie, and his mouth appeared to tremble and he blinked rapidly. In fact, Adam could swear he saw tears, although the lawyer quickly blew his nose with a used paper towel on his desk.
"Fucking New York Times," he grumbled.
"Not the best PR," Adam agreed.
"And you! What the fuck were you thinking?"
"Please, Mr. Goldreckt," said Ann Marie. "Cursing is counterproductive. And not at all attractive. Professor Wright is here to help; he has a plan."
Goldreckt's face fell, and he looked for a moment like a child caught with his hand in the cookie jar.
"You're not still angry with me?"
"Not if you keep quiet and listen to Professor Wright."
Goldreckt clamped his lips shut tight and raised his eyes beseechingly at Ann Marie. She let him sit there in silence for a full minute before she nodded at Adam to continue.
"You always told me the best time to make a deal is when the market collapses," Adam began. "Prices are low; sellers are desperate; there are no white knights."
"They've fled for the hills," agreed Goldreckt.
"That's why now is the time to buy Manhattan Law School."
"What? Are you crazy?"
"I'm just following your advice. The school's on the verge of closing and no one will touch it, but the fundamentals are sound. The building and faculty aren't going anywhere, and the students are stuck, too. Sure, some of them will drop out, but most have already invested too much to leave. And going to school is better than scrambling for a job anyway. But their parents need a reason to keep paying tuition. That's where you come in. You swoop in with a plan to save the place, clean up its reputation, and preserve their investment. You'll be a hero."
"You'd be my hero," Ann Marie echoed.
Goldreckt looked from Ann Marie to Adam to Ann Marie again. It was a little disgusting, really, the way they were toying with his affections, but he deserved it. Plus, Adam had crunched the numbers, and he knew the deal could work if certain bloated faculty salaries were allowed to sink into the canal.
"Where do I get that kind of money?"
"You can get it."
"Not all at once."
"A structured transaction. Pay out the bondholders over time, stepping down the interest rate for certain milestones like bar passage rate and job to homelessness ratio."
"How much you think they'll take?"
"Pennies on the dollar. It's a fire sale. They don't have a choice."
Goldreckt gnawed on his lip for a moment while his brain did the math. "You think I should do it?" he asked Ann Marie.
She gave him her sweetest smile. "Yes, Howie. I think you should."
If a man could turn to jelly, Goldreckt would have been a quivering blob. He sighed hopelessly.
"But what about the bad press? Even if we can keep the school open, applications will dry up. No one's going to apply to the school after the hatchet job in the Times."
Adam withdrew the tape recorder from his pocket and set it on Goldreckt's desk. "Leave that to me," he said.
Then he pressed "play" and told them about the rest of his plan.
## 25
## A Light Shines in Brooklyn
ON A CLEAR DAY YOU could see New Jersey.
It made Laura happy—knowing there were worse places in the world. Gowanus might have been a dump, but at least it didn't pretend to be the "Garden State." And, as hard as it was to admit, there were things she liked about the place. In fact, the other day she had noted an art exhibition at a new gallery in a converted waste transfer station. Black-clad hipsters stood smoking outside, while a velvet light strobed through the open door. "Hey there, law lady," one of them had called. "Buy me a drink?"
She was too old to be offended by the catcall, too young to need to buy a guy a drink. Besides, she had Adam to do the buying. Very unfeminist of her, she knew, but a girl needed to feel like a woman every now and then.
She smiled at the memory as she gazed out the window of her apartment. The sun was a fat fruit sinking on the horizon, and pink clouds skidded across the sky. Across the rooftops she could see flowers growing, trees in bloom, and what seemed to be a mugging in progress—but even that couldn't spoil the view. She loved this time of summer. Thick August heat. A lush darkness settling. August was the reason she became a professor. Long summer days with nothing to do but read and lounge until nightfall; late mornings lingering over strong coffee; a good afternoon nap. And no one trying to schedule committee meetings.
Monk barked, drawing her attention away from the view.
"Okay, little man. I'm done mooning."
She fastened his leash and left her apartment. In the lobby she ran into a neighbor, an older woman who was struggling with the door while pushing her shopping cart. Laura held the door while the woman slipped inside, and then the two of them chatted while Monk yipped at the woman's ankles. After a few minutes, satisfied that her neighbor could make it upstairs in the elevator, Laura said goodbye and headed outdoors.
A warm breeze eddied in currents along the sidewalk, blowing bits of paper and leaves in tight circles. Laura headed west, toward the river, while Monk more or less cooperated. At the corner she saw another familiar face, and stopped for another moment. Across the street a shopkeeper waved to her, and a man on a bicycle rang his bell as he zipped through the changing light. They said New York was a town of strangers, but she knew better. New York was thousands of different towns made up of people of every shape and flavor who came together, split apart, overlapped, reconfigured, reinvented, grew, thrived, changed, were broken by scandal but recovered, eventually, fully—and stronger for having had the experience. It was the magic of the city, the dynamic that powered it. Nothing that could be imagined was impossible; no problem unsolvable; no condition that couldn't be cured. It was what she loved about New York, and what inspired her and kept her. If she could make it here, she could make it anywhere. And the truth was, she didn't have much of a choice anyway.
She wasn't surprised to get Rosie's letter. After the exposé of Pay for an A she assumed her teaching career was over. Although not guilty of participating, she was guilty of turning a jaundiced eye, and truly believed she deserved her fate. Her penance would be banishment. She was prepared for it.
The light changed, and she crossed Columbia Heights, the smell of salt and sea air carrying in the drafts off the water. Manhattan was an island, but so were all the boroughs except the Bronx, part of an archipelago whose deep channels and calm harbors brought adventurers to their shores. It was easy to forget New York's nautical history while in the shadow of skyscrapers and brownstones, but a walk to the waterfront—or a hurricane—quickly brought it back.
She turned down Doughty Street and was briefly sheltered from the wind, then the promenade opened before her. Once again she was struck by its beauty—both man-made and natural. Directly in front of her were deciduous trees and rows of blooming flowers. Beyond them, the slate gray calm of the Hudson River. On its far shore the skyscrapers and spires of Wall Street rose like conquerors, citadels to wealth, vanity, and power. Finally, to her north was the Brooklyn Bridge, its twined cables glinting silver in the setting sun. Over 130 years old, it was the first land link between Manhattan and Brooklyn, and led to the creation of New York City. Without it, there would have been no New York Marathon, no New York bagel, and no Manhattan Law School. And she, Laura, would have been jobless, which, technically, she wasn't, at least not yet.
Grateful for small favors, she tightened her grip on Monk and crossed into the park. Elderly couples strolled hand in hand while kids ran unattended as if dodging an obstacle course. Young lovers swooned on benches while old friends gathered over food and concealed beers. A man with a steel string guitar played a folk song badly while a woman read lines for a dramatic monologue like a professional. All in all, another brilliant evening in Brooklyn.
Laura made her way to the river and stood at the railing. She took deep steady breaths of the sweet fresh air. At the far end of the promenade a man with curly brown hair and a lanky walk waved to her. She would have recognized him anywhere. Monk started yapping, and Laura released his leash. The dog ran swiftly toward Adam, and Laura followed.
## 26
## Snap, Crackle, and Pop
ADAM SURVEYED THE ROWS OF students sitting in front of him. Laptops open, hats on backwards, body parts exposed. There was a boy with a tattoo of Homer Simpson on his forearm next to a girl with the lyrics to a Taylor Swift song hennaed across her midriff. In the rear of the room, the backbenchers had their feet on the seats and giant buckets of soda perched precariously at the edges of their desks. All in all, a typical morning in the life of the beleaguered twenty-first century law professor.
And yet so much had changed since he and Ann Marie left Howell Goldreckt's office with the rough outlines of an offer in their hands. Three days later, with few other options, the Board of Trustees overrode the faculty and sold the law school to Goldreckt's ragtag group of investors, a hastily organized collection of persuadable clients, indebted colleagues, and intrigued alumni, all willing to give Howell the chance to turn a school from a laughingstock into something better. A small band of faculty members, led by Jasper Jeffries, threatened to sue, but they were quickly dissuaded when Clopp's long-suffering (and frequently forgotten) secretary, Lucinda Morris, produced several volumes of ledger entries detailing all the unreported (and untaxed) payments that would surely be Exhibits 1–100 at their depositions. As it was, the ensuing IRS and DOE investigations into how Pay for an A funds were accrued and accounted for would consume their time, money, and lawyers for years to come.
With only a handful of innocent faculty remaining, the law school struggled to close out the academic year while fighting off an audit, an accreditation review, and a class action filed by students who weren't involved in Pay for an A. In the end, the court dismissed their claims because, it held, students had no expectation of any particular grade and therefore no standing to challenge the allocation of grades based on financial payments. "It may not be fair," the court wrote, "but neither is our educational system. The wealthiest have always been able to afford superior schools, hire tutors for their children, and buy access to better information. If something is rotten, it's rotten throughout the kingdom."
Despite their victories on and off the court, it was touch and go whether the school would open in the fall, and whether any matriculated students would actually attend. The copy center shut down when it ran out of toner, then paper clips, then staples. Faculty were forced to piggyback on the unsecured Wi-Fi of someone named "BritneyBitch" when Internet service was canceled. To save money, the school outsourced its health-care services to a webcam clinic in India and sold off its prized collection of Thomas E. Dewey memorabilia. It rented out the student cafeteria for a scene from Law & Order that called for "INT. DILAPIDATED DINING HALL IN STATE MENTAL INSTITUTION." Still, it was weeks since anyone had received a paycheck, and some joker taped a sign on the lobby doors that read: "Last one to go pleaz shut off the lights."
The survivors sat in limbo, between a rock and the Gowanus Canal, when The New York Times published a follow-up to its scathing exposé of the school. Written by D. Campbell Wiley, the article appeared above the fold on the front page of the Business Day section with the headline, "Law School Back From The Brink—Rumors of its Death Greatly Exaggerated."
In May, Lucinda Morris, assistant to the dean of Manhattan Law School, tried to make a reservation for a conference the dean planned to attend in Denver. The school's credit card was rejected. It had just suffered through the worst scandal in its history—indeed, one of the worst scandals in the history of higher education—after it was disclosed that professors had been selling grades for money in a scheme called "Pay for an A." Most of the faculty resigned or were fired in disgrace, and the school was bombarded with lawsuits and demands for tuition refunds. "We were surviving on fumes," said Professor Adam Wright, who was named the school's 32nd dean by an investment group that purchased the school in the wake of the scandal. But in a textbook example of crisis management, the new administration instituted a series of concrete changes in teaching policies, including giving more responsibility to students for their own learning, under a program designed by Professor Laura Stapleton. The result has been a leaner, more vibrant curriculum that has begun to draw notice and raves. Even some potential employers have reached out, hoping that a set of better-educated law school graduates might be exactly the young associates they would need.
Although the article struck some as cloyingly deferential, it was the opinion of the Times legal department that a single (non-precedential) puff piece was a small price to pay to resolve an expensive (and embarrassing) claim for breach of contract. Of course Adam never threatened to sue the Times—he was too big a fan of the First Amendment to be so blunt—but the lawyers paid attention when he played his tape recording of D. Campbell Wiley's promise of confidentiality. "An oral contract is binding with valid consideration," he reminded them. "See Cohen v. Cowles Media." The lawyers didn't need the citation; they knew the case law. But it was a nice flourish to accompany a stack of solid evidence, and it clinched the deal.
Within hours following the article's publication, students started emailing the school to ask whether their acceptance letters were still valid. Long-overdue course selection forms suddenly appeared in the registrar's office. Parents who had stopped payment on their deposits called to say it had all been a terrible bank mistake. The stink of scandal began to lift, and a bit of light shined through the bird-shit stained windows. Even the canal smelled fresher.
In the end, although it had gone down to the wire, things unfolded just as Adam planned. He had done enough deals with Goldreckt to know the lawyer could seize an opportunity when he saw one. Of course it helped that it was Ann Marie who made the offering. Corraling the Times was more difficult, but Adam knew his old chum Wiley would act true to character, and his bosses would prefer a quiet resolution to a public shaming. Once Adam convinced the in-house lawyers that a follow-up article was both newsworthy and would release all claims whether known or unknown throughout the universe in perpetuity, the tuition dollars followed. Wiley disappeared soon afterward, and he was last seen tending bar in the former British colony of Tasmania.
But the best part of Adam's plan was persuading Laura to stay. After Berkeley withdrew its offer she had contemplated leaving New York anyway, going on a long-deferred Himalayan trek or African safari, anywhere without Wi-Fi and running water. "You said you wanted to change the system," he reminded her. "So, let's change it."
They were at dinner at his brother's apartment, an Upper East Side co-op fit for a successful young doctor and his girlfriend-of-the-moment. It was during one of the couple's lovey-dovey moments when Adam cornered Laura on the couch. "The school needs you; the students need you; I need you," he pleaded.
"I bet you say that to all the girls."
"Only the ones who can teach Civil Procedure."
Laura pulled him close for a lengthy kiss that was only interrupted by Sam's loud and theatric coughing fit when he and his girlfriend returned to the room.
Adam remembered that moment fondly as he faced the class. Lost in his daydream, he was interrupted by a knock on the door and the entrance of the Law Review's new Editor in Chief, Ann Marie Kowalski. These first-year students in Adam's classroom had been her peers until they were stripped of their credits and threatened with expulsion. Now they were starting again. Granted amnesty under the condition that they return to school and retake their classes, they were also given tuition credit for their previously completed semesters. Yet not everyone had returned. Some, better suited for lives as strippers, squeegee men, or three-card monte dealers, returned to the streets, joining the hustle and flow of the big city and providing future clientele for their former classmates.
Because it was the first day of a new school year, Adam had invited Ann Marie to welcome the new students, and to say some encouraging words about the journey that awaited them if they worked hard, stayed straight, and persevered. She had brought her managing editor with her, a young man Adam never expected to see alive, let alone in a clean shirt with his teeth brushed. But the meds and Gary's girlfriend had straightened him out, and Laura's tutoring helped him write his way onto Law Review. As for Ann Marie, with Goldreckt's assistance she was applying for federal court clerkships, and had promised to give his firm another look the following year. In private she told Adam she didn't think she was meant for Big Law, but he told her to keep her options open, grateful that the school still had students who actually had options.
The class listened to Ann Marie's short speech, and one student even raised a hand to ask a question about future issues of the Law Review. Ann Marie said they were still soliciting manuscripts and organizing a symposium in the spring with Professor Stapleton.
"If you need any help with the planning or anything, let me know," added Asher Herman. "I have some experience with that kind of stuff."
Ann Marie smiled sweetly and said she would keep it in mind, while Gary fixed his former rival with an indignant glare. Then the two Law Review editors left, and Adam turned to face his new students.
This is it, he thought. A new beginning. A fresh start. The first day of the rest of their lives. It was a frightening concept, but it was also an exciting and empowering one. He would show them how the law worked, unfold it, and demonstrate its ticking parts, and then they could put it together in new and interesting ways, adapting it for different times and shaping it to serve—dare he say—truth and justice.
"Palsgraf v. Long Island Railroad," he began. "Who wants to tell us about the facts of that case?"
Ten hands shot in the air, overeager and ramrod straight. Adam looked over the room, suddenly crackling with energy, alive and engaged. He chose a young woman sitting in front who was taking notes by hand in an old-fashioned spiral binder.
"Ms. Fortunato. Please, come up here and tell us the story."
Then Adam yielded the lectern while Staci stood before the class and began to teach.
## Acknowledgments
We wish to thank: Jesseca Salky, for her unflagging support and enthusiasm, her insightful notes, and her kindred spirit; Leigh Eisenman at HSG Agency; Jon Malysiak and his team at Ankerwycke, including Jill Nuppenau, Kaitlyn Bitner, Elizabeth Kulak, and the designer of our wonderful cover, Elmarie Jara; David Lat, for his advice and encouragement; Lenny Beckerman at Lotus Entertainment; Christine, Simon, Lulu, Nina, and Micah, for their good humor and patience; and each other, for making this book something more than either one of us would have written alone.
Cover and interior design by Elmarie Jara/ABA Publishing.
This is a work of fiction. Names, characters, places, and events either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events or locales is entirely coincidental.
© 2016 Jeremy Blachman and Cameron Stracher. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. For permission contact the ABA Copyrights & Contracts Department, copyright@americanbar.org, or complete the online form at <http://www.americanbar.org/utility/reprint.html>.
Library of Congress Cataloging-in-Publication Data
Names: Blachman, Jeremy, author. | Stracher, Cameron, author.
Title: The curve : a novel / Jeremy Blachman ; Cameron Stracher.
Description: Chicago, Illinois : ABA Publishing, American Bar Association, [2016]
Identifiers: LCCN 2016000279| e-ISBN: 978-1-63425-327-7
Subjects: LCSH: Law teachers—Fiction. | Law schools—Fiction. | GSAFD: Legal stories. | Satire.
Classification: LCC PS3602.L23 C87 2016 | DDC 813/.6—dc23 LC record available at <http://lccn.loc.gov/2016000279>
Discounts are available for books ordered in bulk. Special consideration is given to state bars, CLE programs, and other bar-related organizations. Inquire at Book Publishing, ABA Publishing, American Bar Association, 321 N. Clark Street, Chicago, Illinois 60654-7598.
www.ShopABA.org
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Q: Url data in table row being pulled from Inner html, trying to figure out how to make it open the URL in a new Tab on click I have some code below which pulls a URL data from a csv that is linked and places it inside a table row and converts it to a URL link on each row of the table,
What I am trying to figure out is how to make it also open the URL in a new Tab of the browser.
$('td:nth-child(13)').html(function() {
return "<a href='" + this.innerHTML + "'>Local Time URL</a>";
});
i have tried
$('td:nth-child(13)').html(function() {
return "<a href='" + this.innerHTML + " target="_blank"'>Local Time URL</a>";
});
but that then failed to load any of the csv data into the table
A: Nevermind i figured it out when i looked at this code written here I realised this was the answer
return "<a href='" + this.innerHTML + "'target='_blank'>Local Time URL";
| {
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Q: Solr detects the language. How do I search now across multiple description_* fields? I am trying to make multi-language stemming working with the Solr. I have setup language detection with LangDetectLanguageIdentifierUpdateProcessorFactory as per official Solr guides. The language is recognized and now I have a whole bunch of dynamic fields like:
*
*description_en
*description_de
*description_fr
*...
which are properly stemmed.
The question now is how do I search across so many fields? Making a long query every time that will search across dozens possible language fields doesn't seem like a smart option. I have tried using copyField like:
<copyField source="description_*" dest="text"/>
but stemming is being lost in the text field when I do that.
The text field is defined as solr.TextField with solr.WhitespaceTokenizerFactory. Maybe I am not setting up the text field properly or how is this supposed to be done?
A: You have multiple options:
*
*search over all the fields you mentioned. There always will be some overhead: the more fields you use, the slower search will be (gradually)
*try to recognise query language and search over only necessary fields: for example recognised and some default one. Here you can find library for this
*develop custom solution with multiple languages in one field, which is possible and could work in production according to Trey Graigner
The question is a bit old, but maybe that answer will help other people.
| {
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Extraordinary bespoke garden opens to enhance hospital experience for ICU patients
Posted Friday, 2 December 2022
University Hospitals Plymouth NHS Trust (UHP) has today officially opened its newly transformed Intensive Care Unit (ICU) Secret Garden, the first of its kind in the south of England.
Nestled away on the Derriford hospital site, this bespoke outdoor space will now enable ICU patients to breathe fresh air in the company of their families, loved ones and carers, and will even provide the opportunity for them to be reunited with their beloved pets, in a private and safe environment outside of a typical hospital setting.
The benefits of patients being able to experience fresh, clean air coupled with natural light are well documented. Not only does it improve patient experience, but it can also have a significant impact on delirium, mood and sleep. Every year, up to 3,000 patients and 6,000 relatives will come through the doors of the ICU at UHP and access to the garden will greatly enhance the hospital stays for many.
In 2018, Andrew Heveran, a then ICU patient, asked to go outside to be reunited with his dogs and loved ones during a prolonged hospital stay. This brought to light the fact that UHP didn't have a suitable space to take ventilated and very unwell patients outside, nor a private space to do so. Inspired by Andrew, ICU Specialist Sister Kate Tantam set about finding a way to transform a former grey courtyard on level 4 into a functional, outdoor ICU garden space.
"I felt like UHP needed a dedicated space to be adapted for the needs of all ICU patients, to not only provide them with the opportunity to reconnect with the outdoors whilst mitigating the weather, but also improve capacity and prolong visiting times," explains Kate. "Thanks to the efforts of my colleagues and a wonderful team of volunteers, we began to use the space in 2018 for patient rehabilitation and it has witnessed some really special events, such as birthdays and weddings. The area was particularly important in helping patients to gain a sense of normality during the COVID-19 pandemic, as well as providing staff with somewhere to recover from the pressures of looking after critically unwell patients, whilst also wearing full PPE."
Now, a bespoke garden room has been constructed, fitted with specialist equipment including piped oxygen, power, skylights, floor-to-ceiling glass walls and bi-fold doors, which extend the room's use to the outside area. Harnessing the positive healing power of nature, the garden itself has been created by Mark Lane, garden designer and the first BBC gardening presenter in a wheelchair. He himself spent months in a critical care unit after a serious car crash. The garden is designed to be fully inclusive and accessible for all.
The garden's woodland theme will now provide patients with the opportunity to be amongst tailored plants and trees in a private, controlled yet non-clinical looking environment, whilst still being connected to the medical equipment that they require. Wide, smooth paths, custom benches, screens and raised flower beds provide a sensory space and the skylights and glass walls will allow very unwell patients to be exposed to nature. This will reduce the level of risk to all ICU patients whilst increasing their safety and most importantly improve the hospital stays of patients when they want to spend time outside.
All of this has been made possible by generous grants and donations to UHP's official NHS Charity, including money from NHS Charities Together, as well as a substantial contribution from the Trust. An external team also came together and donated valuable time and resources to help make this project a reality, including the Trust's Capital Projects team, lead designers Bailey Partnership, engineering consultancy Hoare Lea and Nevada Construction. Despite facing some challenges, all worked collaboratively to design, deliver and build this amazing place within the heart of the hospital.
"This project has been close to our hearts for many years now, so it is wonderful to see it finally come to fruition, and we are all really delighted with the outcome," adds Kate. "These outdoor ICU beds will play a critical role in supporting ICU patients in a secure manner, as well as end-of-life patients too. This garden is set to make a real difference to patients and their loved ones as well as colleagues for years to come."
Andrew [pictured right, with Kate and his family] explains what the garden means to him: "My stay in Derriford's ICU lasted about 6 months. When you're in the ICU you really lose track of all time and are essentially stuck inside one room. I just wanted to get out. I met Kate who arranged for me to be wheeled outside in my hospital bed and it was a real tonic for me. To be able to go outside and feel the fresh air with my family and dogs was really uplifting. The opening of this garden will mean a lot to patients. A bespoke space like this, with fresh air and greenery would have definitely aided my mental state."
Wilma Heveran, Andrew's wife, adds, "It wasn't a nice time, but to be able to visit Andrew outside of the same four walls we were used to was really nice. Being outside meant that we could visit as a whole family, dogs included of course. Back then, the space we could access was on a different level to the ICU, so it was more difficult and taxing on the staff. This new garden is purpose built and more accessible for a wider range of patients."
Dan Paige Cocks [pictured left, with Kate and his wife Viki] was reunited with his wife and young family in the garden before its complete transformation. He had been placed in an induced coma after contracting COVID-19 and found that the outdoor environment played a big part in his recovery.
"The garden, even in its former state, was better for me mentally than any drug. It was a massive undertaking to be taken outside in my bed but it made such a huge difference and made me feel alive again. Being reunited with my children and for them to be able to visit me in the garden was just incredible. It gave me the willpower to recover and was so much better for my children to experience. They now think of it as a beautiful place that made Daddy better instead of having to see me in the perhaps overwhelming environment of the ICU ward."
For further information about the development of the ICU Secret Garden, please visit: https://www.plymouthhospitals.nhs.uk/derriford-secret-garden
* Spam Guard: Does a dog tweet or bark? | {
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\section{Introduction}
\subsection{Functoriality}
The goal of this paper and its continuation,\footnote{Any references to sections or equations numbered 6 or above refer to \cite{SaTransfer2}.} \cite{SaTransfer2}, is to demonstrate a local theory ``beyond endoscopy'', which indicates that functorial transfer between trace formulas is governed by ``transfer operators'' that have a certain structure: First, in the low-rank cases that we are examining, these operators can be explicitly described in terms of Fourier transforms and other relatively innocuous operators (such as multiplication by an automorphic character) which, \emph{in principle}, are amenable to application of the Poisson summation formula and, hence, to a global comparison. Secondly, these transfer operators are deformations of completely understood transfer operators that one obtains when letting the spaces degenerate to their \emph{asymptotic cones} (or \emph{boundary degenerations}). In the process, we develop the local theory behind ``non-standard'' comparisons of trace formulas that have appeared in the literature, namely in Rudnick's and Venkatesh's theses \cite{Rudnick, Venkatesh}, revealing a structure that is not evident in the analytic number theory approach. Our results should also be related to Herman's trace-formula-theoretic proof of the functional equation of the standard $L$-function of $\operatorname{GL}_2$ \cite{Herman} (which should correspond to the local Godement--Jacquet theory on the Kuznetsov formula, developed by Jacquet in \cite{Jacquet}), and Altug's theory of Poisson summation for the stable trace formula of $\operatorname{SL}_2$ \cite{Altug1,Altug2,Altug3}. Most of the results of this paper were announced, without proofs, in \cite{SaHanoi}.
``Functoriality'', here, is understood in the generalized sense, in the tradition of Jacquet, that includes \emph{spherical varieties} in addition to reductive groups (which are a special case). Thus, the trace formulas compared are \emph{relative trace formulas}, which include the trace formula in the group case. With the exception of Venkatesh's thesis, that we revisit, the functorial lifts that underlie our comparisons correspond to an \emph{isomorphism of $L$-groups}. To the reader eager to see non-trivial cases of functoriality between reductive groups, this might seem, and is, a disappointment. However, the ``relative'' generalization demonstrates that the problem of functoriality is highly non-trivial already for the identity maps of $L$-groups. What hopes do we have to tackle the general problem if we don't understand this basic case? Moreover, the examples collected here demonstrate the central role played by the \emph{Kuznetsov formula} in any successful ``beyond endoscopy'' comparison that I know. This fact is reinforced by the upcoming paper \cite{SaRankone}, which will compare any relative trace formula of rank one to the Kuznetsov formula. Although it is too early to pass verdict on this, it may just be that the idea of directly comparing stable Arthur--Selberg trace formulas is infeasible, and one has to move to the ``relative'' setting of the Kuznetsov formula in order to prove the functoriality conjecture; this idea appeared early on in the short history of ``beyond endoscopy'', in a letter of Sarnak to Langlands \cite{Sarnak}.
Our methods of proof are quite classical, and rely heavily on Rankin--Selberg theory. Not surprisingly, in retrospect, several different methods to study the same problem converge to the same, when viewed as a comparison between the appropriate trace formulas. The hope is that this new trace formula-theoretic approach will generalize to cases where Rankin--Selberg theory and other techniques do not. Again, in rank one this will be confirmed in the upcoming paper \cite{SaRankone}.
\subsection{Overview}
The relative trace formula is a distribution (with a geometric and a spectral expansion) on the adelic points of a stack. In this paper we work over a local field $F$, and only with relative trace formulas attached to quotients of the form $[X\times X/ G^{\operatorname{diag}}]$, where $X$ is a homogeneous spherical $G$-variety of low rank. We will allow the case $X= (N,\psi)\backslash G$, that is, the Whittaker case of the variety $N\backslash G$, equipped with a generic character of the maximal unipotent subgroup $N$, except that in that case we will take the character to be $\psi^{-1}$ on the second copy of $X$. The Arthur--Selberg trace formula corresponds to the case $X=H$, a reductive group, under the action of $G=H\times H$ by left and right multiplication; in that case, $[X\times X/G^{\operatorname{diag}}] = [\frac{H}{H}]$, by which we denote the quotient of $H$ by $H$-conjugacy. In the general case where $X=H\backslash G$, we have an isomorphism $[X\times X/G^{\operatorname{diag}}]=[H\backslash G/H]$, and we will use these two ways to represent this quotient interchangeably.
In the entire paper we work over a local field $F$. The variety $X$ is assumed to be quasi-affine, and let us denote by $\mathfrak{C}_X$ the invariant-theoretic quotient $X\times X\sslash G = \operatorname{spec} F[X\times X]^G$ (from now on, $G$ will always be understood to be acting diagonally on such a quotient). The space $$ \mathcal S(X\times X/G)$$
of \emph{stable test measures} for the associated relative trace formula is the push-forward of the Schwartz space of measures on $X\times X$ to $\mathfrak{C}_X$. (In the Whittaker case, we need to define a ``twisted push-forward''; see \S \ref{sstwistedpf}.)
There is a notion of \emph{stable relative character} $J_\Pi^X$ attached to an ($L$-packet $\Pi$ of) irreducible representation(s) of $G$, generalizing the stable character of a reductive group; it is a functional on $\mathcal S(X\times X/G)$. If there is a non-zero such relative character, we say that $\Pi$ is \emph{$X$-distinguished}. A morphism of $L$-groups
$$ {^LX}_1\to {^LX}_2$$
of two spherical varieties $X_1, X_2$ should, according to the relative (local) functoriality conjecture, induce a map
$$ \{X_1\mbox{-distinguished $L$-packets}\} \longrightarrow \{X_2\mbox{-distinguished $L$-packets}\},$$
at least for those $L$-packets that participate in the corresponding Plancherel formulas (the \emph{$X_i$-tempered} ones). A basic proposition of ``beyond endoscopy'' is that the resulting map of stable relative characters
$$ J_{\Pi_1}^{X_1} \mapsto J_{\Pi_2}^{X_2}$$
should be realized as the adjoint of a ``transfer operator'' between test measures:
$$ \mathcal T: \mathcal S(X_2\times X_2/G_2) \to \mathcal S(X_1\times X_1/G_1).$$
In the group case, this operator has been studied by Langlands \cite{Langlands-ST} and Johnstone \cite{Johnstone} when $X_1$ is a torus and $X_2$ is $\operatorname{GL}_n$. Ideally, this operator should be used in a global comparison of trace formulas, to establish the corresponding functorial transfer of \emph{automorphic} representations, but it is not clear at this moment how to do that in the context of the Arthur--Selberg trace formula.
The goal of this paper is to study such transfer operators $\mathcal T$, and to prove that they have some structure. In our comparisons, $X_2$ will always be the Whittaker model for a group $G^*$ (so that ${^LX}_2 = {^LG}^*$, and $\mathcal S(X_2\times X_2/G^*) = \mathcal S(N,\psi\backslash G^*/N,\psi)$ is the space of test measures for the Kuznetsov formula), and $X_1$ will be a different spherical variety $X$ for a group $G$. In all cases but one (the study of Venkatesh's thesis in Section \ref{sec:Venkatesh}), we will have ${^LX} = {^LG}^*$. The transfer operator that we construct originates from an \emph{enlargement} $\mathcal S^-_{L_X}(N,\psi\backslash G^*/N,\psi)$ of the space of test measures for the Kuznetsov formula. This is to be expected, because in a global comparison where the following diagram would commute:
\begin{equation}\label{commRTF} \xymatrix{ \mathcal S^-_{L_X}(N,\psi\backslash G^*/N,\psi) \ar[rr]^{\mathcal T}\ar[dr]_{{\operatorname{RTF}}} & & \ar[dl]^{{\operatorname{RTF}}}\mathcal S(X\times X/G)\\
& \mathbb{C} &}\end{equation}
(where ${\operatorname{RTF}}$ denotes the relative trace formula functional), the spectral side of the relative trace formula for $X$ is, roughly (and conjecturally, cf.\ \cite[\S 17]{SV}), an integral over $X$-distinguished automorphic representations $\pi$ of certain quotients of $L$-values of the form $\frac{L_X(\sigma_\pi)}{L(\sigma_\pi,\mathrm{Ad}, 1)}$, where $\sigma_\pi$ is an automorphic representation of $G^*$ of which $\pi$ is a lift, and $L_X$ is a certain $L$-value of $\sigma_\pi$, while the corresponding term for the Kuznetsov formula with \emph{standard} test functions is just $\frac{1}{L(\sigma_\pi,\mathrm{Ad}, 1)}$ --- thus, the factor $L_X(\sigma_\pi)$ has to be inserted. The enlargement of the standard space of test measures should also be expected for representation-theoretic reasons (both global and local), because the spectrum of the space $X$ is typically larger, containing, for example, the trivial representation, which is not present on the Kuznetsov side with usual test functions, but is added in this larger space. (This feature already appeared in \cite{SaBE1, SaBE2}, where the first comparison of this type was performed, both locally and globally.) This enlargement of the Schwartz space (see \S \ref{ssnonstandard}) will be done in a somewhat ad-hoc way; I do not yet know a good way\footnote{Although Rapha\"el Beuzart-Plessis has informed me that he does!} to characterize the enlarged space $\mathcal S^-_{L_X}(N,\psi\backslash G^*/N,\psi)$ for any $L$-value $L_X$, but at least, in the non-Archimedean case, it should contain the image of the ``generating Whittaker measure of the local $L$-value $L_X$'', that measure (or function) whose Poincar\'e series, at the unramified level, extracts the desired $L$-value from each automorphic representation (at least, formally). In approaches to ``beyond endoscopy'' expressed in classical language, this non-standard Poincar\'e series corresponds to ``a series of Kuznetsov formulas'' with varying test functions, closely modelling the Dirichlet series of the desired $L$-value $L_X$.
We may also let our varieties degenerate to their \emph{asymptotic cones} (or boundary degenerations). For the space of Whittaker measures $\mathcal S(N,\psi\backslash G^*)$ this means letting the character $\psi$ become trivial. For the spherical variety $X$, the asymptotic cone $X_\emptyset$ is obtained by considering a graded version of its coordinate ring; for example, when $X=\operatorname{SL}_2$, we have $X_\emptyset = $ the variety of $2\times 2$-matrices of rank one. We will explain that, in this case, there is a very natural transfer operator
$$ \xymatrix{ \mathcal S^\pm_{L_X}(N\backslash G^*/N) \ar[rr]^{\mathcal T_\emptyset} & & \mathcal S^\pm(X\times X/G)}$$
between suitable spaces of test measures that we denote by $\mathcal S^\pm$, given as a composition of \emph{multiplicative Fourier convolutions} $\mathscr F_{\check\lambda, s}$:
\begin{equation}\label{Fourierconv} \mathscr F_{\check\lambda, s}f(\xi):= \int_{\mathbb{G}_m} f( \check\lambda(x)^{-1} \xi) \psi(x) |x|^s d^\times x.\end{equation}
Here, $f$ is a measure on a ``universal Cartan'' $A_X = A_{G^*}$, and $\check\lambda$ is a cocharacter into that torus. Notice that we habitually denote the set of $F$-points of a variety $X$ simply by $X$, so an integral over $\mathbb{G}_m$ just means an integral over $F^\times$.
To summarize, we have a good understanding of ``degenerate'' transfer operators $\mathcal T_\emptyset$, and we would like to study the transfer operators $\mathcal T$ of the original problem. The findings of this paper can be summarized as follows:
\begin{mainconclusions*}
\begin{enumerate}
\item In a range of examples, the transfer operators $\mathcal T$ can be calculated explicitly, and are linear isomorphisms
$$\mathcal T:\xymatrix{ \mathcal S^-_{L_X}(N,\psi\backslash G^*/N,\psi) \ar[rr]^{\sim} & & \mathcal S(X\times X/G)}$$
satisfying the fundamental lemma (i.e., sending the ``basic vector'' of one space to the ``basic vector'' of the other, and also the ``fundamental lemma for the Hecke algebra'').
\item The transfer operators computed are indeed the ``correct'' ones for functoriality: the pullbacks of relative characters for $X\times X/G$ are relative characters for the Kuznetsov trace formula of $G^*$.
\item They are \emph{deformations} of the degenerate transfer operators $\mathcal T_\emptyset$, in the sense that they are given by similar-looking formulas.
\item They are \emph{amenable to a global Poisson summation formula}, in the sense that they are given by multiplicative Fourier convolutions and multiplication by factors that are trivial on global rational points, thus they seem to fit in a commutative diagram like \eqref{commRTF}.
\end{enumerate}
\end{mainconclusions*}
I do not perform such a global comparison here, but I expect that the methods employed in \cite{SaBE2} would apply to all cases discussed here.
The above conclusions are not restricted to the transfer operators that should be responsible for the functoriality map corresponding to
$$ {^LX}_2\to {^LX}_1,$$
but also to the so-called \emph{Hankel transforms} that are responsible for the functional equation of certain $L$-functions. These are addressed in the continuation to this paper, \cite{SaTransfer2}. While transfer operators have the property that they pull back relative characters to relative characters normalized in a distinguished way:
\begin{equation}
\mathcal T^* J_{\Pi_1}^{X_1} = J_{\Pi_2}^{X_2},
\end{equation}
the Hankel transforms are transforms between enlarged spaces of test measures of the \emph{same} relative trace formula (for us, always the Kuznetsov formula), but with the enlargement corresponding to dual $L$-functions:
$$\mathcal H_r:\xymatrix{ \mathcal S^-_{L(r)}(N,\psi\backslash G/N,\psi) \ar[rr]^{\sim} & & \mathcal S^-_{L(r^\vee)}(N,\psi\backslash G/N,\psi)}$$
and having the property that they act on relative characters \emph{by the gamma factor of the local functional equation}:
\begin{equation}
\mathcal H_r^* J_{\Pi} = \gamma(r) J_{\Pi}.
\end{equation}
For example, when $G= \operatorname{GL}_n$ and $r={\operatorname{Std}}$, the standard representation of the dual group, the space $\mathcal S^-_{L(r)}(N,\psi\backslash G/N,\psi)$ is just the image of the Schwartz space $\mathcal S(\operatorname{Mat}_n)$ of the variety of $n\times n$-matrices, and the Hankel transform $\mathcal H_{{\operatorname{Std}}}$ is the descent of Fourier transform, computed by Jacquet in \cite{Jacquet}. In this paper, we will compute the Hankel operator $\mathcal H_{\operatorname{Sym}^2}$ for $r=$ the \emph{symmetric square $L$-function} of $\operatorname{GL}_2$, and verify that both $\mathcal H_{{\operatorname{Std}}}$ and $\mathcal H_{\operatorname{Sym}^2}$ satisfy the Main Conclusions listed above.
\subsection{Index of main results}
References to chapters \ref{sec:intro2} and above refer to the continuation of this paper, \cite{SaTransfer2}. The main results of this series of two papers are:
\begin{itemize}
\item Theorem \ref{thmRudnick}, on the comparison between the Kuznetsov formula and the stable trace formula for $\operatorname{SL}_2$. It is complemented by Theorem \ref{groupdegen} which performs the same comparison for their asymptotic cones.
\item Similarly, Theorem \ref{torusdegen} performs the asymptotic cone comparison for the transfer between the Kuznetsov formula for $\operatorname{PGL}_2$ and the relative trace formula for the quotient $\mathbb{G}_m\backslash \operatorname{PGL}_2/\mathbb{G}_m$. The actual transfer between these two quotients was performed earlier in \cite{SaBE1}.
\item Theorem \ref{thmSym2}, on the Hankel transform responsible for the functional equation of the symmetric-square $L$-function of $\operatorname{GL}_2$ (or, more correctly, of $\mathbb{G}_m\times\operatorname{PGL}_2$). It is complemented by Theorems \ref{Sym2degen} and \ref{Stddegen} that compare this Hankel transform (and the one for the standard $L$-function) to the corresponding transforms on the asymptotic cone.
\item Theorems \ref{transfertokappa} and \ref{twistedRudnick}, which describe the transfer from the Kuznetsov formula to the endoscopic parts of the trace formula for $\operatorname{SL}_2$, and Theorem \ref{thmVenkatesh}, which describes the transfer from the Kuznetsov formula of $\operatorname{SL}_2$ to the Schwartz space of a torus.
\end{itemize}
\subsection{Example: Comparison between the Kuznetsov formula and the stable trace formula for $\operatorname{SL}_2$.} \label{ExRudnick}
In Section \ref{sec:Rudnick} (with some proofs to be postponed until Section \ref{sec:sym2}) I develop the local theory behind Rudnick's thesis \cite{Rudnick} (and its potential generalization to non-holomorphic automorphic forms). Here, the goal is to obtain a transfer operator
$$\mathcal T: \mathcal S^-_{L_X}(N,\psi\backslash G/N,\psi) \xrightarrow\sim \mathcal S(\frac{G}{G})$$
between the spaces of test measures for the Kuznetsov formula and the stable trace formula for $G=\operatorname{SL}_2$. The operator should have the property that its adjoint takes the stable character $\Theta_\Pi$ of any tempered $L$-packet $\Pi$ to the relative character (Bessel distribution) $J_\Pi$ of the Kuznetsov formula, thus realizing the functoriality map corresponding to
$$ {^LX}_1 \xrightarrow\sim {^LX}_2,$$
where $X_1 =\operatorname{SL}_2$ and $X_2=$the Whittaker model of $\operatorname{SL}_2$.
I show that such an operator exists, if we take an enlarged space of test measures $\mathcal S^-_{L_X}(N,\psi\backslash G/N,\psi)$ corresponding to $L_X=$the adjoint $L$-function of $\operatorname{SL}_2$ evaluated at $1$; what this means, at the very least, is that $\mathcal S^-_{L_X}(N,\psi\backslash G/N,\psi)$ will contain the image of the unramified Whittaker measure which corresponds to the coefficients of the local Dirichlet series for $L(\mathrm{Ad},1)$ --- see \S \ref{ssnonstandard}. Explicitly, if we choose representatives $\begin{pmatrix} & -\zeta^{-1} \\ \zeta \end{pmatrix}$ for generic $N\times N$-orbits on $G$, this is the usual space of test measures for the Kuznetsov formula, except that, instead of being of rapid decay as $\zeta\to \infty$, the test measures will be equal to $C(\zeta^{-1}) d^\times \zeta$, where $C$ is a smooth function at zero.
Moreover, I show that this operator has a very simple form: Notice that the elements of $\mathcal S^-_{L_X}(N,\psi\backslash G/N,\psi)$ are measures on the one-dimensional affine space $N\backslash G\sslash N$ (where we use the coordinate $\zeta$ above). Similarly, the elements of $\mathcal S(\frac{G}{G})$ will also be measures on an one-dimensional space, namely the Chevalley quotient $\Dfrac{G}{G}$, where we take the coordinate to be the trace. It turns out that the transfer operator $\mathcal T$ is given by the multiplicative Fourier convolution $\mathscr F_{\operatorname{Id}, 1}$, discussed above, applied to measures on the affine line (under the action of the multiplicative group on the coordinates that we fixed).
The Fourier transform of Rudnick's thesis also appears, in a slightly different and certainly more general form, in the thesis of Altug \cite{Altug1,Altug2,Altug3}. It also appears in a theorem of Soundararajan and Young \cite[Theorem 1.3]{SoundYoung} that is very close to the result that I prove here (it is a global version of the comparison, restricted to hyperbolic conjugacy classes). On the other hand, the spectral side of the comparison between the Kuznetsov formula and the trace formula appears, in the setting of complex loop groups and $D$-modules, features in recent work of Ben-Zvi and Gunningham in \cite{BZvi-Gunningham}.
\subsection{Scattering theory and asymptotics}
Why should Fourier transforms on the line be of any relevance? Are the spaces $N\backslash G\sslash N$ or $\Dfrac{G}{G}$ vector spaces, in any sense? This is probably a misleading point of view, which does not seem to lead to correct conclusions in higher rank. What I show here is that there is another, conceptual way to understand these quotient spaces, or at least their ``limits'' when we replace the spaces by their asymptotic cones. Most of the present, first part of this series of two papers is devoted to developing the harmonic analysis necessary in order to produce this conceptual explanation of the transfer operators.
That is, we will replace the non-trivial character $\psi$ of $N$ by the trivial character, and we will replace the quotient $\frac{G}{G} = G^{\operatorname{diag}}\backslash (G\times G)/G^{\operatorname{diag}}$ by $G^{\operatorname{diag}}\backslash (G_\emptyset\times G_\emptyset)/G^{\operatorname{diag}}$, where $G_\emptyset$, the asymptotic cone of $\operatorname{SL}_2$, is the variety of $2\times 2$ matrices of determinant one. Notice that the general setting of the relative trace formula is essential here, even when we are studying the usual trace formula: to study its ``asymptotics'', we need to replace the group by a different space, and view the adjoint quotient of the trace formula as a special case of a quotient space for the relative trace formula.
When we do that, the quotients $N\backslash G/N$ and $G^{\operatorname{diag}}\backslash (G_\emptyset\times G_\emptyset)/G^{\operatorname{diag}}$ naturally become embeddings of the \emph{same} torus $A\simeq \mathbb{G}_m$ (the universal Cartan of $\operatorname{SL}_2$). Character theory on these degenerate spaces is particularly simple (because they are essentially induced from a torus), and we use local harmonic analysis (scattering theory, see Section \ref{sec:scattering}) to explain that the correct transfer operator (the one that behaves in a prescribed way with respect to characters) between suitable spaces of test measures
$$\mathcal T_\emptyset: \mathcal S^\pm_{L_X}(N\backslash G/N) \xrightarrow\sim \mathcal S(G^{\operatorname{diag}}\backslash (G_\emptyset\times G_\emptyset)/G^{\operatorname{diag}})$$
is given by the multiplicative Fourier convolution $\mathscr F_{\operatorname{Id}, 1}$. Thus, geometrically $\mathcal T$ is given by the same formula as $\mathcal T_\emptyset$, if we choose appropriate coordinates. (I should mention that I do not have a conceptual reason for this on-the-nose equality; in fact, it fails in other cases, as we will see, although one operator is still a deformation of the other.)
I mention that exactly the same phenomenon is true for the comparison between the Kuznetsov formula for $\operatorname{PGL}_2$ and the relative trace formula for $\mathbb{G}_m\backslash \operatorname{PGL}_2/\mathbb{G}_m$, which was developed in \cite{SaBE1}. I briefly revisit this case in Section \ref{sec:Waldspurger}.
\subsection{Open problems}
I hope that the present paper will raise more questions than it settles. Let me list some of them:
\begin{enumerate}
\item Let $G$ be a quasi-split group, and $\psi$ a generic character of a maximal unipotent subgroup $N$. Let $r:{^LG}\to \operatorname{GL}(V)$ be a representation of its $L$-group, not necessarily irreducible. It does not harm to assume that there is a character $\partial$ of $G$ whose dual, composed with $r$, is the canonical (central) cocharacter $\mathbb{G}_m\to\operatorname{GL}(V)$. Attached to these data there should be a canonical space $\mathcal D^-_{L(r,\frac{1}{2})}(N,\psi\backslash G/N,\psi)$ of test half-densities for the Kuznetsov formula, containing (in the non-Archimedean case) the image of the generating series of the local unramified $L$-value $L(r,\frac{1}{2})$, and such that the integrals
$$ \int f J_\pi |\partial|^s$$
of elements $f\in \mathcal D^-_{L(r,\frac{1}{2})}(N,\psi\backslash G/N,\psi)$ against the relative characters $J_\pi$ of irreducible representations admit a Godement--Jacquet theory: for example, their quotients by the local $L$-function $L(r,\frac{1}{2}+s)$ should be entire in $s$, and there should be a ``Hankel transform''
$$ \mathcal H_r: \mathcal D^-_{L(r,\frac{1}{2})}(N,\psi\backslash G/N,\psi) \xrightarrow\sim \mathcal D^-_{L(r^\vee,\frac{1}{2})}(N,\psi\backslash G/N,\psi)$$
satisfying a local functional equation involving the associated local $\gamma$-factors.
None of these properties completely characterizes the space. It would be desirable to have a spectral characterization (possibly by means of a local relative trace formula), as well as a geometric one.
If $r$ is irreducible, this would be the image of the Schwartz space $\mathcal D(G_r)$ or the $L$-monoid of Ng\^o \cite{Ngo-PS}, which also has not been defined yet. It is interesting to ask whether it is easier to define this space at the level of the Kuznetsov formula. Although the definitions of such spaces that I give in this paper are somewhat ad-hoc, one may observe that at least for $r=$ many copies of the standard or the symmetric-square representation of $\operatorname{GL}_2$, the definition of $\mathcal D^-_{L(r,\frac{1}{2})}(N,\psi\backslash G/N,\psi)$ seems to be quite straightforward (see \S \ref{ssnonstandard}), while monoids are not well-suited to handle multiplicity.
\item Once local transfer operators or Hankel transforms are available, and have a form that ``in principle'' satisfies a Poisson summation formula, it would be desirable to develop such a summation formula, which would amount to an identity of relative trace formulas. Such an application was developed in \cite{SaBE2}, and I do not see serious obstacles to adapting the methods to the transforms of the present paper; however, streamlining the arguments for the global application would be important in light of future developments, and would enhance our understanding of the nature of orbital integrals close to singularities (and the behavior of those under non-standard transfer operators). For example, one could try to upgrade the local Hankel transform of \eqref{Hankel-Sym2-intro} to a trace formula-theoretic proof of the functional equation of the symmetric-square $L$-function.
\item Although the fundamental lemma for the Hecke algebra is proven in this paper for all transfer operators and Hankel transforms considered, it would be desirable to have a geometric proof of the fundamental lemma, as in the endoscopic case \cite{Ngo-FL}. Such a proof would apply, in particular, to the more general transfer operators considered in the upcoming paper \cite{SaRankone}, where there is ongoing work of Johnstone and Krishna on the fundamental lemma.
\item Most important of all, though, is to understand the nature of transfer operators and Hankel transforms, and how they generalize to higher rank. In this paper, I show that these operators are ``deformations'' of abelian Fourier convolutions of the corrresponding transforms on the horospherical boundary degenerations, which are completely understood. In examples of rank one, discussed in Sections \ref{sec:Rudnick} and \ref{sec:Waldspurger}, they are actually \emph{equal} to those Fourier convolutions, for a suitable choice of coordinates; this is generalized to all rank one varieties in the upcoming paper \cite{SaRankone}. The Hankel transforms, however, discussed in Section \ref{sec:sym2}, require intermediate ``correction factors'' that I do not understand. This is also the case for transfer operators in higher rank, which I have computed for some more examples jointly with Chen Wan, generalizing the calculation of Section \ref{sec:Rudnick}. Understanding the nature of these deformations is, in my mind, the quintessential issue in order to make progress towards ``beyond endoscopy'' in higher rank.
Regarding Hankel transforms, Ng\^o has recently formulated a conjecture, stunning in its simplicity, about the kernel of the transform giving rise to the functional equation of \emph{any} $L$-function, as an invariant distribution on the group \cite{Ngo-Takagi}. It would be desirable to know what transformation it induces at the level of the Kuznetsov formula.
\end{enumerate}
\subsection{Notation}\label{ssnotation}
We work over a local (locally compact) field $F$. Whenever no confusion arises, I denote the set of $F$-points of a variety $X$ simply by $X$; for example, an integral of the form $\int_{\mathbb{G}_m}$ denotes an integral over the group $F^\times$. When discussing (Langlands) dual groups, I will similarly denote them as algebraic groups, e.g.: $\mathbb{G}_m$ is the dual group of $\mathbb{G}_m$; it will be clear from the context if we are referring to a group or its dual.
The categorical quotient $\operatorname{spec} F[X]^G$ of an affine or quasi-affine variety $X$ by a $G$-action will be denoted by $X\sslash G$. When $X$ is a group and $G$ a subgroup acting on $G$ by conjugacy, I will denote this quotient by $\Dfrac{X}{G}$ (and will use $\frac{X}{G}$ as a formal notation for what is denoted by $X/G$ below).
The notation $X\times^G Y$ will denote the quotient of a product $X\times Y$ of two $G$-varieties by the diagonal action of the group $G$. The notation will be used when $G$ acts freely, and the quotient exists as a variety; only when explicitly discussing stacks will this notation be used for the quotient in the stacky sense.
For a product of spaces $X\times Y$, I will sometimes use the notation $\mathcal L_X\boxtimes \mathcal L_Y$ to indicate the tensor product of two vector bundles, one pulled back from $X$ and the other from $Y$. I will sometimes use similar notation for operators (e.g., $S\boxtimes T$), to stress that each is applied to a different variable. ``Vector bundles'' (and especially ``line bundles'') will sometimes refer to complex vector bundles over the $F$-points of a variety; they correspond to the $l$-sheaves of Bernstein and Zelevinsky, in the non-Archimedean case, and to complex vector bundles for the smooth topology, in the Archimedean case; in particular, in both cases the notion of smooth sections is defined. Typically, at least in the Archimedean case, there will also be a natural notion of ``polynomial growth'' for sections of these bundles (i.e., they will be Nash bundles on Nash manifolds, cf.\ \cite{AGSchwartz}). When this structure is clear from the setting, I will be using it without explanation.
The space of ($\mathbb{C}$-valued) Schwartz measures on the $F$-points of a smooth variety $X$ will be denoted by $\mathcal S(X)$. These are just smooth, compactly supported measures in the non-Archimedean case; in the Archimedean case, they are smooth measures which decay rapidly, together with their polynomial derivatives, cf.\ \cite{AGSchwartz}. (For ease of language, we will often not differentiate between the Archimedean and non-Archimedean cases, and say ``rapid decay'' for both; the reader should interpret this as ``compact support'' in the non-Archimedean case.) These are sections of a cosheaf for the semi-algebraic topology on $X$ (or, for the usual topology in the non-Archimedean case); at a few points, I will talk about the stalks over a closed subset $Y$, which are simply defined as the quotient $\mathcal S(X)/\mathcal S(X\smallsetminus Y)$. Moreover, at a few points I will need to work in the more general context of stacks, instead of varieties. The appropriate notion of a Schwartz space of measures, in this context, was introduced in \cite{SaStacks}, but I make an effort to describe them explicitly in the examples at hand, so that the reader will not require this background.
For smooth varieties, it also makes sense to talk about spaces of Schwartz functions or half-densities; those will be denoted by $\mathcal F(X)$, resp.\ $\mathcal D(X)$. Of course, if $dx$ is a nowhere vanishing smooth positive measure of polynomial growth (such as a Haar measure), we have $\mathcal S(X) = \mathcal F(X) dx = \mathcal D(X) (dx)^\frac{1}{2}$.
The image of the push-forward map from $\mathcal S(X)$ to measures on $X\sslash G$ will be denoted by $\mathcal S(X/G)$. This is a slight departure from notation used in \cite{SaStacks} for the Schwartz space of the quotient stack $[X/G]$, so, whenever I actually need a more stacky version of such a Schwartz space, I will be using notation of the form $\mathcal S([X/G])$ (and will explain what I mean by it, in each case). Notice that $\mathcal S(X/G)$ is different from $\mathcal S(X\sslash G)$ --- the latter is simply the usual Schwartz space of the affine variety $X\sslash G$ (assumed smooth), and it is typically, but not always, contained in $\mathcal S(X/G)$. One can typically translate from measures to functions (by choosing appropriate Haar measures), and then the space $\mathcal S(X/G)$ corresponds to the space of ``stable orbital integrals'' for the $G$-action on $X$.
We broaden these notions (and notations), to include spaces of the form $\mathcal F((H,\chi)\backslash G)$, when $\chi$ is a complex character of a subgroup $H$ of $G$; in this case, this notation means $(H,\chi)$-equivariant functions on $G$, which are smooth and of compact support (in the non-Archimedean case) or rapid decay (in the Archimedean case) modulo $H$. The character $\chi$ being of ``polynomial growth'' means that the notion of rapid decay modulo $H$ makes sense, by choosing semi-algebraic local sections of the map $G\to H\backslash G$. Elements of $\mathcal F((H,\chi)\backslash G)$ can also be thought of as Schwartz sections of a complex line bundle $\mathcal L_\chi$ over $H\backslash G$; we can, similarly, consider Schwartz measures or half-densities. For a character $\psi$ of a maximal unipotent group $N$ of $G$, we will be using the notation $\mathcal S(N,\psi\backslash G/N,\psi)$ for the space of ``Schwartz test measures for the Kuznetsov quotient of $G$'', see \S \ref{sstwistedpf}. The appropriate way to think of those is as measures valued in a complex line bundle over the stack $[N\backslash G/N]$, but here we just understand them as measures on the affine quotient $N\backslash G\sslash N$, by some conventions that we explain in \S \ref{sstwistedpf}. Sometimes, we will treat the symbol $(N,\psi\backslash G/N,\psi)$ as a ``space'', e.g., we will be talking about push-forward of measures to that ``space'', meaning the twisted push-forward to $N\backslash G\sslash N$ that is described in that section.
All of these Schwartz spaces will be viewed as abstract vector spaces, in the non-Archimedean case, and as (nuclear) Fr\'echet spaces, in the Archimedean case. The Fr\'echet structure is the usual one (see \cite{AGSchwartz}) for $\mathcal S(X)$, while $\mathcal S(X/G)$ will inherit the quotient topology. By $\hat\otimes$ I denote the completed tensor product of nuclear Fr\'echet spaces; the completion should be ignored in the non-Archimedean case, as should any references to topology. (For convenience of language, I do not always differentiate between the Archimedean and the non-Archimedean case.) The space $V_G$ of coinvariants of a Fr\'echet representation of a group $G$ will, by definition, be the completion of the algebraic coinvariant space, that is, the quotient of $V$ by the \emph{closure} of the set of vectors of the form $v-g\cdot v$.
In the Archimedean case, the appropriate category of Fr\'echet $G$-repre\-sentations to consider is that of \emph{Fr\'echet representations of moderate growth}, or \emph{$F$-representations}, in the language of \cite{BeKr}: these are countable inverse limits of $G$-representations, i.e., they have a complete system of seminorms for which the $G$-action is continuous. I point the reader to \cite{BeKr} for more details, and for the corresponding ``smooth'' notion of \emph{$SF$-representations}.
We fix throughout a non-trivial unitary character $\psi$ of the additive group $F$. If $F$ is non-Archimedean, we will be assuming that its conductor is the ring of integers $\mathfrak o$. We also fix a measure $dx$ on $F$ which is self-dual with respect to $\psi$; this induces a measure $|\omega|$ on $X(F)$, for every volume form $\omega$ on a smooth variety $X$ over $F$. We will also use the measure $d^\times x := \frac{dx}{|x|}$ on the multiplicative group $F^\times$. When $F$ is non-Archimedean and unramified over the base field $\mathbb Q_p$ or $\mathbb F_p((t))$, $dx(\mathfrak o)=1$. The absolute value of any local field is defined to be compatible with the one of the base field under the norm map; in particular, the absolute value on $\mathbb{C}$ is the square of the usual one.
We write $f\ll g$ for two positive functions on a space $X$ to indicate that there is an absolute constant $C$ such that $f(x)\le C g(x)$.
For a representation $V$ of a group $G$, and a measure $h$ on $G$ (or, more general distributions in some settings, see \S \ref{ssmultipliers}), we write $h\cdot v$ for the integral of the $G$-action against $h$:
$$h\cdot v := \int_G h(g) (g\cdot v),$$
whenever this makes sense. When $G$ acts on a space $X$ (say, on the right), the action on functions, half-densities or measures will be the regular one (which is a left action); in particular,
$$ h\cdot f(x) = \int_G h(g) f(xg).$$
There is also another, right action in this case, the convolution action, corresponding to the push-forward under the action map $X\times G\to X$. The two are related by
$$ h\star f(x) = \int_G h(g^{-1}) f(xg) = h^\vee\cdot f(x),$$
where $h^\vee (g) = h(g^{-1})$.
For a torus $T$, I denote by $\hat T$ the group of its unitary (complex) characters (that it, characters of $T(F)$), and by $\hat T_\mathbb{C}$ its \emph{complexification}, the group of all complex characters. The notion of ``complexification'' makes sense, here, because $\hat T$ is naturally a real algebraic variety (with infinitely many components, in general), if $F$ is non-Archimedean, and a real analytic subgroup of a complex analytic group $\hat T_\mathbb{C}$, when $F$ is Archimedean. I point the reader to \S \ref{ssMellin} for more details.
When the torus $T$ acts on a space $X$, at various points in the paper I define Mellin transforms $\hat T_\mathbb{C}\ni \chi\mapsto \check f(\chi)$ of functions, half-densities or measures $f$ on $X$. I note that my parametrization of Mellin transforms is such that the map $f\to \check f(\chi)$ is $(T,\chi)$-equivariant (sometimes, for a normalized action); this is inverse to the classical definition of Mellin transform of a function on ${\RR^\times_+}$ as $\check f(s) = \int f(x) x^s d^\times x$. By this convention, if $h$ is a measure on $T$ with Mellin transform $\check h$, and $f$ is a measure on the space $X$, we have
$$ \widecheck{h\star \varphi}(\chi) = \check h(\chi) \check f(\chi)$$
for the convolution action, and
$$ \widecheck{h\cdot\varphi}(\chi) = \check h(\chi^{-1}) \check f(\chi)$$
for the regular action. I caution that this convention for Mellin transforms is also inverse to the conventions about Satake transforms of split tori over non-Archimedean fields; if we identify the identity component of $\hat T_\mathbb{C}$ with the (complex points of the) Langlands dual torus $\check T$, then the Satake transform of a measure $h$ on $T(F)/T(\mathfrak o)$ is equal to what we denote by $\check h(\bullet^{-1})|_{\check T}$.
For a reductive group $G$, its \emph{universal Cartan}, or simply its Cartan $A_G$ is not a subgroup, but an abstract torus, defined as the quotient of any Borel subgroup $B$ by its unipotent radical $N$. It is unique up to unique isomorphism, and defined over $F$ even when $B$ is not. It comes equipped with a based root datum; in particular, the ``positive'' and ``negative'' roots of $A_G$ are well-defined subsets of its character group, and similarly for coroots. We will typically use additive notation for weights and coweights of a torus, so when we need think of them as morphisms to or from the multiplicative group $\mathbb{G}_m$, we may use exponential notation, like $e^\alpha$.
The Cartan of $G$ is the appropriate basis for the definition of the dual group: the connected dual group $\check G$ of $G$ contains a canonical Cartan which is dual to $A_G$, and the $L$-group ${^LG}$ contains the $L$-group of $A_G$.
Given a representation $r$ of ${^LG}$, the associated local Langlands $L$-function of an irreducible representation $\pi$ of $G$ will be denoted by
$$ L(\pi, r, s).$$
For example, when $G$ is a torus and $\check\lambda:\mathbb{G}_m\to G$ a cocharacter (thought of as a character of the dual torus), $L(\chi,\check\lambda, s)$ stands for the local Dirichlet $L$-function $L(\chi\circ e^{\check\lambda}, s)$. This notation for Dirichlet $L$-functions will also be used in this paper, and we will denote the local Dedekind zeta function of $F$ simply by $\zeta(s)$. We will sometimes also use alternative, standard notation for some $L$-functions, e.g.,
$ L(\pi_1\times \pi_2, s)$, $L(\operatorname{Sym}^2(\pi), s)$ etc., for the Rankin--Selberg, resp.\ the symmetric-square $L$-function of $\operatorname{GL}_n$. If $\chi$ is a multiplicative character, we may also use expressions of the form $L(\chi\times \operatorname{Sym}^2(\pi), s)$ for the $L$-function that arises from the stated ($\operatorname{Sym}^2$, in this example) representation of the dual group of $G$, tensored by the scalar action of $\mathbb{G}_m=$ the dual group of $\mathbb{G}_m$. We denote by $|\bullet|$ the absolute value character of $\mathbb{G}_m$ (that is, of $F^\times$), so, in this notation,
$L(|\bullet|^s\times \operatorname{Sym}^2(\pi),0)$ is the same as $L(\operatorname{Sym}^2(\pi),s)$.
For a large part of the paper (from Section \ref{sec:RS} on) we will consistenly be using $A$ to denote the Cartan of $\operatorname{SL}_2$, and $A_{\operatorname{ad}}$ for the Cartan of $\operatorname{PGL}_2$. We will consistently be identifying these groups with $\mathbb{G}_m$, the former via the positive half-root character, and the latter via the half-root character. These identifications translate the natural map $A\to A_{\operatorname{ad}}$ to the square map $\mathbb{G}_m\to \mathbb{G}_m$. Sometimes, when it is clear which of these two groups we are talking about, we will be using these identifications to write their $L$-functions as Dirichlet $L$-functions, that is:
\begin{description}
\item for a character $\chi$ of the group $A$, $L(\chi,s):=L(\chi,\check\alpha,s)$;
\item for a character $\chi$ of the group $A_{\operatorname{ad}}$, $L(\chi,s):= L(\chi,\frac{\check\alpha}{2},s)$.
\end{description}
Since these notations are not compatible with the pullback map of characters, we will be careful not to use them when both groups are in play.
The notation $\gamma(\pi, r, s, \psi)$ will be used to denote the factors of the local functional equation of an $L$-function, cf.\ \S \ref{ssTate} and \S \ref{ssHankelrelchars}. It is related to the $L$- and epsilon-factors by
\begin{equation}\label{epsilon}
\gamma(\pi, r, s, \psi) L(\pi, r, s) = \epsilon(\pi, r, s, \psi) L(\pi, r^\vee, 1-s),
\end{equation}
where $r^\vee$ is the dual representation of $r$, so $L(\pi, r^\vee, 1-s) = L(\tilde\pi, r, 1-s)$. Here and throughout, $\tilde\pi$ denotes the admissible dual (contragredient) of a smooth, admissible representation $\pi$.
The notion of ``universal Cartan'' generalizes from groups to spherical varieties. If $X$ is a spherical variety (i.e., a normal connected variety with an open orbit $\mathring X$ for the Borel subgroup $B$) under the action of a reductive group $G$, the quotient $\mathring X\sslash N$ (where $N$ is the unipotent radical of $B$) has an action of $A_G=B/N$, that factors through a faithful action of a quotient $A_X$ of $G$ --- this is the Cartan of $X$. For example, for $X=H$ a reductive group under the $G=H\times H$-action (which we define to be a right action, i.e., $x\cdot (h_1, h_2):= h_1^{-1} x h_2$), by the Bruhat decomposition one sees that $A_X$ is the quotient of $A_G=A_H\times A_H$ by the subgroup of elements $({^wa}, a)$, where $a\in A_H$ and $w$ is the longest element of the Weyl group.
This definition creates some inconvenience for a \emph{horospherical} variety (=one where stabilizers contain maximal unipotent subgroups), like $X=N\backslash G$, because the natural action of $A=B/N$ by $G$-automorphisms on $X$ is not compatible with the above map $A\to A_X$ (which in this case is just an isomorphism), but is \emph{conjugate to it by the longest element of the Weyl group}. To resolve this notational problem, we consistenly define the $A$-action on $N\backslash G$ to be the \emph{twist} of the obvious one by the longest Weyl group element, that is, if $a\in A$ then we define
$$ a\cdot Nx := Nb x,$$
where $b\in B$ \emph{represents the element ${^wa}$} in $A=B/N$. We extend this convention to all horospherical varieties. This is an unfortunate nuissance, but it is more benign than having a different action of $A$ on the variety $X$, and a different on its universal Cartan $A_X$.
Finally, I mention that actions of groups on spaces of functions or measures on $G$-spaces will often be normalized, in order to be unitary. (Actions on half-densities need no such normalization, which makes them particularly convenient.) However, in order for the reader not to have to keep track of normalizations, I have made sure that they are not needed in the statements of the main theorems (unless explicitly stated otherwise). Similarly, the notation is adapted to unnormalized actions, even if we are working with normalized ones. For example, in \S \ref{ssHankel} we introduce certain spaces of test half-densities and measures, $\mathcal D^-_{L(\operatorname{Sym}^2, \frac{1}{2})}(N,\psi\backslash G/N,\psi)$ and $\mathcal S^-_{L(\operatorname{Sym}^2, 1)}(N,\psi\backslash G/N,\psi)$, for the Kuznetsov formula of the group $G=\mathbb{G}_m\times\operatorname{SL}_2$; the difference in notation, $L(\operatorname{Sym}^2, \frac{1}{2})$ vs.\ $L(\operatorname{Sym}^2, 1)$, has to do with their images under \emph{unnormalized} push-forward (integration over $\mathbb{G}_m$) to the Kuznetsov formula of $\operatorname{SL}_2$, despite the fact that in \S \ref{ssdescent}, and elsewhere, we work with a normalized version of this push-forward. The analog of this in a more the more familiar setting of Tate's thesis would be the spaces of Schwartz half-densities and measures on $\mathbbm A^1$, where a Tate zeta integral
$$ \int f(a) |a|^s$$
against a Schwartz measure $f$ would be a holomorphic multiple of the local zeta function $\zeta(s+1)$, while a Tate zeta integral
$$ \int \varphi(a) |a|^s (d^\times a)^\frac{1}{2}$$
against a Schwartz half-density $\varphi$ would be a holomorphic multiple of $\zeta(s+\frac{1}{2})$; hence, the analogous notation would be $\mathcal S_{L(\operatorname{Id}, 1)} (\mathbb{G}_m)$ for $\mathcal S(\mathbbm A^1)$, and $\mathcal S_{L(\operatorname{Id}, \frac{1}{2})}$ for $\mathcal D(\mathbbm A^1)$.
\subsection{Acknowledgements}
This paper would not have been possible without the constant encouragement, numerous conversations, and many references and ideas provided by Ng\^o Bao Ch\^au, who invited me to spend the winter and spring quarters of 2017 at the University of Chicago. I also thank Daniel Johnstone for a presentation of Venkatesh's thesis which initiated my understanding of it. I thank Valentin Blomer for various references on related results in analytic number theory. Last but not least, I am deeply indebted to the Institute for Advanced Study for providing me with the perfect environment in order to complete this work, during the academic year 2017--2018. In fact, my earlier papers \cite{SaBE1, SaBE2}, which gave birth to this cycle of ideas, were conceived during my previous stay at the Institute in the spring of 2011, hence this paper owes to the IAS in multiple ways.
This work was supported by NSF grant DMS-1502270, and by a stipend to the IAS from the Charles Simonyi Endowment.
\section{Basic tools: multiplicative Fourier convolutions, non-standard Kuznetsov test measures} \label{sec:background}
The tools presented in this section were also introduced in \cite{SaHanoi}. I summarize them briefly for the sake of completeness.
\subsection{Mellin transforms and multiplicative Fourier convolutions}
\subsubsection{Mellin transforms} \label{ssMellin}
Let $T$ be a torus over $F$. The unitary dual of $T=T(F)$ will be denoted by $\widehat T$, and its entire character group by $\widehat T_\mathbb{C}$. The character group has a natural structure of a complex manifold and, if $F$ is non-Archimedean, of a complex algebraic variety (with infinitely many components). The structure is automatically determined by its restriction to the identity component, which is the character group of $\Lambda_T:= $ the image of the map
\begin{eqnarray}\nonumber
\log_T: T(F) \to \mathfrak t_\mathbb{R} := \operatorname{Hom}_\mathbb{Z}(\operatorname{Hom}(T,\mathbb{G}_m), \mathbb{R})\\
t \mapsto (\chi \mapsto \log|\chi(t)|).\label{logtorus}
\end{eqnarray}
This group is discrete, in the non-Archimedean case, so the identity component of $\widehat T_\mathbb{C}$ is the complex torus with coordinate ring equal to the group ring of $\Lambda_T$.
In the Archimedean case,
we have $\Lambda_T = \mathfrak t_\mathbb{R}$, and the identity component of $\widehat T_\mathbb{C}$ can be identified with the dual $\mathfrak t_\mathbb{C}^* = \operatorname{Hom}(T,\mathbb{G}_m)\otimes \mathbb{C}$, by sending an element $s$ in the latter to the character $\chi_s (t) = e^{\left< \log_T t, s\right>}$. The space $\mathfrak t_\mathbb{C}^*$ is the direct sum of $\mathbb{R}$-vector subspaces:
$$ \mathfrak t_\mathbb{C}^* = \mathfrak t_\mathbb{R}^* \oplus i \mathfrak t_\mathbb{R}^*,$$
with the imaginary summand corresponding to the identity component of $\widehat T$. By a ``bounded vertical strip'' we will mean the preimage of a compact subset of the real subspace under the projection map.
In the Archimedean case there is a canonical splitting of the sequence
\begin{equation}\label{logArchim} 1\to T_0 \to T(F) \to \mathfrak t_\mathbb{R} \to 1,
\end{equation}
where $T_0$ is the compact group $\ker \log_T$, whose image coincides with the subgroup generated by the subgroups $\check\lambda({\RR^\times_+})$, where $\check\lambda$ ranges over the cocharacters into $T$. This identifies the character group
\begin{equation}\label{dualArchimedean}
\widehat T_\mathbb{C} \simeq \widehat{T_0} \times \mathfrak t_\mathbb{C}^*.
\end{equation}
Notice that the character group $\widehat{T_0}$ is discrete.
The \emph{Mellin transform} of a measure $f\in \mathcal S(T)$ is the function
\begin{equation}\label{Mellin-torus}\widehat T_\mathbb{C}\ni \chi\mapsto \int_T f(a) \chi^{-1}(a).
\end{equation}
We recall the Paley--Wiener theorem:
\begin{theorem}\label{PW-torus}
In the non-Archimedean case, Mellin transform defines an isomorphism between $\mathcal S(T)$ and the space of polynomial functions on $\widehat T_\mathbb{C}$ supported on a finite number of components.
In the Archimedean case, under the isomorphism \eqref{dualArchimedean}, it defines an isomorphism between $\mathcal S(T)$ and the completed tensor product of Fr\'echet spaces
$$ \mathbb H^{\operatorname{PW}}(\widehat T_\mathbb{C}):= \mathscr C(\widehat{T_0}) \hat\otimes \mathbb H^{\operatorname{PW}}(\mathfrak t_\mathbb{C}^*).$$
Here, $\mathscr C(\widehat{T_0})$ denotes the dual of $\mathcal S(T_0)$, that is, the space of functions $\varphi$ on the discrete abelian group $\widehat{T_0}$ such that, for any norm $\Vert \bullet \Vert$ on the vector space $\widehat{T_0}\otimes_{\mathbb{Z}} \mathbb{R}$, and any $N\ge 0$, the function $\Vert n \Vert^N \varphi(n)$ is bounded; and
$\mathbb H^{\operatorname{PW}}(\mathfrak t_\mathbb{C}^*)$ is the Paley--Wiener space of entire functions on $\mathfrak t_\mathbb{C}^*$ which are of rapid decay on bounded vertical strips, that is, on every bounded vertical strip $V$ and for every $N$ satisfy
\begin{equation}\label{vertstrip} \sup_{s\in V} |f(s)| (1+\Im(s)|)^N <\infty,\end{equation}
where $|\bullet|$ denotes any norm on the imaginary subspace $i\mathfrak t_\mathbb{R}^*$.
\end{theorem}
This theorem is very classical, but since in most references the Paley--Wiener theorem is stated for compactly supported smooth functions, the reader can consult \cite[\S 3.1]{SaSelberg} for a proof on Schwartz spaces.
Finally, I introduce the notion of \emph{average volume} of a torus with respect to the logarithmic map \eqref{logtorus}: The space $\mathfrak t_\mathbb{R}$ has a canonical lattice, dual to the character lattice of $T$, hence a canonical Haar measure, induced from the standard measure on $\mathbb{R}$. Given a measure $dt$ on $T$, we define
\begin{equation}\label{avgvol}{\operatorname{AvgVol}}(T)= {\operatorname{AvgVol}}(T,dt) = \lim_{c\to\infty} \frac{dt(\log_T^{-1}(cB))}{\operatorname{Vol}(cB)},
\end{equation}
where $B$ is any ball around zero in $\mathfrak t_\mathbb{R}$. If the character group is trivial, we take $\operatorname{Vol}(\mathbb{R}^0)=1$, so ${\operatorname{AvgVol}}(T)=\operatorname{Vol}(T)$.
For example, if $T=F^\times$, with $F$ an unramified extension of $\mathbb Q_p$ with ring of integers $\mathfrak o$ and residual degree $q$, and we take $dt = d^\times x= \frac{d x}{x}$ with $dx(\mathfrak o)=1$, we have ${\operatorname{AvgVol}}(F^\times) = \frac{1-q^{-1}}{\log q}$. In general, if $dx$ is the self-dual measure with respect to the additive character $\psi$ on $F$, we have
\begin{equation}\label{AvgVolFtimes}
{\operatorname{AvgVol}}(F^\times, d^\times x) = \mathrm{Res}_{s=0} \gamma(1-s,\psi),
\end{equation}
see \cite[(2.26)]{SaBE2}, where $\gamma$ is the gamma factor of the local functional equation of Tate integrals, to be recalled below in \eqref{gammaZeta}.
\subsubsection{Multipliers} \label{ssmultipliers}
Let $\mathcal M(T)$ denote the following categories of modules for $T$:
\begin{itemize}
\item in the non-Archimedean case, smooth representations;
\item in the Archimedean case, \emph{smooth representations of moderate growth on Fr\'echet spaces}, or, equivalently, countable inverse limits of Banach representations, cf.\ \cite{BeKr}.
\end{itemize}
The action of $T$ on any $V \in \mathcal M(T)$ extends to an action
$$ \mathcal S(T)\hat\otimes V \to V.$$
Here, $\hat\otimes$ denotes completed tensor product in the Archimedean case (where both spaces are Fr\'echet, and $\mathcal S(T)$ is nuclear), and should be identified with $\otimes$ in the non-Archimedean case.
If $V$ denotes a space of measures on $T$, and the Mellin transform \eqref{Mellin-torus} can be extended by a convergent integral to $V$, for $\chi$ in some region in $\widehat T_\mathbb{C}$, then for such $\chi$ we have
$$ \widecheck{(f \star \varphi)}(\chi) = \check f(\chi)\check\varphi(\chi),$$
for $f\in \mathcal S(T), \varphi\in V$, where $\star$ denotes convolution.
In fact, the action on any object of $\mathcal M(T)$ extends to a larger algebra
$ \widehat{\mathcal S(T)}$,
consisting of those (tempered) distributions on $T$ which after convolution by elements of $\mathcal S(T)$, become elements of $\mathcal S(T)$. Indeed, for every $V\in \mathcal M(T)$, the map $\mathcal S(T)\otimes V\to V$ (uncompleted tensor product!) is surjective --- in the Archimedean case, this is the Dixmier--Malliavin theorem. Thus, for every object $V\in \mathcal M(T)$ and $v = \sum h_i\cdot v_i\in V$, we can define the action of an element $h \in \widehat{\mathcal S(T)}$ as
$$ h\cdot v = \sum_{i=1}^n (h\star h_i)\cdot v.$$
To see that it is well-defined, use projectors $e_\chi \in \mathcal S(T)$ to the various $T_0$-types $\chi$ (i.e., characters of $T_0$, the maximal compact subgroup), and notice that the map $V\ni v \mapsto (v_\chi)_{\chi \in \widehat{T_0}}:= (e_\chi\cdot v)_\chi$ is injective. If $\sum h_i\cdot v_i=0$ then $\sum_i (h\star h_i) \cdot v_i = 0$, because
$$ \left(\sum_i (h\star h_i) \cdot v_i\right)_\chi = \sum_i (e_\chi\star h) \star (e_\chi\star h) \cdot v_i =(e_\chi\star h) \cdot \sum_i (e_\chi\star h) \cdot v_i .$$
In the non-Archimedean case, $\widehat{\mathcal S(T)}$ coincides with the completed Hecke algebra (Bernstein center) of \emph{essentially compact} distributions, i.e., those distributions which become compactly supported after smoothing; these are the distributions whose Mellin transforms are polynomial on $\widehat T_\mathbb{C}$, without the assumption of support on a finite number of components.
In the Archimedean case, this includes, of course, the enveloping algebra of the complexified Lie algebra of $T$, and in particular the enveloping algebra of the image of $\mathfrak t_\mathbb{R}$ under the splitting of \eqref{logArchim}, which via Mellin transform is identified with the polynomial algebra on $t_\mathbb{C}^*$, pulled back to $\widehat T_\mathbb{C}$ via the projection to the second factor of \eqref{dualArchimedean}.
\subsubsection{Tate zeta integrals} \label{ssTate}
Recall that we have fixed a non-trivial unitary character $\psi$ of the additive group $F$, and a measure $dx$ which is self-dual with respect to $\psi$. We set $d^\times x := \frac{dx}{|x|}$.
The \emph{Tate zeta integral} of a Schwartz function on the line, $\Phi \in \mathcal F(F)$, is the integral
$$ Z(\Phi, \chi, s) = \int_{F^\times} \Phi(x) \chi(x) |x|^s d^\times x.$$
Thus, the Tate zeta integral is the Mellin transform of the measure $\Phi d^\times x$, evaluated at the character $\chi'=\chi^{-1}|\bullet|^{-s}$. It is defined convergent when $\Re(s)\gg 0$, and extends to $\chi' \in \widehat{F^\times}_\mathbb{C}$ as a rational function, in the non-Archimedean case, and a meromorphic one, in the Archimedean case. It is a holomorphic multiple of the $L$-factor $L(\chi, s)$, and of rapid decay in bounded vertical strips (away from the poles).
Defining the Fourier transform of the function as $\hat\Phi(t) = \int \Phi(u) \psi(ut) du$, the local functional equation of Tate \cite{Tate-Corvallis} defines a \emph{gamma factor} by:
\begin{equation}\label{gammaZeta}\gamma(\chi,s,\psi) Z(\varphi,\chi,s) = Z(\hat\varphi, \chi^{-1},1-s).
\end{equation}
The gamma factor can be written as
\begin{equation}\label{gammafactor}\gamma(\chi,1-s,\psi) = \frac{\epsilon(\chi,1-s,\psi) L(\chi^{-1},s)}{L(\chi,1-s)},\end{equation}
where the epsilon factor is entire.
We recall that, in the Archimedean case, the $L$-factor is as follows:
\begin{itemize}
\item If $F=\mathbb{R}$ and $\chi = (\mathrm{sgn})^\epsilon |\bullet|^s$, where $\mathrm{sgn}$ is the sign character and $\epsilon = 0$ or $1$, we have
$$ L(\chi, 0) = \pi^{-\frac{s+\epsilon}{2}} \Gamma(\frac{s+\epsilon}{2}).$$
\item
If $F=\mathbb{C}$ and $\chi(z) = e^{i \cdot m \arg(z)} |z|^s$ (the absolute value here is the square of the usual one, i.e., the absolute value of the norm to $\mathbb{R}^\times$), then
$$ L(\chi, 0) = 2\cdot (2\pi)^{-s} \Gamma(s + \frac{|m|}{2}).$$
\end{itemize}
Notice that the poles of the $L$-function are contained in the poles of the function
$$\mathfrak G(s):= \begin{cases}
\Gamma(s) ,\mbox{ if } F=\mathbb{R}; \\
\Gamma(2s), \mbox{ if } F=\mathbb{C}.
\end{cases}$$
We let $\mathbb H^{\operatorname{PW}}_{\mathfrak G}(\mathbb{C})$ denote the Fr\'echet space of holomorphic multiples of $\mathfrak G$, which are of rapid decay in bounded vertical strips, away from the poles.
We have the following description of the image and residues of Tate zeta integrals:
\begin{proposition}\label{Tateimage}
The Tate zeta integral $\Phi\mapsto Z(\Phi,\chi,0)$ defines an isomorphism between the space $\mathcal F(F)$ and the space of polynomial multiples, in the non-Archimedean case, and holomorphic multiples, in the Archimedean case, of the function $L(\chi,0)$ on $\widehat{F^\times}_\mathbb{C}$ which have the following properties:
\begin{itemize}
\item in the non-Archimedean case, they are supported on a finite number of connected components of $\widehat{F^\times}$;
\item in the Archimedean case, factoring the character group of $F^\times$ as in \eqref{dualArchimedean}, with $T_0=$the maximal compact subgroup of $F^\times$, they belong to the completed tensor product
$$ \mathscr C(\widehat{T_0}) \hat\otimes \mathbb H^{\operatorname{PW}}_{\mathfrak G}(\mathbb{C}).$$
\end{itemize}
Moreover, the residue at the trivial character is given by the formula
\begin{equation}\label{Tateresidue} \mathrm{Res}_{s=0} Z(\Phi, 1, s) = \Phi(0) {\operatorname{AvgVol}}(F^\times),\end{equation}
where ${\operatorname{AvgVol}}$ is the average volume defined in \eqref{avgvol}.
\end{proposition}
\begin{remark}\label{remarkPWspace}
We will adopt the convention, both in the Archimedean and non-Archimedean cases, that the above space of functions on $\widehat{F^\times}_\mathbb{C}$ will be denoted by $\mathbb H^{\operatorname{PW}}_{L(\bullet,0)}(\widehat{F^\times}_\mathbb{C})$. This notion of ``Paley--Wiener functions'' (or sections) will recur, in a more general context, later in this paper.
\end{remark}
\begin{proof}
I only sketch the proof for the image of the Tate integral in the Archimedean case; the rest of the results are found in any reference on Tate's thesis. Consider the multiplication map $T_0\times \mathbb{R}_{\ge 0} \to F$. The Schwartz space of rapidly decaying smooth functions on $T_0\times \mathbb{R}_{\ge 0}$ can be written
$$\mathcal F(T_0\times \mathbb{R}_{\ge 0}) = C^\infty(T_0)\hat\otimes \mathcal F(\mathbb{R}_{\ge 0}),$$
and Mellin transform identifies the second factor on the right with the Fr\'echet space $\mathbb H^{\operatorname{PW}}_{\mathfrak G}(\mathbb{C})$. (The difference in the definition of $\mathfrak G$ is due to the fact that in the complex case we are using the square of the usual norm.) Thus,
$$ \mathcal F(T_0\times \mathbb{R}_{\ge 0}) = \mathscr C(\widehat{T_0})\hat\otimes \mathbb H^{\operatorname{PW}}_{\mathfrak G}(\mathbb{C}).$$
The elements of this space which descend to elements of $\mathcal F(F)$ are those belonging to the closed subspace of holomorphic multiples of $L(\chi, 0)$.
\end{proof}
\subsubsection{Fourier convolutions} \label{sssFourierconv}
Let $T$ be a torus, and $V$ a space of measures, functions or half-densities on $T=T(F)$ (with properties to be specified). For any $s\in \mathbb{C}$, consider the distribution
\begin{equation} \label{DS} D_s:=|x|^s \psi(x) d^\times x
\end{equation}
on $F^\times$. Any cocharacter $\check\lambda: \mathbb{G}_m\to T$ induces, by push-forward, a distribution $\check\lambda_*D_s$ on $T$. The equivariant Fourier transform $\mathscr F_{\check\lambda,s}$ is defined as the operator of convolution by $\check\lambda_* D_s$, on the given space $V$. More generally, for any character $\chi$ of $T(F)$, we define $\mathscr F_{\check\lambda,\chi, s}$ as the operator of multiplicative convolution by the measure $\check\lambda_*\left(\chi(x)|x|^s \psi(x) d^\times x\right)$; of course, by definition, $\mathscr F_{\check\lambda,\chi, s} = \mathscr F_{\check\lambda,\chi|\bullet|^s, 0}$.
If $V = \mathcal S(T)$, the convolution is convergent. In the general case, we will typically need to regularize it. For example, if $T=F^\times$ and $V = \mathcal S(F)$ (considered by restriction as measures on $F^\times\subset F$), with $\check\lambda$ the identity cocharacter, we formally have:
$$ \mathscr F_{\check\lambda,s} f (\xi) = \int_{F^\times} |x|^s \psi(x) f(x^{-1} \xi) d^\times x = |\xi|^{-s} \int_{F^\times} |x|^{s-1} f(x^{-1}) \psi(x\xi) dx.$$
As is well-known, the association $x\mapsto |x|^{s-1} f(x^{-1})$ is a finite measure when
$\Re(s)>-1$, and makes sense by analytic continuation as a tempered distribution for all but a countable set of values of $s$ (corresponding to the poles of a Tate zeta integral). Thus, the multiplicative Fourier convolution above will be interpreted as the Fourier transform of this distribution (valued, again, in distributions, since we have fixed the self-dual measure $dx$). Whenever we say that a multiplicative Fourier convolution should be interpreted ``in a regularized sense'', we will mean as the Fourier transform of a distribution.
As was explained in \cite[Proposition 2.1]{SaHanoi}, multiplicative Fourier convolution acts on Mellin transforms as follows:
\begin{equation}\label{FE} \widecheck{(\mathscr F_{\lambda , s} f)}(\chi) = \gamma(\chi,\check\lambda,1-s,\psi) \check f(\chi),\end{equation}
where $\gamma(\chi,\check\lambda,1-s,\psi)$ is the gamma factor \eqref{gammafactor} of the local functional equation for the character $\chi\circ e^{\check\lambda}$.
The equation \eqref{FE} is literally true for $f\in \mathcal S(T)$. For more general spaces of functions, where $\mathscr F_{ {{{\check\lambda}}} , s} f$ will be defined as the Fourier transform of a distribution, it will require some justification.
\subsection{Non-standard test measures for the Kuznetsov formula}
\subsubsection{Twisted push-forward} \label{sstwistedpf}
Let $G$ be any of the groups in the sequence
$$ \operatorname{SL}_2 \overset{\hookrightarrow}\leftarrow\mathbb{G}_m\times\operatorname{SL}_2 \twoheadrightarrow\operatorname{GL}_2\twoheadrightarrow \operatorname{PGL}_2,$$
where the map $\mathbb{G}_m\to \operatorname{GL}_2$ is the canonical map into the center. Let $N$ denote the subgroup of upper triangular unipotent matrices, $\mathfrak{C}$ the quotient $N\backslash G\sslash N$, and $\mathfrak{C}^0$ its open subset corresponding to the open Bruhat cell.
Let $A$ be the universal Cartan of $G$, identified with the subgroup of diagonal matrices (or $\mathbb{G}_m \times$ that subgroup, in the second case) by choosing the upper triangular Borel subgroup $B$. Let $w=\begin{pmatrix} & -1 \\ 1 \end{pmatrix}$. We identify $A$ with $\mathfrak{C}^0$ via the map $a\mapsto [wa]$.
We identify $N\simeq \mathbb{G}_a$ in the usual way, so that $\psi$ becomes a character of $N$. Let $C^\infty((N,\psi^{-1})\backslash BwB/(N,\psi^{-1}))$ the space of smooth functions on the open Bruhat cell which varies by the character $\psi^{-1}$ under left and right multiplication by $N$. The section $$\mathfrak{C}^0\simeq A \ni a \mapsto wa \in BwB$$
allows us to identify, by restriction,
\begin{equation}C^\infty((N,\psi^{-1})\backslash BwB/(N,\psi^{-1})) \simeq C^\infty(\mathfrak{C}^0).\end{equation}
Dual to this restriction map, we have well-defined \emph{twisted push-forward maps} of measures,
\begin{equation}\label{pushfsingle} \mathcal S(G)\twoheadrightarrow \mathcal S(N,\psi\backslash G) \xrightarrow{p_!} {\operatorname{Meas}}(\mathfrak{C}^0).
\end{equation}
The image of the last map will be denoted by $\mathcal S(N,\psi\backslash G/N,\psi)$ --- it is the space of standard test measures for the Kuznetsov formula. The last map can also be identified with a $G^{\operatorname{diag}}$-invariant map (to be denoted by the same symbol)
\begin{equation}\label{pushftwocopies} \mathcal S(N,\psi\backslash G)\hat\otimes \mathcal S(N,\psi^{-1}\backslash G) \xrightarrow{p_!} \mathcal S(N,\psi\backslash G/N,\psi),
\end{equation}
again dual to the pullback of $(N,\psi^{-1})$-equivariant functions, this time through the map
$$ (N\backslash G)^2\ni (g_1,g_2)\mapsto [g_1g_2^{-1}]\in \mathfrak{C}.$$
It is well-known that both the maps \eqref{pushfsingle}, \eqref{pushftwocopies} can be identified with the corresponding coinvariant quotients:
\begin{equation}\label{Whittakercoinvariants} \mathcal S(N,\psi\backslash G/N,\psi) = \mathcal S(G)_{(N,\psi)^2} = \mathcal S(N,\psi\backslash G)_{(N,\psi)} = \left(\mathcal S(N,\psi\backslash G)\hat\otimes \mathcal S(N,\psi^{-1}\backslash G)\right)_{G^{\operatorname{diag}}}.
\end{equation}
Indeed, the isomorphisms among the various coinvariant spaces follow from the construction of Schwartz spaces on stacks in \cite[\S 3.4]{SaStacks} (with trivial modifications to incorporate the line bundle defined by the character $\psi$), and the isomorphism with $\mathcal S(N,\psi\backslash G/N,\psi)$ is equivalent to the density of regular orbital integrals for the Kuznetsov formula, which is well-known --- see references in the proof of Theorem \ref{density} in the next section.
Since the map $N\backslash G\twoheadrightarrow N\backslash G\sslash N$ is smooth, the untwisted push-forward map: $\mathcal S(N\backslash G)\to {\operatorname{Meas}}(N\backslash G\sslash N)$ has image in Schwartz measures; in particular, elements of $\mathcal S(N,\psi\backslash G/N,\psi)$ are bounded by Schwartz measures on $\mathfrak{C}$, and hence extend to finite measures on $\mathfrak{C}$. In reality, we will never use this fact, but it is convenient to refer to elements of $\mathcal S(N,\psi\backslash G/N,\psi)$ as measures on $\mathfrak{C}$.
Let $f$ belong to any of the spaces
$$\mathcal S(G), \, \, \mathcal S(N,\psi\backslash G), \mbox{ or } \, \mathcal S(N,\psi\backslash G)\hat\otimes \mathcal S(N,\psi^{-1}\backslash G),$$ and write it in the form $\Phi dx$, where $\Phi$ is a (Whittaker, in the second and third cases) Schwartz function and $dx$ is an invariant measure on the corresponding space. For compatible choices of invariant measures, we have the integration formula
\begin{equation}\label{integration}
p_! f(a) = \delta(a) O_a (\Phi) da,
\end{equation}
where $\delta$ is the modular character of the Borel subgroup, considered as a function on $A\subset \mathfrak{C}$, $O_a$ is the orbital integral
$$ O_a(\Phi) = \int_{N\times N} \Phi(n_1 wa n_2) \psi^{-1}(n_1 n_2) dn_1 dn_2$$
in the first case, and similarly in the others, and $da$ is a (multiplicative) Haar measure on $A$.
For a Schwartz function $\Phi$ on $G$ (or one of the other spaces), we define its twisted push-forward by
$$ p_! \Phi(a) = O_a(\Phi),$$
and for a Schwartz half-density $\varphi = \Phi d^\frac{1}{2}x$ we define
\begin{equation}\label{pushf-densities} p_! \varphi(a) = \delta^\frac{1}{2}(a) O_a(\Phi) d^\frac{1}{2}a.
\end{equation}
These push-forwards depend on the choice of a Haar measure on $N$, which however we have fixed to be the self-dual measure with respect to our character $\psi$. The data $d^\frac{1}{2}x$, $d^\frac{1}{2}a$ are then proportional to each other by the integration formula \eqref{integration}, and hence these twisted push-forwards are determined by the choice of Haar measure on $N$. The image of the space of Schwartz Whittaker half-densities under the twisted push-forward will be denoted by
$$\mathcal D(N,\psi\backslash G/N,\psi).$$
Note that these are \emph{densely defined} half-densities on $\mathfrak{C}$; more precisely, they are defined on the open subset $\mathfrak{C}^0\simeq A$.
Finally, I mention that for $G=\operatorname{PGL}_2$ or $\operatorname{SL}_2$ we will be identifying the space $\mathfrak{C}$ with $\mathbb{G}_a$ by choosing the coordinate on $A\subset\mathfrak{C}$ which for $\operatorname{PGL}_2$ is the positive root character, and for $\operatorname{SL}_2$ is the positive half-root character. Thus, explicitly, we have identified $\mathbb{G}_m$ as a subset of $\mathfrak{C}$, and moreover we have a section obtained from the embedding $A\mapsto wA$, as follows:
\begin{eqnarray}\label{sections}\nonumber\mbox{For $\operatorname{PGL}_2$: }\tilde \xi = w e^{\frac{\check\alpha}{2}}(\xi) = \begin{pmatrix} & -1 \\ \xi \end{pmatrix},\,\, \xi\in \mathbb{G}_m;\\
\mbox{For $\operatorname{SL}_2$: }\tilde\zeta = w e^{\check\alpha}(\zeta) = \begin{pmatrix} & -\zeta^{-1} \\ \zeta \end{pmatrix},\,\, \zeta\in \mathbb{G}_m.\end{eqnarray}
\subsubsection{An invariant formulation} \label{ssinvariant}
It will be necessary to have a more invariant description of the Whittaker model and its Kuznetsov quotient.
Let $V$ be a two-dimensional vector space, and $G=\operatorname{SL}(V)$. We take the action of $G$ on $V$ to be a \emph{right} action.
Let $V^\vee$ denote the dual vector space, and $\tilde V = \{ (v, v^\vee)\in V\times V^\vee| \left<v,v^\vee\right>=1\}$. Then $\tilde V$ is a torsor over $V^*:=V\smallsetminus\{0\}$ for the group scheme $\mathbb S$ of stabilizers, in $G$, of the points of $V^*$ (and similarly over $V^{\vee *} := V^\vee\smallsetminus\{0\}$).
The action of $G$ on this group scheme exhibits it as a constant group scheme, and we can fix a $G$-equivariant isomorphism $\mathbb S\simeq \mathbb{G}_a \times V^*$. We will call such an isomorphism a \emph{Whittaker structure} for $V$. Equivalently, this turns $\tilde V$ into a $G$-equivariant $\mathbb{G}_a$-torsor over $V$.
One way to fix such a structure, is to endow $V$ with an invariant symplectic form $\omega$. This defines an isomorphism $\iota_\omega: V\xrightarrow\sim V^\vee$ by $\left<u,\iota_\omega(v)\right> = \omega(u,v)$, the space $\tilde V$ becomes the space of pairs $(v,w)\in V^*\times V^*$ with $\omega(v, w)=1$, and the $\mathbb{G}_a$-action on $\tilde V$ is given by
\begin{equation}\label{Gaaction}
x\cdot (v, w) = (v, w - xv).
\end{equation}
It is immediate to see that this is a bijection between the sets of
\begin{itemize}
\item Whittaker structures on $V^*$, and
\item non-zero alternating forms on $V$.
\end{itemize}
Any two choices of a Whittaker structure are conjugate by a \emph{unique} $G$-automorphism (i.e., scalar automorphism) of $V$; in that sense, the resulting $\mathbb{G}_a$-bundle $\tilde V\to V^*$ is rigid.
From this point on, we fix a Whittaker structure, and use it to view $\tilde V$ as a subvariety of $V^*\times V^*$, so we have two projection maps $\tilde V\underset{t}{\overset{s}\rightrightarrows} V^*$.
The name ``Whittaker structure'' is due to the fact that,
through the additive character $\psi$ of $F$, a $\mathbb{G}_a$-bundle induces a ($\mathbb{C}^\times$-bundle and hence a) complex line bundle $\mathcal L_\psi$ on the $F$-points of $V^*$, whose sections are the Whittaker functions for $G$. The Whittaker line bundle $\mathcal L_\psi$ comes with a trivialization of its pullback to $\tilde V$, arising from the canonical isomorphism (obtained from the projection and action maps): $\mathbb{G}_a\times \tilde V\simeq \tilde V\times_V \tilde V$. Explicitly, Whittaker functions are functions on $\tilde V$ which satisfy
$$ \Phi(v,u- xv) = \psi(x) \Phi(v,u).$$
If we choose the symplectic form $\omega$ and coordinates $(x,y)$ on $V$ so that the symplectic form is $\omega = dx \wedge dy$, the distinguished $G$-orbit on $V^*\times V^*$ is that of the ordered pair $((1,0), (0,1))$. The stabilizer $N^-$ of $(1,0)$ is identified with $\mathbb{G}_a$ by
$$ x\mapsto \begin{pmatrix} 1 & \\ x & 1 \end{pmatrix},$$
and this identifies its orbit through $(0,1)$ as a $\mathbb{G}_a$-torsor by $x\cdot (y,1) = (-x+y,1)$. If we identify an ordered pair $((a,b),(c,d))$ with the element $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2$, Whittaker functions are functions on $\operatorname{SL}_2$ which satisfy
$$ \Phi \left( \begin{pmatrix} 1 \\ x & 1 \end{pmatrix} g\right) = \psi^{-1}(x) \Phi(g).$$
The function $\Phi'(g) = \Phi(w^{-1} g)$ is then a Whittaker function which transforms under the character $\psi$ of the upper triangular subgroup $N\simeq \mathbb{G}_a$, as before, and the translation from $\Phi'$ back to the abstract description of the function $\Phi$ is that $\Phi'\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \Phi((c,d),(-a,-b))$.
The section of the map $G\to \mathfrak{C}$ (over the open $\mathfrak{C}^0$), which allowed us to define a twisted push-forward in the previous subsection, now admits the following description; more precisely, let us describe the section giving rise to \eqref{pushftwocopies}, which now can be written as a map
\begin{equation}\label{pushftwocopies-canonical} \mathcal S(V^*,\mathcal L_\psi) \otimes \mathcal S(V^*,\mathcal L_{\psi^{-1}}) \to \mathcal S(N,\psi\backslash G/N,\psi).
\end{equation}
Since elements of $\mathcal S(V^*,\mathcal L_{\psi^{\pm 1}})$ are scalar-valued functions on $\tilde V$,
we consider the maps
$$ \tilde V \times \tilde V \to V^*\times V^*\to \mathfrak{C}$$
and will describe a distinguished $G$-orbit on $\tilde V\times \tilde V$ that can be used to trivialize push-forwards.
First of all, we have an isomorphism, which can be taken as the definition
$$ \mathfrak{C} = V\times V \sslash G^{\operatorname{diag}}.$$
Moreover, the map $(v_1, v_2)\mapsto \omega(v_1, v_2)$ identifies $\mathfrak{C} \simeq \mathbb{G}_a$. (This is compatible with the map $\mathbb{G}_m \to \mathfrak{C}$ obtained from \eqref{sections}).
Now, over $\mathfrak{C}^0=\mathfrak{C}\smallsetminus\{0\}$ we have a distinguished $G$-stable subset $\widetilde{\mathbb V}\subset \tilde V\times \tilde V$, consisting of those pairs $(v_1, w_1)$ and $(v_2, w_2)$ such that $w_1$ and $v_2$ are colinear, and $v_1$ and $w_2$ are colinear. The map $\widetilde{\mathbb V} \to V^*\times V^*$, $(v_1,w_1,v_2,w_2)\mapsto (v_1,v_2)$, is an isomorphism, and it allows us to pull back $\mathcal L_\psi\boxtimes \mathcal L_{\psi^{-1}}$-valued measures on $V^*\times V^*$ to actual measures on $\widetilde{\mathbb V}$, and push them forward to $\mathfrak{C}$. This gives rise to the twisted push-forward \eqref{pushftwocopies-canonical}.
\subsubsection{Generating series for unramified $L$-values}
Assume here that $F$ is a non-Archimedean field, with ring of integers $\mathfrak o$ and residual degree $q$. Let $K=G(\mathfrak o)$, and $\mathcal H(G,K)$ the Hecke algebra of $K$-biinvariant, compactly supported measures on $G$.
Unramified characters of the universal Cartan $A$ of $G$ are in canonical bijection with points of the complex dual torus $\check A$; we will denote this bijection as $\chi \leftrightarrow \check \chi$. It is characterized by $\chi({\check\lambda}(\varpi)) = {\check\lambda}(\check\chi)$ for any coweight $\check\lambda$ into $A$, and $\varpi$ a uniformizer in $\mathfrak o$.
The Satake isomorphism
$$ \mathcal H(G,K)\ni h \mapsto \check h \in \mathbb{C}[\check G]^{\check G} = \mathbb{C}[\check A]^W$$
is characterized by the property that, for the principal series representation $\pi_\chi$ obtained by normalized induction from the character $\chi$ of a Borel subgroup $B$ (through the quotient $B\twoheadrightarrow A$), we have
$\pi_{\chi}(h) v_{K,\chi} = \check h(\check\chi) v_{K,\chi}$.
If $X$ is a smooth, quasi-affine $G$-space over $\mathfrak o$, we call the characteristic function $1_{X(\mathfrak o)}$ of $X(\mathfrak o)$ the \emph{basic function} of $X$. We would like to focus on the Whittaker model, so we will use $X$ to denote the ``space'' $X = (N,\psi)\backslash G$, that is, the space $N\backslash G$, but endowed with the complex line bundle defined by the character $\psi$. In that case, the basic function (still to be denoted by $1_{X(\mathfrak o)}$ will be the left-$(N,\psi)$-equivariant function on $G$ which is supported on $NK$ and equal to $1$ on $K$.
Let $r: \check G\to \operatorname{GL}(V)$ be an algebraic representation, and $s\in \mathbb{C}$. The \emph{local unramified $L$-value}
$L(r,s)$
is the element
$$L(r,s):= \frac{1}{\det(I-q^{-s}r)} \in \mathbb{C}(\check G)^{\check G}.$$
It can be written as a formal Taylor series in $q^{-s}$:
$$L(r,s) = \sum_{n=0}^\infty q^{-ns} \operatorname{tr}(\operatorname{Sym}^n r).$$
If $r$ is reducible, $r = \bigoplus_{i=1}^m r_i$, we can allow $s$ to denote an $m$-tuple $(s_i)_{i=1}^m$ of complex numbers, and define $L(r,s)=\prod_{i=1}^m L(r_i,s_i)$. I proceed with a single $s$, and the adjustments for the general case are obvious.
The \emph{generating series of the local $L$-value} $L(r,s)$ on $X$ is the Whittaker function
\begin{equation}\label{Lseries}\Phi_{L(r,s)} = \sum_{n=0}^\infty q^{-ns} h_{\operatorname{Sym}^n r} \star 1_{X(\mathfrak o)},
\end{equation}
whenever this series converges, where $h_{\operatorname{Sym}^n r}\in \mathcal H(G,K)$ is the element with
$$ \check h_{\operatorname{Sym}^n r} = \operatorname{tr}(\operatorname{Sym}^n r).$$
It appears more appropriate ensure that there is a character $\partial: G\to \mathbb{G}_m$, whose dual $\partial^*: \mathbb{G}_m\to \mathcal Z(\check G)$, followed by $r$, gives rise to the canonical cocharacter to the center of $\operatorname{GL}(V)$. For example, when $G=\operatorname{PGL}_2$, the standard representation of $\check G= \operatorname{SL}_2$ should be extended to the group $\operatorname{GL}_2$, which corresponds to replacing $G$ by $\operatorname{GL}_2$. Similarly, for the symmetric-square representation of $\operatorname{SL}_2$, which factors through the adjoint representation of $\operatorname{PGL}_2$, one should take $\mathbb{G}_m\times\operatorname{PGL}_2$ to be the dual group, so that $G=\mathbb{G}_m \times \operatorname{SL}_2$. In that case, the complex parameter $s$ of the $L$-value becomes a red herring, and can be fixed to be $0$ (or $\frac{1}{2}$), since we have an equality of rational functions on $\check G$:
$$ L(r, s)(x) = L(r, 0) (x\partial^*(q^{-s})).$$
For Langlands $L$-functions of representations, this is the statement that $L(\pi, r, s) = L(\pi\otimes |\partial|^s, r, 0)$.
Moreover, in that case, the series \eqref{Lseries} makes sense for every $s$, since the $n$-th summand is compactly supported on the subset with ${\operatorname{val}} (\partial )= n$.
For our present purposes, however, we would like to allow ourselves to take \eqref{Lseries} in $\operatorname{SL}_2$ or $\operatorname{PGL}_2$, which corresponds to integrating it over the fibers of the map $\mathbb{G}_m\times\operatorname{SL}_2\to\operatorname{SL}_2$ or $\operatorname{GL}_2\to\operatorname{PGL}_2$. This integral converges when $\Re(s)\gg 0$; in the examples of interest, it will follow from the results that we prove that it admits meromorphic continuation to all $s$, rational in the parameter $q^{-s}$.
Finally, we would like to work with measures instead of functions. For that purpose, we choose the invariant measure $dx$ on $N\backslash G$ such that $\operatorname{Vol}(N\backslash G(\mathfrak o))=1$. The product $\Phi_{L(r,s)} dx$ will be called the ``generating measure of $L(r,s)$''.
\subsubsection{Non-standard test measures} \label{ssnonstandard}
Now let $G$ be one of the four groups above, and consider the twisted push-forward $p_!:\mathcal S(N,\psi\backslash G)\to \mathcal S(N,\psi\backslash G/N,\psi)$ of \S \ref{sstwistedpf}. It is easy to see that, restricted to $K$-invariants, where $K = G(\mathfrak o)$, it is locally finite, in the sense that for any $c\in \mathfrak{C} = N\backslash G\sslash N$ there is only a finite number of $K$-orbits on $N\backslash G$ such that the elements of $\mathcal S(N,\psi\backslash G)^K$ which are supported on those $K$-orbits have non-zero push-forward in a neighborhood of $c$, cf.\ \cite[\S 6.3]{SaBE1}.\footnote{There is a typo on the last line of \cite[(6.2)]{SaBE1}: $m=1$ should read $m=0$. The property of local finiteness of the push-forward explained here is not special to $K$ --- it holds for invariants of any compact open subgroup.} Thus, we can extend the twisted push-forward to \emph{any} $K$-invariant Whittaker measure. In particular, the twisted push-forward of $\Phi_{L(r,s)} dx$ is well-defined whenever $\Phi_{L(r,s)}$ is, and will be denoted by
$$ f_{L(r,s)} \in {\operatorname{Meas}}(N,\psi\backslash G/N,\psi).$$
Explicitly, using the integration formula \eqref{integration}, we have
$$ f_{L(r,s)}(a) = (1-q^{-2})^{-1} \left(\int_N \Phi_{L(r,s)}(wan) \psi^{-1}(n) dn\right) \cdot \delta(a) da,$$
when the measure on $N(\mathfrak o)$ is taken to be $1$, and $da(A(\mathfrak o))=(1-q^{-1})$; indeed, the Haar measure on $G$ which gives total mass $1$ to $G(\mathfrak o)$ restricts on the open Bruhat cell $Nw A N$, in coordinates $n_1wan_2$, to the measure $(1-q^{-2})^{-1} dn_1 \delta(a) da dn_2$, and $da$ is the Haar measure with $da(A(\mathfrak o))=1-q^{-1}$.
Now we specialize to $G=\operatorname{PGL}_2$ and $r=$copies of the standard representation of $\check G=\operatorname{SL}_2$, or $G=\operatorname{SL}_2$ and $r=$copies of the adjoint representation of $\check G=\operatorname{PGL}_2$. So, we write $r = \bigoplus_i r_i$ with all $r_i$'s equal to the standard representation (for $\operatorname{PGL}_2$) or the adjoint representation (for $\operatorname{SL}_2$), and correspondingly $s=(s_i)_i$ denotes a collection of complex numbers, one for every $r_i$. Then $f_{L(r,s)} $ lives in a natural space of test measures
$$ \mathcal S^-_{L(r,s)} (N,\psi\backslash G/N,\psi) = \mathcal S^-_{\prod_i L(r_i,s_i)} (N,\psi\backslash G/N,\psi)$$
for the Kuznetsov formula, described as follows; here, we allow again the field $F$ to be Archimedean:
We let $ \mathcal S^-_{L(r,s)} (N,\psi\backslash G/N,\psi)$ be the space of measures on $\mathfrak C$ which on any compact set coincide with elements of $\mathcal S(N,\psi\backslash G/N,\psi)$, while in a neighborhood of infinity, in the coordinates of \eqref{sections}, when all of the $s_i$'s are \emph{distinct}, they are of the form
\begin{equation}\label{expansionSL2}
\sum_i C_i(\zeta^{-1}) |\zeta|^{1-s_i} d^\times \zeta,
\end{equation}
in the case of $G=\operatorname{SL}_2$, and
\begin{equation}\label{expansionPGL2}
\sum_i C_i(\xi^{-1}) |\xi|^{\frac{1}{2}-s_i} d^\times \xi,
\end{equation}
in the case of $G=\operatorname{PGL}_2$, where the $C_i$'s are smooth functions in a neighborhood of zero. When two or more of the $s_i$'s coincide with some complex number $s_0$, the corresponding summands at infinty will be replaced by $(C_1(\zeta^{-1}) + C_2(\zeta^{-1})\log|\zeta| + C_3(\zeta^{-1}) \log^2|\zeta| +\dots) |\zeta|^{1-s_0} d^\times \zeta$ (as many summands as occurences of the exponent $s_0$), and similarly for $\operatorname{PGL}_2$.
More generally, to accommodate possible push-forwards from the groups $\operatorname{GL}_2$ and $\mathbb{G}_m\times \operatorname{SL}_2$ to $\operatorname{PGL}_2$, resp.\ $\operatorname{SL}_2$, we can define, for any collection of characters $\chi=(\chi_i)_i$ of $\mathbb{G}_m$, a space
$$ \mathcal S^-_{L(r,\chi)} (N,\psi\backslash G/N,\psi)$$
whose elements
coincide with elements of $\mathcal S(N,\psi\backslash G/N,\psi)$ away from infinity, and are of the form
\begin{equation}\label{expansionSL2-chi}
\sum_i C_i(\zeta^{-1}) |\zeta| \cdot \chi_i^{-1}(\zeta) d^\times \zeta,
\end{equation}
in the case of $G=\operatorname{SL}_2$, and
\begin{equation}\label{expansionPGL2-chi}
\sum_i C_i(\xi^{-1}) |\xi|^{\frac{1}{2}} \cdot \chi_i^{-1}(\xi) d^\times \xi,
\end{equation}
with the analogous modifications when some of the $\chi_i$'s coincide.
The following was stated as \cite[Proposition 3.2]{SaHanoi}, and can be proven as in \cite[Lemma 5.3]{SaBE1}:
\begin{proposition}
If $F$ is non-Archimedean and $\Phi_{L(r,s)} dx$ is well-defined (for example, when $\Re(s)\gg 0$), its image $ f_{L(r,s)} $ is contained in $\mathcal S^-_{L(r,s)} (N,\psi\backslash G/N,\psi)$.
\end{proposition}
We will call $f_{L(r,s)}$ the \emph{basic vector} of $\mathcal S^-_{L(r,s)} (N,\psi\backslash G/N,\psi)$.
\begin{example}
Let $G=\operatorname{SL}_2$, considered as a reductive group over $\mathfrak o$, $K=G(\mathfrak o)$, $dx$ the invariant measure on $N\backslash G$ with $dx(N\backslash G(\mathfrak o))=1$, and take $r=\mathrm{Ad}$. Let $\Lambda^+= - \mathbb N \check\alpha$ be the set of anti-dominant elements in the coweight lattice of the universal Cartan $A$, and, for every $\check\lambda\in \Lambda^+$, let $e^{\check\lambda}$ stand for the element in $\mathcal S(N,\psi\backslash G)^K$ which is equal to the product of $dx$ by the Whittaker function which is
zero of the coset $N e^{-\check\lambda}(\varphi) K$, and equal to $q^{\left<\rho,\check\lambda\right>}$ on $e^{-\check\lambda}(\varphi)$. We use rational functions of the form $\frac{1}{1-q^{-s}e^{\check\lambda}}$ to denote the series $\sum_{i\ge 0} q^{-is} e^{i\check\lambda}$, but caution that $\frac{1}{1-q^{-s}e^{\check\lambda}}$ and $\frac{-q^s e^{-\check\lambda}}{1-q^s e^{-\check\lambda}}$ denote different series.
Then, \cite[Theorem 7.7]{SaSatake} states that, for $\Re(s)\gg 0$, the element $\Phi_{L(r,s)} dx$ coincides with the restriction to $\Lambda^+$ of the ``series''
\begin{equation}\label{basicadjoint} \frac{1-e^{\check\alpha}}{(1-q^{-s} e^{\check\alpha})(1-q^{-s} )(1-q^{-s} e^{-\check\alpha})}.
\end{equation}
Moreover, \cite[(6.2)]{SaBE1} computes the orbital integrals of the Whittaker function $\frac{q^{-\left<\rho,\check\lambda\right>}e^{\check\lambda}}{dx}$; we multiply by
$$q^{\left<\rho,\check\lambda\right>}\cdot (1-q^{-2})^{-1} \delta(a) da = q^{\left<\rho,\check\lambda\right>}\cdot (1-q^{-2})^{-1} |\zeta|^2 d^\times \zeta,$$
to deduce that the twisted push-forward of $e^{\check\lambda}$, when $\check\lambda \ne 0$, is equal to the measure
$$q^{\left<\rho,\check\lambda\right>}\cdot (1-q^{-2})^{-1} |\zeta|^2 d^\times \zeta \cdot \left(1_{|\zeta|=q^{-\left<\rho,\check\lambda\right>}}- 1_{|\zeta|=q^{-\left<\rho,\check\lambda\right>-1}}\right)=$$
$$= (1-q^{-2})^{-1} |\zeta| d^\times \zeta \cdot \left(1_{|\zeta|=q^{-\left<\rho,\check\lambda\right>}}- q^{-1} \cdot 1_{|\zeta|=q^{-\left<\rho,\check\lambda\right>-1}}\right).$$
We can let $\epsilon^{\check\lambda}$ denote the restriction of the measure $|\zeta| d^\times\zeta$ to the set $|\zeta|=q^{-\left<\rho,\check\lambda\right>}$, then the twisted push-forward of $e^{\check\lambda}$ reads
$$ (1-q^{-2})^{-1} (\epsilon^{\check\lambda} - q^{-1} \epsilon^{\check\lambda + \check\alpha}),$$
and the twisted push-forward of the series \eqref{basicadjoint}, restricted to the set $|\zeta|>1$, will be equal to the ``series''
\begin{equation}\label{basicadjointpushf} (1-q^{-2})^{-1} \frac{(1-\epsilon^{\check\alpha})(1-q^{-1}\epsilon^{\check\alpha})}{(1-q^{-s} \epsilon^{\check\alpha})(1-q^{-s} )(1-q^{-s} \epsilon^{-\check\alpha})}
\end{equation}
restricted to negative multiples of $\check\alpha$.
The series $\frac{1}{1-q^{-s} \epsilon^{-\check\alpha}}$, just by itself, is equal to the measure $|\zeta|^{1-s} d^\times\zeta$. I leave it to the reader to check that the asymptotics of \eqref{basicadjointpushf} are now obtained by setting $\epsilon^{\check\alpha}=q^{-s}$ in the remaining factors; we find that $f_{L(\mathrm{Ad}, s)}$ is equal to
\begin{equation}\label{asymptoticsbasicadjoint}
\frac{1-q^{-1-s}}{(1-q^{-2})(1-q^{-2s})} |\zeta|^{1-s} d^\times\zeta
\end{equation}
for large $|\zeta|$.
\end{example}
We will also work with half-densities, thus we similarly define a space $\mathcal D^-_{L(r,s)} (N,\psi\backslash G/N,\psi)$ which contains $\mathcal D(N,\psi\backslash G/N,\psi)$. In this case, the analogous to \eqref{expansionSL2}, \eqref{expansionPGL2} expansions at infinity are:
\begin{equation}\label{expansionSL2-densities}
\sum_i C_i(\zeta^{-1}) |\zeta|^{-s_i} (d^\times \zeta)^\frac{1}{2},
\end{equation}
in the case of $G=\operatorname{SL}_2$, and
\begin{equation}\label{expansionPGL2-densities}
\sum_i C_i(\xi^{-1}) |\xi|^{-s_i} (d^\times \xi)^\frac{1}{2},
\end{equation}
by comparing \eqref{integration} and \eqref{pushf-densities}. The \emph{basic vector} of these spaces, in the non-Archimedean case, will be the quotient of the measure $f_{L(r,s)} $ by $\delta^\frac{1}{2}(a) d^\frac{1}{2}a$, that is, by $|\zeta| (d^\times \zeta)^\frac{1}{2}$ in the case of $\operatorname{SL}_2$ and $|\xi|^\frac{1}{2}(d^\times \xi)^\frac{1}{2}$ in the case of $\operatorname{PGL}_2$. It will again be denoted by $f_{L(r,s)} $, when it is clear that we are referring to half-densities, instead of measures.
Finally, we mention that the regular action of the unramified Hecke algebra $\mathcal H(G,K)$ on $\mathcal S(N,\psi\backslash G)^K$ descends to its image in $\mathcal S(N,\psi\backslash G/N,\psi)$; indeed, the action of $\mathcal H(G,K)$ coincides with the action of the unramified component of the Bernstein center, which descends to the coinvariant space $\mathcal S(N,\psi\backslash G)_{(N,\psi)} = \mathcal S(N,\psi\backslash G/N,\psi)$. Thus, we will feel free to write $h\cdot f$ for $h\in \mathcal H(G,K)$ and $f\in \mathcal S(N,\psi\backslash G/N,\psi)$ in the image of $\mathcal S(N,\psi\backslash G)^K$ (or a series of such elements).
\section{Scattering theory} \label{sec:scattering}
A main theme of the present paper is the comparison between transfer operators involving such a variety $X$, and transfer operators involving its \emph{boundary degeneration} or \emph{asymptotic cone} $X_\emptyset$, as horospherical $G$-space which is, roughly, responsible for the continuous spectrum of $X$.
\subsection{Asymptotic cone}
The asymptotic cone can be defined using coordinate rings: Suppose that $X$ is quasi-affine, and $k[X]= \bigoplus_{\lambda} V_\lambda$ \emph{as a $G$-module}, a multiplicity-free direct sum of irreducible submodules $V_\lambda$, with highest weight $\lambda$ varying over some submonoid $ \Lambda_X^{++}$ of the character group of the universal Cartan $A$ of $G$. Notice that, fixing a Borel subgroup, for two highest-weight vectors $v_\lambda \in V_\lambda$, $v_\mu\in V_\mu$ we have
\begin{equation}\label{highestweight}v_\lambda\cdot v_\mu = v_{\lambda+\mu}\end{equation} for some highest weigt vector $v_{\lambda+\mu}\in V_{\lambda+\mu}$, but in general $V_\lambda \cdot V_\mu\nsubset V_{\lambda+\mu}$. We define $k[X_\emptyset^a] := \bigoplus_{\lambda} V_\lambda$ \emph{as an algebra}, where the algebra structure is defined by projecting the product of $V_\lambda$ and $V_\mu$ to the direct summand $V_{\lambda+\mu}$, and take $X_\emptyset$ to be the open $G$-orbit in $X_\emptyset^a$. For the Whittaker case, we define the boundary degeneration by retaining the same space $N\backslash G$, but making the character on $N$ trivial.
Here is a list of the spaces $X$ that we are using in this paper, and the isomorphism classes of their asymptotic cones:
\begin{equation}\label{tableX} \begin{array}{|c|c|c|}
\hline G & X & X_\emptyset \\
\hline
\operatorname{PGL}_2, \operatorname{SL}_2, \operatorname{GL}_2 \mbox{ or } \mathbb{G}_m\times \operatorname{SL}_2 & (N,\psi)\backslash G & N\backslash G \\
\operatorname{PGL}_2 & \mathbb{G}_m\backslash \operatorname{PGL}_2 & N\backslash \operatorname{PGL}_2 \\
\operatorname{SL}_2^2/\{\pm 1\}^{\operatorname{diag}}\simeq {\operatorname{SO}}_4 & \operatorname{SL}_2 & T^{\operatorname{diag}}(N\times N^-)\backslash \operatorname{SL}_2^2\\
\hline
\end{array}
\end{equation}
In the last case, $N$ and $N^-$ are the unipotent radicals of two opposite Borel subgroups of $\operatorname{SL}_2$, and $T$ denotes their intersection. In that case, $X_\emptyset$ can be identified with the variety of $2\times 2$-matrices of rank one.
In every case, the asymptotic cone $X_\emptyset$ has an action of a torus $A_X$ of $G$-automorphisms; the character group of $A_X$ is generated by the monoid $\Lambda_X^{++}$ of weights appearing in the highest weight decomposition above, and the action is equivalent to the grading of the coordinate ring. Notice, however, that \eqref{highestweight} translates to a \emph{canonical} isomorphism
\begin{equation}\label{samehoro} X\sslash N_G = X_\emptyset \sslash N_G,
\end{equation}
where $N_G$ is the unipotent radical of a Borel subgroup of $G$. This identifies general $N_G$-orbits on $X$ and $X_\emptyset$, and rigidifies $X_\emptyset$, in the sense that the action of a non-trivial element of $A_X$ would not preserve this isomorphism.
In the examples above, for the Whittaker model and the variety $\mathbb{G}_m\backslash \operatorname{PGL}_2$, we have $A_X = A_G$, the Cartan of $G$, while for $X=\operatorname{SL}_2$ we have $A_X=$ the Cartan $A$ of $\operatorname{SL}_2$. There is a canonical finite reflection group $W_X$ acting on $A_X$, the \emph{little Weyl group} of $X$; for the examples above, this Weyl group is isomorphic to $\mathbb{Z}/2$.
By definition, the action of $A_X$ on $X_\emptyset$ is compatible with its action on $X_\emptyset\sslash N_G = X\sslash N_G$. Unfortunately, this creates some confusion for a variety such as $X=N_G\backslash G$, because $A_X = A_G$ in this case is also identified with the quotient $B_G/N_G$, but \emph{this is not the way it acts on $X$}; indeed, it would be preferable to represent $X$ as $N_G^-\backslash G$, where $N_G^-$ is \emph{opposite to the Borel subgroup used to define the Cartan $A_G$}; equivalently, if $x\in X$ is a point with stabilizer $N_G$, \emph{the quotient $A_G=B_G/N_G$ acts on $X$ by twisting the natural action by $w_0$, the longest element of the Weyl group}. To avoid excessive notation, we will usually be denoting a maximal unipotent stabilizer by $N_G$ or $N$, not by $N_G^-$, \textbf{but the reader should keep this convention about the $A_X$-action in mind}, to avoid confusion.
Moreover, for a space of the form $X=S\backslash G$, where $N_G\subset S \subset \ker(e^{2\rho}) \subset G$, where ${2\rho}$ is the sum of positive roots, so that $X$ admits a $G$-invariant measure $dx$, we \emph{define a normalized action of $A_X = (B_G/S)^{w_0}$ on functions and measures on $G$}, as follows:
\begin{equation}\label{action-normalized-functions}
(a\cdot \Phi)(x) = \delta^{\frac{1}{2}}(a) \Phi( a\cdot x)
\end{equation}
on functions, and
\begin{equation}\label{action-normalized-measures}
(a\cdot \mu)(x) = \delta^{-\frac{1}{2}}(a) \mu( a\cdot x)
\end{equation}
on measures, where $\delta = |e^{2\rho}|$ is the modular character of $B_G$. This action is unitary on the $L^2$-spaces of functions or measures. On half-densities, no normalization is needed.
\subsection{Scattering operators}
From now on until the end of this section, let $F$ be non-Archimedean. I describe some of the results of \cite{DHS}. For each of the varieties $X$ of Table \eqref{tableX}, there is a space $\mathcal S^+(X_\emptyset)$ of smooth measures on $X_\emptyset$, with an $A_X$-semilinear action of $W_X$ by $G$-automorphisms, the \emph{scattering morphisms}
$$\mathfrak S_w: \mathcal S^+(X_\emptyset) \xrightarrow\sim \mathcal S^+(X_\emptyset),$$
such that $\mathcal S^+(X_\emptyset)$ is generated by $\mathcal S(X_\emptyset)$ under these scattering morphisms. The support of elements of $\mathcal S^+(X_\emptyset)$ has compact closure in the affine variety $X_\emptyset^a$ that we saw above. Moreover, there is a
canonical ``asymptotics'' morphism
\begin{equation}\label{asymptotics} e_\emptyset^*: \mathcal S(X)\to \mathcal S^+(X_\emptyset),
\end{equation}
defined in \cite[Section 5]{SV} which, according to the Paley--Wiener theorem \cite[Theorem 1.8]{DHS} has image precisely in the subspace of $W_X$-invariants under the scattering morphisms.
I will not repeat here what makes the asymptotics morphism canonical; roughly speaking, it is the only morphism such that a measure $\varphi$ is ``equal'' to $e_\emptyset^* \varphi$ ``close to infinity'', see \cite[Section 5]{SV} for details. The scattering operators are characterized by the above properties, but we need a more explicit description of them, in order to compute them. This description is given by \cite[(9.4) and Proposition 10.18]{DHS}, and we repeat it here; unfortunately, the theoretical description is quite involved; the reader may want to skip directly to our computation of scattering operators in the three examples of Table \eqref{tableX}, which is performed in the following subsections and is quite straightforward, and return to the definitions as needed.
We assume, for simplicity, that $X$ admits a $G$-invariant measure; by \cite[\S 4.2]{SV}, any such measure induces a $G$-invariant measure on $X_\emptyset$. We can work with functions instead of measures: any element of $\mathcal S^+(X_\emptyset)$ can be written as the product of a $G$-invariant measure $dx$ by a function in a space $\mathcal F^+(X_\emptyset)$, and once the scattering operators have been defined for functions, they are defined for measures by multiplying by $dx$. The normalizations \eqref{action-normalized-functions}, \eqref{action-normalized-measures} ensure that multiplication by $dx$ is $A_X\times G$-equivariant.
An element $\Phi\in \mathcal F^+(X_\emptyset)$ can be reconstructed from its Mellin transform
\begin{equation}\label{Mellin-Splus} \check\Phi(\chi)(x) = \int_{A_X} a\cdot \Phi(x) \chi^{-1}(a) da \in C^\infty(A_X,\chi\backslash X_\emptyset)
\end{equation}
Here, $C^\infty(A_X,\chi\backslash X_\emptyset)$ means that the function is $\chi$-equivariant with respect to the \emph{normalized} action of $A_X$. If we choose a base point on $X_\emptyset$, with stabilizer contained in a Borel subgroup $B$, the space $C^\infty(A_X,\chi\backslash X_\emptyset)$ becomes equal to the normalized induced representation $I_B({^{w_0}\chi})$, where $w_0$ is the longest element of the Weyl group. (Recall the conventions about the $A_X$-action on $X_\emptyset$, described above.) This Mellin transform is convergent once $\chi^{-1}$ vanishes fast enough on the complement of $X_\emptyset$ in $X_\emptyset^a$ (that is, it vanishes fast enough on the boundary of an $A_X$-orbit), extends rationally to the variety of all complex characters of $A_X$, and choosing such a character $\omega$ that vanishes fast enough on the boundary of an $A_X$-orbit, the \emph{inverse Mellin transform} is:
$$\Phi(x) = \int_{\omega^{-1} \widehat{A_X}} \check\Phi(\chi)(x) d\chi.$$
The Haar measure $da$ on $A_X$ chosen to define the Mellin transform is not important here, but one has to choose the dual Haar measure $d\chi$ on its unitary dual $\widehat{A_X}$.
The scattering operator $\mathfrak S_w$ can be accordingly decomposed:
\begin{equation}\label{scatteringMellin} \mathfrak S_w \Phi = \int_{\omega^{-1} \widehat{A_X}} \mathscr S_{w,\chi}\check\Phi({^{w^{-1}}\chi})(x) d\chi,
\end{equation}
where\footnote{There is a slight difference here from the notation of \cite{DHS}: scattering operators are indexed by the character \emph{in their image}, not in their source. This is to ensure compatibility with other morphisms, like the $\mathfrak N_\chi$'s below, which have only a character in appearing their image, not in their source.}
\begin{equation}\label{fiberwisescattering}\mathscr S_{w,\chi}: C^\infty(A_X,{^{w^{-1}}\chi}\backslash X_\emptyset) \to C^\infty(A_X,\chi\backslash X_\emptyset)\end{equation}
are the \emph{fiberwise scattering operators}, varying rationally in $\chi$, that are characterized by the commutativity of the following diagram:
\begin{equation}\label{fiberscatteringdiagram} \xymatrix{ && C^\infty(A_X,{^{w^{-1}}\chi}\backslash X_\emptyset^h) \ar[rr]^{\mathfrak M_{{^{w^{-1}}\chi}}^{-1}} && C^\infty(A_X,{^{w^{-1}}\chi}\backslash X_\emptyset) \ar[dd]^{\mathscr S_{w,\chi}}
\\ \mathcal F(X) \ar[urr]^{\mathfrak N_{^{w^{-1}}\chi}}\ar[drr]_{\mathfrak N_{\chi}}&&& \\
&& C^\infty(A_X,{\chi}\backslash X_\emptyset^h) \ar[rr]^{\mathfrak M_{\chi}^{-1}} && C^\infty(A_X,{\chi}\backslash X_\emptyset).}
\end{equation}
The notation here is as follows: The space $X_\emptyset^h$ is the space of \emph{generic horocycles} on $X$, or on $X_\emptyset$. It classifies pairs $(B,Y)$, where $B$ is a Borel subgroup of $G$, with unipotent radical $N$, and $Y$ is an $N$-orbit in the open $B$-orbit on $X$, or on $X_\emptyset$; by \eqref{samehoro}, $X$ and $X_\emptyset$ give canonically isomorphic spaces by this construction. If $B$ and $B^-$ are two opposite Borel subgroups of $G$ with $B\cap B^-\simeq T$, and we represent $X_\emptyset$ as $SN^-\backslash G$, where $S\subset T$, then $X_\emptyset^h\simeq SN\backslash G$. Although we will not stick with it, it is very useful here to represent the unipotent radical of the stabilizer of a point on $X_\emptyset$ by $N^-$, and the unipotent radical of the stabilizer of a generic horocycle through that point by $N$. The action of $A_X$ on $X_\emptyset^h$ is induced by its action on $X\sslash N$, as suggested by this notation: denoting by $SN$ the stabilizer of a point on $X_\emptyset^h$, the universal Cartan acts on that point via its \emph{defining} identification with $B/N$, unlike the case of $X_\emptyset$. The action of $A_X$ on functions and measures on this space is again defined to be unitary, i.e., as in \eqref{action-normalized-functions}, \eqref{action-normalized-measures} but with $\delta$ replaced by $\delta^{-1}$.
The operator $\mathfrak M_{\chi}$, which can be thought of as the ``standard intertwining operator'', is the operator which, in a region of convergence, takes a function in $C^\infty(A_X,{\chi}\backslash X_\emptyset)$ and integrates it over generic horocycles. \emph{Because there is no canonical measure on those horocycles, this operator depends on a choice of such measures}, and more canonically has image in a certain line bundle over $X_\emptyset^h$ (the line bundle dual to the line bundle whose fiber over a horocycle is the set of invariant measures on it --- see \cite[\S 15.2]{SV}). However, in the cases of Table \eqref{tableX} that we are interested in in this paper, such a choice can be made $G$-equivariantly, and it will not matter for the commutativity of the diagram --- the important point here being that horocycles in $X_\emptyset$ and $X$ are identified by \eqref{samehoro}, and the choices of Haar measures must be made compatibly.
The operator $\mathfrak N_{\chi}$ is, similarly, the integral over the horocycle on $X$, followed by an averaging over horocycles in the same $B$-orbit, against the character $\chi^{-1}$ of $A_X$; that is, for a horocycle $Y$, considered both as a point in $X_\emptyset^h$ and as a subspace of $X$,
$$ \mathfrak N_\chi \Phi (Y) = \int_{A_X} \left(\int_{aY} \Phi(y) dy \right) \chi^{-1}\delta^{-\frac{1}{2}}(a) da.$$
The measure used on $A_X$ is here the same as in the definition of Mellin transform on $X_\emptyset$, so that $\mathfrak M_\chi$ composed with Mellin transform is equal to $\mathfrak N_\chi$ when $X=X_\emptyset$.
We will now calculate the fiberwise scattering operators $\mathscr S_w$ of \eqref{fiberwisescattering} for the non-trivial element $w$ of $W_X\simeq \mathbb{Z}/2$, for the cases of Table \eqref{tableX}.\footnote{For the first line of Table \eqref{tableX}, we assume that $G$ is split of semisimple rank one; the formula for the general split case, which we will not need, can be deduced from this, and the fact that scattering operators compose as in $W_X$, i.e., define an action of $W_X$.} The final result will have the form
$$ \mathscr S_{w,\chi} = \gamma(\chi) \mathfrak R_\chi,$$
where $\mathfrak R_\chi$ is as standard intertwining operator between principal series (essentially, the same as $\mathfrak M_\chi$ after some non-canonical identifications of the spaces involved), and $\gamma(\chi)$ a constant depending on $\chi$ (and expressed in terms of abelian gamma factors of the local functional equation of Tate integrals).
The calculation is quite elementary, on one hand; on the other, it is quite fine to normalize operators such as ``the'' standard intertwining operators between principal series. The constructions needed to formulate a precise result are essentially the constructions needed to prove it; therefore, breaking with the principles of good mathematical exposition, I will formulate the result in the end; the reader can jump ahead to Theorem \ref{thmscattering} to read it.
The calculations that follow hold both over non-Archimedean and over Archimedean fields; thus, despite the fact that the results on asymptotics do not hold as stated in the Archimedean case, we take diagram \eqref{fiberscatteringdiagram} as the defining diagram for the fiberwise scattering operators, and work over an arbitrary local field.
\subsection{The Whittaker case}\label{scatWhittaker}
In order to compute the scattering maps in the Whittaker case, we will adopt the abstract point of view on the Whittaker model, that was introduced in \S \ref{ssinvariant}. Hence, we take $G=\operatorname{SL}(V)$, where $V$ is a two-dimensional \emph{symplectic} vector space. In particular, it is endowed with a Whittaker structure, and Whittaker functions are functions on the $\mathbb{G}_a$-torsor $\tilde V =\{ (v, v^\vee)\in V \times V^\vee| \left<v,v^\vee\right>=1\}$ over $V^* = V\smallsetminus\{0\}$, which vary by the character $\psi$ of $\mathbb{G}_a$.
We identify the dual $V^\vee$ with $V$ through the isomorphism $\iota_\omega: V\xrightarrow\sim V^\vee$ by $\left<u,\iota_\omega(v)\right> = \omega(u,v)$.
Notice that $V^{\vee*}$, the complement of zero in $V^\vee$, can be identified with the variety $V^h$ of ``generic horocycles'' on $V$, that is, affine lines which do not contain the origin; the correspondence sends a functional $v^\vee$ to the affine line of those $v\in V$ with $\left<v,v^\vee\right>=1$, that is, the fiber of $\tilde V$ over $v^\vee$. Thus, we can identify $\tilde V$ with the tautological $G$-orbit on $V^*\times V^h$, consisting of pairs $(v,V_u)$, where $V_u\subset V$ is a generic affine line containing $v$.
Having fixed the symplectic form $\omega$, we get isomorphisms $V^*\simeq V^{\vee*}\simeq V^h$,
and we can identify $\tilde V$ as the subset of $V^*\times V^*$ consisting of pairs $(v,u)$ with $\omega(v,u)=1$. This is a \emph{generic} $G$-orbit in $V^*\times V^*$, that is, one where stabilizers of the two points do not belong to the same Borel subgroup. On the other hand, the identification $V^*\xrightarrow\sim V^h$ corresponds to a \emph{special} $G$-orbit on $V^*\times V^h$, i.e., the stabilizers of two points belong in the same Borel subgroup. Let us write $\mathcal B$ for the flag variety of Borel subgroups of $G$, then it is immediate that, by these associations, a Whittaker structure is also equivalent to the following:
\begin{itemize}
\item a \emph{generic} $G$-orbit on $V^*\times V^*$;
\item a $G$-orbit on $V^*\times_{\mathcal B} V^h$.
\end{itemize}
Now we define \emph{Fourier transform}, \emph{Radon transform} and the \emph{Jacquet integral}.
Fourier transform $\mathfrak F$ will be defined as an $\operatorname{SL}(V)$-equivariant endomorphism of the Schwartz space of functions $\mathcal F(V)$ by the formula
\begin{equation}\label{Fourierconvention}\mathfrak F \Phi(v^*) = \int_V \Phi(v) \psi(\omega(v,v^*)) |\omega|(v).\end{equation}
Radon transform is the map
$$ \mathfrak R: \mathcal F(V)\to C^\infty(V^*)$$
given by the pull-push construction under the above maps, i.e.,
$$ \mathfrak R = t_! s^*,$$
where $s^*$ denotes pullback of functions under the second projection, and $t_!$ is integration over the fibers of $t$, \emph{which are $\mathbb{G}_a$-torsors and hence are endowed with the fixed Haar measure of $F$}.
Explicitly, in coordinates $(x,y)$ with $\omega = dx \wedge dy$,
\begin{equation}\label{Radonincoordinates} \mathfrak R\Phi(0,1) = \int \Phi(1,x) dx.
\end{equation}
The Jacquet integral\footnote{Usually, this name is given to its adjoint $\mathfrak J^*$ that appears below.} is the map
$$ \mathfrak J: \mathcal F(V^*, \mathcal L_\psi)\to C^\infty(V^*)$$
given by
$$ \mathfrak J = t_! s_\psi^*,$$
where $s_\psi^*$ is the pullback of Whittaker sections to $\tilde V$ (where, again, the pullback of $\mathcal L_\psi$ is equipped with a trivialization). In coordinates, it is given by the same formula \eqref{Radonincoordinates} as the Radon transform above, as long as the argument of $\Phi$ inside of the integral is replaced by the pair $((1,x), (0,1))$.
The measure $|\omega|$ on $V$ gives rise to a duality pairing between functions, or between sections of the line bundle $\mathcal L_\psi$ and sections of the line bundle $\mathcal L_{\psi^{-1}}$ defined by the inverse character, and
the adjoint of the Jacquet integral (a morphism $\mathcal J^*: \mathcal F(V^*)\to C^\infty(V^*,\mathcal L_{\psi^{-1}})$) can be written
$$ \mathfrak J^* = s_{\psi,!} t^*,$$
where the twisted push-forward $s_{\psi,!}$ is the dual map to $s_\psi^*$ with respect to the fixed Haar measure on the fibers of the $\mathbb{G}_a$-torsor $\tilde V\xrightarrow{s} V$.
We are interested in \emph{functional equations (scattering operators) for the Jacquet integral}.
Consider the $L^2$-normalized action of $\mathbb{G}_m$ on $C^\infty(V^*)$:
$$ a\cdot \Phi(v) = |a| \Phi(av).$$
The Radon transform is anti-equivariant with respect to this action:
$$\mathfrak R(a\cdot \Phi) = a^{-1} \cdot \mathfrak R\Phi.$$
We have the following relation between Fourier and Radon transforms:
\begin{equation}\label{RadonFourier} \mathfrak F\Phi(v) = \int \mathfrak R(a\cdot \Phi)(v) \psi^{-1}(a) |a|d^\times a.\end{equation}
Indeed, for $v\in V$, choose a section $a\mapsto u_a = a u_1$ of the quotient map $V\ni u \mapsto \omega(u,v)\in \mathbb{G}_a$; then, by definition, the value of Radon transform at $v$ is
$$ \mathfrak R(\Phi)(v) = \int \Phi(u_1 - z v) dz,$$
Hence
$$ \int \mathfrak R(a\cdot \Phi)(v) \psi(a) |a|d^\times a = \int |a| \int \Phi(a u_1 + zav) dz \psi(a) da = \int \int \Phi(a u_1 + z' v) dz' \psi(a) da.$$
By definition, the measure $|\omega|$ on $V$ is equal to $da dz'$ when $(a,z)$ are the coordinates in the basis $(u_1, v)$, so the last integral can be written
$$ \int_V \Phi(u) \psi(\omega(u,v)) |\omega|(u) = \mathfrak F\Phi(v).$$
On the other hand, the adjoint of the Jacquet integral, evaluated on $(v,u)\in \tilde V$, can be written
$$ \mathfrak J^*\Phi(v,u) = \int \Phi(u-zv) \psi(z) dz.$$
Hence, if $\mathfrak F^*$ denotes Fourier transform defined with the character $\psi^{-1}$, instead of $\psi$,
$$\mathfrak J^*\mathfrak F^*\Phi(v,u) = \int \iint \Phi(av+bu) \psi(-a-bz+z) da db dz =
\int \Phi(av+u) \psi(-a) da = \mathfrak J^*\Phi(v,u).$$
We have shown:
$$\mathfrak J^* \circ \mathfrak F^* = \mathfrak J^*,$$
or, taking adjoints and noticing that the adjoint of $\mathfrak F^*$ is $\mathfrak F$,
\begin{equation}\label{JacquetFourier} \mathfrak F \circ \mathfrak J = \mathfrak J.\end{equation}
Now we project to coinvariants with respect to various characters of $\mathbb{G}_m$. Mellin transform:
$$ \check\Phi(\chi) (v) = \int a\cdot \Phi(v) \chi^{-1}(a) d^\times a$$
is a morphism
$$\mathcal F(V^*) \to C^\infty(\mathbb{G}_m,\chi\backslash V^*),$$
where the notation $C^\infty(\mathbb{G}_m,\chi\backslash V^*)$ means $(\mathbb{G}_m,\chi)$-equivariant functions with respect to the normalized action. For $\Phi\in \mathcal S(V)$, or for $\Phi$ in the image of Radon transform and the Jacquet integral, it converges for $\chi$ in some domain, and admits rational continuation to all $\chi$. Because of their equivariance properties with respect to the $\mathbb{G}_m$-action, all the above transforms descend to meromorphic families of transforms between the coinvariant spaces, that will be denoted by the index $\chi$:
$$ \xymatrix{
\mathcal F(V) \ar[d]\ar[r]^{\mathfrak F} &\mathcal F(V)\ar[d] \\
C^\infty(\mathbb{G}_m,\chi^{-1}\backslash V^*) \ar[r]^{\mathfrak F_\chi} &C^\infty(\mathbb{G}_m,\chi\backslash V^*);}$$
$$ \xymatrix{
\mathcal F(V) \ar[d]\ar[r]^{\mathfrak R} & \mathfrak R(\mathcal F(V))\ar[d] \\
C^\infty(\mathbb{G}_m,\chi^{-1}\backslash V^*) \ar[r]^{\mathfrak R_\chi} &C^\infty(\mathbb{G}_m,\chi\backslash V^*);}$$
$$ \xymatrix{
\mathcal F(V^*, \mathcal L_\psi) \ar[dr]_{\mathfrak J_\chi} \ar[r]^{\mathfrak J} & \mathfrak J(\mathcal F(V))\ar[d] \\
&C^\infty(\mathbb{G}_m,\chi\backslash V^*).}$$
\begin{remark}\label{remarkcaution}
Some caution with the notation is needed here when comparing with the operators $\mathfrak N_\chi, \mathfrak M_\chi$ of \eqref{fiberscatteringdiagram}, when $X_\emptyset=V^*$: the identification $V^h\simeq V^*$ that we have here is \emph{anti-equivariant} with respect to the action of $\mathbb{G}_m$. Thus, the space $C^\infty(\mathbb{G}_m,\chi\backslash V^*)$ should be denoted by $C^\infty(\mathbb{G}_m,\chi^{-1}\backslash V^h)$, if we replaced $V^*$ by $V^h$, and what is denoted here with $\mathfrak R_\chi$ would be $\mathfrak M_{\chi^{-1}}$ in the notation of \eqref{fiberscatteringdiagram}, and $\mathfrak J_\chi$ would be $\mathfrak N_{\chi^{-1}}$.
\end{remark}
The relation \eqref{RadonFourier} translates to
\begin{equation}\label{RadonFourierspectral}
\mathfrak F_\chi= \gamma(\chi,0,\psi) \mathfrak R_\chi.
\end{equation}
Indeed, we have
$$ \mathfrak F_\chi(\check\Phi(\chi^{-1}))(v) = \widecheck{\mathfrak F \Phi}(\chi)(v) = \int z\cdot \mathfrak F\Phi (v) \chi^{-1}(z) d^\times z = $$
$$= \int \int z\cdot \mathfrak R(a\cdot \Phi)(v) \psi(a) |a|d^\times a \chi^{-1}(z) d^\times z $$
$$ = \int |z| \int x^{-1}\cdot \mathfrak R\Phi(v) \psi(xz) dx \chi^{-1}(z) d^\times z,$$
which is the Tate integral
$$ Z(\hat\varphi,\chi^{-1}, 1) = \int \hat\varphi(z) |z|\chi^{-1}(z) d^\times z$$
of the function $\hat\varphi(z) = \int \varphi(x) \psi(zx) dx$, where $\varphi (x) = x^{-1}\cdot \mathfrak R\Phi(v)$.
By the functional equation \cite{Tate-Corvallis}:
$$ \gamma(\chi,s,\psi) Z(\varphi,\chi,s) = Z(\hat\varphi, \chi^{-1},1-s).$$
we get
$$\mathfrak F_\chi(\check\Phi(\chi^{-1}))(v) = \gamma(\chi,0,\psi) \int z^{-1}\cdot \mathfrak R\Phi(v) \chi(z) d^\times z = \gamma(\chi,0,\psi) \mathfrak R_\chi \Phi(v).$$
Similarly, \eqref{JacquetFourier} translates to
\begin{equation}\label{JacquetFourierspectral}
\mathfrak F_{\chi^{-1}} \circ \mathfrak J_{\chi} = \mathfrak J_{\chi^{-1}}.
\end{equation}
If we identify $V^*$ with $N\backslash \operatorname{SL}_2$, under our conventions the universal Cartan $A$ of $\operatorname{SL}_2$ acts on $V^*$ by the character $e^\frac{\alpha}{2}$. Thus, for a character $\tilde\chi$ of $A$, we set $\chi = \tilde\chi\circ e^{\check\alpha}$. Then the operators $\mathfrak N_{\tilde\chi}$, $\mathfrak M_{\tilde\chi}$ of \eqref{fiberscatteringdiagram} correspond to $\mathfrak J_{\chi^{-1}}$, $\mathfrak R_{\chi^{-1}}$, respectively (see Remark \ref{remarkcaution}), and we have the corresponding commutative diagram, with \eqref{JacquetFourierspectral} added to it:
$$ \xymatrix{ && C^\infty(\mathbb{G}_m,\chi\backslash V^*) \ar[dd]^{\mathfrak F_{\chi^{-1}}}\ar[r]^{\mathfrak R^{-1}_\chi} & C^\infty(\mathbb{G}_m,\chi^{-1}\backslash V^*) \ar[dd]^{\mathscr S_{w,\tilde\chi}}
\\ \mathcal F(V^*,\mathcal L_\psi) \ar[urr]^{\mathfrak J_{\chi}}\ar[drr]_{\mathfrak J_{\chi^{-1}}}&&& \\
&& C^\infty(\mathbb{G}_m,\chi^{-1}\backslash V^*) \ar[r]^{\mathfrak R_{\chi^{-1}}^{-1}} & C^\infty(\mathbb{G}_m,\chi\backslash V^*) .}$$
Thus, we get
$$ \mathscr S_{w,\tilde\chi} = \mathfrak R_{\chi^{-1}}^{-1} \circ \mathfrak F_{\chi^{-1}} \circ \mathfrak R_\chi,$$
and, invoking \eqref{RadonFourierspectral}, this is equal to
$$ \gamma(\chi^{-1},0,\psi) \mathfrak R_{\chi},$$
or, in other words (writing now $\tilde\chi = \chi\circ e^{\check\alpha}$ as $\chi$):
\begin{equation}\label{scattering-Whittaker} \mathscr S_{w,\chi} = \gamma(\chi, -\check\alpha,0,\psi) \cdot \mathfrak R_{\chi}.\end{equation}
Although we have worked with $\operatorname{SL}(V)$ up to now, this formula remains valid for the Whittaker model of any split group $G$ of semisimple rank one; indeed, given a non-trivial unipotent subgroup $N$ of $G$ with an identification $N\simeq\mathbb{G}_a$, and compatible maps $\operatorname{SL}(V)\to G$, $V^*\to Y:=N\backslash G$, this induces a Whittaker structure on $V$, and hence a distinguished $\operatorname{SL}_2(V)$-orbit in $\subset V^*\times_{\mathcal B} V^h$, whose image determines a distinguished $G$-orbit on $Y\times_{\mathcal B} Y^h$, which, it is immediate to confirm, does not depend on choices. Morover, generic horocycles on $V^*$ map isomorphically to generic horocycles on $Y$, so we can transfer the measures induced by the Whittaker structure, and define the operator $\mathfrak R_{\chi}$ accordingly. The fiberwise scattering maps $\mathscr S_{w,\chi}$ should then be rational multiples of $\mathfrak R_\chi$, and the rational scalar can be computed by pullback to $V^*$; thus, equation \eqref{scattering-Whittaker} remains valid for $G$.
I add the following corollary to \eqref{scattering-Whittaker}, which will be used later:
\begin{corollary}\label{corWhittakerasymptotics}
Let $F$ be a non-Archimedean field, $G$ a split group of semisimple rank one, and $X=(N,\psi)\backslash G$ a symbol for the Whittaker model of $G$. Let $A=A_X$ be the universal Cartan of $G$, acting on measures on $X_\emptyset$ by the normalization described in \eqref{action-normalized-measures}. Let $h$ be the measure on $A$ with Mellin transform
$$\check h(\chi) = L(\chi,\check\alpha, 1)^{-1}.$$ (It belongs to the Bernstein center, i.e., the completed Hecke algebra of $A$.) Then, for any $\varphi\in \mathcal S^+(X_\emptyset)$, the element
$$ h\cdot \varphi (x) = \int_A a\cdot \varphi(x) h(a) $$
belongs to $\mathcal S(X_\emptyset)$.
\end{corollary}
\begin{proof}
Indeed, $\mathcal S^+(X_\emptyset)$ is generated by $\mathcal S(X_\emptyset)$ under the action of the scattering operator $\mathfrak S_w$, which is expressed in terms of the Mellin transform by \eqref{scatteringMellin}. By \eqref{scattering-Whittaker},
$$ \mathscr S_{w,\chi} (h\cdot \varphi) = \gamma(\chi, -\check\alpha,0,\psi) \cdot \mathfrak R_{\chi} \widecheck{(h\cdot \varphi)({^w\chi})} =
\gamma(\chi, -\check\alpha,0,\psi) \check h({^w\chi^{-1}}) \cdot \mathfrak R_\chi \check\varphi({^w\chi}) =$$
$$ = \gamma(\chi, -\check\alpha,0,\psi)L(\chi,\check\alpha, 1)^{-1} \cdot \mathfrak R_{\chi} \check \varphi({^w\chi}) = \epsilon(\chi, -\check\alpha,0,\psi)L(\chi, -\check\alpha,0)^{-1} \cdot \mathfrak R_{\chi} \check \varphi({^w\chi})$$
(see \eqref{gammafactor}).
The factor $\epsilon(\chi, -\check\alpha,0,\psi)L(\chi, -\check\alpha,0)^{-1}$ is polynomial in $\chi$, hence corresponds to another element $h'$ of the completed Hecke algebra of $A$. Thus, applying inverse Mellin transform \eqref{scatteringMellin},
\begin{equation}\label{ShR} \mathfrak S_w h\cdot \varphi = h'^\vee \cdot \mathfrak R \varphi,\end{equation}
where $h^\vee (a) = h(a^{-1})$.
The support of the measure $\mathfrak S_w (h\cdot \varphi)$ has compact closure in the affine completion $X_\emptyset^a = \operatorname{spec} F[N\backslash G]$. On the other hand, $\mathfrak R \varphi$ is supported away from the ``cusp'' $X_\emptyset^a\smallsetminus X_\emptyset$, as is very easy to see from the definition. (When we identify the space $X_\emptyset$ with its horocycle space, a point approaching the cusp corresponds to a horocycle approaching ``infinity'' in $X_\emptyset^a$.)
Thus, \eqref{ShR} implies that $\mathfrak S_w (h\cdot \varphi) \in \mathcal S(X_\emptyset)$.
\end{proof}
\subsection{The case of $\mathbb{G}_m\backslash\operatorname{PGL}_2$} \label{scattorus}
Now let $G=\operatorname{PGL}(V)$, where $V$ is a two-dimensional vector space. We assume that $V$ is endowed with an alternating form, and let $X=$ the $G$-variety of quadratic forms of discriminant $-\frac{1}{4}$ (so that in some coordinates $(y,z)$ for a standard symplectic basis, such a form is given by $yz$). Notice that we think of $G$ as $\operatorname{SL}(V)/\{\pm 1\}$ in order to define a $G$-action that fixes the discriminant. Thus, $X\simeq \mathbb{G}_m\backslash \operatorname{PGL}_2$.
The boundary degeneration $X_\emptyset\simeq N\backslash G$ can be identified with the space of degenerate quadratic forms of rank one, which is canonically isomorphic to $(V^{\vee*})/\{\pm 1\}$, by sending a linear functional to its square. Having fixed the symplectic form $\omega$, and hence the isomorphism $\iota_\omega: V\xrightarrow\sim V^\vee$ as in the previous subsection, we will identify $X_\emptyset$ with $V^*/\{\pm 1\}$.
The evaluation map gives rise to canonical isomorphisms
\begin{eqnarray} (X\times V)\sslash \operatorname{SL}(V)\xrightarrow\sim \mathbb{G}_a, \nonumber \\
\label{isomorphisms} (X_\emptyset\times V)\sslash \operatorname{SL}(V)\xrightarrow\sim \mathbb{G}_a,\end{eqnarray}
which are the ones inducing the canonical bijection of horocycles \eqref{samehoro}. The preimage of $1$, in each case, is a distinguished $G$-orbit, which projects to distinguished $G$-orbits
$$ \tilde X\subset X\times X_\emptyset,$$
$$ \tilde X_\emptyset \subset X_\emptyset.$$
Notice that, in the case of $\tilde X_\emptyset$, this is the projection of the canonical $G$-orbit $\tilde V\subset V\times V$, induced by the symplectic form on $V$ (as in the previous subsection).
Fourier transform on $V$ descends to Fourier transform on $X_\emptyset$, but we have to be careful, because the map $V^*\to X_\emptyset$ is not surjective at the level of $F$-points. One therefore needs to treat $X_\emptyset$ as an open subset of the stack $[V/\{\pm 1\}]$, and define Fourier transform on functions (or measures) on the $F$-points of this stack. Specifically, this means the following: For any $\alpha\in H^1(F, \mathbb{Z}/2)$ (corresponding to a quadratic extension $E^\alpha$ of $F$, including the trivial one $F\oplus F$), let $R_\alpha$ be the corresponding $\mathbb{Z}/2$-torsor over $F$ (isomorphic to a pair of distinct conjugate points of $E^\alpha$), and let $V^\alpha \simeq V\times^{\mathbb{Z}/2} R^\alpha$. It is an $F$-vector space which can be identified with $V\otimes_F \Im(E^\alpha)$, where $\Im(E^\alpha)$ is the ``imaginary'' line of elements of $E^\alpha$ which are conjugate to their opposite. Then we have
$$ X_\emptyset (F) = \bigoplus_{\alpha \in H^1(F,\mathbb{Z}/2)} V^{\alpha *}(F)/\{\pm 1\}.$$
The symplectic form $\omega: V\times V\to \mathbb{G}_a$ is invariant under the diagonal $\mathbb{Z}/2$-action, and hence induces a symplectic form on $V^\alpha$. Explicitly, if we choose an element $e\in \Im(E^\alpha)$ to write any element of $V^\alpha$ as $v\otimes e$, we have
$$ \omega(v_1\otimes e, v_2\otimes e) = e^2 \omega(v_1, v_2).$$
This defines Fourier transform on the Schwartz space $\mathcal F(V^\alpha)$, and in particular defines a Fourier transform $\mathfrak F$ on
$$ \mathcal F(X_\emptyset^a):= \bigoplus_\alpha \mathcal F(V^\alpha(F))^{\mathbb{Z}/2}.$$
(The notation $X_\emptyset^a$ stands for the affine closure of $X_\emptyset$.) It is easy to see that $\mathfrak F$ is a $G$-equivariant and $A_X$-anti-equivariant, endomorphism
$$\mathfrak F: \mathcal F(X_\emptyset^a) \to \mathcal F(X_\emptyset^a),$$
where, here, $A_X=$ the universal Cartan of $G$, and the action of $A_X$ is the normalized one, as in \eqref{action-normalized-functions}.
Similarly, the correspondence $\tilde X_\emptyset \underset{t}{\overset{s}\rightrightarrows} X_\emptyset$, together with the Haar measure on horocycles (i.e., fibers of the map $t$) descending from that on $V^*$, gives rise to Radon transform
$$ \mathfrak R: \mathcal F(X_\emptyset^a) \to C^\infty (X_\emptyset),$$
defined as before.
The analog of the Jacquet integral here is the morphism
$$ \mathfrak I: \mathcal F(X)\to C^\infty(X_\emptyset)$$
obtained by the correspondence
$$\tilde X \subset X\times X_\emptyset.$$
again with the measure on generic horocycles on $X$ obtained by their identification with generic horocycles of $X_\emptyset$.
The adjoint of $\mathfrak I$ (with respect to invariant measures, which we do not necessarily need to fix)
$$ \mathfrak I: \mathcal F(X_\emptyset) \to C^\infty(X)$$
given by the integral
\begin{equation}\label{dualI} \mathfrak I^*\Phi(x) = \int_{(X_\emptyset)_x} \Phi(v) \mu(v),\end{equation}
where $(X_\emptyset)_x\subset X_\emptyset$ is the fiber of $\tilde X$ over $x$, and $\mu$ is a $G_x$-invariant measure on it.
As before, the composition of $\mathfrak I$ with Mellin transform will be denoted
$$\mathfrak I_\chi: \mathcal F(X) \to C^\infty(A_X,\chi\backslash X_\emptyset).$$
Dualizing, we get a morphism
$$\mathfrak I_\chi^*: C^\infty(A_X,\chi^{-1}\backslash X_\emptyset) \to C^\infty(X),$$
which in some domain of convergence is given again by \eqref{dualI}.
The composition of this with Mellin transform
$$\tilde{\mathfrak I}_\chi^*: \mathcal F(X_\emptyset) \to C^\infty(A_X,\chi^{-1}\backslash X_\emptyset) \xrightarrow{\mathfrak I_\chi^*} C^\infty(X)$$
can be written as
\begin{equation}\label{dualII} \tilde{\mathfrak I}_\chi^*\Phi(x) = \int_{X_\emptyset} \Phi(v) \tilde\chi\circ p((x,v)) dv,\end{equation}
where $\tilde\chi(z)=|z|^{-\frac{1}{2}}\chi\circ e^{\frac{\check\alpha}{2}}(z)$ and
$$ p: (X\times X_\emptyset)\sslash G \xrightarrow\sim \mathbb{G}_a$$
is the canonical isomorphism induced from the evaluation map \eqref{isomorphisms}.
Notice that, if we denote by $A$ the universal Cartan of $G'= \operatorname{SL}(V)$ which acts on $V^*$, the map $A\backslash V^* \to A_X\backslash X_\emptyset$ is an isomorphism. Thus, we can think of $\mathfrak I_\chi^*$ as a morphism from $C^\infty(A,\chi'^{-1}\backslash V^*)$, where $\chi'$ is the pullback of $\chi$ to $A$. This morphism, though, will depend on $\chi$ itself, not just $\chi'$, and if we unfold the definitions and compose with Mellin transform, we will see that the resulting functional
$$\tilde{\mathfrak I}'^*_\chi :\mathcal F(V^*) \to C^\infty(A,\chi'^{-1}\backslash X'_\emptyset) \xrightarrow{\mathfrak I_\chi^*} C^\infty(X)$$
is given by the formula
$$ \tilde{\mathfrak I}'^*_\chi\Phi(x) = \int_{V^*} \Phi(v') \tilde\chi\circ p((x,v')) dv',$$
where we are using the same letter ($p$) for the evaluation map on $X\times V^*$.
We explicate this functional: Given $x\in X$, choose a standard symplectic basis $(u,v)$ on $V$ (i.e., $\omega(u,v)=1$) such that the quadratic form associated to $x$ is
$$ y u + z v \mapsto yz.$$
In these coordinates, the above integral is equal to
$$\int \Phi(z,y) \tilde\chi(yz) dy dz,$$
and adjoint Fourier transform on $V$ is biven by
$$ \mathfrak F^*\Phi(y,z) = \int_V \Phi(a, b) \psi^{- 1}(az-by) da db.$$
We compute their composition.
Assuming $\Phi(y,z) = \Phi_1(y) \Phi_2(z)$ for convenience of notation, we have
$$ \tilde{\mathfrak I}'^*_\chi \mathfrak F^*\Phi(x) = \int \int \Phi_1(a) \Phi_2(b) \psi^{-1}(az-by) da db \tilde\chi(yz) dy dz =$$
$$ = Z(\hat\Phi_1^{\psi^{-1}}, \tilde\chi, 1) Z(\hat\Phi_2^{\psi}, \tilde\chi, 1),$$
where the exponent of Fourier transforms denotes the character they are defined by. Applying the local functional equation, we get that this is equal to
$$ \gamma(\tilde\chi^{-1}, 0, \psi) \gamma(\tilde\chi^{-1},0, \psi^{-1}) \int \Phi(z,y) \tilde\chi^{-1}(zy) d^\times z d^\times y = $$
$$ = \gamma(\tilde\chi, 1, \psi^{-1})^{-1} \gamma(\tilde\chi,1, \psi)^{-1} \tilde{\mathfrak I}'^*_{\chi^{-1}} \Phi(x) = $$
$$ = \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1})^{-1} \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi)^{-1} \tilde{\mathfrak I}'^*_{\chi^{-1}} \Phi(x). $$
Dualizing,
$$\mathfrak F_{\chi^{-1}} \circ \tilde{\mathfrak I}'_\chi = \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1})^{-1} \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi)^{-1} \tilde{\mathfrak I}'_{\chi^{-1}}.$$
Thus, the diagram \eqref{fiberscatteringdiagram} now reads
$$ \xymatrix{ && C^\infty(A_X,\chi\backslash X_\emptyset) \ar[dd]^{c_\chi\cdot \mathfrak F_{\chi^{-1}}}\ar[r]^{\mathfrak R_{\chi^{-1}}^{-1}} & C^\infty(A_X,\chi^{-1}\backslash X_\emptyset) \ar[dd]^{\mathscr S_{w,\chi}}
\\ \mathcal F(X) \ar[urr]^{\mathfrak I_{\chi}}\ar[drr]_{\mathfrak I_{\chi^{-1}}}&&& \\
&& C^\infty(A_X,\chi^{-1}\backslash X_\emptyset) \ar[r]^{\mathfrak R_{\chi}^{-1}} & C^\infty(A_X,\chi\backslash X_\emptyset),}$$
where $c_\chi = \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1}) \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi)$,
and invoking \eqref{RadonFourierspectral} again, we get:
\begin{equation}\label{scattering-torus}\mathscr S_{w,\chi} = \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1}) \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi) \gamma(\chi,-\check\alpha, 0,\psi) \mathfrak R_{\chi},\end{equation}
\subsection{The group case} \label{ssscatteringgroup}
Let us now study the group case, $X=H = \operatorname{SL}(V)$, where $V$ is a two-dimensional vector space, and $G=H\times H$. (Again, $H$ acts on the right on $V$.) The asymptotic cone $X_\emptyset=H_\emptyset$ can be identified with the subspace of $\operatorname{End}(V)$ of elements of rank one.
The identification \eqref{samehoro} of generic horocycles, then, is as follows: a generic horocycle, both in $H$ and in $H_\emptyset$, is equivalent to a pair $(L, L')$ of lines in $V$, together with a non-zero homomorphism $\tau: L\to V/L'$; these data correspond to the horocycle of the pair $(B\times B', Y)$, where $B, B'$ are the stabilizers of $L, L'$, and $Y\subset H$ or $Y\subset H_\emptyset$ is the subvariety of endomorphisms which induce $\tau$. Notice that these data are also equivalent to an element of $\mathbb{G}_m\backslash V^*\times V^h$, namely the class of $(v, \tau(v))$, where $v$ is a non-zero element of $L$. Thus, $H_\emptyset^h = \mathbb{G}_m\backslash V^*\times V^h$, canonically.
For a pair $(v,Y) \in V^*\times V^h$, we will write $[v:Y]\in \mathbb{G}_a$ for the scalar $\lambda$ such that $\lambda v \in Y$.
We can also identify $H_\emptyset$, canonically, with the space $\mathbb{G}_m^{\operatorname{diag}}\backslash (V^h\times V^*)$ (notation as before), by mapping a pair $(Y,v)$ consisting of a generic horocycle and a non-zero vector to the rank-one endomorphism that sends $Y$ to $v$, in other words for the endomorphism $u\mapsto [u:Y]\cdot v$.
For any pair
$$\tau_1: V/L_1\to L_1'\subset V$$
$$\tau_2: V/L_2\to L_2'\subset V$$
of elements of $H_\emptyset$ we obtain, by restriction, homomorphisms
$$ \tau_2|_{L_1} : L_1\to L_2', \mbox{ and }$$
$$ \tau_1|_{L_2}: L_2\to L_1'.$$
As long as $L_1\ne L_2$ and $L_1'\ne L_2'$, there is a unique element $M_{\tau_1, \tau_2}\in \operatorname{GL}(V)$ whose restrictions also induce these endomorphisms, and the pairs $(\tau_1,\tau_2)$ with $M_{\tau_1,\tau_2}\in \operatorname{SL}(V)$ form a distinguished $G^{\operatorname{diag}}$-orbit $\tilde H_\emptyset \subset H_\emptyset\times H_\emptyset$. Equivalently, the distinguished $G^{\operatorname{diag}}$-orbit is characterized by the property that
\begin{equation}\label{groupcondition} v_1\wedge v_2 = \tau_2(v_1) \wedge \tau_1(v_2)
\end{equation}
for all $v_1\in L_1, v_2\in L_2$.
The correspondence between generic $G^{\operatorname{diag}}$-orbits on $H_\emptyset\times H_\emptyset$ and $G^{\operatorname{diag}}$-orbits on $H_\emptyset\times_{\mathcal B_H \times\mathcal B_H} H_\emptyset^h$ sends a pair $(\tau_1,\tau_2)$ as above to the pair $(\tau_1, \tau_2^\circ)$, where $\tau_2^\circ\in H_\emptyset^h$ is the horocycle represented by the composition
$$ L_1 \xrightarrow{\tau_2|_{L_1}} L_2' \to V/L_1'.$$
The distinguished $G^{\operatorname{diag}}$-orbit corresponds to the set of pairs
$$ (V/L\xrightarrow{\tau_1} L'\subset V, V\supset L\xrightarrow{\tau_2^\circ} V/L') \in H_\emptyset\times_{\mathcal B_H \times\mathcal B_H} H_\emptyset^h$$
such that any lift of $(\tau_1,\tau_2^\circ)$ to an endomorphism of $V$ has determinant $1$.
Equivalently, representing
$$H_\emptyset\times_{\mathcal B_H \times\mathcal B_H} H_\emptyset^h =\mathbb{G}_m^{\operatorname{diag}}\backslash(V^h\times V^*) \times_{\mathcal B_H\times \mathcal B_H} \mathbb{G}_m^{\operatorname{diag}}\backslash (V^* \times V^h),$$
the distinguished $G^{\operatorname{diag}}$-orbit is obtained as follows: choose \emph{any} $H$-orbit on $V^*\times_{\mathcal B} V^h$, say with a representative $(v, Y)$, and take the $G^{\operatorname{diag}}$-orbit represented by $(Y, v, -v, Y)$. Indeed, for any $y\in Y$, the endomorphism defined by $y\mapsto v, -v\mapsto y$ has determinant one.
This gives rise to a \emph{canonical} Radon transform
$$\mathfrak R : \mathcal F(H_\emptyset) \to C^\infty(H_\emptyset),$$
which is given by
$$ \mathfrak R \Phi(V/L\xrightarrow\tau L') = \int_{(\P(V)\smallsetminus\{L\})\times (\P(V)\smallsetminus\{L'\})} \Phi(\sigma^\tau_{(L_1, L_1')}) d(L_1, L_1'),$$
where $\sigma^\tau_{(L_1, L_1')}: V/L_1 \to L_1'\subset V$ is the unique such morphism with the property that \eqref{groupcondition} is satisfied when $\tau_1=\sigma$ and $\tau_2=\tau$. The measure on $(\P(V)\smallsetminus\{L\})\times (\P(V)\smallsetminus\{L'\})$ is the following: choose \emph{any} pair of vectors $(v\in L, v'\in L')$, let $Y =\tau^{-1}(v')\subset V$ and let $Y'\subset V$ be the horocycle of all vectors $y'$ with $y\wedge v = v'\wedge y'$ for all $y\in Y$. Then $Y\times Y'$ is a $\mathbb{G}_a\times \mathbb{G}_a$-torsor by $(x,x')\cdot (y, y') = (y-x v, y'-x' v')$, hence carries a measure induced from our fixed measure on $F^2$, and under the projection map it can be identified with $(\P(V)\smallsetminus\{L\})\times (\P(V)\smallsetminus\{L'\})$. It is immediate that the resulting measure on $(\P(V)\smallsetminus\{L\})\times (\P(V)\smallsetminus\{L'\})$ does not depend on the choice of $(v,v')$.
Another way to explicate this Radon transform is to choose any Whittaker structure (equivalently, a symplectic form $\omega$) on $V^*$. The $\mathbb{G}_m$-anti-equivariant identification $V^*\xrightarrow\sim V^h$ that it induces (which we have normalized so that $v$ corresponds to the horocycle $\{u\in V| \omega(u,v)=1\}$) allows us to identify
$$H_\emptyset =\mathbb{G}_m^{\operatorname{diag}}\backslash (V^h\times V^*) \simeq \mathbb{G}_m^{\operatorname{adiag}}\backslash (V^*\times V^*),$$
and defines two correspondences $\tilde V \times V^* \rightrightarrows V^*\times V^*$ and $V^*\times \tilde V \rightrightarrows V^*\times V^*$, where $\tilde V = \{(u,v)|\omega(u,v)=1\}$. The product of the resulting Radon transforms (where we denote by an index the variable that we apply the transform to):
$${\mathfrak R = \mathfrak R_1 \boxtimes \mathfrak R_2}: \mathcal F(V^*\times V^*) \to C^\infty(V^*\times V^*)$$
descends to $\mathbb{G}_m^{\operatorname{adiag}}\backslash V^*\times V^*$, i.e., to a map
$$\mathfrak R: \mathcal F(V^*\times V^*) \to C^\infty(V^*\times V^*).$$
This depends on the choice of a symplectic form, but its pullback to $H_\emptyset$ does not, and coincides with the Radon transform described above, as a simple calculation shows.
More explicitly, take a Borel subgroup of $G$ of the form $B\times B^-$, where $B, B^-$ are two opposite Borel subgroups of $H$ with unipotent radicals $N, N^-$ and intersection $B\cap B^-=T$, and identify $H_\emptyset = T\times^{B\times B^-} G$ in such a way that the embeddings of $T$ into $H$ and $H_\emptyset$ are compatible with the isomorphism \eqref{samehoro}: $N\backslash H\sslash N^- = H_\emptyset\sslash (N\times N^-)$. If we identify $H$ with $\operatorname{SL}_2$, $B=$ the upper triangular subgroup and $B^-=$ the lower triangular subgroup, then we have
$$ \mathfrak R\Phi(1) = \int_{F^2} \Phi\left( \begin{pmatrix} & -1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1 \end{pmatrix}, \begin{pmatrix} & -1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 & \\ y& 1 \end{pmatrix} \right) dx dy.$$
By \cite[Proposition 15.2]{DHS}, the scattering operator for the non-trivial element $w\in W_X=W_H$ is given by
$$ \mathscr S_{w,\chi}= \mathfrak R_{1,\chi} \boxtimes \mathfrak R_{2,\chi^{-1}}^{-1}: C^\infty(A_H,\chi^{-1}\backslash H_\emptyset) \to C^\infty(A_H,\chi\backslash H_\emptyset).$$
The individual factors of this depend on the choice of a Whittaker structure, but the product does not. Moreover, the expression is symmetric in the two factors, i.e., we have $\mathfrak R_{1,\chi} \boxtimes \mathfrak R_{2,\chi^{-1}}^{-1} = \mathfrak R_{1,\chi^{-1}}^{-1}\boxtimes\mathfrak R_{2,\chi}$.
We now compute this as a multiple of the operator $\mathfrak R_\chi= \mathfrak R_{1,\chi}\boxtimes R_{2,\chi}$. This is the calculation of the Plancherel measure, but it follows immediately from \eqref{RadonFourierspectral}: Invoking Fourier transform on $V^*$, and the fact that $\mathfrak F^* \circ \mathfrak F = 1$, where $\mathfrak F^*$ is Fourier transform defined with the character $\psi^{-1}$, instead of $\psi$, and setting $\tilde\chi = \chi\circ e^{\check\alpha}$ (so that it corresponds to the character $\chi$ of \eqref{RadonFourierspectral}),
we have
$$ 1 = \mathfrak F_{\chi^{-1}} \circ \mathfrak F^*_\chi = \gamma(\tilde\chi^{-1},0,\psi) \mathfrak R_{2,\chi^{-1}} \circ \gamma(\tilde\chi,0,\psi^{-1}) \mathfrak R_{2,\chi} \Rightarrow$$
$$ \mathfrak R_{2,\chi^{-1}}^{-1} = \gamma(\tilde\chi^{-1},0,\psi) \gamma(\tilde\chi,0,\psi^{-1}) \mathfrak R_{2,\chi},$$
and we get
\begin{equation}\label{scattering-group}
\mathscr S_{w,\chi}= \gamma(\chi,\check\alpha, 0,\psi^{-1}) \gamma(\chi,-\check\alpha,0,\psi) \cdot \mathfrak R_{1,\chi} \boxtimes \mathfrak R_{2,\chi} = \gamma(\chi,\check\alpha, 0,\psi^{-1}) \gamma(\chi,-\check\alpha,0,\psi) \cdot \mathfrak R_\chi.
\end{equation}
\vspace{2cm}
We summarize the formulas \eqref{scattering-Whittaker}, \eqref{scattering-torus} and \eqref{scattering-group} for the scattering operators.
\begin{theorem}\label{thmscattering}
For the cases of Table \eqref{tableX}, in terms of the canonical\footnote{To summarize: Radon transform depends on a choice of $G$-orbit on $X_\emptyset\times_{\mathcal B} X^h_\emptyset$, and there is a \emph{canonical} choice in any one of the three cases.} spectral Radon transforms $\mathfrak R_\chi$ that were described in the previous subsections,
$$\mathfrak R_\chi: C^\infty(A_X,\chi^{-1}\backslash X_\emptyset) \to C^\infty(A_X,\chi\backslash X_\emptyset),$$
the scattering operator $\mathscr S_{w,\chi}$ for the non-trivial element $w$ of $W_X$ is given by the following formulas:
\begin{itemize}
\item For the Whittaker case, $X = (N,\psi)\backslash G$,
\begin{equation} \mathscr S_{w,\chi} = \gamma(\chi, -\check\alpha,0,\psi) \cdot \mathfrak R_{\chi};\end{equation}
\item for the variety $X = \mathbb{G}_m\backslash \operatorname{PGL}_2$,
\begin{equation}\mathscr S_{w,\chi} = \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1}) \gamma(\chi, \frac{\check\alpha}{2}, \frac{1}{2}, \psi) \gamma(\chi,-\check\alpha, 0,\psi) \cdot \mathfrak R_{\chi};\end{equation}
\item for the group case, $X = H=\operatorname{SL}_2$ under the $G=H\times H$-action,
\begin{equation}
\mathscr S_{w,\chi}= \gamma(\chi,\check\alpha, 0,\psi^{-1}) \gamma(\chi,-\check\alpha,0,\psi) \cdot \mathfrak R_\chi .
\end{equation}
\end{itemize}
\end{theorem}
\subsection{Relative characters}\label{ssrelchars}
The quotient $\mathfrak{C}_\emptyset:=(X_\emptyset \times X_\emptyset)\sslash G$ has an $A_X$-action, which we normalize so that it descends from the action on the first copy of $X_\emptyset$, and the canonical generic $G$-orbit $\tilde X_\emptyset\subset X_\emptyset\times X_\emptyset$, that was also used to define Radon transform, defines in every case an isomorphism
$$ (X_\emptyset \times X_\emptyset)^\circ\sslash G \xrightarrow\sim A_X \subset \mathfrak{C}_\emptyset,$$
where $(X_\emptyset \times X_\emptyset)^\circ$ is the open subspace of pairs of points whose stabilizers belong to distinct Borel subgroups.
Characters of $F^\times \subset F$ pull back, generically, to $G$-invariant \emph{generalized functions} on $X_\emptyset\times X_\emptyset$, in fact, to functionals on the extended Schwartz space $\mathcal S^+(X_\emptyset\times X_\emptyset)$ obtained by applying scattering operators (on both variables) to $\mathcal S(X_\emptyset\times X_\emptyset)$,
as the following proposition shows. To formulate it,
let $p:(X_\emptyset \times X_\emptyset)^\circ \to A_X$
be the quotient map; its fiber over $1\in A_X$ is the distinguished $G$-orbit $\tilde X_\emptyset$. Fix a $G$-invariant measure on $X_\emptyset$, and take the Haar measure $da$ on $A_X$ which disintegrates the Haar measure $dx\times dx$ on $X_\emptyset \times X_\emptyset$ as the product of $\delta(a) da$ with the $G$-invariant measure on $\tilde X_\emptyset$ for which the following formula holds:
$$ \int_{\tilde X_\emptyset} \Phi(u,v) d(u,v) = \int_{X_\emptyset} \mathfrak R_1\Phi(v,v) dv.$$
Here, as before, $\mathfrak R_1$ denotes Radon transform in the first variable.
\begin{proposition}
Given a measure $\varphi(u,v)= \Phi(u,v) du dv \in \mathcal S^+(X_\emptyset\times X_\emptyset)$, the ``open'' Mellin tranform of its push-forward $f$ to $\mathfrak{C}_\emptyset$ (an element of the space of push-forward measures that we should denote by $\mathcal S^+(X_\emptyset\times X_\emptyset/G)$)
\begin{equation}\label{Mellin-boundary} \check f(\chi) = \int_{A_X} f(\xi) \chi^{-1}(\xi)
\end{equation}
converges when $\chi^{-1}$ vanishes sufficiently fast on the complement of $A_X$ in $\mathfrak{C}_\emptyset$, and admits rational continuation.
The measure $f\in \mathcal S^+(X_\emptyset\times X_\emptyset/G)$ can be reconstructed by inverse Mellin transform:
\begin{equation}\label{invMellin}
f = \left(\int_{\omega\widehat{A_X}} \check f(\chi) d\chi\right) da,
\end{equation}
where $\omega$ is a character such that $\omega^{-1}$ vanishes sufficiently fast on the complement of $A_X$ in $\mathfrak{C}_\emptyset$, and $(da, d\chi)$ is a pair of dual Haar measures on $A_X$ and $\widehat{A_X}$.
Finally, the ``open'' Mellin transform can be written in terms of ``closed'' Mellin transforms and Radon transforms as follows:
\begin{equation}\label{closed}
\check f(\chi)= \int_{X_\emptyset} \mathfrak R_{1, {^w\chi} \delta^{\frac{1}{2}}} \Phi(v, v) dv,
\end{equation}
where $\mathfrak R_{1, {^w\chi} \delta^{\frac{1}{2}}}: \mathcal F(X_\emptyset\times X_\emptyset) \to \mathcal F((A_X, {^w\chi} \delta^{\frac{1}{2}} \backslash X_\emptyset)\times X_\emptyset)$ is the spectral Radon transform in the first variable, encountered in the previous subsections, defined with that Haar measure $da$ on $A_X$ described above.
\end{proposition}
The terms ``open'' and ``closed'', here, refer to the ``Bruhat cell'' $X_\emptyset \times X_\emptyset$ on which the integration takes place. Notice that only the ``open'' Mellin transform is a functional on the space of push-forward measures $\mathcal S^+(X_\emptyset\times X_\emptyset/G)$; the closed Bruhat cell lives over a proper subvariety of $\mathfrak{C}_\emptyset$, and information about it is lost when we take push-forwards.
The spectral Radon transform $\mathfrak R_{1,\chi}$, here, is (by abuse of notation) the composition of what was denoted before by $\mathfrak R_\chi$ with Mellin transform \eqref{Mellin-Splus}, applied to the first variable; when $\Phi(u,v) = \Phi_1(u) \Phi_2(v)$ the integral \eqref{closed} can also be written in terms of the Mellin transforms of $\Phi_1$, $\Phi_2$ as
\begin{equation}\label{closed-alt}\int_{A_X\backslash X_\emptyset} \mathfrak R_{1, {^w\chi} \delta^{\frac{1}{2}}} \left(\check\Phi_1(\chi \delta^{-\frac{1}{2}})\right) \cdot \check\Phi_2({^w\chi^{-1}} \delta^{-\frac{1}{2}}).
\end{equation}
Notice that, unlike the Mellin transform \eqref{Mellin-Splus} on $X_\emptyset$, in the definition of Mellin transform for $\mathcal S^+(X_\emptyset\times X_\emptyset/G)$, we do not normalize the action of $A_X$ on measures or functions on $\mathfrak{C}_\emptyset$, which is why, as we will see, the Mellin transform defined here is a relative character for $C^\infty (A_X, \chi \delta^{-\frac{1}{2}} \backslash X_\emptyset) \simeq I(\chi^{-1} \delta^{\frac{1}{2}})$, the principal series representation obtained by normalized induction from the character $\chi^{-1} \delta^{\frac{1}{2}}$ of $A_X$.
\begin{proof}
We compute:
$$ \check f(\chi) = \int_{X_\emptyset\times X_\emptyset} \Phi(u, v) \chi^{-1}(p(u,v)) du dv = $$
$$= \int_{A_X} \left(\int_{\tilde X} \Phi(a\cdot u, v) d(u,v)\right) \chi^{-1}(a) \delta(a) da=$$
$$ = \int_{A_X} \int_{X_\emptyset} \mathfrak R_1\Phi(a^{-1} v, v) dv \chi^{-1}(a) da = \int_{X_\emptyset} \mathfrak R_{1, {^w\chi} \delta^{\frac{1}{2}}} \Phi(v, v) dv.$$
This is rational in $\chi$, as the expression \eqref{closed-alt} shows; recall that an element of $\mathcal S^+(X_\emptyset)$ can be written as $\varphi_1\oplus \mathfrak S_w \varphi_2$, where both $\varphi_1$ and $\varphi_2$ belong to $\mathcal S(X_\emptyset)$, and that the scattering map $\mathfrak S_w$ is decomposed \eqref{scatteringMellin} in terms of the fiberwise scattering maps $\mathscr S_{w,\chi}$, which are rational in $\chi$.
The formula for the inverse Mellin transform \eqref{invMellin} follows from the fact that elements of $\mathcal S^+(X_\emptyset\times X_\emptyset)$ are of moderate growth, and up to a rapidly decaying measure are supported on a compact subset of an affine embedding $X^a_\emptyset\times X^a_\emptyset$; thus, their push-forwards will be of moderate growth on $A_X$ and, up to rapid decay, supported on a compact subset of the affine embedding $\mathfrak{C}_\emptyset^a:= X^a_\emptyset\times X^a_\emptyset\sslash G$.\footnote{In the examples of Table \eqref{tableX}, this coincides with $\mathfrak{C}_\emptyset$.} Thus, the measure $\omega^{-1}f$ will be in $L^2(A_X)$, for $\omega$ as in the statement of the proposition, and the result follows from standard Fourier analysis.
\end{proof}
Let $M_\chi$ denote the functional $f\mapsto \check f(\chi \delta^{\frac{1}{2}})$ on $\mathcal S^+(X_\emptyset\times X_\emptyset/G)$.
As is clear from \eqref{closed-alt}, it is a \emph{relative character} for the representation
$$\pi_{\chi}:=C^\infty (A_X, \chi \backslash X_\emptyset) \simeq I(\chi^{-1}),$$
i.e., its pullback to $\mathcal S^+(X_\emptyset\times X_\emptyset)$ factors through a morphism
$$ \mathcal S^+(X_\emptyset\times X_\emptyset) \to \pi_{\chi} \otimes \widetilde{\pi_{\chi}} \xrightarrow{\left<\,\, , \,\,\right>} \mathbb{C}.$$
Now we restrict to the case when $F$ is non-Archimedean.\footnote{Replacing the Schwartz space with the Harish-Chandra Schwartz space, and restricting to $\chi$ unitary, the results that follow extend to the Archimedean case, using asymptotics of admissible generalized matrix coefficients. We will not need them, so we avoid introducing more material.}
We would like to consider its further pullback to $\mathcal S(X\times X)$ via the asymptotics map $e_\emptyset^*\otimes e_\emptyset^*$, and determine its role in the Plancherel formula of $X$. For that purpose, denote by $M_\chi^{\operatorname{cl}}$ the corresponding ``closed'' Mellin transform that we saw in \eqref{closed} (without the composition with Radon transform), that is,
$$ M_\chi^{\operatorname{cl}}(f) = \int_{X_\emptyset} \check\Phi_1({\chi} ) \cdot \check\Phi_2({\chi^{-1}} ),$$
when $f$, here, denotes the image of $\Phi_1(u)\Phi_2(v) du dv$ \emph{in the $G$-coinvariant space $\mathcal S^+(X_\emptyset\times X_\emptyset)_G$}. As mentioned above, here we cannot identify $f$ with its push-forward, because this would lose the information about ``closed'' Mellin transforms.
Let $I_\chi$, $I_\chi^{\operatorname{cl}}$ be the pullbacks of $M_\chi$, $M_\chi^{\operatorname{cl}}$, respectively, to $\mathcal S(X\times X)$ via the asymptotics map $e_\emptyset^*\boxtimes e_\emptyset^*$.
The following theorem, proven in \cite[\S 14.1]{SV}, states that the relative characters $I_\chi^{\operatorname{cl}}$ decompose the ``most continuous summand'' of the space $L^2(X)$, in the sense of the Plancherel decomposition. The ``most continuous summand'' is a canonical subspace $L^2(X)_\emptyset\subset L^2(X)$, whose definition I will not repeat here. Fix an invariant measure $dx$ on $X$ and a compatible measure on $X_\emptyset$ \cite[\S 4.2]{SV}, and use them to consider $L^2(X)$, $L^2(X_\emptyset)$ as spaces of measures.
\begin{theorem}\label{Plancherel}
For $\varphi_1, \varphi_2\in L^2(X)_\emptyset$, we have
$$\int_X \frac{\varphi_1 \cdot \varphi_2}{dx} = \frac{1}{|W_X|}\int_{\widehat{A_X}} I_\chi^{\operatorname{cl}}(\varphi_1\otimes\varphi_2) d\chi,$$
where the Haar measure $d\chi$ on the unitary dual $\widehat{A_X}$ is the one dual to the Haar measure $da$ on $A_X$.
\end{theorem}
Thus, the relative characters $I_\chi^{\operatorname{cl}}$ are, in some sense, the canonical characters which decompose the most continuous spectrum of $X$ against Haar--Plancherel measure $d\chi$.
The whole point of the present section was to compare the pullbacks $I_\chi$ of the ``open'' Mellin transforms $M_\chi$ to these relative characters:
\begin{theorem} \label{thmpullbackfrombd}
For each of the cases of Table \eqref{tableX}, define a rational function $\mu_X(\chi)$ on $\widehat{A_X}_\mathbb{C}$, as follows:
\begin{itemize}
\item For the Whittaker case, $X = (N,\psi)\backslash G$,
\begin{equation} \mu_X(\chi) = \gamma(\chi, \check\alpha,0,\psi);\end{equation}
\item for the variety $X = \mathbb{G}_m\backslash \operatorname{PGL}_2$,
\begin{equation}\mu_X(\chi) = \gamma(\chi, -\frac{\check\alpha}{2}, \frac{1}{2}, \psi^{-1}) \gamma(\chi, -\frac{\check\alpha}{2}, \frac{1}{2}, \psi) \gamma(\chi,\check\alpha, 0,\psi);\end{equation}
\item for the group case, $X = H=\operatorname{SL}_2$ under the $G=H\times H$-action,
\begin{equation}
\mu_X(\chi) = \gamma(\chi,-\check\alpha, 0,\psi^{-1}) \gamma(\chi,\check\alpha,0,\psi).
\end{equation}
\end{itemize}
Then the pullbacks of the ``open'' and ``closed'' Mellin transforms are related by the formula
\begin{equation}\label{relationopenclosed}
I_\chi(\Phi_1 \otimes \Phi_2) = \mu_X(\chi)^{-1} I_\chi^{\operatorname{cl}}(\Phi_1\otimes\Phi_2).\end{equation}
In particular, the relative characters $I_\chi$ decompose the space $L^2(X)_\emptyset$ \emph{with Plancherel measure $\mu_X(\chi)d\chi$}:
$$\int_X \frac{\varphi_1 \cdot \varphi_2}{dx} = \frac{1}{|W_X|}\int_{\widehat{A_X}} I_\chi(\varphi_1\otimes\varphi_2) \mu_X(\chi) d\chi.$$
\end{theorem}
\begin{proof}
Recall that the image of the asymptotics map is invariant under the scattering operators. In particular, if $\Phi_i = e_\emptyset^* \varphi_i$ ($i=1,2$) then $\mathscr S_{w,\chi}\check\Phi_i({^{w^{-1}}\chi}) = \check\Phi_i(\chi)$.
On the other hand, by Theorem \ref{thmscattering}, $\mathscr S_{w,\chi} = \mu_X({^w\chi}) \cdot \mathfrak R_\chi$. The $G$-equivariant scattering operator, applied to the first variable:
$$\mathfrak S_{w,1}: \mathcal S^+(X_\emptyset\times X_\emptyset) \to \mathcal S^+(X_\emptyset\times X_\emptyset)$$
descends to $G^{\operatorname{diag}}$-coinvariants, and hence \eqref{closed} reads
$$ M_\chi = \mu_X({\chi})^{-1} M^{\operatorname{cl}}_{^w\chi} \circ \mathfrak S_{w,1}.$$
Therefore, the pullbacks to $\mathcal S(X\times X)$ satisfy \eqref{relationopenclosed}.
The final statement follows from the Plancherel formula of the previous theorem.
\end{proof}
In our applications of this theorem, we will want to compare the transfer operator or Hankel transform for two relative trace formulas:
$$ \mathcal T: \mathcal S(X\times X/G) \to \mathcal S(Y\times Y/G')$$
(typically with non-standard spaces of test measures, which do not appear in our notation here),
with an abelian transfer operator for the corresponding degenerations
$$ \mathcal T_\emptyset: \mathcal S^+(X_\emptyset\times X_\emptyset/G) \to \mathcal S(Y_\emptyset\times Y_\emptyset/G'),$$
which is chosen so that the following diagram commutes:
\begin{equation}\label{Bcommute} \xymatrix{
\mathcal S(X\times X/G) \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*}\ar[d]^{\mathcal T} && \mathcal S^+(X_\emptyset\times X_\emptyset/G) \ar[d]^{\mathcal T_\emptyset} \\
\mathcal S(Y\times Y/G') \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*} && \mathcal S^+(Y_\emptyset\times Y_\emptyset/G') }.
\end{equation}
Here, by abuse of notation, we denote by $e_\emptyset^*\otimes e_\emptyset^*$ the descent of the morphism
$$ e_\emptyset^*\otimes e_\emptyset^*: \mathcal S(X\times X)\to \mathcal S^+(X_\emptyset\times X_\emptyset)$$
to the spaces of push-forward measures. The fact that it descends follows from the following fact, true for each of the spaces $X$ of Table \ref{tableX}, but not for their degenerations $X_\emptyset$:
\begin{theorem}\label{density}
For each of the spaces $X$ of Table \ref{tableX}, the map
$$\mathcal S(X\times X)_G\to \mathcal S(X\times X/G),$$
from coinvariants for the diagonal $G$-action to the space of push-forward measures, is an isomorphism.
\end{theorem}
\begin{proof}
For the Kuznetsov case, this is proven in more generality by Jacquet, Aizenbud and Gourevitch in \cite[Theorem 1.1]{Jacquet-density}, \cite[Corollary 6.0.4]{AGsmoothtransfer}.
The quotient stack $[\mathbb{G}_m\backslash \operatorname{PGL}_2/\mathbb{G}_m]$ looks locally like $[\mathbbm A^2/\mathbb{G}_m]$ (with action $(x,y)\cdot a = (ax, a^{-1}y)$ \cite[Lemma 3.2]{SaBE1}. The density of regular orbital integrals is then \cite[Lemma 2.3]{SaBE1}, which we will revisit in Lemma \ref{coinvariantA2} below to fill in some missing details in the proof.
The theorem on the group is a well-known theorem of Harish-Chandra. Notice that $[\operatorname{SL}_2 \times \operatorname{SL}_2 /{\operatorname{SO}}_4^{\operatorname{diag}}] \simeq [\frac{\operatorname{SL}_2}{\operatorname{PGL}_2}]$, so we can invoke the density of regular semisimple orbital integrals on the open subset $\operatorname{SL}_2(F)/\{\pm 1\}$ of $\operatorname{PGL}_2(F)$.
\end{proof}
The way to verify that diagram \eqref{Bcommute} commutes is to examine pullbacks of Mellin transforms. Theorem \eqref{thmpullbackfrombd} enables us to do that. For example, if $A_X=A_Y$, $W_X=W_Y$ and the map $\mathcal T$ is designed to respect Plancherel measures, that is (for the most continuous spectrum), to pull back the relative character $I_\chi^{Y,{\operatorname{cl}}}$ for $Y$ to the relative character $I_\chi^{X,{\operatorname{cl}}}$ for $X$ (see Theorem \ref{Plancherel}), then
\begin{quote}
\emph{the pullback of Mellin transform $M_\chi^{Y_\emptyset}$ on $Y_\emptyset$ via $\mathcal T_\emptyset$ should be}
\begin{equation}\mathcal T_\emptyset^* M_\chi^{Y_\emptyset} = \frac{\mu_X(\chi)}{\mu_Y(\chi)}\cdot M_\chi^{X_\emptyset}.
\end{equation}
\end{quote}
By the explicit form of the scalars $\mu_X(\chi)$, this corresonds to a composition of the \emph{multiplicative Fourier convolutions} that we defined in \S \ref{sssFourierconv} --- compare with \eqref{FE}. This is the conceptual explanation that I can presently give for all the transfer operators $\mathcal T$ and Hankel transforms that will appear in the rest of this paper: their geometric expressions are equal or, at least, \emph{deformations} of the geometric expressions for the transfer operators $\mathcal T_\emptyset$ of the boundary degenerations. These are given by Fourier convolutions determined by the scattering operators which, in turn, are closely related to the $L$-functions of the associated global period integrals, by \cite[\S 17]{SV}. Hence, the examples discussed in this paper suggest that \emph{the $L$-functions attached to spherical varieties inform the way that their relative trace formulas will be geometrically compared}.
\section{Transfer between the Kuznetsov formula and the stable trace formula for $\operatorname{SL}_2$} \label{sec:Rudnick}
\subsection{Relative characters}\label{ssrelchars-Kuz}
Here we discuss the local comparison behind Rudnick's thesis \cite{Rudnick}. Let $G=\operatorname{SL}_2$. Both $G$ as a $G\times G$-variety and the Whittaker space $(N,\psi)\backslash G$ have the same dual group, namely, $\operatorname{PGL}_2$. We will construct a local transfer map
$$ \mathcal T: \mathcal S^-_{L(\mathrm{Ad},1)}((N,\psi)\backslash G/(N,\psi)) \xrightarrow\sim \mathcal S(\frac{G}{G}),$$
which gives rise to stable functoriality between the Kuznetsov and the Selberg trace formula. The non-standard space $\mathcal S^-_{L(\mathrm{Ad},1)}((N,\psi)\backslash G/(N,\psi)) $ of orbital integrals was defined in \S \ref{ssnonstandard}. In terms of the representatives
$$ \tilde\zeta =\left(\begin{array}{cc}
& -\zeta^{-1} \\
\zeta &
\end{array}\right)$$
of regular orbits for the Kuznetsov formula, it consists of measures which, in a neighborhood of $\zeta=0$, coincide with the usual test measures $\mathcal S((N,\psi)\backslash G/(N,\psi))$ for the Kuznetsov formula, while in a neighborhood of $\zeta =\infty$ they are of the form
$$ C(\zeta^{-1}) d^\times \zeta,$$
where $C$ is a smooth function in a neighborhood of zero.
We can think of $N\backslash G$ as $V^*$, the complement of zero in a two-dimensional vector space, and the identification $N\simeq \mathbb{G}_a$ as a Whittaker structure. Then $N\backslash G\sslash N\simeq (V^*\times V^*)\sslash G$ was canonically identified in \S \ref{ssinvariant} with $\mathbb{G}_a$ through the symplectic pairing. This identification is compatible with the section $\zeta\mapsto \tilde\zeta$ over $F^\times$.
Let $\pi$ denote an irreducible tempered representation of $\operatorname{SL}_2$, and $\Pi$ its $L$-packet (the restriction of an irreducible tempered representation of $\operatorname{GL}_2$). We assume that $\pi$ is the unique generic element of $\Pi$ with respect to the character $\psi$ of $N$.
Define a morphism
$$ \tilde\pi\hat\otimes \pi\to C^\infty((N,\psi^{-1})\backslash G \times (N,\psi)\backslash G)$$
so that evaluation at the coset represented by $(1,1)$ is given by
\begin{equation}\label{IIWhittaker} \tilde v \otimes v\mapsto \int^*_{N} \left<\pi(n) v, \tilde v\right> \psi(n) dn,\end{equation}
with the measure on $N$ induced by its identification with $\mathbb{G}_a$.
This regularized integral is understood as the value at $\lambda =1$ of the Fourier transform (defined as $\int \Phi(x)\psi(\lambda x) dx$) of the $L^2$-function $n\mapsto \left<\pi(n) v, \tilde v\right>$ (where $N$ is identified again with $\mathbb{G}_a$). I point the reader to \cite[\S 2]{LM} and \cite[\S 6.3]{SV} for details.
The space $C_{\rm mod}^\infty((N,\psi^{-1})\backslash G \times (N,\psi)\backslash G)$ of smooth Whittaker functions of moderate growth is in canonical duality with $\mathcal S(((N,\psi)\backslash G \times (N,\psi^{-1})\backslash G)$. The restriction ``moderate growth'' only applies to the Archimedean case, and the image of the morphism defined by \eqref{IIWhittaker} automatically lands in it; however, from now on, for notational simplicity, we will be abusing notation and writing $C^\infty$ for $C^\infty_{\rm mod}$.
The adjoint of the map above is a morphism
$$\mathcal S((N,\psi)\backslash G \times (N,\psi^{-1})\backslash G) \to \pi\hat\otimes \tilde\pi.$$
Its composition with the canonical pairing $\pi\hat\otimes\tilde\pi \to \mathbb{C}$ is a $G^{\operatorname{diag}}$-invariant functional on $\mathcal S((N,\psi)\backslash G \times (N,\psi^{-1})\backslash G)$, which factors through the coinvariant space
$\mathcal S(N,\psi\backslash G/ N,\psi)$ (see \eqref{Whittakercoinvariants}), will be denoted by $J_\pi$ or $J_\Pi$:
$$J_\Pi: \mathcal S(N,\psi\backslash G/ N,\psi) \to \mathbb{C}.$$
This is the \emph{relative character} (or Bessel distribution) attached to the packet $\Pi$.
Explicitly,
$$ J_\Pi(\Phi_1\otimes\Phi_2) = \sum_{(v,\tilde v)} \int_{(N\backslash G)^2} \Phi_1(x_1)\Phi_2(x_2) \int^*_{N} \left<\pi(nx_1) v, \tilde\pi(x_2)\tilde v\right> \psi(n) dn,$$
where $(v,\tilde v)$ runs over dual pairs in dual bases of $\pi$ and $\tilde\pi$. Notice that it does not make a difference whether we sum over a dual basis for $\pi$ or for the entire $L$-packet $\Pi$; since the other elements of the packet are not generic, their contribution will be automatically zero.
\begin{example}
Let $F$ be non-Archimedean, with ring of integers $\mathfrak o$, residual degree $q$, and a uniformizer $\varpi$.
Suppose that $\pi = I(\chi)$ is a $K=G(\mathfrak o)$-unramified principal series representation, unitarily induced from an unramified character $\chi$ of the upper triangular Borel subgroup (identified with a character of $F^\times$). Let $\phi_{K,\chi}\in \pi$, $\phi_{K,\chi^{-1}}\in\tilde \pi$ be $K$-invariant vectors satisfying $\left<\phi_{K,\chi},\phi_{K,\chi^{-1}}\right>=1$.
It takes an elementary calculation (or an application of Macdonald's formula on zonal spherical functions) to show that the unramified matrix coefficient
$$\Phi(y) = \left< \pi \begin{pmatrix} 1 & y \\ & 1 \end{pmatrix} \phi_{K,\chi}, \phi_{K,\chi^{-1}}\right>$$
depends only on the absolute value of $y$, and satisfies $\Phi(y)=1$ on $\mathfrak o$ and $\Phi(y) = \frac{q^{-1}-q^{-2}+q^{-1}\chi(\varpi) + q^{-1}\chi(\varpi)^{-1}}{1+q^{-1}}$ when $y\in \varphi^{-1}\mathfrak o^\times$.
Thus,
if $f\in \mathcal S(N,\psi\backslash G/N,\psi)$ is the image of the identity element of the Hecke algebra, then
$$ \left< f, J_\pi\right> = \int \Phi(y) \psi(y) dy = $$
\begin{equation}\label{Bessel-basic}1-\Phi(\varpi^{-1}) = \frac{(1-q^{-1}\chi(\varpi))(1-q^{-1}\chi(\varpi)^{-1})}{1+q^{-1}} = \frac{\zeta(2)}{L(\pi,\mathrm{Ad},1)}.
\end{equation}
\end{example}
On the other hand, consider the space of test measures $\mathcal S(\frac{G}{G})$ for the stable trace formula of $G$. The \emph{stable character} $\Theta_\Pi$ of the $L$-packet $\Pi$ is a functional on this space, descending from the sum of the characters of the elements of $\Pi$, which are generalized functions on the group (that is, functionals on $\mathcal S(G)$).
It was proven in \cite[Theorem 6.3.4]{SV} that the characters $J_\Pi$ and $\Theta_\Pi$ satisfy the Plancherel formula for their respective spaces, for the \emph{same} Plancherel measure. More precisely, let $\widehat{G}_{\operatorname{st}}^{\operatorname{temp}}$ denote the ``stable tempered dual'' of $G$, i.e., the set of tempered representations modulo the equivalence of belonging to the same $L$-packet (or, equivalently, to the restriction of the same tempered representation of $\operatorname{GL}_2$). Then, fixing a measures $dg$ on $G$ to define $L^2(G)$ as a space of measures, there is a unique measure $\mu_G(\Pi)$ on $\widehat{G}_{\operatorname{st}}^{\operatorname{temp}}$ such that, for any $\varphi_1, \varphi_2\in \mathcal S(G)$ we have:
\begin{equation}\label{Plancherel-group}
\int_G \frac{\varphi_1\cdot \varphi_2}{dg} = \int_{\widehat{G}_{\operatorname{st}}^{\operatorname{temp}}} \Theta_\Pi(\varphi_1\otimes \varphi_2) \mu_G(\Pi).
\end{equation}
Let $dx$ be the measure on $N\backslash G$ which factorizes the measure $dg$ with respect to the fixed Haar measure on $N\simeq\mathbb{G}_a$. Then, Theorem 6.3.4 of \cite{SV} states that
for any $\varphi_1, \varphi_2\in \mathcal S(N,\psi\backslash G)$ we have:
\begin{equation}\label{Plancherel-Whittaker}
\int_G \frac{\varphi_1\cdot \varphi_2}{dg} = \int_{\widehat{G}_{\operatorname{st}}^{\operatorname{temp}}} J_\Pi(\varphi_1\otimes \varphi_2) \mu_G(\Pi)
\end{equation}
for \emph{the same} measure $\mu_G$.
\subsection{The main theorem}
Fix the isomorphism $\Dfrac{G}{G} \simeq \mathbb{G}_a$ via the trace map --- hence, both $\mathcal S(\frac{G}{G})$ and the non-standard test measures $\mathcal S^-_{L(\mathrm{Ad},1)}((N,\psi)\backslash G/(N,\psi))$ for the Kuznetsov formula are understood as measures on $\mathbb{G}_a$. In this section we will prove part \eqref{two} of the following theorem, assuming the other statements, which will be proven in Section \ref{sec:sym2}.
\begin{theorem}\label{thmRudnick}
Consider the equivariant Fourier transform $\mathcal T:=\mathscr F_{\operatorname{Id},1}$ of multiplicative convolution with the measure $D_1 = \psi(\zeta) d\zeta = \psi(\zeta) |\zeta|d^\times\zeta$ on $\mathbb{G}_m$.
Then:
\begin{enumerate}
\item \label{one} The convolution makes sense on $\mathcal S(N,\psi\backslash G/N,\psi)$ as the Fourier transform of a distribution, and maps it into $\mathcal S(\frac{G}{G})$.
\item \label{two} For every tempered packet $\Pi$ and $J_\Pi$ as above, \begin{equation} \label{pullbackchar}
\mathcal T^*\Theta_\Pi = J_\Pi.
\end{equation}
\item \label{three} The transform extends to an isomorphism, given by the same convolution understood, again, as the Fourier transform of a(n $L^2$-)distribution:
\begin{equation}\label{transferRudnickeq}
\mathcal T: \mathcal S_{L(\mathrm{Ad},1)}^-(N,\psi\backslash G/N,\psi) \xrightarrow\sim \mathcal S(\frac{G}{G}).
\end{equation}
\item \label{four} At non-Archimedean places, unramified over the base field $\mathbb Q_p$ or $\mathbb F_p((t))$, it satisfies the fundamental lemma for the Hecke algebra up to a factor of $\zeta(2)=(1-q^{-2})^{-1}$, namely: for all $h\in \mathcal H(G,K)\subset S(G)$, it takes the element
$$ h\cdot f_{L(\mathrm{Ad}, 1)} \in \mathcal S_{L(\mathrm{Ad},1)}^-(N,\psi\backslash G/N,\psi)$$ to the image of $\zeta(2) h$ in $\mathcal S(\frac{G}{G})$.
\end{enumerate}
\end{theorem}
\begin{remarks}
\begin{enumerate}
\item The part of Statement \eqref{four} about non-trivial elements of the Hecke algebra is quite redundant, since it can be deduced for the fundamental lemma for the identity element of the Hecke algebra, and Statement \eqref{two}. In any case, in Section \ref{sec:sym2} we will obtain it ``for free''. In generalizations of this paper, one would expect to prove an analog of Statement \eqref{four} locally, and then use it to deduce, by a global-to-local argument involving comparisons of relative trace formulas, an analog of Statement \eqref{two}.
\item The factor $\zeta(2)$ in Statement \eqref{four} is compatible with the calculation of the relative character applied to the standard basic function of $\mathcal S(N,\psi\backslash G/N,\psi)$ in \eqref{Bessel-basic}.
\item Theorem 1.3 in \cite{SoundYoung} can be seen as a global application of the isomorphism \eqref{transferRudnickeq}, restricted to hyperbolic conjugacy classes.
\item As Valentin Blomer pointed out to me, it seems that \eqref{pullbackchar} can be directly obtained, in the real case, from classical identities such as \cite[(6.611.1)]{Gradshteyn-Ryzhik}, expressing the Fourier transform of Bessel functions in terms of exponentials; I have not checked the details.
\item In a different, but related, setting, David Ben-Zvi and Sam Gunningham have recently established what can be seen as a comparison between the quotient spaces associated to the Kuznetsov formula and the trace formula, in the setting of loop groups of arbitrary complex reductive groups. Their theorem \cite[Theorem 6.16]{BZvi-Gunningham} constructs, by spectral arguments, what can be informally be described as a map from ``$D$-modules on the quotient space $(N,\psi)\backslash G/(N,\psi)$'' to ``$D$-modules on the adjoint quotient of $G$''. It would be interesting to see an explicit geometric description of their comparison, similar to the Fourier transform of the above theorem.
\end{enumerate}
\end{remarks}
We will now prove Statement \eqref{two}, assuming the rest (more precisely assuming Statement \eqref{one}). Statements \eqref{one} and \eqref{three} will be proven at the end of \S \ref{Pfpart3}, and Statement \eqref{four} will be proven at the end of \S \ref{Pfpart4}.
\begin{proof}[Proof of Statement \eqref{two}, assuming the rest]
Let $f(\zeta) = \Phi(\zeta) d^\times\zeta$. Since the map $N\backslash G\twoheadrightarrow \mathfrak{C} := N\backslash G\sslash N$ is smooth, the untwisted push-forward map
$$\mathcal S(N\backslash G)\to {\operatorname{Meas}}(\mathfrak{C})$$
has image in locally bounded measures (in fact, Schwartz measures); a fortiori, the same is true for the twisted push-forward. Hence, $\Phi(\zeta)\ll 1$.
We write, formally,
$$ D_1\star f (\zeta) = d^\times \zeta\cdot D_1 \star (|\bullet|\Phi(\bullet))(\zeta) = $$
$$= d^\times\zeta\cdot \int_{F^\times} |z^{-1}\zeta|\Phi(z^{-1} \zeta) \psi(z) |z| d^\times z = d \zeta\cdot \int_F \Phi(z^{-1}) \psi(z\zeta) dz,$$
and hence interpret $\mathcal Tf$ as the product of the measure $d\zeta$ with the Fourier transform of the $L^2$-function $\zeta\mapsto \Phi(\zeta^{-1})$.
The extension of this to an isomorphism
$$\mathcal T: \mathcal S_{L(\mathrm{Ad},1)}^-(N,\psi\backslash G/N,\psi) \xrightarrow\sim \mathcal S(\frac{G}{G}),$$
together with the fundamental lemma, will be postponed until \S \ref{sec:sym2}, where they will be obtained as special cases of a more general theorem.
We prove the statement on relative characters. It relies on the following formal relation:
\begin{equation}\label{transfer-trace}\mathcal T(f) = r_!(f \times dn),\end{equation}
where $r$ is the quotient map
$$r: \frac{G}{N} \to \frac{G}{G},$$
and $f\times dn$ denotes the ``pullback'' of $f$ to the $N$-torsor $\frac{G}{N} \to N\backslash G/N$ defined by the fixed Haar measure on $N$. \emph{Here we do not think of $f$ as a scalar-valued measure on $N\backslash G\sslash N$} (by the trivialization described in \S \ref{sstwistedpf}), \emph{but as an $(N,\psi)$-equivariant measure on $\Dfrac{G}{N}$, divided by the Haar measure of $N$}, where $n\in N$ acts by sending the class of $g\in G$ to the class of $ng$. If we fix coordinates $\begin{pmatrix} a & b \\ c & d \end{pmatrix}\mapsto (\zeta = c, t=\operatorname{tr})$ for $\Dfrac{G}{N}\simeq \mathbbm A^2$, the trivialization of $f$ introduced in \S \ref{sstwistedpf} is the one obtained by restricting it to the line $t=0$. If $f_1$ is the push-forward to $\Dfrac{G}{N}$ of a Schwartz measure on $G$, and $f$ its twisted push-forward to $\mathcal S(N,\psi\backslash G/N,\psi)$.
and if $R_n$ denotes the translation action of $N$ on measures on $\Dfrac{G}{N}$, we have
\begin{equation}\label{ff1} f \times dn= \int R_n f_1 \cdot \psi^{-1}(n) dn.
\end{equation}
In the coordinates $(\zeta,t)$ above, the group $N=\mathbb{G}_a$ acts as $(\zeta,t)\cdot x = (\zeta, t+\zeta x)$.
We will need to give a rigorous meaning to the formal relation \eqref{transfer-trace}, because the push-forward $r_!$ does not converge absolutely. But, before we do that, let us justify this relation in a formal manner, pretending that all integrals were convergent.
First of all, let us show why \eqref{transfer-trace} should imply \eqref{pullbackchar}: With $f_1$ and $f$ as above we have, formally,
$$ \left< \mathcal T f ,\Theta_\Pi\right> \xlongequal{\eqref{transfer-trace}} \left<f\times dn, p^*\Theta_\Pi\right> = $$
$$=\left<\int_N R_x f_1 \cdot \psi^{-1} (x) dx \right> = \int_N \left< R_x f_1, \Theta_\Pi\right> \psi^{-1}(x) dx = $$
$$=\left<f_1, \int R_x \Theta_\Pi \psi(x) dx \right> = \left< f_1 , J_\Pi\right> = \left< f, J_\Pi\right>,$$
the last equality because the generalized function $J_\Pi$ is already $(N,\psi^{-1})$-equivariant on both sides.
To formally justify \eqref{transfer-trace}, we claim that, for a measure $f \in \mathcal S(N,\psi\backslash G/N,\psi)$ which, after trivialization, is written $f(\zeta) = \Phi(\zeta) d\zeta$, the pullback measure $f\times dn$ on $\frac{G}{N}$ is $\Phi(\zeta)|\zeta|^{-1} \psi(\frac{t}{\zeta}) d \zeta dt$. Indeed, this relies on the calculation that a matrix in $\operatorname{SL}_2$ with given $(\zeta = c,t)$ and $c\ne 0$ can be written as
$$ \begin{pmatrix} 1 & x \\ & 1 \end{pmatrix} \begin{pmatrix} & -c^{-1} \\ c \end{pmatrix} \begin{pmatrix} 1 & y \\ & 1 \end{pmatrix}$$
with $x+y = \frac{t}{c}$.
Thus, the push-forward $r_!(f \times dn)$ is
\begin{equation}\label{rshriek} r_!(f \times dn)(t) = dt \cdot \left(\int_F \Phi(c)|c|^{-1} \psi(\frac{t}{c}) d c \right) = \int f(\frac{t}{z}) \psi(z) |z| d^\times z.
\end{equation}
Now let us rigorously prove \eqref{pullbackchar}: As before, consider a Schwartz measure on $G$, and let $f_1$ be its push-forward to $\Dfrac{G}{N}$, and $f$ its twisted push-forward to $\mathcal S(N,\psi\backslash G/N,\psi)$. The proof relies on the (rigorous) relation
\begin{equation}
\int^* \left<r_! (R_x f_1),\Theta_\Pi\right> \psi^{-1}(x) dx = \left<\mathcal T f,\Theta_\Pi\right>,
\end{equation}
as measures on $\Dfrac{G}{G}$, where the left hand side should be interpreted, as we have done for the right hand side when defining $\mathcal Tf$, as the Fourier transform of a distribution; more precisely, as the value of the generalized function $u\mapsto \int_F \left<\pi_! (R_x f_1),\Theta_\Pi\right> \psi(ux) dx $ at $1$, after we show that this generalized function is a continuous function in a neighborhood of $1$.
Let us first show the corresponding identity for generalized functions, so choose a Schwartz measure $\varphi(u) du$ on $F$. We denote Fourier transforms by $\varphi\mapsto \hat\varphi$, whether they are defined with the character $\psi$ or $\psi^{-1}$, and leave it to the curious reader to figure out where each character is being used. We also write $\Theta$ for $\Theta_\Pi$. We compute:
$$ \int \varphi(u) \int^* \left<r_! (R_x f_1),\Theta\right> \psi^{-1}(ux) dx du \xlongequal{\mbox{by def.}} \int \hat\varphi(x) \left<\pi_! (R_x f_1),\Theta\right> dx = $$
$$ = \iiint \hat\varphi(x) f_1(\zeta,t+\zeta x) \Theta(t) dx,$$
where we have used the fact that the triple integral is absolutely convergent. Indeed,
writing $f_1 = \Phi_1 d\zeta dt$, the product
$$\hat\varphi(x) \Phi_1(\zeta,t+\zeta x) \Theta(t)$$ has local $L^1$-seminorms (i.e., $L^1$-seminorms over additive translates of any fixed compact subset in the variables $(x,\zeta,t)$) of rapid decay in the variable $\min(|x|, |\zeta|, |t|)$, because the function $(\zeta,t)\mapsto \Theta(t)$ has local $L^1$-seminorms of polynomial growth, $\hat\varphi$ is of rapid decay, and $\Phi_1(\zeta,t)$ has rapidly decaying local-$L^1$ seminorms in both variables.
We can now write it as an iterated integral, with the variable $x$ to the interior. Consider the function $x\mapsto \Phi_1(\zeta, t+\zeta x)$. For fixed $\zeta\ne 0$ and $t$, the function is rapidly decaying in $x$. Its Fourier transform against the character $\psi^{-1}$, evaluated at a point $u$, is ${^u\Phi}(\zeta) \psi(u\frac{t}{\zeta})$, where ${^u\Phi}$ is the Fourier transform of $x\mapsto \Phi_1(\zeta, \zeta x)$ (so that, by \eqref{ff1}, $f = {^1\Phi} d \zeta$). Thus, the last integral can be written:
$$ \int_t \Theta(t) \int_\zeta \int_u \varphi(u) {^u\Phi}(\zeta) \psi(u\frac{t}{c}) du d\zeta dt.$$
We now want to switch the order of integration over $\zeta$ and $u$, interpreting the integral over $\zeta$ as a Fourier transform in the sense of distributions:
\begin{equation}\label{toprove}\int_\zeta \int_u \varphi(u) {^u\Phi}(\zeta) \psi(u\frac{t}{\zeta}) du d \zeta = \int_u \varphi(u) \int_\zeta^* {^u\Phi}(\zeta) \psi(u\frac{t}{\zeta}) du d \zeta.\end{equation}
To show this, let $\mapsto F_u$ be the Fourier transform of the function $\zeta\mapsto {^u\Phi}(\frac{1}{\zeta})$ \emph{against the character $x\mapsto \psi(ux)$}. Notice that for $u=1$ the measure $F_1(\zeta) d\zeta$ is precisely the image of ${^1\Phi} d\zeta$ under the transfer operator $\mathcal T$, and hence belongs to $\mathcal S(\frac{G}{G})$. As will be clear from the proof of Statement \eqref{one} (in \S \ref{proofRudnick}), there is nothing special about $u=1$; more precisely, for $u$ in a neighborhood of $1$, the map $u\mapsto F_u d\zeta$ is a continuous section of $\mathcal S(\frac{G}{G})$.
The inverse map $F_u \mapsto {^u\Phi} $ is given by Fourier transform using the character $\psi^{-1}$, instead of $\psi$, which converges absolutely. Indeed, it is straightforward to see that measures in $\mathcal S(\frac{G}{G})$ are bounded by Schwartz measures, and hence $F_u$ is bounded (and of rapid decay). As $u$ varies in a neighborhood of $1$, the Fourier transforms of the functions $F_u$ converge uniformly, since $u\mapsto F_ud\zeta \in \mathcal S(\frac{G}{G})$ is continuous.
Thus, if $\kappa$ is another Schwartz function, and we compute both sides of the desired equality \eqref{toprove} as distributions in the variable $t$, we have
$$ \int_t \hat \kappa(t) \int_u \varphi(u) \int_c^* {^u\Phi}(c) \psi(u\frac{t}{c}) du d^\times c = \int_t \hat\kappa(t) \int_u \varphi(u) F_u(t) dt =$$
$$ = \int_u \varphi(u) \int_t \hat\kappa(t) F_u (t) dt du = \int_u \varphi(u) \int_z \kappa(z) \widehat{^u F}(z) dz du,$$
where $\widehat{F_u}$ is defined using the character $\psi^{-1}$, so the last expression is
$$ \int_u \varphi(u) \int_z \kappa(z) \int_t F_u(t) \psi^{-1}(tz) dt dz du$$
But this converges absolutely as a triple integral, so we can write it as
$$ \int_z \kappa(z) \int_u \varphi(u) \int_t F_u(t) \psi^{-1}(tz) dt du dz = $$
$$ =\int_z \kappa(z) \int_u \varphi(u) \int_t F_u(t) \psi^{-1}(u t\frac{z}{u}) dt du dz = $$
$$ = \int_z \kappa(z) \int_u \varphi(u) {^u\Phi}(\frac{u}{z}) du dz.$$
Applying Fourier transform in the variable $z$ again, this is equal to
$$ \int_t \hat\kappa(t) \int_c \int_u \varphi(u) {^u\Phi}(\frac{u}{\zeta}) \psi(\zeta t) du d\zeta dt$$
$$ = \int_t \hat\kappa(t) \int_\zeta \int_u \varphi(u) {^u\Phi}(\zeta) \psi(\frac{ut}{\zeta}) du d\zeta dt,$$
completing the proof of \eqref{toprove}.
Putting all together, we have shown that
$$ \int \varphi(u) \int^* \left<r_! (R_x f_1),\Theta\right> \psi^{-1}(ux) dx du = \int_t \Theta(t) \int_u \varphi(u) F_u(t) du dt =$$
$$ = \int_u \varphi(u) \int \Theta(t) F_u(t) dt du,$$
the last step because of the continuity of the section $u\mapsto F_u\in \mathcal S(\frac{G}{G})$, and the rapid decay of elements of $\mathcal S(\frac{G}{G})$. Now, the inner integral on the right hand side, as a generalized function of $u$, is represented by a continuous function, hence so is the inner integral on the left hand side, and the equality holds for $u=1$.
\end{proof}
\subsection{Comparison with the degeneration}
In this subsection, $F$ is a non-Archimedean field.
Let $X$ be a symbol for the Whittaker model $(N,\psi)\backslash G$, and let $X_\emptyset = N\backslash G$ be its asymptotic cone. Set $Y=G=\operatorname{SL}_2$, $G'=G\times G/\{\pm 1\}^{\operatorname{diag}}$, and $Y_\emptyset$ the asymptotic cone of $Y$. We recalled in \S \ref{ssrelchars} that there is a canonical ``asymptotics'' morphism
$$ e_\emptyset^*\otimes e_\emptyset^*: \mathcal S(X\times X) \to \mathcal S^+(X_\emptyset\times X_\emptyset),$$
and saw that it descends to spaces of push-forward measures.
Together with the transfer operator $\mathcal T$, they give rise to most of the diagram \eqref{Bcommute}, which we repeat here:
$$ \xymatrix{
\mathcal S(X\times X/G) \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*}\ar[d]^{\mathcal T} && \mathcal S^+(X_\emptyset\times X_\emptyset/G) \ar[d]^{\mathcal T_\emptyset} \\
\mathcal S(Y\times Y/G') \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*} && \mathcal S^+(Y_\emptyset\times Y_\emptyset/G') }.
$$
What is missing is the transfer operator $\mathcal T_\emptyset$ making the diagram commute.
\begin{theorem}\label{groupdegen}
Identify $\mathfrak{C}_\emptyset:= Y_\emptyset\times Y_\emptyset\sslash G' = X_\emptyset\times X_\emptyset\sslash G = \mathbb{G}_a$ as in \S \ref{ssrelchars}, namely, sending the distinguished $G'$-orbit, resp.\ $G$-orbit, to $1$ (and the singular one to $0$). There is a unique $A_X$-equivariant operator $\mathcal T_\emptyset$ making the above diagram commute, given by the multiplicative Fourier convolution $\mathscr F_{\check\alpha, 1}$ --- again, understood as the Fourier transform of a distribution.
\end{theorem}
\begin{remark}
The reader will notice that the only property of the transfer operator $\mathcal T$ used in the proof is that it satisfies \eqref{pullbackchar}, in other words, that it relates the relative characters that correspond to \emph{the same} Plancherel measure, cf.\ \eqref{Plancherel-group} and \eqref{Plancherel-Whittaker}. At no point will we use the explicit expression for the transfer operator as $\mathscr F_{\operatorname{Id},1}$. Thus, this theorem, with a simple comparison of coordinates that follows, gives a \emph{conceptual reason} why the transfer operator $\mathcal T$ is given by this formula: it is ``the same'' as the operator $\mathcal T_\emptyset$! In higher rank, in some examples computed together with Chen Wan, things are similar, but not so simple: the operator $\mathcal T$ tends to be a \emph{deformation} of the boundary operator $\mathcal T_\emptyset$; we currently do not understand the nature of this deformation.
\end{remark}
\begin{proof}
Notice that for both $X$ and $Y$ the character $\delta$ on $A_X=A_Y$ coincides --- in the coordinate $\zeta$ as above, $\delta^\frac{1}{2} = |\zeta|$.
For what follows, denote by $J_\Pi$, $\Theta_\Pi$ by $J_\chi$, $\Theta_\chi$, respectively, when $\Pi$ is the principal series representation of $\operatorname{SL}_2$ obtained by unitary induction from the character $\chi$ of $A_X$.
Let $M_\chi$ denote the Mellin transform $f\mapsto \check f(\chi\delta^\frac{1}{2}) = \check f(\chi|\bullet|)$ on either of the spaces $\mathcal S^+(X_\emptyset\times X_\emptyset/G)$ and $\mathcal S^+(Y_\emptyset\times Y_\emptyset/G')$.
Let $I_\chi^X$, resp.\ $I_\chi^Y$ be its pullback via the asymptotics map to $\mathcal S(X\times X/G)$, resp.\ $\mathcal S(Y\times Y/G')$.
Comparing the Plancherel formulas \eqref{Plancherel-group}, \eqref{Plancherel-Whittaker} and Theorem \ref{thmpullbackfrombd},
we deduce that
$$ I_\chi^Y \mu_Y(\chi) = \Theta_\chi \mu_G(\chi)$$
and
$$ I_\chi^X \mu_X(\chi) = J_\chi \mu_G(\chi),$$
where $\mu_G(\chi) d\chi$ is the Plancherel measure for $\operatorname{SL}_2$. (It can be shown to be equal to $\mu_Y(\chi) d\chi$ for a correct choice of Haar measures, hence $I_\chi^Y = \Theta_\chi$, but we don't need this here.)
Since $\mathcal T^* \Theta_\chi = J_\chi$, for the diagram to commute we would need
$$ \mathcal T^*_\emptyset M_\chi =\frac{\mu_X(\chi)}{\mu_Y(\chi)} M_\chi = \gamma(\chi,-\check\alpha, 0,\psi^{-1})^{-1} M_\chi,$$
or, in other words,
$$ \widecheck{\mathcal T_\emptyset f}(\chi) = \gamma(\chi,-\check\alpha, 1,\psi^{-1})^{-1} \check f(\chi) = \gamma(\chi, \check\alpha, 0,\psi) \check f(\chi).$$
Now consider the multiplicative Fourier convolution $\mathscr F_{\check\alpha, 1}$ of an element $f\in \mathcal S^+(X_\emptyset\times X_\emptyset/G)$. As in the beginning of the proof of Theorem \ref{thmRudnick}, if $f(\zeta) = \Phi(\zeta) d^\times\zeta$ then $\mathscr F_{\check\alpha, 1}$ will be understood as the Fourier transform of the distribution $D(\zeta):= \Phi(\zeta^{-1}) d\zeta = |\zeta| f(\zeta^{-1})$. Notice that $\Phi$ is smooth away from $0$, and of compact support on $F$, therefore this distribution is smooth; its Fourier transform is a compactly supported distribution on $F$. We will now argue that $\mathscr F_{\check\alpha, 1} f$ can be reconstructed from its Mellin transform, precisely as in \eqref{invMellin}; moreover, that \eqref{FE} holds for its Mellin transform:
$$ \widecheck{\mathscr F_{\alpha, 1}f}(\chi) = \gamma(\chi, \check\alpha, 0,\psi) \check f(\chi).$$
This will imply, by \eqref{invMellin}, that $\mathcal T_\emptyset = \mathscr F_{\check\alpha,1}$.
To see this, we apply Corollary \ref{corWhittakerasymptotics}: Suppose that $f$ is the push-forward of an element $\varphi \in \mathcal S^+(X_\emptyset\times X_\emptyset)$. Acting on $\varphi$, in both variables, by the element $h\in \widehat{\mathcal S(A_X)}$ (completed Hecke algebra) whose Mellin transform is $\check h(\chi) = L(\chi,\check\alpha, 1)^{-1}$, we obtain an element $\varphi'\in \mathcal S(X_\emptyset\times X_\emptyset)$, whose push-forward we denote by $f'$. Taking into account the normalization of the action of $A_X$ on measures on $X_\emptyset$, which we did not adopt on the quotient space $\mathfrak{C}_\emptyset$,
$$ f' = (|\bullet|^{-1} h) \cdot (|\bullet|^{-1} h) \cdot f,$$
where now $A_X$ is identified with $\mathbb{G}_m$ through the positive root character.
The morphism $X_\emptyset\times X_\emptyset \to \mathfrak{C}_\emptyset$ is smooth, hence $f'$ is a Schwartz measure on the line. Acting on it once more by the measure $h$:
$$ f'' := h\cdot (|\bullet|^{-1} h) \cdot (|\bullet|^{-1} h) \cdot f,$$
it becomes supported away from zero.
Hence, the distribution
$$ D'' := (|\bullet| h^\vee)\cdot h^\vee \cdot h^\vee \cdot D,$$
where $h^\vee (a) = h(a^{-1})$, is (smooth and) of compact support on $F^\times$. In particular, the theory of Tate zeta integrals applies to it: $\mathscr F_{\check\alpha, 1} f''=$ the Fourier transform of $D''$ is a smooth, compactly supported measure on the affine line, both $D''$ and $\mathscr F_{\check\alpha, 1} f''$ can be reconstructed from their Mellin transforms, as in \eqref{invMellin}, and their Mellin transforms satisfy the functional equation \eqref{gammaZeta}, which can be written as in \eqref{FE}:
$$ \widecheck{\mathscr F_{\check\alpha, 1}f''}(\chi) = \gamma(\chi, \check\alpha, 0,\psi) \check f''(\chi).$$
But, by construction,
$$\check f''(\chi) = L(\chi,-\check\alpha, 1)^{-2} L(\chi,-\check\alpha, 2)^{-1} \cdot \check f(\chi),$$
and, by the equivariance of Fourier convolution,
\begin{equation}\label{equiv} \mathscr F_{\check\alpha, 1}f'' = h\cdot (|\bullet|^{-1} h) \cdot (|\bullet|^{-1} h) \mathscr F_{\check\alpha, 1}f.
\end{equation}
It is now easy to see that, since the factor $L(\chi,-\check\alpha, 1)^2 L(\chi,-\check\alpha, 2)$ has no pole at $\chi=1$, the inverse Mellin transform \eqref{invMellin}, applied to
$$\gamma(\chi, \check\alpha, 0,\psi) L(\chi,-\check\alpha, 1)^2 L(\chi,-\check\alpha, 2) \check f''(\chi) = \gamma(\chi, \check\alpha, 0,\psi) \check f(\chi),$$
represents the unique compactly supported distribution $\mathscr F_{\check\alpha, 1}f$ on $F$ (in fact, in this case, a measure represented by an $L^1$-function) which satisfies \eqref{equiv}.
This implies the claim.
Finally, we argue that $\mathscr F_{\check\alpha, 1} f \in \mathcal S^+(Y_\emptyset\times Y_\emptyset)$. By the Mellin inversion formula \eqref{invMellin}, again, it suffices to show that its Mellin transform is contained in the space of Mellin transforms of elements of $\mathcal S^+(Y_\emptyset\times Y_\emptyset)$. If $f$ is obtained as the asymptotics of some $\tilde f \in \mathcal S(X\times X/G)$, we have already seen that
$$M_\chi \mathscr F_{\check\alpha, 1} f = \gamma(\chi,-\check\alpha, 0,\psi^{-1})^{-1} M_\chi f = \gamma(\chi,-\check\alpha, 0,\psi^{-1})^{-1} I^X_\chi \tilde f = J_\chi \tilde f,$$
but also
$$ J_\chi\tilde f = \Theta_\chi (\mathcal T \tilde f) = M_\chi (e_\emptyset\otimes e_\emptyset(\mathcal T\tilde f)).$$
Therefore,
$\mathscr F_{\check\alpha, 1} f = e_\emptyset^*\otimes e_\emptyset^*(\mathcal T\tilde f) \in \mathcal S^+(Y_\emptyset\times Y_\emptyset)$.
\end{proof}
Finally, let us notice that the coordinates that we have been using on the spaces $Y\times Y\sslash G'$ and $Y_\emptyset\times Y_\emptyset\sslash G'$ are compatible, in the following sense: There is a family $\mathcal Y \to \mathbb{G}_a$, namely,
$$\mathcal Y=\operatorname{Mat}_2\xrightarrow{\det}\mathbb{G}_a,$$ the space of $2\times 2$ matrices, with general fiber $G'$-equivariantly isomorphic to $Y$, and special fiber (over $0\in \mathbb{G}_a$) equal to $Y_\emptyset^a$. The identification of $Y_\emptyset$ with the open $G'$-orbit in the fiber over $0$ was described in \eqref{ssscatteringgroup}.
In coordinates, pick the Borel subgroup $B^-\times B\subset G'$, where $B^-=$ lower triangular matrices and $B=$ upper triangular; then $N^-\backslash \operatorname{SL}_2\sslash N$ is identified with $Y_\emptyset\sslash(N^-\times N)$ via the top left entry of a matrix --- this is the isomorphism \eqref{samehoro}.
Now we will see that there is an isomorphism
$$(\mathcal C, \det): \mathcal Y\times_{\mathbb{G}_a}\mathcal Y\sslash G \simeq \mathbb{G}_a\times \mathbb{G}_a,$$
such that the restriction of $\mathcal C$ to $Y\times Y\sslash G'=$ the fiber of $1$ is the trace map:
$$(g_1,g_2)\mapsto \operatorname{tr}(g_1 g_2^{-1}),$$
while its restriction to the fiber of $0$ is our preferred coordinate for $Y_\emptyset\times Y_\emptyset\sslash G'$.
Indeed, let $w = \begin{pmatrix} & -1 \\ 1\end{pmatrix}$ and take $\mathcal C(g_1,g_2) = \operatorname{tr}(g_1 w g_2^t w^{-1})$. For $g_2\in \operatorname{SL}_2$ we have $g_2^{-1} = wg_2^t w^{-1}$. To check that this coincides with our distinguished coordinate for $Y_\emptyset\times Y_\emptyset\sslash G'=Y^a_\emptyset\times Y^a_\emptyset\sslash G'$, it suffices to observe that the zero matrix is mapped to $0$, and the pair $(g_1,g_2) = (\begin{pmatrix} 1& 0 \\ 0& 0\end{pmatrix}, \begin{pmatrix}0 & 0 \{(0)} & 1\end{pmatrix})$, which belongs to the distinguished $G'$-orbit on $Y_\emptyset\times Y_\emptyset$, is mapped to $1$.
For $X$ and $X_\emptyset$ the analogous statement is a tautology, since the underlying spaces are the same (and we have been using the same coordinate both for $X$ and its degeneration).
\section{Transfer between the Kuznetsov formula and the relative trace formula for torus periods} \label{sec:Waldspurger}
Now consider the case of $Y=T\backslash G$, where $G=\operatorname{PGL}_2$ and $T\simeq \mathbb{G}_m$ is a split torus. One could also consider a non-split torus, but would need to slightly modify the equivariant Fourier transforms that we defined in \S \ref{sssFourierconv}. We review the local transfer operator $$ \mathcal T: \mathcal S^-_{L({\operatorname{Std}},1)^2}((N,\psi)\backslash G/(N,\psi)) \xrightarrow\sim \mathcal S(T\backslash G/T),$$
constructed in \cite{SaBE1, SaBE2}. As in \S \ref{scattorus}, we will identify $X$ with the space of quadratic forms of discriminant $-\frac{1}{4}$ on a two-dimensional symplectic space $V$.
Recall that the non-standard space $\mathcal S^-_{L({\operatorname{Std}},\frac{1}{2})^2}((N,\psi)\backslash G/(N,\psi)) $ of orbital integrals was defined in \S \ref{ssnonstandard}. In terms of the representatives
$$ \tilde\xi =\left(\begin{array}{cc}
& -1 \\
\xi &
\end{array}\right)$$
of regular orbits for the Kuznetsov formula, it consists of measures which, in a neighborhood of $\xi=0$, coincide with the usual test measures $\mathcal S((N,\psi)\backslash G/(N,\psi))$ for the Kuznetsov formula, while in a neighborhood of $\xi =\infty$ they are of the form
$$ (C_1(\xi^{-1})+C_2(\xi^{-1}) \log|\xi|) d^\times \xi,$$
where $C_1$, $C_2$ are smooth functions in a neighborhood of zero.
The role of the character $\Theta_\Pi$, here, will be played by a relative character $I_\pi$ for an irreducible tempered representation of $\operatorname{PGL}_2$, for the quotient space $\mathbb{G}_m\backslash \operatorname{PGL}_2/\mathbb{G}_m$. The definition of the relative character $I_\pi$ is completely analogous to that of the Kuznetsov relative character $J_\pi$: It is given as the composition
$$ \mathcal S(Y\times Y)\to \pi\hat\otimes\tilde\pi\to\mathbb{C},$$
where the dual of the map to $\pi\hat\otimes\tilde\pi$ is the morphism
$$ \tilde\pi\otimes\pi\to C^\infty(Y\times Y)$$
that, composed with evaluation at $T1 \times T1$ is given by:
$$\tilde v \otimes v\mapsto \int_{T} \left<\pi(t) v, \tilde v\right> dt.$$
Here the integral is convergent (for tempered representations), and no normalization is needed. Moreover, the $L$-packets for the group $\operatorname{PGL}_2$ are singletons (if we do not consider its inner forms, which we should have, in the case of a non-split torus), therefore there is no need to distinguish, notationally, between $\pi$ and its $L$-packet $\Pi$.
The measure on $T$ is fixed to be the multiplicative Haar measure $d^\times x$ on $F^\times$, after identifying $T\simeq\mathbb{G}_m$ --- there are two inverse ways to perform this identification, and they give rise the same measure.
\begin{theorem}
Consider the equivariant Fourier transform $\mathcal T:=\mathscr F_{\operatorname{Id},1} \circ \mathscr F_{\operatorname{Id},1}$ of multiplicative convolution, twice, with $D_{1} = \psi(\bullet) |\bullet| d^\times\bullet$ on measures on $\mathbb{G}_m$.
Then:
\begin{enumerate}
\item The convolution makes sense on $\mathcal S(N,\psi\backslash G/N,\psi)$ as the Fourier transform of a distribution, and maps it into $\mathcal S(T\backslash G/T)$.
\item For every tempered representation $\pi$, \begin{equation}
\mathcal T^*I_\pi = J_\pi.
\end{equation}
\item The transform extends to an isomorphism, given by the same convolution understood, again, as the Fourier transform of a(n $L^2$-)distribution:
\begin{equation}
\mathcal T: \mathcal S_{L({\operatorname{Std}},\frac{1}{2})^2}^-(N,\psi\backslash G/N,\psi) \xrightarrow\sim \mathcal S(T\backslash G/T).
\end{equation}
\item At non-Archimedean places, it satisfies the fundamental lemma for the Hecke algebra, up to a factor of $\zeta(1)^2=(1-q^{-1})^{-2}$, namely: for all $h\in \mathcal H(G,K)$, it takes the element
$$ h\cdot f_{L({\operatorname{Std}}, \frac{1}{2})^2} \in \mathcal S_{L({\operatorname{Std}},\frac{1}{2})^2}^-(N,\psi\backslash G/N,\psi)$$ to the image of $\zeta(1)^2\cdot h$ in $\mathcal S(T\backslash G/T)$.
\end{enumerate}
\end{theorem}
For precise references to \cite{SaBE1, SaBE2}, and an explanation of how the formulas there relate to the above transfer operator (given the different coordinates that we are using), I point the reader to \cite{SaHanoi}. Here, I would like to discuss the relation to transfer operators on the asymptotic cone.
First, let us fix compatible coordinates for $Y\times Y\sslash G$ and for $Y_\emptyset\times Y_\emptyset \sslash G$. Consider the family $\mathcal Y\to \mathbb{G}_a$ whose fiber $\mathcal Y_t$ over $t\in \mathbb{G}_a$ is the space of quadratic forms of discriminant $-\frac{t^2}{4}$ on $V$ (so that all non-zero fibers correspond to split non-degenerate forms). The fiber over $0$ contains the boundary degeneration $Y_\emptyset$, i.e., the space of rank-one quadratic forms, as was explained in \S \ref{scattorus}. As we did there, we identify the quotient $\mathcal Y_t\times V\sslash G\simeq \mathbb{G}_a$ by the evaluation map.
We can fix an isomorphism
$$(\mathcal C, t): \mathcal Y\times_{\mathbb{G}_a} \mathcal Y\sslash G \xrightarrow \sim \mathbb{G}_a\times \mathbb{G}_a,$$
where $t$ is the defining morphism to $\mathbb{G}_a$, and $\mathcal C$ is as follows: Consider a quadratic form $q$ on $V$; it defines a morphism $V\to V^\vee$, which combined with the fixed isomorphism $\iota_\omega^{-1}: V^\vee \to V$ induced by the symplectic form, gives rise to an endomorphism
$$ \iota_q: V\to V.$$
Explicitly, $\omega (u,\iota_q(v)) = q(u,v)$ for all $u, v\in V$. We now define $\mathcal C (q_1, q_2) = \operatorname{tr}(\iota_{q_1}\circ \iota_{q_2}) + \frac{t}{2}$.
The reader can check that, on the fiber over $t=1$ ($=Y\times Y\sslash G$), the coordinate $\mathcal C$ is the one that we fixed above, while on the fiber over $t=0$ ($=Y_\emptyset\times Y_\emptyset\sslash G$) the coordinate descends from the map
$$ V\times V \ni (u,v)\mapsto -\omega(u,v)^2\in \mathbb{G}_a$$
under the isomorphism $Y_\emptyset\simeq V^*/\{\pm 1\}$. Therefore, this is \emph{opposite} to the ``canonical'' isomorphism $Y_\emptyset\times Y_\emptyset\sslash G\to \mathbb{G}_a$ that was discussed in \S \ref{scattorus} --- we will need to take this difference into account. Notice that the choice of $\mathcal C$ seems quite arbitrary, and indeed, it is only justified because this turns out to give, over $t=1$, the coordinate that works for comparison to the Kuznetsov formula, for the representatives of $N\backslash G/N$ cosets that we have chosen. However, we could not have preserved the coordinate at $t=1$ and multiplied the one at $t=0$ by $(-1)$, if we want to have a coordinate that extends over the whole family.
We fix the coordinate $\xi(v,u)= \omega(v,u)$ for $V \times V\sslash G$, as in \S \ref{ssinvariant}. Then we have the following:
\begin{theorem} \label{torusdegen}
There is a unique $A_X=A_Y$-equivariant operator $\mathcal T_\emptyset$ making the following diagram commute:
$$ \xymatrix{
\mathcal S(X\times X/G) \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*}\ar[d]^{\mathcal T} && \mathcal S^+(X_\emptyset\times X_\emptyset/G) \ar[d]^{\mathcal T_\emptyset} \\
\mathcal S(Y\times Y/G) \ar[rr]^{e_\emptyset^*\otimes e_\emptyset^*} && \mathcal S^+(Y_\emptyset\times Y_\emptyset/G) }.
$$
The operator $\mathcal T_\emptyset$ is given by the multiplicative Fourier convolutions $\mathscr F_{\frac{\check\alpha}{2}, 1}\circ \mathscr F_{\frac{\check\alpha}{2}, 1}$ --- again, understood as the Fourier transform of a distribution.
\end{theorem}
\begin{proof}
The proof is the same as for Theorem \ref{groupdegen}. We only need to explain why, for the above choices of coordinates on $X_\emptyset\times X_\emptyset\sslash G$ and $Y_\emptyset\times Y_\emptyset\sslash G$, the operator $\mathcal T_\emptyset$ must act on Mellin transforms as follows:
$$\widecheck{(\mathcal T_\emptyset f)} (\chi) = \gamma(\chi,\frac{\check\alpha}{2}, 0,\psi)^2 \check f(\chi).$$
If we were following the arguments of Theorem \ref{groupdegen} using the coordinates for $X_\emptyset\times X_\emptyset\sslash G$ and $Y_\emptyset\times Y_\emptyset\sslash G$ that we used in \S \ref{scattorus}, we would, instead, have the factor
$$\gamma(\chi,\frac{\check\alpha}{2}, 0,\psi) \gamma(\chi,\frac{\check\alpha}{2}, 0,\psi^{-1})$$
instead (originating in Theorem \ref{thmpullbackfrombd}). Now, however, that we are using the \emph{negative} of this coordinate for $Y_\emptyset\times Y_\emptyset\sslash G$ (but not for $X_\emptyset\times X_\emptyset\sslash G$!), we have to mutiply this factor by $\chi(-1)$. This turns the factor $\gamma(\chi,\frac{\check\alpha}{2}, 0,\psi^{-1})$ to $\gamma(\chi,\frac{\check\alpha}{2}, 0,\psi)$.
\end{proof}
\bibliographystyle{alphaurl}
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Although October numbers actually were down slightly this year because of an historic rain event the first weekend in October, numbers year-to-date are up 6.64 percent, with just more than 13 million visitors through October this year.
"Lower gas prices, restoring services in some additional areas, and the National Park Service Centennial beginning to be on the radar are some of the possible reasons for the increase," Brandon said.
The Smokies, the parkway, and the Carl Sandburg Home Historic Site in Flat Rock, also part of the National Park Service, all have free entrances, unlike many national park sites in other parts of the country. But a recent study showed the parkway and the Smokies are two huge economic drivers in the region.
Economic drivers
The National Park Service report showed that almost 17 million visitors to national parks in North Carolina spent $1.1 billion in the state in 2014. That spending resulted in 18,528 jobs and had a cumulative benefit to the state economy of $1.5 billion.
Visitors spent more money while visiting the Blue Ridge Parkway last year than in any other national park.
The report showed the Blue Ridge Parkway, which runs north to Shenandoah National Park in Virginia, brought in the biggest bang for the buck.
In 2014, park visitors spent an estimated $863.5 million in local gateway regions while visiting the parkway. They supported 14,000 jobs, $408.8 million in labor income, $678.6 million in value added (a measure of contribution of National Park Service visitor spending to the Gross Domestic Product of a regional economy) for a total of $1.2 billion in economic output in communities surrounding the parkway.
The spending report also shows that the 10.1 million visitors to Great Smoky Mountains National Park last year spent almost $807 million in communities near the park. That spending supported 12,759 jobs in the local area.
Olga Pader, president of the Nantahala Hiking Club, believes that people are seeking the kind of adventure that offers a thrill and adrenaline rush, without too much danger.
"I think that in general we may be seeing an uptick in people going back into the woods. We are so stressed out by our electronic society. I think people are craving leaving something behind. The pull of solitude, the pull of nature, the pull of wilderness, is becoming more prevalent," she said.
She has hiked towering mountains such as the 14,500-foot Mount Whitney in California, and hiked 648 of the 900 miles of trails in the Smokies.
"I think another other thing that may be pulling people: In the front country, you see families and older people, but in the backcountry, it's more younger people. I think they're looking for adventure. Especially for girls and women, I think that 40 years ago, when I was growing up, women were socialized not to do something risky. To me, what's surprising is the number of women in the woods alone or with a buddy in the woods.
"They're finding that looking over the edge of a mountain is not something to be feared, but something to take you out of your boundaries. It's a rush, a thrill. You can get this natural rush, without drugs."
Thrive Q&A: Wrestling bears, running from elk
Food scarcity challenging Smokies bears
Great Smoky Mountain visitors through October 2015 by entrance:
Entrance, 2015 YTD, 2014 YTD, Percent change
Gatlinburg, Tennessee: 3,238,237 3,020,992 7.2 percent
Townsend, Tennessee: 1,395,409 1,295,609 7.7 percent
Oconaluftee, North Carolina: 1,956,434 1,670,900 17.1 percent
Outlying areas (Cataloochee, Abrams Creek, Big Creek, etc.): 2,864,620 2,964,851 -3.4 percent
Total areas October 2015: 1,370,640
Total October increase: 5.2 percent
Total areas 2015 YTD: 9,454,699
Total areas YTD increase: 5.6 percent | {
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{"url":"https:\/\/www.nag.co.uk\/numeric\/dt\/nagdotnet_dtw02\/html\/M_NagLibrary_G07_g07gb.htm","text":"g07gb returns a flag indicating whether a single data point is an outlier as defined by Peirce's criterion.\n\n# Syntax\n\nC#\n```public static bool g07gb(\nint n,\ndouble e,\ndouble var1,\ndouble var2,\nout double x,\nout double lx,\nout double ux,\nout int ifail\n)```\nVisual Basic\n```Public Shared Function g07gb ( _\nn As Integer, _\ne As Double, _\nvar1 As Double, _\nvar2 As Double, _\n<OutAttribute> ByRef x As Double, _\n<OutAttribute> ByRef lx As Double, _\n<OutAttribute> ByRef ux As Double, _\n<OutAttribute> ByRef ifail As Integer _\n) As Boolean```\nVisual C++\n```public:\nstatic bool g07gb(\nint n,\ndouble e,\ndouble var1,\ndouble var2,\n[OutAttribute] double% x,\n[OutAttribute] double% lx,\n[OutAttribute] double% ux,\n[OutAttribute] int% ifail\n)```\nF#\n```static member g07gb :\nn : int *\ne : float *\nvar1 : float *\nvar2 : float *\nx : float byref *\nlx : float byref *\nux : float byref *\nifail : int byref -> bool\n```\n\n#### Parameters\n\nn\nType: System..::..Int32\nOn entry: $n$, the number of observations.\nConstraint: ${\\mathbf{n}}\\ge 3$.\ne\nType: System..::..Double\nOn entry: $\\stackrel{~}{e}$, the value being tested.\nvar1\nType: System..::..Double\nOn entry: ${\\sigma }^{2}$, the residual variance on fitting model $M$\u00a0to $y$.\nConstraint: ${\\mathbf{var1}}>0.0$.\nvar2\nType: System..::..Double\nOn entry: ${\\stackrel{~}{\\sigma }}^{2}$, the residual variance on fitting model $M$\u00a0to $\\stackrel{~}{y}$.\nConstraints:\n\u2022 ${\\mathbf{var2}}>0.0$;\n\u2022 ${\\mathbf{var2}}<{\\mathbf{var1}}$.\nx\nType: System..::..Double%\nOn exit: an estimated value of $x$, the cutoff that indicates an outlier.\nlx\nType: System..::..Double%\nOn exit: $l$, the lower limit for $x$.\nux\nType: System..::..Double%\nOn exit: $u$, the upper limit for $x$.\nifail\nType: System..::..Int32%\nOn exit: ${\\mathbf{ifail}}={0}$\u00a0unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).\n\n#### Return Value\n\ng07gb returns a flag indicating whether a single data point is an outlier as defined by Peirce's criterion.\n\n# Description\n\ng07gb tests a potential outlying value using Peirce's criterion. Let\n\u2022 $e$\u00a0denote a vector of $n$\u00a0residuals with mean zero and variance ${\\sigma }^{2}$\u00a0obtained from fitting some model $M$\u00a0to a series of data $y$,\n\u2022 $\\stackrel{~}{e}$\u00a0denote the largest absolute residual in $e$, i.e., $\\left|\\stackrel{~}{e}\\right|\\ge \\left|{e}_{i}\\right|$\u00a0for all $i$, and let $\\stackrel{~}{y}$\u00a0denote the data series $y$\u00a0with the observation corresponding to $\\stackrel{~}{e}$\u00a0having been omitted,\n\u2022 ${\\stackrel{~}{\\sigma }}^{2}$\u00a0denote the residual variance on fitting model $M$\u00a0to $\\stackrel{~}{y}$,\n\u2022 $\\lambda$\u00a0denote the ratio of $\\stackrel{~}{\\sigma }$\u00a0and $\\sigma$\u00a0with $\\lambda =\\frac{\\stackrel{~}{\\sigma }}{\\sigma }$.\nPeirce's method flags $\\stackrel{~}{e}$\u00a0as a potential outlier if $\\left|\\stackrel{~}{e}\\right|\\ge x$, where $x={\\sigma }^{2}z$\u00a0and $z$\u00a0is obtained from the solution of\n $R=\u03bb1-nn-1n-1nn$ (1)\nwhere\n $R=2expz2-121-\u03a6z$ (2)\nand $\\Phi$\u00a0is the cumulative distribution function for the standard Normal distribution.\nUnlike g07ga, both ${\\sigma }^{2}$\u00a0and ${\\stackrel{~}{\\sigma }}^{2}$\u00a0must be supplied and therefore no assumptions are made about the nature of the relationship between these two quantities. Only a single potential outlier is tested for at a time.\nThis method uses an algorithm described in e04ab to refine a lower, $l$, and upper, $u$, limit for $x$. This refinement stops when $\\left|\\stackrel{~}{e}\\right|\u00a0or $\\left|\\stackrel{~}{e}\\right|>u$.\n\n# References\n\nGould B A (1855) On Peirce's criterion for the rejection of doubtful observations, with tables for facilitating its application The Astronomical Journal 45\nPeirce B (1852) Criterion for the rejection of doubtful observations The Astronomical Journal 45\n\n# Error Indicators and Warnings\n\nErrors or warnings detected by the method:\n${\\mathbf{ifail}}=1$\nOn entry, ${\\mathbf{n}}<3$.\n${\\mathbf{ifail}}=3$\nOn entry, ${\\mathbf{var1}}\\le 0.0$.\n${\\mathbf{ifail}}=4$\nOn entry, ${\\mathbf{var2}}\\le 0.0$\u00a0or ${\\mathbf{var2}}\\ge {\\mathbf{var1}}$.\n${\\mathbf{ifail}}=-9000$\nAn error occured, see message report.\n\nNot applicable.\n\nNone.\n\nNone.\n\n# Example\n\nThis example reads in a series of values and variances and checks whether each is a potential outlier.\nThe dataset used is from Peirce's original paper and consists of fifteen observations on the vertical semidiameter of Venus. Each subsequent line in the dataset, after the first, is the result of dropping the observation with the highest absolute value from the previous data and recalculating the variance.\n\nExample program (C#): g07gbe.cs\n\nExample program data: g07gbe.d\n\nExample program results: g07gbe.r","date":"2017-11-20 17:37:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 59, \"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\": 0.9729416966438293, \"perplexity\": 2828.962765988414}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806086.13\/warc\/CC-MAIN-20171120164823-20171120184823-00295.warc.gz\"}"} | null | null |
package io.codearte.jfairy;
import io.codearte.jfairy.data.DataMaster;
import io.codearte.jfairy.producer.VATIdentificationNumberProvider;
import io.codearte.jfairy.producer.company.locale.en.EmployerIdentificationNumberProvider;
import io.codearte.jfairy.producer.person.NationalIdentityCardNumberProvider;
import io.codearte.jfairy.producer.person.PassportNumberProvider;
import io.codearte.jfairy.producer.person.locale.en.EnPassportNumberProvider;
import io.codearte.jfairy.producer.person.locale.en.SocialSecurityCardNumberProvider;
import io.codearte.jfairy.producer.util.CharConverter;
import io.codearte.jfairy.producer.util.locale.NonOpCharConverter;
import java.util.Random;
/**
* @author Olga Maciaszek-Sharma
* @since 08.03.15
*/
public class EnFairyModule extends FairyModule {
public EnFairyModule(DataMaster dataMaster, Random random) {
super(dataMaster, random);
}
@Override
protected void configure() {
super.configure();
bind(NationalIdentityCardNumberProvider.class).to(SocialSecurityCardNumberProvider.class);
bind(VATIdentificationNumberProvider.class).to(EmployerIdentificationNumberProvider.class);
bind(PassportNumberProvider.class).to(EnPassportNumberProvider.class);
bind(CharConverter.class).to(NonOpCharConverter.class);
}
}
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Dave Farrish (né le à Wingham, dans la province de l'Ontario au Canada) est un joueur professionnel de hockey sur glace.
Carrière de joueur
Il a été repêché en , au total par les Rangers de New York au repêchage amateur de 1976.
Statistiques
Pour les significations des abréviations, voir statistiques du hockey sur glace.
| 1973-1974|| Wolves de Sudbury || LHO ||58 ||11 ||20 ||31 ||205 ||- ||- ||- ||- ||-
|-
| 1974-1975|| Wolves de Sudbury || LHO ||60 ||20 ||44 ||64 ||258 ||- ||- ||- ||- ||-
|-
| 1975-1976|| Wolves de Sudbury || LHO ||66 ||27 ||48 ||75 ||155 ||- ||- ||- ||- ||-
|-
| 1976-1977|| Rangers de New York || LNH ||80 ||2 ||17 ||19 ||102 ||- ||- ||- ||- ||-
|-
| 1977-1978|| Nighthawks de New Haven|| LAH ||10 ||0 ||3 ||3 ||4 ||- ||- ||- ||- ||-
|-
| 1977-1978|| Rangers de New York || LNH ||66 ||3 ||5 ||8 ||62 ||3 ||0 ||0 ||0 ||0
|-
| 1978-1979|| Rangers de New York || LNH ||71 ||1 ||19 ||20 ||61 ||7 ||0 ||2 ||2 ||14
|-
| rowspan="2"|1979-1980|| Hawks du Nouveau-Brunswick || LAH ||20 ||3 ||1 ||4 ||22 ||- ||- ||- ||- ||-
|-
|| Firebirds de Syracuse || LAH ||14 ||4 ||10 ||14 ||17 ||- ||- ||- ||- ||-
|-
| rowspan="2"|1979-1980|| Nordiques de Québec || LNH ||4 ||0 ||0 ||0 ||0 ||- ||- ||- ||- ||-
|-
|| Maple Leafs de Toronto || LNH ||20 ||1 ||8 ||9 ||30 ||3 ||0 ||0 ||0 ||10
|-
| 1980-1981|| Maple Leafs de Toronto || LNH ||74 ||2 ||18 ||20 ||90 ||1 ||0 ||0 ||0 ||0
|-
| 1981-1982|| Hawks du Nouveau-Brunswick || LAH ||67 ||13 ||24 ||37 ||80 ||15 ||4 ||5 ||9 ||20
|-
| 1982-1983|| Saints de St. Catharines || LAH ||14 ||2 ||12 ||14 ||18 ||- ||- ||- ||- ||-
|-
| 1982-1983|| Maple Leafs de Toronto || LNH ||56 ||4 ||24 ||28 ||38 ||- ||- ||- ||- ||-
|-
| 1983-1984|| Saints de St. Catharines || LAH ||4 ||0 ||2 ||2 ||6 ||7 ||0 ||1 ||1 ||4
|-
| 1983-1984|| Maple Leafs de Toronto || LNH ||59 ||4 ||19 ||23 ||57 ||- ||- ||- ||- ||-
|-
| 1984-1985|| Saints de St. Catharines || LAH ||68 ||4 ||12 ||16 ||56 ||- ||- ||- ||- ||-
|-
| 1985-1986|| Bears de Hershey|| LAH ||74 ||5 ||17 ||22 ||78 ||18 ||0 ||4 ||4 ||24
|-
| 1986-1987|| SC Riessersee|| 2. Bundesliga ||26 ||7 ||10 ||17 ||64 ||- ||- ||- ||- ||-
|-
| 1987-1988|| Nighthawks de New Haven || LAH ||30 ||4 ||14 ||18 ||26 ||- ||- ||- ||- ||-
|-
| 1987-1988 || EC VSV || EBEL ||33 ||8 ||43 ||51 ||33 ||- ||- ||- || -|| -
|-
| 1988-1989|| Skipjacks de Baltimore || LAH ||60 ||2 ||13 ||15 ||62 ||- ||- ||- ||- ||-
|-
| 1989-1990|| Hawks de Moncton || LAH ||1 ||0 ||0 ||0 ||2 ||- ||- ||- ||- ||-
|430 ||17 ||110 ||127 ||440 ||14 ||0 ||2 ||2 ||24
Références
Naissance en août 1956
Joueur canadien de hockey sur glace
Choix de repêchage des Rangers de New York
Choix de repêchage des Mariners de San Diego
Joueur des Wolves de Sudbury
Joueur des Rangers de New York
Joueur des Nordiques de Québec
Joueur des Maple Leafs de Toronto
Joueur des Nighthawks de New Haven
Joueur des Hawks du Nouveau-Brunswick
Joueur des Firebirds de Syracuse
Joueur des Saints de Saint Catharines
Joueur des Bears de Hershey
Joueur des Skipjacks de Baltimore
Joueur des Hawks de Moncton
Champion de la Coupe Calder
Champion de la Coupe Stanley
Défenseur de hockey sur glace
Gagnant du trophée Eddie-Shore | {
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El propósito de este libro es educar, por tanto no se han escatimado esfuerzos para darle la mayor precisión posible. Esta es una revisión de la evidencia científica que se presenta para propósitos informativos.
Ninguna persona debe usar la información contenida en esta obra con el fin de autodiagnosticarse, tratarse o justificarse para aceptar o rechazar cualquier terapia médica por problemas de salud o enfermedad. No se quiere instar a nadie a no buscar asesoría y tratamiento médico profesional, este libro no brinda asesoría médica alguna.
Cualquier aplicación de la información aquí contenida es a la sola discreción y riesgo del lector. Por lo tanto, cualquier persona con algún problema de salud específico o que esté tomando medicamentos debe primero buscar asesoría de su médico o proveedor de asistencia sanitaria antes de comenzar algún programa alimenticio. El autor y Grupo Nelson, Inc., no tendrán obligación ni responsabilidad alguna hacia cualquier persona o entidad con respecto a pérdida, daño o lesión causados o que se alegue que han sido causados directa o indirectamente por la información contenida en este libro. No asumimos responsabilidad alguna por los errores, inexactitudes, omisiones o cualquier incongruencia aquí contenida.
En vista de la naturaleza compleja e individual de los problemas de la salud y del buen estado físico este libro, así como las ideas, los programas, los procedimientos y las sugerencias aquí contenidos no pretenden reemplazar el consejo de profesionales médicos capacitados. Todos los aspectos con respecto a la salud de una persona requieren supervisión médica. Se debe consultar a un médico antes de adoptar cualquiera de los programas aquí descritos. El autor y la editorial niegan cualquier responsabilidad que surja, directa o indirectamente del uso de esta obra.
© 2006 por Grupo Nelson
Una división de Thomas Nelson, Inc.
Nashville, Tennessee, Estados Unidos de América
www.gruponelson.com
Título en inglés: _The Great Physician's Rx for a Healthy Heart_
© 2006 por Jordan Rubin y Joseph Brasco
Publicado por Nelson Books
Una división de Thomas Nelson, Inc.
Traducción: _Rolando Cartaya_
Diseño interior: _Grupo Nivel Uno, Inc._
ISBN-10: 0-88113-200-4
ISBN-13: 978-0-88113-200-7
Reservados todos los derechos. Ninguna parte del presente libro puede reproducirse, guardarse en sistema de almacenamiento alguno, ni transmitirse en cualquier forma ni por cualquier medio, sea electrónico, mecánico, fotocopiado, grabado, escaneado o cualquier otro, a excepción de citas breves en revisiones o artículos de crítica, sin el permiso escrito por anticipado de parte de la casa editorial.
Impreso en Estados Unidos de América
_A abuelito Al y abuelo Jerry, que nos dejaron demasiado pronto a mi familia y a mí, al sufrir mortales infartos cardiacos a los cincuenta y cinco y sesenta y dos años, respectivamente. Los tiempos que pasamos juntos constituyen algunos de los mejores recuerdos de mi infancia._
CONTENIDO
_Introducción:_ Asuntos del corazón
Llave # 1: Coma para vivir
Llave # 2: Complemente su dieta con alimentos integrales, nutrientes vivos y superalimentos
Llave # 3: Practique una higiene avanzada
Llave # 4: Acondicione su cuerpo con ejercicios y terapias corporales
Llave # 5: Reduzca las toxinas en su ambiente
Llave # 6: Evite las emociones mortales
Llave # 7: Viva una vida de oración y con propósito
Plan de batalla de _La receta del Gran Médico para_ _un corazón saludable_
_Notas_
_Acerca de los autores_
[INTRODUCCIÓN
_Asuntos del corazón_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch0037)
Mi hijo, Joshua, acaba de cumplir dos años y lo encuentro, en su simpático estilo de chiquilín, divertido y animado, entusiasta y emotivo, y en ocasiones, desafiante y desobediente.
Todo eso le hace un párvulo perfectamente normal, ¿cierto?
Joshua es nuestro único hijo. Aunque mi esposa, Nicki, y yo estamos seguros de ser los padres del niño más especial del planeta, ahora tenemos una mejor comprensión de lo que quería decir el comediante Bill Cosby cuando sentenció: «Denme doscientos chiquillos activos de dos años y conquistaré el mundo». Creo que Joshua podría encabezar la ofensiva de ese ejército, aunque los «dos años terribles» no lo han sido tanto para Nicki y para mí. Y eso se debe a que tenemos una mejor idea de cómo manejar al pequeñín después de haber leído el libro fundamental del doctor James Dobson sobre cómo disciplinar a los hijos, _Atrévete a disciplinar._
Como millones de padres jóvenes, hemos descubierto de repente que los niños crecen con voluntades propias, lo que quiere decir que necesitamos un plan para criar a un hijo sano, respetuoso y feliz.
Después de leer el libro del doctor Dobson, comprendemos mejor cómo enseñarle a Joshua a distinguir el bien del mal, así como el arte del autocontrol.
Mientras nuestro enérgico hijo atraviesa corriendo desaforadamente su primera infancia, Nicki y yo ponemos en práctica de manera minuciosa los conceptos de una disciplina razonable y coherente articulados por Dobson en su obra.
La razón por la que le cuento esta historia es que _Atrévete a disciplinar_ , edición revisada y reescrita del éxito de librería de 1970 _Atrévete a disciplinar,_ estuvo cerca de no publicarse nunca. El doctor Dobson sufrió un grave infarto cardiaco el 15 de agosto de 1990, unos meses antes de la fecha programada para que empezara a revisar la obra original.
Un buen amigo mío que trabajó por mucho tiempo con Enfoque a la Familia, la organización fundada por Dobson a fines de los años setenta, me contó que aquella mañana el doctor se levantó temprano para jugar una «guerrilla» de baloncesto en su congregación, la Primera Iglesia del Nazareno de Pasadena, en el sur de California, cuyo pastor era su primo, H.B. London.
La iglesia poseía un buen gimnasio, y al doctor Dobson le encantaba el espíritu competitivo y la camaradería de esos partidos de tres contra tres en media cancha. Alrededor de una docena de amigos y empleados de Enfoque a la Familia solían acompañarle para esas contiendas al rayar el alba.
Aquel día estival de 1990 el doctor Dobson tenía cincuenta y cuatro años, y cualquiera que le observara encestando canastas tres veces a la semana le habría encontrado en magnífica forma física para su edad.
Sin embargo, no exhibía aquella mañana su mejor juego. En el argot del básquetbol, estaba lanzando pedradas contra el tablero de vidrio y permitiendo que los demás jugadores le pasaran impunes por el lado, como si sus pies estuvieran clavados al pulido tabloncillo de dura madera.
Después de perder un rebote fácil debajo del aro, Dobson sintió un agudo dolor en el centro del pecho. Mientras trataba de recobrar el aliento se dio cuenta instantáneamente de que algo andaba mal. Quizás recordara que este era el mismo tabloncillo donde dos años antes había sostenido en sus brazos, moribundo, a una leyenda del baloncesto, Pete Maravich, de los Pistols, cuando se derrumbó, mortalmente herido por un infarto cardiaco.
Al percatarse de que el dolor en el pecho no cedía, Dobson recogió las llaves de su automóvil y se despidió: «Lo siento, muchachos, tengo que irme», dijo mientras salía.
No era propio de él abandonar un partido antes de las ocho de la mañana.
Uno de los jugadores corrió tras él y le preguntó si se sentía bien. «Eso creo», dijo, pero algo en su interior le decía que no lo estaba al ciento por ciento.
En lugar de irse a duchar a su casa, manejó hasta un hospital cercano, el St. Luke's de Pasadena, donde estacionó el auto y puso en orden sus pensamientos. Entrar a una sala de emergencias y anunciar que estaba sintiendo dolores en el pecho abriría una sensible brecha en su superocupado calendario: reuniones con el personal ejecutivo, grabaciones para transmitir, responder a las decenas de mensajes telefónicos recibidos por sus ayudantes... El doctor se enorgullecía de estar al día con su «pila», una montaña de memoranda y correspondencia de no menos de treinta centímetros de alto, que le llegaban en aquellos días anteriores al correo electrónico. También entendía —desde sus días en la facultad de medicina de la Universidad del Sur de California— que entrar al hospital y decir: «Creo que tengo un ataque cardiaco» le comprometería a tres días de exámenes y observación médica.
El doctor Dobson permaneció sentado en su automóvil casi treinta minutos, ponderando las consecuencias de cruzar las puertas de la sala de emergencias. «Señor ¿qué quieres que haga?», oró. «Tengo cincuenta y cuatro años y un dolor en el pecho».
El doctor Dobson hizo bien al permitir que le internaran aquella mañana en St. Luke's, pues los exámenes revelaron que había sufrido un infarto cardiaco de leve a moderado. Aparentemente, una de las cinco arterias coronarias que conducen la sangre al corazón se le bloqueó. Gracias a una rápida intervención, recibió la atención médica que necesitaba para no morir. En muchas formas fue un hombre afortunado: su cardiólogo determinó que varias arterias colaterales compensaron el déficit de la arteria bloqueada, lo cual impidió que sufriera daños permanentes o la muerte.
Esas arterias colaterales se desarrollaron gracias a los vigorosos partidos de baloncesto que el doctor había jugado a través de los años. No obstante, fue como un llamado de alerta para él, que dispuso de bastante tiempo para reflexionar durante su permanencia de diez días en el hospital. Y sin duda pensó cuán cerca había estado de morir como su padre, James Dobson, que sucumbiera a un infarto trece años antes, en diciembre de 1977, sólo setenta y un días después de sobrevivir al primero.
Cuando estaba en una transmisión, y también en sus escritos, el doctor Dobson siempre decía que su padre había sido su mejor amigo, su más confiable consejero y la persona que había guiado a su hijo a optar por sus valores, sus sueños y su Dios.
¿Estaba también él destinado a morir joven? Para el doctor Dobson, un infarto cardiaco a los cincuenta y cuatro años fue más que el proverbial llamado de alerta: motivó al fundador de Enfoque a la Familia a operar importantes cambios en su estilo de vida, especialmente en su dieta.
CAMBIOS RADICALES
De joven, James Dobson fue criado conforme a la dieta estándar estadounidense de los años cincuenta: «Carne con papas y pásame el pan».
Nacido en Shreveport, Louisiana, su familia se mudó varias veces hasta que se estableció en San Benito, Texas, durante los años de James en la escuela secundaria.
Él se acostumbró a paladear lo que se considera una deliciosa comida en el sur de los Estados Unidos: pollo frito, puré de papas anegado en el jugo de la carne, bistecs empanizados al estilo campesino y pastel de manzanas con helado. Después de cursar la universidad en el área del sur de California, se casó con Shirley y concluyó su doctorado en la Universidad del Sur de California.
Allí se creó una rutina: su desayuno eran dos rosquillas y dos tazas de café; el almuerzo, una hamburguesa con chile y papas fritas; y la cena, cualquier cosa. Y siempre quedaba espacio para el postre en la heladería Baskin Robbins, donde pedía noche tras noche un _sundae_ coronado con chocolate caliente o un helado de caramelo.
En esa época, el doctor Dobson desarrolló una predilección por Tommy's, una hamburguesería de tejado rojo y en forma de A, famosa por una especie de catedral de grasa de quince centímetros de alto conocida como el Chili Burger: dos porciones de carne molida, rematadas con salsa de chiles, cebollas frescas, una rebanada de tomate y queso americano doble grueso. Un solo Chili Burger de Tommy's contenía cuatrocientos noventa calorías y veintidós gramos de grasa, pero en aquella época ¿a quien le importaban los datos nutricionales? Después de su ataque cardiaco, sin embargo, el doctor Dobson renunció a sus escapadas a Tommy's en el horario de almuerzo. Y también a las deliciosas comidas fritas culminadas con empalagosos postres. En realidad, cambió por completo su dieta: a partir de entonces sólo comía pollo o pescado a la parrilla (cero carnes rojas), ensaladas de hortalizas frescas, muchas frutas y vegetales y ningún alimento frito, ni pan blanco, ni nada que contuviera azúcar, mantequilla o crema agria. Durante sus transmisiones radiales bromeaba diciendo que estaba condenado «a comer alpiste» por el resto de su vida.
Otra cosa que hizo fue incrementar un poco más su nivel de ejercicios físicos. Como se consideró que los rápidos movimientos del baloncesto podían ser demasiado extenuantes para su corazón, instaló en su casa una caminadora fija para estirar las piernas durante una hora mientras veía el noticiero de la mañana, antes de partir hacia el campus de Enfoque a la Familia. Tomó tan en serio el ejercicio, que durante años no dejó de hacerlo un día.
Desde que sufriera su infarto en 1990, el doctor Dobson ha hecho mucho más que revisar _Atrévete a disciplinar_. En los últimos dieciséis años ha grabado probablemente mil quinientos programas de radio, escrito o revisado una docena de libros, aparecido veintenas de veces en programas como _Larry King Live_ y _Hannity & Colmes_, y disertado sobre temas familiares ante la clase política en Washington, DC. Y todo porque sobrevivió a un feo encuentro con el «asesino silencioso», un cese del flujo de sangre al corazón. Sabía que podía considerarse afortunado por estar vivo. En su boletín mensual, declaró que «Cerca de un millón de oraciones salvaron mi vida».
Hablando en términos estadísticos, el doctor Dobson precisó una intervención divina para sobrevivir a su ataque cardiaco. De los alrededor de setecientos mil estadounidenses que sufren un repentino ataque coronario cada año, cerca de ciento ochenta mil no sobreviven.1 Los infartos del miocardio y otras enfermedades cardiovasculares (ECV) son la causa número uno de muerte en este país, según la más reciente compilación de «Estadísticas de Enfermedades Cardiacas y Accidentes Cerebrovasculares» de la American Heart Association (Asociación americana del corazón).
Un total de novecientos diez mil seiscientos catorce americanos fallecieron a consecuencia de diversas enfermedades cardiovasculares en 2003, el año más reciente sobre cuál existían datos en el momento de imprimirse este libro.3 Las enfermedades cardiovasculares incluyen a la hipertensión arterial, problemas coronarios (infartos y angina), fallo cardiaco congestivo, accidentes cerebrovasculares y defectos congénitos del corazón.
La percepción pública usual acerca de un ataque al corazón es: _¡Boom! ¡Se fue!_ Tal vez esto provenga de haber visto demasiadas veces _El Padrino_ , con Don Corleone de rodillas ante sus plantas de tomate para encontrarse con su Hacedor y escapar de las represalias de otros jefes del crimen organizado en Nueva York. Esto se ajusta al estilo de Hollywood, donde los protagonistas que sufren un infarto cardiaco se llevan las manos al pecho, viran los ojos en blanco, se deslizan a la inconsciencia y exhalan su último suspiro segundos después de caer al suelo.
Si ha notado que me estoy refiriendo a hombres, permítame hacerle esta pregunta: ¿Cuándo fue la última vez que vio a una dama sufrir un dramático ataque cardiaco en la pantalla de un cine o en un canal para ellas? Nunca se ve a mujeres morir de un infarto en Hollywood, pese a que representan casi la mitad de los decesos por ataques cardiacos, y una tercera parte de todas las mujeres mueren a consecuencia de algún tipo de dolencia cardiovascular.
Sea usted hombre o mujer, lo cierto es que dos terceras partes de las víctimas de problemas cardiacos no se arrodillan y mueren como en las películas. Con harta frecuencia, un infarto cardiaco mortal es precedido durante meses, días y horas por episodios de dolor en el pecho, falta de aire y una sensación de hormigueo en el brazo izquierdo. Otras personas, como el doctor Dobson, experimentan un agudo dolor en el pecho, o náuseas, un dolor que se irradia al cuello y los brazos, y frialdad en la piel.
La terminología médica que se aplica a un ataque al corazón es infarto al miocardio. Quizás suene a vocabulario de comediante, pero no hay nada simpático en un infarto al miocardio, pues los médicos estiman que existe una ventana de supervivencia de sólo cinco minutos para llamar a la ambulancia y que los paramédicos se presenten en el lugar.
Esta frase polisilábica se deriva de tres raíces latinas: _myo_ , que significa músculo; _cardio_ , que quiere decir corazón; e _infarctus,_ que se refiere a la muerte de tejidos debido a la falta de suministro de sangre.
El corazón es probablemente el órgano más asombroso que Dios ha creado. Late a un ritmo estable de sesenta, setenta o más veces por minuto —más de cien mil veces al día— y envía un flujo continuo de sangre a los músculos y órganos del cuerpo sin perder un latido. Este músculo del tamaño de un puño es en realidad dos bombas en una: el lado derecho recibe la sangre del cuerpo y la bombea hacia los pulmones; en tanto que el lado izquierdo recibe la sangre de los pulmones y bombea estos nutrientes ricos en oxígeno hasta los más lejanos rincones del organismo veinticuatro horas al día y siete días a la semana.
Los problemas cardiovasculares empiezan a manifestarse cuando las arterias coronarias que transportan la sangre desde el corazón se obstruyen con materia grasa o placa. Los depósitos de placa, que son duros por fuera pero blandos por dentro, atraen a los componentes de la sangre, que se adhieren al revestimiento de la pared de las arterias. Esta acumulación de grasa puede desprenderse y formar un coágulo sanguíneo, reduciendo o bloqueando el flujo de la sangre. Cuando ocurre una pequeña reducción se experimenta dolor en el pecho, también conocido como _angina pectoris_. Pero cuando un coágulo interrumpe repentinamente todo el suministro de sangre o la mayor parte de este, el músculo cardiaco no recibe oxígeno. En pocos minutos sus células empiezan a morir, lo cual marca el inicio oficial de un infarto al miocardio, o lo que se conoce popularmente como ataque al corazón.
A veces la muerte es instantánea, como le sucedió a Pete Maravich; otras veces se sobrevive, como el final feliz en el caso del doctor Dobson. La magnitud del daño sufrido por el corazón depende del área que dependía de la arteria bloqueada, y del tiempo transcurrido desde que aparecen los síntomas hasta que se recibe atención médica.
TRATAMIENTO CONVENCIONAL
Cuando los paramédicos responden a una llamada de emergencia saben que están corriendo contra el reloj. Si su corazón deja de bombear, y usted de respirar, en cuestión de cuatro a seis minutos se producen daños cerebrales irreversibles. Los paramédicos suelen emplear un defibrilador para resucitar a las víctimas de infartos cardiacos. Acercan un par de paletas al área del corazón, mediante las cuales envían choques eléctricos, tratando de estimular al corazón para que vuelva a latir.
Una vez que le han estabilizado, le conducen al hospital, donde los médicos de la sala de emergencias entran inmediatamente en acción. Le administran un fármaco tras otro para disolver los coágulos sanguíneos y con frecuencia realizan una angioplastia urgente, un procedimiento en el cual introducen un catéter dentro de las arterias bloqueadas y usan un pequeño balón inflable para despejar el bloqueo. Mientras más rápido puedan los médicos ejecutar esta operación de desatascamiento, mayores serán las probabilidades de sobrevivir del paciente.
El proceso entero, conocido como reperfusión, puede hacerse en menos de una hora, y a veces hasta en treinta minutos. Pero digamos que usted experimenta, como el doctor Dobson, los síntomas clásicos de un ataque al corazón: dolor pectoral, falta de aire, una presión prolongada en el centro del pecho u hormigueo en los brazos. Se siente lo bastante mal como para llamar a la ambulancia, o a su médico de familia.
Como le sucedió al doctor Dobson, una vez que usted le menciona a un profesional de la salud que tiene dolor en el pecho, realizará una visita no programada a la sala de emergencias de un hospital, donde los médicos le someterán a una serie de exámenes, entre ellos, los siguientes:
• Un electrocardiograma, o EKG, que verifica el ritmo cardiaco y puede localizar el área posible del infarto.
• Varios análisis de sangre, incluyendo la búsqueda en esta de sustancias llamadas biomarcadores, que sirven a los médicos para determinar si han sido dañadas células del corazón.
• Pruebas de estrés, que localizan las obstrucciones en los vasos sanguíneos.
• Tomografías computarizadas (TC o TAC), procedimientos diagnósticos mediante imágenes que utilizan rayos X y tecnología de computadoras para permitir a los médicos observar las áreas dañadas del corazón, así como detectar problemas en la capacidad de este para bombear sangre al organismo.
Una vez realizadas esas pruebas, los galenos revisan los resultados. Pueden optar por prescribir ciertos medicamentos para disolver o prevenir los coágulos de sangre y evitar que las plaquetas se aglutinen y se adhieran a la placa. O pueden decidir que usted es un seguro candidato a la mencionada angioplastia o cirugía de desvío de las arterias coronarias, la operación del corazón que más se realiza en todo el mundo, con más de trescientos mil procedimientos realizados solamente en los Estados Unidos cada año.
En este procedimiento, los cirujanos utilizan un segmento de vena (tomado generalmente de una pierna del paciente) o algún material sintético similar al tejido de las arterias para conectar la aorta, en un extremo, con la arteria coronaria más allá del área de la obstrucción. En Estados Unidos, estas operaciones suelen costar más de veinte mil dólares.
Los críticos arguyen que las cirugías de desvío coronario consumen más dinero de los fondos dedicados a la medicina que cualquier otro tratamiento o procedimiento.
La operación de desvío de las arterias coronarias es «puntera en materia de equipo y personal, espacios de hospital e ingresos totales asociados», señala el doctor Thomas A. Preston, jefe del. departamento de cardiología de Pacific Medical Center en Seattle, estado de Washington. «La operación es anunciada por la prensa popular, magnificada por la profesión médica y procurada activamente por el público consumidor».
No obstante, no todos los que padecen enfermedades cardiovasculares están destinados al bisturí. El manual convencional para combatir las enfermedades del corazón sin cirugía invasiva comprende el tratamiento de la hipertensión arterial y fármacos para la reducción del colesterol.
A los sobrevivientes de infartos al miocardio se les prescribe de manera rutinaria tomar aspirina, lo cual reduce la tendencia de las plaquetas en la sangre a aglutinarse y formar coágulos. Otras «terapias farmacológicas» incluyen la nitroglicerina, relajadora de la pared muscular de los vasos sanguíneos; inhibidores de la enzima de conversión de la angiotensina o ACE, como el Vasotec, Zestril y Prinvil, que reducen la carga de estrés sobre el corazón; y agentes betabloqueadores que aminoran el riesgo de futuros ataques cardiacos. Una vez que usted ha recuperado su fuerza —y los médicos confían en que ya no es candidato a un nuevo infarto— le envían a su casa con instrucciones de tomarlo tranquilo, lo que siempre se dice más fácilmente de lo que se hace.
Las operaciones de desvío coronario, los procedimientos de angioplastia y los fármacos para reducir el colesterol se han convertido en importantes industrias de la salud en Estados Unidos desde que el país está a la cabeza del mundo en los índices de mortalidad por enfermedades cardiovasculares. Se estima que actualmente más de sesenta millones de estadounidenses —alrededor de la cuarta parte de la población adulta— padecen algún tipo de enfermedad cardiovascular. El doctor Preston está en lo cierto: la cuenta médica total por tratamiento de enfermedades cardiovasculares y accidentes cerebrovasculares en 2005 fue de trescientos noventa y cinco millones de dólares, según cifras de la American Heart Association y el Instituto Nacional para el Corazón, los Pulmones y la Sangre. Debe aclararse siempre que el costo de la atención médica, los servicios de hospitales y hogares de ancianos, las medicinas y otros bienes y servicios médicos no es comparable con el costo en sufrimiento humano y vidas perdidas infligido por las enfermedades del corazón a familias e individuos.
TRATAMIENTOS ALTERNATIVOS
Una de las alternativas más populares a la angioplastia y la cirugía coronaria convencionales es la quelación, en la cual la sustancia química orgánica ácido etilenediaminatetraacético (EDTA) se introduce en el cuerpo en forma intravenosa o de suplemento para eliminar los depósitos de placa y calcio de las paredes arteriales. El material nocivo es luego excretado con la orina.
La principal organización que promueve la terapia de quelación es el Colegio Estadounidense para el Progreso de la Medicina (ACAM), que afirma que es eficaz contra la ateroesclerosis y las enfermedades coronarias y vasculares periféricas.
Mi amigo el doctor Garry Gordon, que tiene una especialización en Osteopatía y es considerado el padre de la terapia de quelación moderna, afirma: «Basta con ver un grupo de mis pacientes en cualquier fase de la enfermedad, y comprobar cuántos de ellos están vivos cinco años después, comparados con otros que se han sometido a terapias convencionales. Me costaba trabajo creer que importantes cardiólogos, en eventos médicos en los que he participado, consideran el diagnóstico de fallo cardiaco congestivo una virtual sentencia de muerte, pues más del sesenta por ciento de sus pacientes mueren en el primer año después del diagnóstico ¡Yo no he perdido un paciente de fallo cardiaco congestivo en diez años! Me asombra ver cuánta diferencia existe, dependiendo de la escuela médica a que uno se adscriba».
La American Heart Association dice, sin embargo, no haber encontrado evidencias científicas que demuestren los beneficios de esta variante terapéutica. «Hasta ahora, no hemos visto estudios científicos adecuados, controlados ni publicados que utilicen una metodología científica aprobada para respaldar el uso de esta terapia contra las enfermedades cardiovasculares», expresa una nota de prensa de la asociación. «Concuerdan en este punto con la American Heart Association, la Administración de Alimentos y Fármacos (FDA), los Institutos Nacionales de la Salud (NIH) y el American College of Cardiology».
Otro tratamiento alternativo para las enfermedades cardiovasculares es la terapia de oxígeno hiperbárica (HBOT), que algunos en la profesión médica estiman que merece consideración. Sus promotores sostienen que colocar a los pacientes en una cámara hiperbárica reduce a la mitad el tiempo que requiere el corazón para reanudar su actividad eléctrica normal y elevar la «facción de eyección», la capacidad del corazón para bombear sangre. En la terapia hiperbárica, el paciente se acuesta sobre una camilla acolchada que se desliza dentro de un tubo plástico transparente de unos dos metros de largo. La cámara es gradualmente presurizada con oxígeno puro hasta que alcanza el doble de la presión atmosférica normal. Los pacientes, a quienes se les recuerda que deben relajarse y respirar normalmente durante el tratamiento, pueden experimentar dolor en los oídos o un leve malestar, lo cual desaparece generalmente reduciendo un poco la presión.
La terapia de oxígeno hiperbárica todavía no figura en el equipo reglamentario de las salas de emergencias debido al costo de cada cámara (más de cien mil dólares) y la escasa confianza de la comunidad médica en cuanto a su eficacia. La Administración de Alimentos y Fármacos solamente ha aprobado trece posibles usos de la terapia de oxígeno hiperbárica, y las enfermedades cardiovasculares e infartos al miocardio no figuran en la lista.
A DONDE VAMOS DESDE AQUÍ
¿Incrementan o reducen el riesgo de enfermedades coronarias e infarto los factores relacionados con el estilo de vida? La American Heart Association afirma que su significación y preponderancia no han sido determinadas con exactitud, lo cual estoy dispuesto a aceptar, pero estoy seguro de que muy pocos en la comunidad médica discreparían de la siguiente máxima: _Cuide su corazón y él cuidará de usted_.
Es cierto que la genética podría desempeñar un papel, y alrededor de uno de cada cien recién nacidos nace con defectos cardiacos cada año en los Estados Unidos. Sus corazones laten como bombas de tiempo hasta que, súbitamente... dejan de latir... como le sucedió a Pete Maravich, que nació con un raro defecto congénito: un solo sistema de arterias coronarias en lugar de dos. Después de su muerte repentina, los médicos se preguntaban asombrados cómo había vivido tantos años practicando un deporte tan exigente.
Pero si usted come como un cerdo, bebe como un pez y fuma como una chimenea —para utilizar tres clichés conocidos— debe ser también, como el pobre Pete, una bomba de tiempo ambulante.
Si su idea del ejercicio se limita a caminar hasta un restaurante McDonald's en lugar de ir por la ventanilla de los autos, debería asegurarse de tener al día los pagos de su seguro de vida, pues cada año, cada mes y cada mañana es un juego de azar.
A mi juicio, las enfermedades cardiovasculares —exceptuando, claro está, a los nacidos con defectos congénitos— son en alto grado evitables para quienes sigan _La receta del Gran Médico para un corazón saludable_. Mi enfoque se basa en siete llaves que utilizamos para liberar el potencial de salud del cuerpo, las cuales establecí en mi libro fundamental, _La receta del Gran Médico para tener salud y bienestar extraordinarios_ :
Estas son:
• Llave # 1: Coma para vivir.
• Llave # 2: Complemente su dieta con alimentos integrales, nutrientes vivos y superalimentos.
• Llave # 3: Practique una higiene avanzada.
• Llave # 4: Acondicione su cuerpo con ejercicios y terapias corporales.
• Llave # 5: Reduzca las toxinas en su ambiente.
• Llave # 6: Evite las emociones mortales.
• Llave # 7: Viva una vida de oración y con propósito.
Como está leyendo este libro, supongo que probablemente conozca algo sobre las enfermedades cardiovasculares o los ataques al corazón, bien a partir de una experiencia personal, o porque su cónyuge o algún familiar sobrevivió recientemente a alguno. O quizás no. De ser así, quiero que sepa que para usted escribí _La receta del Gran Médico para un corazón saludable_. Haya usted contemplado su propia mortalidad, o haya visto a un ser querido tratando de recuperarse de un amenazante infarto, estoy seguro de que se siente motivado a hacer algo. Y creo que entiende que esta es la enfermedad de las segundas oportunidades: sobrevivir a un ataque al corazón es como recibir otra oportunidad para montar el carrusel de la vida. Si ese es el caso, bienvenido a _La receta del Gran Médico para un corazón saludable_.
Un corazón sano bombea nutrientes vitales a través de su organismo, suministrándole la energía que necesita para pasar su día, estar disponible para sus hijos y cumplir el propósito de Dios para su vida. Se dice que cambiando el corazón podemos modificar nuestras vidas ¿necesita transformar el suyo antes de adoptar sin reservas _La receta del Gran Médico para un corazón saludable_? Debe entender que este libro no le garantiza la prevención ni el tratamiento de una enfermedad cardiovascular, y no me gustaría que nadie lo presentara como la promesa de una «cura» a esta potencialmente letal dolencia. Mi mayor esperanza es que las siete llaves consigan hacer por usted un par de cosas:
1. Ofrecerle la mejor oportunidad posible de tener una vida larga y saludable, sin llegar a desarrollar una enfermedad cardiovascular.
2. Apoyar cualquier terapia —convencional o alternativa— a la que usted se esté sometiendo para tratar su mal.
Antes de concluir, me alegro de que el doctor Dobson sobreviviera para que Nicki y yo pudiéramos ser beneficiarios de sus consejos sobre la crianza de los hijos. Me agrada saber que transformó su dieta y que convirtió en una prioridad su rutina diaria en el caminador, actos que forman parte de _La receta del Gran Médico para tener salud y bienestar extraordinarios_. Obviamente, Dios tiene un plan para seguir utilizando al doctor Dobson, tal como también tiene un plan para usarle a usted.
Pero para que eso ocurra, Él necesita que usted se encuentre en su óptima forma física, espiritual y emocional.
[LLAVE # 1
_Coma para vivir_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch00725)
Cuando alguien sufre en medio de su vida un infarto cardiaco, como le ocurrió al doctor Dobson, los médicos a menudo le inculcan el temor a Dios en lo que se refiere a su dieta.
_Renuncie a las carnes rojas._
_Basta de pizzas._
_Sacúdase el hábito de comer con sal._
_Deshágase de la mantequilla._
_Y olvídese de las nueces._
La opinión generalizada entre la comunidad médica de nuestros días es que una dieta alta en colesterol promueve las enfermedades coronarias y, a mi modo de ver, este artículo de fe es de los más profundamente incrustados en la medicina moderna. Los cardiólogos citan el «Estudio Framingham sobre el Corazón», que comenzó en 1948, cuando los investigadores reclutaron a cinco mil doscientos nueve hombres y mujeres de la población de Framingham, Massachusetts, en un marco de edades entre treinta y sesenta y dos años, e iniciaron la primera ronda de exhaustivos exámenes físicos y entrevistas sobre estilo de vida, que luego serían analizados a fin de encontrar patrones comunes relacionados con el desarrollo de enfermedades cardiovasculares.
En las últimas cinco décadas, las detalladas historias clínicas, exámenes físicos y pruebas de laboratorio realizadas a los voluntarios del estudio determinaron el alto nivel de colesterol en la sangre como uno de los principales factores de riesgo de las enfermedades coronarias. Este abarcador estudio, realizado por el Instituto Nacional del Corazón (ahora conocido como Instituto Nacional para el Corazón, los Pulmones y la Sangre o por las siglas NHLBI) concluyó que el colesterol endurece las arterias, encargadas de transportar nutrientes y oxígeno vitales hacia y desde el corazón.
El Estudio Framingham sobre el Corazón alentó a la industria farmacéutica a desarrollar las estatinas, que interrumpen la formación de colesterol en el organismo. Este es un tipo de fármaco que se utiliza para reducir el colesterol en la sangre. Las píldoras inducen al hígado a bloquear una sustancia que el cuerpo necesita para producir colesterol, al tiempo que le ayudan a reabsorber el colesterol que se ha acumulado en las placas adheridas a las paredes arteriales. Algunas de las estatinas que más se recetan son:
• Lipitor (atorvastatin)
• Pravachol (pravastatin)
• Lescol (fluvastatin)
• Mevacor (lovastatin)
• Crestor (rosuvastatin calcium)
• Zocor (simvastatin)
Supongo que habrá reconocido la primera de la lista: el Lipitor. Esta droga, que en 2005 recaudó once mil millones de dólares en ventas, es la que más se vende en el mundo, y una de las gallinas de los huevos de oro de Pfizer, la compañía que creó la Viagra. Si usted tiene cuarenta años o más (especialmente si es del sexo masculino) y al someterse a un chequeo médico anual sus muestras de sangre revelan que su nivel total de colesterol es mayor de doscientos, probablemente su médico familiar le recetará Lipitor. Es algo que en la actualidad se hace casi automáticamente, y una de las razones por la que doce millones de estadounidenses, o el cuatro por ciento de la población de este país toman esta droga con su jugo de naranja matutino, a pesar de su alto precio: la mayoría paga entre setecientos y ochocientos dólares al año por su Lipitor.
El Lipitor y otros fármacos de la familia de las estatinas, son potentes medicamentos que por lo general reducen los niveles de colesterol, aunque el paciente tiene que pagar por ello también un precio físico. Entre los efectos colaterales «menores» se encuentran náuseas, diarrea, estreñimiento y dolores musculares. Otros dos, potencialmente graves, son un incremento de las enzimas del hígado (lo cual en lenguaje médico equivale a daño hepático) y miopatia de las estatinas o fallo renal. Hasta el momento no se conocen los efectos colaterales a largo plazo.
No obstante, creo firmemente que si usted se compromete a comer los alimentos que forman parte de _La receta del Gran Médico para un corazón saludable_ , cuando vaya a visitar a su médico y este le ordene análisis de sangre, encontrará que su colesterol se encuentra dentro del rango normal. Entonces podrá deshacerse de todas esas drogas costosas para reducir el colesterol (con la condición, desde luego, de que usted y su médico estén de acuerdo en seguir esa vía de acción).
Nuestros cuerpos necesitan colesterol, una sustancia blanda parecida a la cera que es producida por el hígado y que se encuentra en el torrente sanguíneo y en todas las células. El colesterol digiere las grasas, es la materia prima de las membranas celulares y de varias hormonas y cumple otras funciones indispensables para el organismo. El hígado produce alrededor de mil miligramos de colesterol diarios, pero el cuerpo recibe otros doscientos a quinientos miligramos de alimentos como las carnes, pollo, pescado, huevos, mantequilla, queso y leche entera. Las frutas, vegetales y cereales no contienen colesterol.
El colesterol es transportado en el torrente sanguíneo por las lipoproteínas, o más exactamente por dos tipos diferentes de ellas. Se conoce como «colesterol bueno» a una lipoproteína de alta densidad (HDL), que recolecta el colesterol no utilizado y lo lleva de regreso al hígado, donde es destruido. En cambio, a una lipoproteína de baja densidad (LDL) se la considera el enemigo público número uno, debido a que este «colesterol malo» se acumula y adhiere al interior de las arterias. Los investigadores médicos han descubierto que una cifra elevada de partículas de LDL tiene un fuerte vínculo con los problemas coronarios. Es por eso que los médicos prestan suma atención al índice de LDL, y que en la medicina moderna existe una presión para reducir sus niveles a menos de cien miligramos por decilitro (mg/dl) en los hombres y ciento diez mg/dl en las mujeres, por medio de fármacos a base de estatinas como el Lipitor o el Pravachol.
Sin embargo, he encontrado recientemente un enfriamiento hacia esta terapia en ciertos círculos médicos, encabezado por un médico sueco, el doctor Uffe Ravnskov, que obtuvo un doctorado en filosofía por sus estudios científicos en los Departamentos de Nefrología y Química Clínica del Hospital Universitario de Lund, Suecia.
Ravnskov investigó durante años la literatura científica acerca del colesterol y las enfermedades cardiovasculares, pero no recuerda haber visto en todo ese tiempo un estudio que demostrara que un nivel elevado de colesterol fuese peligroso para el corazón o las arterias coronarias. La opinión médica mundial estaba sumamente influenciada por el Estudio Framingham sobre el Corazón, que levantaba el índice acusador contra el colesterol como culpable de la primera causa de muerte en los Estados Unidos.
Pero desde el punto de vista del doctor Ravnskov, el emperador —el Rey Colesterol— estaba desnudo, pues los estudios médicos no demostraban de manera concluyente la conexión entre niveles altos de colesterol y enfermedades cardiovasculares. El médico sueco escribió cerca de cuarenta ensayos médicos criticando la presunta asociación entre el colesterol y las enfermedades del corazón y los vasos sanguíneos. Señaló, por ejemplo, que en un estudio de seguimiento a la población de Framingham a lo largo de treinta años, el alto índice de colesterol no constituía un pronóstico certero de ataque cardiaco después de los cuarenta y siete años, mientras que aquellos que habían reducido su nivel del colesterol ¡tenían el mayor riesgo de sufrir un infarto al miocardio! La cita fue tomada directamente del estudio Framingham: «Por cada reducción de un mg/dl del colesterol, se registró un incremento del once por ciento en la mortalidad total y por causas coronarias» (las cursivas son mías).
Señalar las incongruencias le ganó al doctor Ravnskov entre la comunidad de médicos cardiólogos una reputación de criticón en el mejor de los casos y de agitador en el peor. Su análisis crítico, pero científico, no despertó gran interés entre los editores de las publicaciones médicas _Journal of the American Medicine Association_ (JAMA) o _New England Journal of Medicine_ (NEJM), y tampoco entre la comunidad médica tradicional, que continuó aconsejando a sus pacientes que limitaran su ingestión de grasas, incluyendo las ricas en ácidos grasos saturados, a fin de reducir el riesgo de morir de enfermedades cardiovasculares.
Frustrado en sus intentos por encontrar apoyo entre sus pares, el doctor Ravnskov publicó sus preocupaciones en un libro titulado _The Cholesterol Myths_ , originalmente en Suecia en 1991, en idioma sueco. El libro fue mayormente ignorado y tuvo poco impacto. Lo que es peor, ¡algunos críticos lo quemaron en un programa de la televisión finlandesa! Pasaron varios años y el doctor Ravnskov pensó que si hacía publicar _The Cholesterol Myths_ en inglés la discusión podría reanudarse. Pero sus gestiones con agentes literarios y casas editoras en Gran Bretaña y los Estados Unidos fueron tajantemente rechazadas.
Entonces, en la segunda mitad de los años noventa, entró en escena la Internet. De repente, el doctor Ravnskov pudo prescindir de las editoriales; publicó en la web algunos capítulos de _The Cholesterol Myths_ y ¡presto!, dejó de ser un paria batallando contra molinos de viento como Don Quijote. Mientras los comentarios aumentaban y los internautas escribían en sus motores de búsqueda «colesterol» y «enfermedades cardiovasculares », el científico sueco recibía correos electrónicos de personas impresionadas por su análisis lúcido y comedido, entre ellos, investigadores que veían con escepticismo la relación dietacorazón que empezaba a permear la comunidad médica.
La obra del doctor Ravnskov confirma mi creencia de que la tan recomendada dieta baja en grasas y en colesterol, alabada como la solución para prevenir las enfermedades cardiovasculares, nunca será la esperada panacea. Hay demasiados médicos recomendando a sus pacientes no comer ciertos alimentos que en realidad pueden ser muy beneficiosos para su salud cardiovascular.
Me estoy anticipando aquí a mí mismo, pero ciertos alimentos «altos en grasa» —carne de res, huevos, mantequilla y productos lácteos— si proceden de animales no confinados y criados con alimentos orgánicos, contienen grasas que su cuerpo necesita para tener una salud óptima. Dios, en su infinita sabiduría, creó ciertas grasas para realizar las siguientes funciones: desempeñar un papel vital en la salud de los huesos; fortalecer el sistema inmunológico; proteger al hígado del alcohol y otras toxinas; y proteger al tracto digestivo de microorganismos dañinos.
Los mejores ejemplos de «grasas beneficiosas» son las grasas saturadas saludables, los ácidos grasos omega-3 poliinsaturados y los ácidos grasos monoinsaturados (omega-9). Usted puede encontrar estas grasas en una amplia variedad de alimentos, incluyendo el salmón y las carnes de cordero y cabra, así como en su leche y los quesos hechos con ella; en el coco, las nueces, aceitunas, almendras y aguacate o palta. Ellas nos suministran una fuente concentrada de energía y son la materia prima de las membranas celulares y varias hormonas.
Usted se preguntará: «¿No sabía el Creador que las grasas y el colesterol son las causas principales de las enfermedades coronarias? »Claro que lo sabía, pero probablemente son los alimentos que contienen grasas _trans_ los que se encuentran en la raíz de nuestra epidemia nacional de enfermedades del corazón.
Las grasas trans, que sí obstruyen las arterias se producen calentando aceites vegetales líquidos en presencia de hidrógeno para hacerlos sólidos a temperatura ambiente, un proceso conocido como hidrogenación. Los conglomerados alimentarios utilizan como rutina aceites hidrogenados en sus plantas de procesamiento, lo que quiere decir que hay grasas trans en casi todos nuestros alimentos procesados, alimentos que Dios claramente no creó.
Hablo de la margarina, comidas congeladas, helados, quesos procesados, papas fritas, galletas dulces, pan blanco, meriendas, rosquillas, caramelos, aliños para ensaladas... la lista es interminable ¿Por qué emplean tanta química los productores de alimentos? Porque les permite elaborar un producto más competitivo en materia de precios y con un tiempo de expiración más largo. Las comidas fritas producidas para el mercado como las papas y los anillos de cebolla se fríen en aceites vegetales poliinsaturados y contienen también gran cantidad de grasas trans.
Un problema mayor de las grasas trans es la manera en que aumentan los niveles de colesterol. El consumo de alimentos que las contienen eleva su colesterol LDL o malo y reduce el HDL o bueno, con lo cual aumenta el riesgo de enfermedades cardiovasculares, así como de la diabetes del tipo 2. Los científicos nos han estado advirtiendo durante años que comer grasas trans puede conducir a problemas cardiacos, razón por la cual la Administración de Alimentos y Fármacos de Estados Unidos comenzó a exigir en 2006 a las compañías alimentarias que precisaran la cantidad de estas grasas como parte de los datos nutricionales en el envase del producto.
ECHE LOS CIMIENTOS
Mi primera llave —«Coma para vivir»— resulta ser la receta más importante en _La receta del Gran Médico para un corazón saludable_ , porque lo que usted escoge para nutrirse afectará positiva o negativamente la salud de su corazón —el más importante de sus músculos— así como la totalidad de su organismo. La mejor manera de «comer para vivir» puede resumirse en estos dos principios básicos:
1. Coma de lo que Dios creó como alimento.
2. Cómalo en una forma que sea sana para su cuerpo.
Comer los alimentos que Dios creó en una forma que sea sana para su cuerpo significa escoger comestibles lo más cercanos a su fuente original que sea posible, lo cual nutrirá su organismo, ayudará a su corazón a latir con fuerza y regularidad, y asegurará en su vida la mejor salud posible. Como ya se habrá dado cuenta, estoy a favor de los alimentos naturales, cultivados orgánicamente, ya que son alimentos que Dios creó en una forma saludable para su cuerpo.
Creo que optimizar la nutrición comienza por una conciencia de lo que hacemos circular por el tracto digestivo. Para empezar, todo lo que usted pone en su boca es una proteína, una grasa o un carbohidrato. Veamos más de cerca estos macronutrientes.
LO BÁSICO SOBRE LAS PROTEÍNAS
Las proteínas, uno de los componentes básicos de los alimentos, son los bloques de construcción esenciales del organismo, y participan en las funciones de toda célula viva. Una de sus tareas fundamentales es proveer material nutritivo específico para generar y reparar células, especialmente las del músculo cardiaco. Científicos del Centro Médico Suroccidental en la filial de Dallas de la Universidad de Texas, están estudiando cómo protege al músculo cardiaco de daños una proteína llamada limosina beta-4, e incluso promueve la reparación del corazón después de un ataque coronario.
Todas las proteínas son combinaciones de veintidós aminoácidos que intervienen en la generación de los órganos, músculos y nervios, por mencionar sólo algunas de sus funciones más importantes.
Nuestros cuerpos, sin embargo, no pueden producir los veintidós aminoácidos que necesitamos para vivir saludables. Los científicos han descubierto que nos faltan ocho aminoácidos esenciales, lo que significa que debemos tomarlos de fuentes externas. Sé que el siguiente dato pone de mal humor a los vegetarianos y semivegetarianos, pero las proteínas de origen animal —pollo, carne de res, cordero, productos lácteos, huevos y otros— son la _única_ fuente completa para obtener los ocho grandes aminoácidos que no fabricamos en las cantidades y proporciones adecuadas.
La mejor alternativa para un corazón sano consiste en consumir las fuentes más sanas y magras de proteína animal disponibles, que son el ganado vacuno, ovino y caprino criado con alimentación orgánica, así como los búfalos y los venados, animales que toman su alimento de los prados. La carne de los vacunos criados con pastos y forrajes tiene menos grasa y es más baja en calorías que la de los que se crían con granos y piensos. Soy también un gran entusiasta de las aves de corral que se crían sueltas y del pescado que se captura en lagos, corrientes fluviales o en las profundidades marinas. Los peces de escama y aleta pescados en su medio natural son fuentes magras de proteína y proveen todos los aminoácidos esenciales. Son, desde el punto de vista nutritivo, muy superiores a los criados por métodos de piscicultura y deben consumirse con frecuencia.
UN REPASO SOBRE LAS GRASAS
Ya he mencionado que las grasas trans obstruyen las arterias como los desechos en una cañería. Sin embargo comer grasas sanas puede tener un efecto protector contra las enfermedades del corazón. Me refiero a alimentos con un alto contenido de lo siguiente:
• grasas poliinsaturadas (ricas en ácidos grasos omega-3)
• ácidos grasos monoinsaturados (omega-9)
• ácido linoleico conjugado (CLA)
• grasas omega-6 claves como el GLA
• grasas saturadas sanas que contienen ácidos grasos de cadena corta y mediana, como la mantequilla y el aceite de coco.
Estas grasas beneficiosas se encuentran en una amplia variedad de alimentos incluyendo el salmón, aceite de hígado de bacalao, carnes de cordero y cabra, huevos ricos en omega-3, semillas de linaza, nueces, aceitunas, macadamias, aguacate, mantequilla de animales alimentados con pastos y productos lácteos derivados de la leche de cabra, de oveja y de vaca.
El problema con la dieta estadounidense normal es que las personas comen demasiados alimentos que contienen grasas perjudiciales y muy pocos de los que contienen grasas saludables. Dos de mis grasas y aceites favoritos son el aceite de coco extravirgen y los aceites de oliva, que son beneficiosos para el cuerpo y ayudan con el metabolismo. Le exhorto a cocinar y hornear con aceite de coco extravirgen, un alimento milagroso del cual pocas personas tienen conocimiento.
CARBOHIDRATOS DE LA VICTORIA
Por definición, los carbohidratos son las féculas y azúcares producidos por los vegetales alimenticios, y se encuentran en la sangre en forma de glucosa. Esta es regulada por la insulina, una hormona que guarda la llave de las puertas nutricionales del cuerpo. Gracias a la dieta baja en carbohidratos que popularizó el doctor Robert Atkins, los estadounidenses desataron en los últimos cinco a diez años una cacería de brujas contra los carbohidratos. Las dietas bajas en carbohidratos se consideran un buen régimen dietético para los pacientes de problemas cardiovasculares, especialmente los que necesitan bajar de peso.
La dieta Atkins, para hablar sólo de la más antigua y más ampliamente practicada de las dietas bajas en carbohidratos, aconseja un alto consumo de carnes procesadas (jamón, tocineta, pepperoni, salami y salchichas) o derivadas de un ganado criado mediante técnicas convencionales, todos con alto contenido de grasas insalubres, lo cual sólo puede incrementar su riesgo de un ataque cardiaco. La dieta Atkins también prohíbe comer generosas cantidades de frutas y vegetales ricos en fécula, que contienen gran cantidad de fibra y antioxidantes, dos ingredientes de una buena salud cardiovascular. Tengo una pregunta: ¿Cómo puede conciliarse ese consejo nutricional con la recomendación de la American Heart Association de que comamos al menos cinco porciones de frutas y vegetales diarias, a fin de reducir tres de los mayores factores de riesgo de infarto cardiaco: colesterol alto, hipertensión arterial y peso corporal excesivo? La AHA también recomienda comer carne de aves sin la piel y carnes magras.
Si usted se está recuperando de un ataque al corazón, no creo que necesite limitar sus carbohidratos. En lugar de ello debe añadir a su dieta carbohidratos no refinados, incluyendo los ya mencionados vegetales y frutas, y también granos integrales como la avena, trigo, centeno, maíz, arroz y cebada. Los carbohidratos no refinados contienen el grano entero, incluyendo el salvado y el germen, por lo que son más ricos en fibra y en ácidos grasos saludables.
Uno de los mejores carbohidratos no refinados para la salud cardiaca es la avena integral, cuya fibra soluble contiene betaglucanos. La Administración de Alimentos y Fármacos de Estados Unidos concluyó que existe un vínculo entre la fibra soluble de los alimentos a base de avena integral y una reducción en el riesgo de enfermedades coronarias, pues la fibra soluble con betaglucanos es el principal componente responsable de reducir los niveles del colesterol malo LDL. Esa es la razón de que usted vea un corazón rojo en las cajas de cereal como la de avena Quaker y Kashi Heart to Heart.
Lamentablemente, gran parte del cereal y del pan que se producen en este país está hecha con trigo y otros granos sometidos a un proceso de «enriquecimiento». Y lo único que «enriquece» esa harina son los niveles del nocivo colesterol LDL, quizás también los triglicéridos en la sangre y, por supuesto, los niveles de insulina, razón por la cual los alimentos hechos con harina enriquecida deben evitarse por completo, especialmente si usted se está recuperando de un ataque cardiaco. Compre en su lugar harina orgánica en cuya etiqueta pueda leerse _molida con piedra_ , _sin levadura_ o _germinada._ Estos productos de granos integrales o la avena tradicional no han sido privados de su fibra, vitaminas y componentes minerales vitales. La Escuela de Salud Pública de Harvard analizó las dietas e historias clínicas de más de veintisiete mil hombres entre las edades de cuarenta a setenta y cinco años, durante un periodo de catorce años. Encontró que los que comían más granos enteros reducían su riesgo de enfermedades cardiovasculares en casi un veinte por ciento.
Cuando se presentan problemas del corazón, muchos tienden a pensar que necesitan limitar sus carbohidratos, pero consumir carbohidratos de alto contenido nutritivo y bajo de azúcar es apropiado para el órgano primo. Estoy hablando de la mayoría de las frutas ricas en fibra (como las bayas), vegetales, nueces, semillas, legumbres, productos lácteos fermentados y granos.
Comer alimentos ricos en fibra mejorará de inmediato los niveles de azúcar en la sangre al desacelerar la absorción de azúcar al torrente sanguíneo. En las mujeres, una dieta alta en fibra puede reducir el riesgo de problemas coronarios hasta en veintitrés por ciento, según un estudio publicado en la revista _Journal of the American Medical Association_. Después de controlar varios factores, los investigadores hallaron que las mujeres que consumían a diario la mayor cantidad de fibra, unos veintitrés gramos, lograban reducir su riesgo de enfermedades coronarias.
RECOMENDACIONES NUTRICIONALES DEL GRAN MÉDICO
Con posterioridad a un infarto cardiaco, la mayoría de los médicos y enfermeras enfatizan la importancia de las frutas, vegetales, pan y cereales integrales, cantidades moderadas de carne y productos lácteos, así como una reducción del consumo de azúcares y grasas.
Si bien estoy de acuerdo con buena parte de lo que la medicina convencional recomienda a los pacientes cardiovasculares con respecto a su nutrición, quiero agregar algunos puntos acerca de las constantes en la mayoría de las dietas, así como ofrecer mis propias recomendaciones.
Permítame en primer lugar recordarle que debe masticar bien sus alimentos. Si quienes le rodean la critican por «inhalar» sus alimentos, tal vez sea porque come muy rápido. Le recomiendo masticar cada bocado de comida entre veinticinco y setenta y cinco veces antes de tragarlo. Quizás este consejo le parezca ridículo, pero sé que un esfuerzo consciente por masticar despacio asegura que se agreguen suficientes jugos digestivos a la comida al iniciar su recorrido a través del tracto digestivo.
Veamos ahora más de cerca lo que usted debe y no debe comer a fin de preservar la salud de su corazón:
**_1. Carnes_**
La sabiduría en la comunidad cardiovascular prescribe que las víctimas de infartos cardiacos deben reducir su consumo de carnes rojas, las cuales contienen más grasas saturadas y colesterol que el pescado o las carnes blancas de aves. La American Heart Association recomienda comer no más de seis onzas diarias de carne magra cocida, pollo o pescado. Los cortes de carne de res que contienen menos grasa son la tapa, el pecho, el solomillo y el lomo.
Se ha escrito mucho acerca de los peligros de la carne —especialmente la de res— como causa de enfermedades cardiovasculares. Pues claro: Si usted come cortes de falda de una res a la que se han inyectado hormonas, alimentada con forrajes rociados con pesticidas y entreverados con antibióticos, de seguro que ello planteará algún problema a su corazón. Una solución mucho mejor sería comer la carne de ganado alimentado orgánicamente, cordero, cabras, bisontes y venados que pastan en los pródigos pastizales de la naturaleza.
Las carnes de animales alimentados con pasto, y el pescado capturado en su medio natural, son ricos en vitaminas B12 y B6, una información significativa en relación con las enfermedades cardiovasculares. En años recientes, los científicos han identificado otro factor de riesgo para las enfermedades del corazón: un nivel elevado de homocisteína. Esta es un aminoácido que se encuentra normalmente en la sangre, pero un nivel elevado afecta la función vasomotora endotelial, lo que se traduce de la jerga médica como la facilidad con que la sangre circula a través de los vasos sanguíneos. Niveles elevados de homocisteína dañan las arterias coronarias y facilitan la agregación de las plaquetas, que resulta en la formación de coágulos. Estos son los precursores de los infartos al miocardio.
Como las vitaminas B12 y B6 están presentes en el metabolismo de la homocisteína, comer carne de animales alimentados con pasto y peces capturados en su medio natural provoca un descenso del nivel de homocisteína en la sangre, beneficioso para el corazón. La trucha silvestre, por ejemplo, es una de las fuentes más ricas en B12 natural, junto con el salmón y el de Alaska especialmente.
De igual modo que hay carnes que usted debe comer para preservar su salud cardiovascular, hay otras que debe evitar. Me refiero a las salchichas para el desayuno, tocineta, lascas de jamón prensado, perros calientes y otros embutidos. Estas utilizan preservantes llamados nitratos que les dan su color rojizo, les añaden sabor y resisten el desarrollo de esporas de botulismo. Los nitratos pueden convertirse en nitritos, los cuales han sido durante décadas estudiados por instituciones públicas y privadas. Se cree que han sido causa de cánceres y tumores en animales de laboratorio.
Tengo otras razones para recomendarle que no coma tocineta, jamón ni otras carnes procesadas. En mis libros anteriores he señalado coherentemente que el cerdo, al cual la publicidad llama en Estados Unidos «la otra carne blanca», debe ser evitado, pues este animal fue llamado «inmundo» en los libros bíblicos de Levítico y Deuteronomio. Dios, en su infinita sabiduría, creó a los cerdos como animales que se alimentan de desechos, y que sobreviven bien con cualquier desperdicio o agua sucia que se les dé. Ellos tienen un tracto digestivo simple en el cual todo lo que comen baja rápidamente al estómago y sale por la puerta trasera en un máximo de cuatro horas. Son capaces de comer hasta sus propios excrementos si están muy hambrientos. Aun si usted decide seguir comiendo carnes producidas por métodos comerciales en lugar de la versión orgánica, le ruego encarecidamente que se abstenga de comer cerdo. Si lee Levítico 11 y Deuteronomio 14 podrá ver lo que Dios estableció en cuanto a comer la carne de animales limpios o inmundos. Las palabras hebreas utilizadas para describir las carnes inmundas pueden traducirse como «pútrido» o «impuro», los mismos términos que se usan para describir el excremento humano.
La más sana de todas las carnes es el pescado capturado en su medio natural, incluyendo el salmón, sardinas, arenque, macarela, atún, pargo, bass y bacalao. Estos peces criados en su ambiente son una rica fuente de ácidos grasos omega-3, que ayudan a preservar la salud cardiovascular. Los científicos han descubierto que cuando su cuerpo no recibe un suministro suficiente de omega-3, utiliza las grasas saturadas para construir las membranas celulares. Estas últimas resultan entonces menos elásticas, lo cual ejerce un efecto negativo sobre el corazón, a cuyo músculo se le dificulta más regresar a un ritmo de descanso.
Una de las primeras asociaciones entre los ácidos grasos omega-3 y la salud humana fue establecida en los años setenta, cuando científicos que estudiaban a la población _inuit_ de Groenlandia descubrieron que ese grupo étnico padecía mucho menos de enfermedades coronarias que los europeos, a pesar de que su dieta, que incluye carne de ballenas, focas y salmón, tiene un alto contenido de grasas.
No obstante, tenga presente que no toda la vida marina que nos sirve de alimento es beneficiosa para la salud humana. Las carnes de los mariscos de concha dura o exoesqueleto quitinoso, así como la de los peces sin aletas ni escamas, tales como el siluro, el tiburón y la anguila, también son descritas en Levítico 11 y Deuteronomio 14 como «inmundas». Dios llamó inmundos a los crustáceos de concha dura como la langosta, el cangrejo, los camarones y las almejas porque se alimentan de lo que yace en el lecho marino, contentándose con las excretas de otras especies acuáticas. Esto podrá purificar el agua, pero no hace nada por la salud de la carne de estos habitantes del mar, y seguramente tampoco por la suya si la ingiere.
Comer alimentos inmundos contamina su cuerpo y puede conducir a enfermedades cardiovasculares y cánceres al introducir toxinas en el torrente sanguíneo. Dios declaró inmundas esas carnes porque conoce las consecuencias de comerlas; también usted debe conocerlas.
**_2. Productos lácteos_**
Los médicos clasifican las grasas saturadas que contienen los productos lácteos en la misma categoría que las carnes rojas, estableciendo la ingestión de grasas como uno de los factores clave que provocan las enfermedades cardiovasculares. En ese sentido, recomiendan no comer productos lácteos con la grasa entera. Nos dicen que si vamos al supermercado a comprar leche, nos aseguremos de escoger una versión baja en grasas como la que tiene dos por ciento, o que llevemos leche desgrasada. Yo no tengo la misma opinión, porque nunca oí hablar de una vaca o cabra que al ordeñarla diera leche con un dos por ciento de grasas o desgrasada.
La leche baja en grasas es menos nutritiva y digerible, y puede provocar alergias. En lo referente a prevenir enfermedades del corazón, le recomiendo comprar productos lácteos derivados de la leche de cabra o de oveja, aunque los que se derivan de reses alimentadas orgánicamente o con pasto también pueden ser excelentes. Prefiero la leche y los quesos de cabra debido a su estructura: sus moléculas de grasa y proteínas son minúsculas, lo cual permite una rápida absorción por el tracto digestivo. La leche de cabra es menos alergénica, pues no contiene las mismas proteínas complejas que se encuentran en la de vaca. También recomiendo consumir leche en sus formas cultivadas o fermentadas, como el yogurt y el kéfir. El proceso de fermentación hace la leche más fácil de digerir, y el cuerpo aprovecha mejor sus nutrientes.
Por último, trataré sobre otro producto perecedero que se relaciona directamente con la salud cardiovascular, aunque quizás no como usted cree: los huevos. Cuando alguien sufre un ataque cardiaco, una de las primeras cosas que se le ordena suprimir en su lista de víveres son los huevos, debido a que los altos niveles de colesterol son supuestamente perjudiciales para el corazón. Para mí este consejo nunca ha tenido sentido, pues los huevos son un alimento maravilloso, con méritos suficientes para figurar en el Salón de la Fama. Este concentrado de nutrientes contiene seis gramos de proteínas, algo de vitamina B12 y E, luteína, riboflavina, ácido fólico, calcio, zinc, hierro y ácidos grasos esenciales, y sólo setenta y cinco calorías. La Escuela de Salud Pública de Harvard coincide conmigo.
Citando investigaciones de científicos de la Universidad de Harvard, la escuela señala que un consumo moderado de huevos —uno al día— no incrementa los riesgos cardiovasculares en personas sanas, en tanto que nutrientes como las vitaminas B12 y D, junto con la riboflavina y el ácido fólico, ayudan a reducirlos. El «Estudio sobre Ateroesclerosis de Los Ángeles» halló que mientras más huevos consumían sus sujetos, mejor se veían sus arterias.
**_3. Frutas y vegetales (y sus jugos)_**
Todos los que tienen que ver con las enfermedades cardiovasculares, desde los más prestigiosos especialistas médicos hasta los que promueven curas alternativas, cantan la misma canción: Usted debe aumentar su consumo de frutas y vegetales para prevenir o combatir las enfermedades cardiacas y circulatorias. Sin embargo el estadounidense promedio come mucho menos de las cinco a nueve porciones diarias recomendadas. Un estudio de la Escuela de Salud Pública de Harvard indica que por cada porción extra de frutas y vegetales que los participantes agregaban a sus dietas, su riesgo de enfermedades cardiovasculares se reducía en cuatro por ciento.
Las frutas y vegetales contienen compuestos que reducen significativamente la presión arterial y los niveles de colesterol, pues están cargadas de vitaminas, minerales, fibra y antioxidantes, beneficiosos para el corazón. Los pigmentos que dan a los vegetales colores anaranjado y verde oscuro se conocen como carotenoides, son precursores de la vitamina A y actúan como poderosos antioxidantes capaces de proteger al organismo del daño celular causado por un tipo específico de moléculas de oxígeno conocidas como radicales libres. Se cree que el daño que estas causan a las células resulta en enfermedades cardiovasculares.
Usted debe comer como mínimo dos o tres frutas frescas diarias, principalmente en horarios de merienda. Para la salud cardiovascular recomiendo los arándanos azules y rojos, frambuesas y uvas, que contienen flavonoides y reseveratrol. Este último es una fitoalexina que ha demostrado ser eficaz en la prevención de enfermedades cardiacas y de los vasos sanguíneos.
Durante décadas se ha estudiado una forma de jugos de uvas fermentado —el vino tinto— en relación con las dolencias del corazón ¿Nos aleja del cardiólogo beber un vaso de vino tinto diario? La American Heart Association reporta que se han publicado muchos estudios en revistas científicas que indican que beber pequeñas cantidades de vino tinto es beneficioso para el corazón, lo cual ha llevado a los editores de periódicos a escribir titulares optimistas tales como: «Un buen tinto es bueno para su corazón, afirman los médicos».
Cualquiera que haya ido de vacaciones a Francia, lo cual he tenido la suerte de hacer, habrá visto de primera mano como cualquier francés que se respete nunca se sentará en un restaurante sin pedir una botella de Burdeos. Los franceses tienen el consumo per cápita de vino más alto del mundo, hacen una dieta abundante en mantequilla, quesos, cremas, carnes y pâtés abundantes en grasa como el _foie gras_ , y muchos de ellos fuman cigarrillos. Sin embargo, tienen una tasa mucho menor de enfermedades coronarias que los estadounidenses. Y lo que es aun peor —o mejor, dependiendo de la perspectiva— los ataques al corazón cobran en Francia la mitad de las víctimas mortales que en los Estados Unidos. No en balde se conoce este fenómeno como «La paradoja francesa».
Aunque como regla no bebo alcohol (quizás medio vaso un par de veces al año) me doy cuenta de los beneficios que emanan para el corazón de beber unas onzas de vino tinto con la cena. Recomiendo vinos orgánicos que no contengan preservantes ni sulfatos. Pero sea cauteloso: si existe alguna espada de dos filos, esa es el alcohol. Su consumo excesivo ha devastado relaciones y destruido millones de familias a través de los años.
**_4. Semillas y granos húmedos y germinados_**
Como las frutas y vegetales, los granos enteros, semillas, nueces y panes hechos con granos germinados o masa agria son saludables para su corazón. Aparentemente la fibra, las vitaminas, las sustancias fotoquímicas y los antioxidantes presentes en granos enteros debidamente preparados —como trigo, kamut, quinoa, amaranto, mijo, alforfón, cebada, maíz, avena y arroz— parecen articularse en la lucha contra las mortíferas enfermedades cardiovasculares. Se entiende por _grano entero_ el que durante su procesamiento conserva tanto el salvado como el germen. Los cereales húmedos y germinados retienen, si no se cocinan, las enzimas de la planta, aunque son más digeribles cocinados u horneados.
**_5. Vegetales encurtidos y fermentados_**
Aunque no suelen ser muy aceptados en la mesa, los vegetales fermentados como la col agria y las zanahorias, remolachas o pepinos encurtidos, ayudan a restablecer el equilibrio natural del sistema digestivo. Los vegetales fermentados como la col agria contienen abundantes vitaminas como la C, y casi cuatro veces la cantidad de nutrientes que contiene la col sin fermentar.
Los japoneses, que tienen la segunda mayor expectativa de vida del mundo, y tasas de enfermedades cardiovasculares inferiores a las de los Estados Unidos, comen vegetales fermentados como la col, berenjena y rábano daikon con sus comidas tradicionales.
**_6. Nueces_**
La Administración de Alimentos y Fármacos de EUA. anunció en 2003 que «Las evidencias científicas sugieren, aunque no demuestran, que comer una onza y media diaria de la mayoría de las nueces, como parte de una dieta baja en grasas saturadas y colesterol, puede reducir el riesgo de enfermedades del corazón».
Conté cinco condicionantes en esta afirmación, pero he leído muchas investigaciones, incluyendo una de la Universidad Estatal de Pennsylvania que muestra que comer almendras, nueces del Brasil, semillas de marañón, avellanas, macadamias, pacanas, pistachos, nueces y el modesto maní o cacahuate, produce un poderoso efecto protector contra las enfermedades coronarias. Las nueces son una fuente abundante de ácidos grasos no saturados y omega-3 —la grasa «buena»— de origen vegetal.
**_7. Especias_**
Una de las primeras cosas que recomiendan los cardiólogos a las víctimas de infartos es deshacerse de la sal de mesa y otros condimentos elaborados con alto contenido de sodio. El sodio hace que el cuerpo retenga líquidos, lo cual obliga al corazón a trabajar más para bombear esos fluidos excedentes. Usted no necesita sal para sazonar sus alimentos: sus papilas gustativas aprenderán a apreciar la amplia variedad de especias saludables que puede emplear para cocinar u hornear.
Dos especias que seguramente hay en su alacena parecen tener propiedades beneficiosas para el corazón. El jengibre, la especia más cultivada en el mundo, contiene sustancias químicas naturales que desestimulan la coagulación de la sangre, reducen el colesterol e incrementan la fuerza del tejido muscular cardiaco. «El jengibre ofrece una profunda acción principal antioxidante y efectos que se han observado, entre ellos el fortalecimiento del músculo cardiaco y la reducción del colesterol en la sangre», escribe Paul Schulick, autor de _Ginger: Common Spice & Wonder Drug_. Schulick agrega que conoce un hospital en Israel donde a los pacientes de enfermedades coronarias se les aconseja tomar media cucharadita diaria de polvo de jengibre. ¡Eso también despejará sus cavidades nasales
Hay otra especia doméstica que usted debe buscar la forma de consumir, pero esta no podrá tomarla a cucharadas. Me refiero a la pimienta roja o cayena. Contiene una gran cantidad de betacaroteno, un antioxidante que reduce el colesterol «malo» LDL.
**_8. Agua_**
Claro que el agua no es un alimento en sí, pero esta sustancia carente de calorías y de azúcar desempeña muchas funciones vitales en el organismo: regular la temperatura corporal, transportar los nutrimentos y el oxígeno a las células, amortiguar las articulaciones, proteger órganos y tejidos, y eliminar toxinas.
En lo que se refiere a la salud cardiovascular, beber suficiente agua debe figurar en el tope de la lista. «En esencia, no beber suficiente agua puede ser tan dañino para su corazón como fumar», advierte Jacqueline Chan, doctora en salud pública e investigadora principal en un importante estudio realizado por la Universidad de Loma Linda, que encontró que beber cinco o más vasos de agua diarios puede reducir significativamente los riesgos de enfermedades coronarias.
El agua aporta la viscosidad de la sangre y el plasma, casi con los mismos efectos lubricantes del aceite en un motor de alto poder. Ayuda a transportar los nutrientes hasta las células y a mantener bajos los niveles de colesterol. No beber suficiente agua es negativo para su cuerpo y su corazón. El doctor F. Batmanghelidj, autor de _You_ ' _re Not Sick, You_ ' _re Thirsty!_ , afirma que un colesterol elevado es consecuencia directa de la deshidratación crónica. «En la deshidratación crónica, las células hepáticas continúan produciendo cantidades adicionales de colesterol», escribe.
Usted debe beber un mínimo de ocho vasos de agua al día para mantenerse hidratado. De acuerdo, hay que ir al baño con más frecuencia, pero ¿qué hay de malo en eso? Beber bastante agua no sólo es saludable para el cuerpo sino que constituye una parte clave del plan de batalla de _La receta del Gran Médico para un corazón saludable_ (ver página 71), así que mantenga siempre una botella de agua cerca y beba del precioso líquido antes, durante y entre las comidas.
Parece un buen momento para hablar de la obsesión de nuestro país con el café y el té, cortesía de la cafetería Starbucks del barrio. La American Heart Association afirma que aún está en estudio la cuestión de si una alta ingestión de cafeína incrementa los riesgos de enfermedades coronarias, pero debo señalar que el café y el té han sido consumidos durante miles de años por algunos de los pueblos más sanos del planeta.
Aunque no soy un gran fanático, debo decir que me parece bien una taza diaria de café cultivado orgánicamente, recién molido, y saboreado con miel de abejas y crema (también orgánica), siempre que se consuma con moderación, o sea, una taza diaria.
Las infusiones de plantas medicinales y especias son harina de otro costal. Podrá ver en mi plan de batalla de _La receta del Gran Médico para un corazón saludable_ que recomiendo una taza de té caliente con miel de abejas con el desayuno, la cena y las meriendas. También aconsejo el té helado, ya que esta infusión se puede beber caliente, fresca y fría. Observe sin embargo que si bien el té ofrece grandes beneficios para la salud, nada puede sustituir como agente hidratante al agua pura. Aunque es saludable consumir de dos a cuatro tazas diarias de té e infusiones de plantas medicinales, aún necesitará beber al menos seis tazas de agua pura por todas las razones que he descrito en esta sección.
LO QUE NO SE DEBE COMER
Bien esté procurando evitar un ataque al corazón, o recuperándose de uno de ellos, le ofrezco a continuación una lista de alimentos que nunca debería llevarse a la boca. Les llamo la «Docena Mortal». Algunos ya los he tratado en este capítulo, y le presento los demás seguidos por un breve comentario:
**• Carnes procesadas y productos del cerdo.** Estas carnes encabezan mi lista porque son comunes de la dieta normal en los Estados Unidos y son en extremo dañinas para la salud.
**• Mariscos de concha dura o peces sin aletas ni escamas, como el siluro, el tiburón y la anguila.** ¿Quiere esto decir que digo au revoir y sayonara a la langosta Termidor y los camarones tempura? Exactamente: eso es lo que quiero decir.
**• Aceites hidrogenados.** La margarina y la manteca deben ser tabú, así como cualquier torta, pastel, postre o cualquier otra cosa que tenga en la etiqueta las palabras _hidrogenado_ o _parcialmente hidrogenado_. Los aceites y grasas hidrogenados contienen ácidos grasos trans, que pueden conducir a la inflamación de las arterias, uno de los mayores factores de riesgo de las enfermedades cardiovasculares
**• Edulcorantes artificiales.** El aspartame (que se encuentra en el NutraSweet y el Equal), la sacarina (Sweet 'N Low), y la sucralosa (Splenda) son sustancias químicas cien veces más dulces que el azúcar. ¿Producen cáncer? Es difícil asegurarlo. Durante un tiempo, las bebidas de dieta y la goma de mascar sin azúcar, endulzadas con sacarina, llevaban la advertencia: «Consumir este producto puede dañar su salud». Pero en el 2000 la FDA borró a la sacarina de la lista de carcinógenos reconocidos. Para mí, sin embargo, los edulcorantes artificiales deben ser siempre evitados, vengan en envases azules, rosados o amarillos.
**• Harina de trigo blanca.** La harina de trigo blanca no es una sustancia química problemática como los edulcorantes artificiales, pero carece virtualmente de elementos nutritivos y no ayuda a su salud.
**• Azúcar blanca.** Si buscaba un culpable de todos esos «salvavidas» que cuelgan sobre la línea de la cintura, su búsqueda acabó aquí.
**• Gaseosas.** No son más que azúcar líquida. Una CocaCola o Pepsicola de veinte onzas equivale a comer quince cucharaditas de azúcar. Las de dieta, cargadas de edulcorantes artificiales, son aun peores.
**• Jarabe de maíz.** Este no es más que otra versión del azúcar, igualmente dañino para usted, si no peor.
**• Leche desgrasada, pasteurizada y homogeneizada.** Como ya he dicho, la leche orgánica, no homogeneizada, es mejor, y la leche de cabra, la mejor de todas.
**• Proteína de soya hidrolizada.** La proteína de soya hidrolizada se encuentra en los productos que imitan la carne, como la imitación de carne de cangrejo. Yo valoraría la proteína de soya hidrolizada del mismo modo que las carnes curadas con nitritos: aléjese de ella. Siempre le sentará mejor comer carnes orgánicas.
**• Sabores y colores artificiales.** Ni en la mejor de las circunstancias son buenos para usted, y mucho menos si está recuperándose de un infarto.
**• Cualquier cosa frita en grasas insalubres.** Las comidas fritas y las víctimas de ataques al corazón se llevan tan bien como el aceite y el vinagre.
¿CUÁLES ALIMENTOS SON EXTRAORDINARIOS, PROMEDIO O PROBLEMÁTICOS?
He preparado una lista que abarca alimentos clasificados en orden descendente según sus propiedades para la salud. Los que la encabezan son más sanos que los que se encuentran al final.
A los mejores alimentos que podemos comer los considero extraordinarios.
Dios los creó para que los comiéramos y le proporcionarán la mejor oportunidad posible de no vivir una vida larga y feliz. Si usted está batallando con una enfermedad cardiovascular es recomendable que consuma más del setenta y cinco por ciento de las veces alimentos de la categoría extraordinaria.
Si padece una dolencia cardiovascular o se está recuperando de ella, los que he situado en la categoría promedio deben constituir menos del veinticinco por ciento de su dieta cotidiana y ser consumidos sólo esporádicamente.
Un paciente con enfermedades del corazón debe evitar totalmente los comestibles de la categoría problemática.
Para encontrar una lista completa de alimentos extraordinarios, promedio y problemáticos visite: www.BiblicalHealthInstitute.com.
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE: COMA
PARA VIVIR **
• Coma sólo los alimentos que Dios creó.
• Coma sus alimentos en una forma saludable para el cuerpo.
• Consuma alimentos ricos en ácidos grasos omega-3.
• Consuma alimentos ricos en fibra.
• Incremente el consumo de frutas y vegetales crudos.
• Incremente el consumo de alimentos ricos en ácido fólico, vitaminas B12 y B6 como verduras de hojas verde oscuro, carnes rojas de reses alimentadas con pasto y huevos ricos en omega-3.
• Haga un ayuno semanal.
• Beba un mínimo de ocho vasos de agua pura diarios.
• Evite alimentos altos en azúcar.
• Evite los alimentos que contienen grasas hidrogenadas.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios del comer para vivir, por favor pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable._
[LLAVE # 2
_Complemente su dieta con alimentos integrales,
nutrientes vivos ysuperalimentos_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch00954)
Si le preguntara a un médico si tomar multivitaminas y suplementos nutricionales es importante para la prevención o el tratamiento de las enfermedades cardiovasculares, probablemente respondería que una dieta balanceada es importante, que ningún régimen ni plan nutricional puede «curar» dolencias cardiacas, y que tomar suplementos vitamínicos o minerales no debe nunca considerarse como un sustituto de la atención médica. El galeno sería respaldado por la American Heart Association, que recomienda comer una variedad de alimentos con moderación en lugar de tomar suplementos, aunque hace una excepción con los ácidos grasos omega-3.
Esta cautelosa recomendación es una forma de medicina defensiva que no reconoce el potencial enorme —ni las evidencias a favor— de los suplementos nutricionales. Tomar vitaminas y suplementos puede hacer más que protegerle: puede mantenerle en el partido contra las enfermedades cardiovasculares, y quizás ayudarle a derrotarlas.
Uno de los mejores suplementos específicos para el corazón que usted puede tomar se llama coenzima Q10, la cual mejora el suministro de oxígeno al miocardio al tiempo que respalda la función cardiaca y la fuerza muscular. La coenzima Q10 se encuentra en todas las células del cuerpo y facilita las actividades de las enzimas, pero cuando al paciente se le prescribe algún fármaco con estatinas como el Lipitor, los niveles de coenzima Q10 se agotan rápidamente. Para contrarrestar los onerosos efectos colaterales de estas potentes drogas reductoras del colesterol, recomiendo tomar tres veces al día cincuenta miligramos de coenzima Q10.
Me gustaría aclarar desde ahora que no soy de los que creen que es posible revertir las afecciones cardiovasculares con un frasco de píldoras. Después de años de estudio de la medicina y la nutrición naturopáticas, puedo entender mejor que la mayoría que los suplementos dietéticos son únicamente lo que dicen ser: suplementos, y no un sustituto para una dieta inadecuada y un estilo de vida malsano.
Sin embargo, los suplementos nutricionales, nutrientes vivos y superalimentos constituyen una parte importante de _La receta del Gran Médico para un corazón saludable._ Encabezando mi lista figura una multivitamina fermentada con un cultivo prebiótico vivo. Estas multivitaminas de alimentos enteros o vivas contienen compuestos diversos, como ácidos orgánicos, antioxidantes y nutrimentos clave. Su costo de producción es mayor, ya que los ingredientes —frutas, vegetales, vitaminas, minerales y otros—, se exponen a un proceso de fermentación similar al digestivo del cuerpo, pero bien valen la diferencia de precio.
Si usted está recibiendo actualmente medicamentos para alguna condición cardiaca las investigaciones sugieren que podría padecer una deficiencia de ácido fólico, vitaminas B12 y B6 y minerales clave como el magnesio, vanadio y cromo. Como ya mencioné en el capítulo anterior el ácido fólico y las vitaminas B12 y B6 desempeñan un papel clave en la reducción de niveles elevados de homocisteína, proteína que contribuye al proceso de obstrucción de las arterias en la ateroesclerosis. Los suplementos nutricionales a base de alimentos enteros que contengan estos importantes nutrientes pueden encargarse de la situación, así como equilibrar sus niveles de azúcar en la sangre, lo cual mejora el metabolismo.
Además de tomar una multivitamina de alimentos enteros o viva, usted debe incorporar a su plan de nutrición diario los siguientes suplementos:
**_1. Aceite de hígado de bacalao, con alto contenido de omega-3_**
¿Recuerda cuando le hablé de comer pescado capturado su medio natural, por su alto contenido de ácidos grasos omega-3? Quienes padecen enfermedades cardiovasculares no sólo tienen altos niveles de grasas en su sangre, sino que también van por la vida con bajos niveles del colesterol «bueno» HDL. Tomar a diario cucharadas o cápsulas líquidas de aceite de hígado de bacalao rico en omega-3 ayuda a los pacientes cardiacos a evitar tener altos niveles de grasa —los llamados triglicéridos— en los glóbulos de su sangre, y puede inhibir el avance de la ateroesclerosis.
Si no soportó la idea de sorber aceite de hígado de bacalao rico en omega-3, también puede tomar este importante nutrimento en cápsulas líquidas, fáciles de tragar. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**_2. Superalimentos verdes_**
La renuencia a comer vegetales, y especialmente verduras, persiste en numerosas personas, incluso adultas. Saben que _deben_ comer más vegetales, pero ven las ensaladas y porciones de vegetales como coloridos ornamentos del plato principal, o sea, la carne y las papas.
Son muchos los que sienten esa aversión: el Departamento de Agricultura de los Estados Unidos calcula que más del noventa por ciento de la población del país no come las tres a cinco porciones diarias de vegetales que se recomiendan.
Si usted ha sufrido un infarto cardiaco me aventuraría a adivinar que no siente predilección por los vegetales, especialmente los de hojas verde oscuro. Si le cuesta trabajo motivarse para comer vegetales, conozco una forma en que su cuerpo puede recibir más alimentos verdes, de suma importancia, debido a que contienen nutrientes que no se encuentran en la típica dieta baja en carbohidratos.
Le recomiendo el consumo de superalimentos verdes en polvo o en cápsulas. Lo único que tiene que hacer es mezclar el polvo con agua o su jugo favorito, o puede optar por tomar un puñado de cápsulas. Un buen suplemento de alimentos verdes es una mezcla orgánica certificada de vegetales verdes secos, vegetales fermentados, vegetales marinos, además de microalgas como la chlorella y la spirulina, y semillas y granos germinados. Cuando usted toma un batido de superalimentos verdes está bebiendo una de las pócimas más densas en nutrientes que existan en este verde planeta y, sin embargo, contiene menos de la vigésima parte de las calorías de una combinación Big Mac de Mc Donald's. Los superalimentos son asimismo una excelente fuente de ácido fólico, que le protege contra el desarrollo de enfermedades del corazón
**_3. Mezcla de fibras de alimentos enteros con semillas de linaza_**
Como mencioné en la llave # 1, la fibra puede ser el mejor amigo de un paciente cardiovascular, puesto que reduce los niveles del colesterol «malo» LDL. Consumir una adecuada cantidad de fibra también mejora la regularidad excretiva, algo que ayuda a eliminar del cuerpo las toxinas de manera eficiente. Como la mayoría de nosotros sólo recibimos en nuestras dietas diarias la quinta parte de la cantidad óptima de fibra, recomiendo tomar suplementos de fibra de alimentos enteros. Busque uno que suministre a su organismo una fuente vegetal altamente aprovechable de fibra dietética.
Si anda buscando un producto de fibra adecuado para usted, escoja alguna marca que utilice semillas, granos y legumbres orgánicos, fermentados o germinados para facilitar la digestión. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**_4. Probióticos_**
Por definición, los probióticos son microbios vivos para alimentación directa, o DFM, que promueven el crecimiento de bacterias beneficiosas en el tracto gastrointestinal. Si está experimentando un constante malestar intestinal, complemente su dieta con probióticos. Los más efectivos contienen microorganismos con base en los suelos (SBO), lactobacilos y bífidobacterias, así como la levadura beneficiosa _saccharomyces boulardii._ (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**_5. Enzimas_**
**C** uando uno ingiere alimentos crudos como las ensaladas y frutas, consume las enzimas que contienen. En cambio, si come alimentos cocinados o procesados como los que sirven en los restaurantes, el páncreas debe producir las enzimas necesarias para digerir la comida. La demanda constante de enzimas ejerce una excesiva presión sobre el páncreas, que debe producir más para satisfacer la demanda. Cuando no recibimos los niveles adecuados de enzimas que se encuentran en los alimentos crudos o fermentados —o en los suplementos— somos susceptibles a la acidez estomacal, una acumulación excesiva de gases, hinchazón, diarrea, estreñimiento y poca energía ¿Le parecen familiares esos síntomas?
Si está tratando de minimizar el consumo de alimentos ricos en enzimas como las bananas, aguacate, semillas y uvas —que también tienen alto contenido de azúcar— puede tomar como suplemento enzimas digestivas de origen vegetal, a fin de ayudar a la digestión de los alimentos. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE:
COMPLEMENTE SU DIETA **
• Tome con cada comida una multivitamina viva basada en alimentos enteros.
• Consuma cada día con la cena entre una y tres cucharaditas, o de tres a nueve cápsulas, de aceite de hígado de bacalao (o pescado) rico en omega-3.
• Tome dos veces al día (al levantarse y antes de acostarse) una mezcla de fibras basadas en alimentos enteros y superalimentos verdes con betaglucanos de fibra soluble de avena.
• Tome con cada comida un producto que contenga antioxidantes y energéticos con vitaminas B, ácido fólico y cromo.
• Para mejorar su digestión, puede tomar con cada comida una fórmula de enzimas y probióticos.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para complementar su dieta con suplementos basados en alimentos enteros, nutrientes vivos y superalimentos, por favor pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable._
[LLAVE # 3
_Practique una higiene avanzada_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch01161)
Soy el primero en admitir que sumergir la cara en un aguamanil con una solución facial, lavarse bajo las uñas con un jabón especial, o lavarse las manos después de ir al sanitario no parece tener mucho que ver con las enfermedades cardiovasculares. Pero hay un aspecto de la buena higiene que resulta relevante para esta discusión, y tiene que ver con el vínculo entre las infecciones respiratorias y varios problemas del corazón.
Primero, regresemos a la clase de biología de la escuela secundaria.
Cada día de su vida, su cuerpo rechaza millones de gérmenes que afectan su sistema inmunológico y le hacen más susceptible a los problemas de la salud. En días alternos de su vida (o así parece) ocurren pequeños percances: un golpe en el dedo gordo del pie, una picadura de mosquito, la piel quemada por el sol, una distensión muscular o una cortadura mientras se afeitaba las piernas (las damas) o la barba (los caballeros). Cada vez que sucede algo de eso, el cuerpo despliega una defensa instantánea, enviando sus células y sustancias químicas naturales a atacar a esos desagradables gérmenes de la gripe o a reparar la pequeña brecha en su piel. En términos científicos esta respuesta se conoce como _inflamación_.
«La inflamación se ha convertido en una de las áreas más estudiadas de la investigación médica», escriben Christine Gorman y Alice Park en la revista _Time_. «Apenas pasa una semana sin que se publique otro estudio para revelar una nueva manera en la que la inflamación crónica perjudica al cuerpo. Esta respuesta desestabiliza los depósitos de colesterol en las arterias coronarias, lo cual conduce a ataques cardiacos...»
La mayoría piensa que inflamación es lo que le ocurre a su espalda después de pasar la mañana del sábado sacando malezas en el jardín. En realidad, la inflamación ocurre también internamente. Cuando los virus invaden el sistema respiratorio, por respirar toxinas en el aire o por engullir un perro caliente en mal estado comprado a algún vendedor callejero, el cuerpo lanza un contraataque que opone desechos a los intrusos y repara cualquier órgano infectado.
Cuando la inflamación ocurre, el hígado produce una prote-ína conocida como proteína de alta sensibilidad Creactiva. Esta sustancia química natural es liberada en el torrente sanguíneo para ayudar al organismo a combatir los gérmenes de la gripe, por ejemplo, o repararse a sí mismo después de sacarse una astilla del dedo índice. Sin embargo, los investigadores médicos han constatado que los niveles de la proteína Creactiva pueden ser una señal de advertencia de un infarto cardiaco inminente. Esto resulta significativo, pues sólo un cincuenta por ciento de las personas que sufren ataques al corazón en los Estados Unidos tienen niveles de colesterol normales o moderadamente elevados. Los altos niveles de la proteína Creactiva pueden explicar además por qué personas que no tienen el colesterol alto pueden desarrollar enfermedades del corazón. Mi llave # 3: «Practique una higiene avanzada», puede proteger a su cuerpo de desarrollar una inflamación crónica, lo cual disminuirá sus niveles de proteína Creactiva, así como su riesgo de desarrollar enfermedades cardiovasculares.
¿Qué quiero decir con la frase «higiene avanzada»?
Me alegro de que se lo pregunte, pues creo firmemente en la necesidad de protegerme de gérmenes dañinos, y he estado practicando rutinariamente un protocolo higiénico durante más de una década. Soy testigo de los buenos resultados que ha tenido en mi vida: En muchos años no he sufrido ni resfriados, ni infecciones de los senos faciales, ni enfermedades respiratorias agudas, y tengo un corazón sano. Sigo un programa desarrollado por un científico australiano, el doctor Kenneth Seaton, que descubrió que los problemas de la nariz, la garganta, los oídos y la piel están en muchos casos vinculados al hecho de que los seres humanos a lo largo del día se tocan la nariz, los ojos y la boca con sus dedos, bajo cuyas uñas se alojan gérmenes.
En términos médicos, esto se llama autoinoculación.
¿Cómo se ensucian las uñas? Por medio del contacto de las manos con superficies o con otras personas.
Si usted creía que la mayoría de los gérmenes se propagan por exposición al aire —como que alguien estornude en su mesa— se equivoca. «Los gérmenes no vuelan, andan de polizontes», declara el doctor Seaton, y tiene razón. Él cree que una vez que esos gérmenes polizontes suben a bordo suyo, se ocultan a hibernar alrededor de las uñas de los dedos de las manos, por más que se las corte.
**_Cómo lavarse las manos_**
1. Mójese las manos con agua tibia. No tiene que estar hirviendo.
2. Aplique abundante jabón en las palmas de ambas manos. Mejor si es un jabón semilíquido en el que pueda hundir las uñas.
3. Frótese vigorosamente las manos una con la otra y restriegue todas las superficies. No descuide la piel entre los dedos, y haga que el jabón penetre en las uñas.
4. Frote y restriegue durante quince a treinta segundos, o el tiempo que tarde en cantar «Feliz Cumpleaños».
5. Enjuáguese bien y séquese las manos con un papel toalla seca o una toalla limpia. Si se encuentra en un sanitario público, sería conveniente cerrar la llave del agua con una toalla. Una idea todavía mejor sería usar la misma toalla para abrir la puerta, pues ese picaporte es el primer lugar que tocan quienes no se lavan las manos después de ir al sanitario.
6. Tenga siempre en su cartera o billetera toallitas higiénicas de bolsillo por si no encuentra agua o jabón en el sanitario. Aunque no son lo mejor, son mejores que nada.
¿Cómo llegan los gérmenes a sus manos? Cuando les estrechamos las manos a otros, o tocamos las cosas que ellos han tocado: pasamanos, perillas, carritos de supermercado, billetes de banco, monedas y alimentos. Sé que esta no es una conversación apropiada para la hora de la cena, pero para mí practicar una higiene avanzada se ha convertido en una rutina cotidiana. Como sé que el noventa por ciento de los gérmenes se domicilian alrededor de las uñas de mis manos, uso un jabón semilíquido cremoso y rico en aceites esenciales.
Cada mañana y en la noche antes de acostarme comienzo metiendo ambas manos en un aguamanil lleno de ese jabón semilíquido y hundo las uñas de las manos en la crema. Luego froto esa crema jabonosa especial en las yemas de mis dedos, las cutículas y las uñas por quince a treinta segundos. Terminado eso, me lavo bien las manos durante quince segundos, antes de enjuagármelas con agua corriente. Una vez que mis manos están bien limpias, tomo otro poco de jabón y me lavo la cara.
**_Cuándo lavarse las manos_**
• Después de ir al baño
• Antes y después de insertar o retirar los lentes de contacto
• Antes y después de preparar los alimentos
• Antes de comer
• Después de estornudar, toser o sonarse la nariz
• Después de tocar a su mascota
• Después de manipular dinero
• Después de cambiar un pañal
• Después de limpiarle la nariz a un niño
• Después de manipular basura
• Después de limpiar el inodoro
• Después de estrechar muchas manos
• Después de hacer las compras en el supermercado
• Después de asistir a un evento en un teatro público
• Antes y después de mantener relaciones sexuales.
Mi segundo paso comprende un procedimiento al que llamo «inmersión facial».
Lleno mi aguamanil o un tazón grande y limpio con agua tibia, no caliente. Le añado una o dos cucharadas de sal común de mesa y dos goteros de una solución facial a base de minerales. Mezclo bien todo con las manos y luego me inclino y sumerjo mi cara en esa mezcla limpiadora, abriendo varias veces los ojos para permitir que también se limpien las membranas oculares. Después de una pausa para respirar vuelvo a meter la cara con los ojos cerrados y la boca fuera del agua, haciendo burbujas a través de la nariz. A esto le llamo «bucear con máscara de oxígeno en una palangana».
Mis dos últimos pasos de higiene avanzada consisten en aplicarme gotas muy diluidas de peróxido de hidrógeno y minerales en los oídos durante treinta a sesenta segundos, para limpiar el canal auditivo; y cepillarme los dientes con una solución dental a base de aceites esenciales, a fin de eliminar de mi boca microbios nocivos. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver los productos de higiene avanzada recomendados.)
Hablando de higiene dental, varias investigaciones médicas han descubierto que las personas con enfermedades de las encías tienen el doble de probabilidades de sufrir enfermedades coronarias. Una hipótesis es que las bacterias de la boca ingresan al torrente sanguíneo y afectan el corazón adhiriéndose a las placas de grasa de las arterias coronarias. Cepillarse bien los dientes y practicar regularmente una higiene avanzada requiere disciplina; usted debe recordárselo hasta que se convierta en una rutina. Encuentro más fácil seguir estos pasos en la mañana cuando estoy fresco que en la noche cuando me siento cansado y con sueño, aunque me esfuerzo por practicar este método por la mañana y por la noche y casi nunca fallo. En cualquier caso, sé que me toma sólo tres minutos completar todos los pasos, y esos podrían ser los mejores tres minutos del día para su corazón.
Por último, hay una proteína que flota en su torrente sanguíneo, de la cual debe estar enterado, pues es la más abundante. Se llama albúmina, y se ocupa de transportar en su sangre hormonas y nutrientes, y recoger los desechos para expulsarlos. Como un camión de volteo repleto de material de relleno, la albúmina conduce las células tóxicas y los desechos hacia el hígado para que sean allí degradados y expulsados del cuerpo.
Los investigadores médicos han descubierto otra interesante noticia acerca de la albúmina: mantener sus niveles altos puede ser crucial para la prevención de enfermedades como las cardiovasculares. El doctor Seaton señala que al someter a análisis de sangre a pacientes de infartos cardiacos, se les encontraron bajos niveles de albúmina. Todavía se investiga qué relación tiene esto con las enfermedades del corazón, pero los médicos creen que quienes tienen bajos niveles de albúmina en el momento de sufrir un ataque cardiaco podrían correr un riesgo mayor.
El doctor Seaton está seguro de que los niveles de esta prote-ína se relacionan con la higiene, no con la dieta, lo cual significa que pueden optimizarse practicando una higiene avanzada. Ellos destacan la importancia de esta llave como parte de _La receta del Gran Médico para un corazón saludable._
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE:
PRACTIQUE UNA HIGIENE AVANZADA **
• Hunda sus dedos en una solución de jabón semilíquido con aceites esenciales y lávese las manos con frecuencia, prestando especial atención a eliminar los gérmenes acumulados bajo sus uñas.
• Limpie a diario sus fosas nasales y las membranas mucosas de los ojos, mediante una inmersión facial.
• Limpie sus canales auditivos al menos dos veces por semana.
• Utilice a diario una solución dental basada en aceites esenciales para eliminar los gérmenes de sus dientes, encías y boca.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para practicar una higiene avanzada, por favor, pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable_.
[LLAVE # 4
_Acondicione su cuerpo con ejercicios
yterapias corporales_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch01369)
Siendo un parvulito, yo corría como el Correcaminos por toda la casa, de una habitación a otra, hasta que mis padres me metían en la cama. Cumplí dos años en 1977, el año en el que recorría Estados Unidos la fiebre de salir a correr. En los alrededores de mi casa, las aceras estaban llenas de gente de la generación de posguerra, por entonces en sus veinte o sus treinta, corriendo en holgadas camisetas, pantalones cortos de colores fosforescentes y zapatillas Nike de la primera generación. (Papá y mamá no siguieron esa moda. El _jogging_ no era para ellos.) El flautista de Hamelin que los guiaba era Jim Fixx, autor de _The Complete Book of Running_. Después de su publicación en 1977, el libro de Fixx, que proclamaba los beneficios de correr para la salud, se convirtió en el más vendido de todos los tiempos en la categoría noficción. Fixx se llevó el mérito de poner a todos a correr.
Siete años después, Jim Fixx cayó abatido por un ataque al corazón mientras corría por un sombreado camino en Vermont, y murió. Por ironías de la vida, tenía sólo cincuenta y dos años. Aunque ya hace dos décadas de su fallecimiento, su nombre todavía aparece en crucigramas (Pista: Corredor fallecido. Cuatro letras). La triste historia de Jim Fixx es también una advertencia: en lo que respecta a la salud cardiovascular un ejercicio vigoroso puede resultar demasiado estresante para un corazón aquejado de una enfermedad coronaria oculta; pero si usted lleva una vida sedentaria, también podría estar condenado. El corazón es el más importante de los músculos, y el ejercicio preserva y protege la calidad de los vasos sanguíneos y previene los ataques cardiacos. Un ejercicio frecuente y regular casi siempre le hace bien al corazón. Usted vivirá más, reducirá su tensión arterial y mejorará su circulación.
En lo que respecta a las enfermedades cardiovasculares y la llave # 4, _Acondicione su cuerpo con ejercicios y terapias corporales_ , no debe esperar a tener dolor en el pecho, colesterol alto o sobrevivir a un primer ataque cardiaco que le llame la atención. Cabe la posibilidad de que no haya una próxima vez. Y si ya sobrevivió a un infarto al miocardio, una serie de estudios indican que las personas que comienzan a desarrollar actividad física frecuente después de un bloqueo coronario tienen mejores tasas de supervivencia y mejor calidad de vida.
Habiendo sido entrenador certificado de forma física, tengo alguna experiencia en este campo. Si usted fuera mi cliente, y su médico le hubiera recomendado comenzar un programa de ejercicios, yo le iniciaría en el de forma física funcional. Este método de ejercicios suaves aumenta su ritmo cardiaco, fortalece los principales músculos del cuerpo y ejercita el sistema cardiovascular mediante la realización de actividades de la vida real, en situaciones reales.
El método de forma física funcional puede realizarse sin equipos o utilizando pesas de mano, minitrampolines y balones de estabilidad. Los gimnasios LA Fitness, Bally Total Fitness y los locales de la Asociación de Jóvenes Cristianos, YMCA, ofrecen clases de forma física funcional y cuentan con el equipo apropiado. Le pedirán que haga cuclillas con las piernas separadas o unidas, y con una atrás y la otra delante. También le pedirán que haga extensiones, planchas contra una pared, y un ejercicio llamado «supermán» que consiste en acostarse en el suelo y levantar totalmente extendidos el brazo derecho y la pierna izquierda. Lo que no le pedirán es que haga ejercicios de impacto fuerte como los que se ven en esas enérgicas sesiones de ejercicios aeróbicos. (Para más información sobre el método de forma física funcional visite www.GreatPhysiciansRx.com.)
Yo también incorporaría estas formas de ejercicios y terapias corporales:
**Entrenamiento con pesas.** Esta es una forma de ejercicio anaeróbico que fortalece y desarrolla el tejido muscular. Levantar pesas en su club de salud local dos veces a la semana durante treinta minutos es un buen comienzo para su corazón, aunque tres veces por semana sería superior, si está cambiando su estilo de vida. Pero no exagere: deje para los más jóvenes esos discos demasiado pesados.
**Déjese «rebotar».** Los minitrampolines son excelentes para hacer ejercicios de poco impacto y quemar más calorías que si saliera a correr. Uno de los beneficios especiales de los minitrampolines es su capacidad para mejorar la circulación en el sistema linfático.
**Practique ejercicios de respiración profunda.** Casi siempre cuando respiramos no llenamos totalmente de aire el diafragma, por desconocer que los pulmones se extienden hasta la altura de las primeras costillas. Le recomiendo sentarse en una silla y concentrarse en llenar totalmente los pulmones. Cuente hasta cinco mientras aspira, y retenga el aliento unos segundos antes de exhalar por la boca, durante unos cuantos segundos más. Visualice el diafragma moviéndose hacia arriba y hacia abajo mientras los pulmones se expanden. Las técnicas de respiración profunda son herramientas poderosas y sedantes para aplacar su sistema nervioso, reducir su ritmo cardiaco y restaurar su energía.
**Camine.** Caminar es especialmente recomendable para aquellos que no han hecho ejercicios durante años. Esta ruta de bajo impacto hacia la forma física somete a un esfuerzo moderado a las caderas y el resto del cuerpo, y cuando se hace rápido, obliga al corazón a trabajar más y genera un mayor gasto energético.
Lo mejor de todo es que usted puede caminar cuando mejor le convenga: antes del trabajo o a la hora del almuerzo; antes o después de cenar. Usted mismo fija su ritmo y decide cuánto quiere dedicar a este ejercicio. Caminar es un ejercicio social que le permite sostener mientras camina una conversación civilizada con un amigo o un ser querido.
**Váyase a la cama temprano _._** El sueño es una terapia corporal que escasea bastante en nuestros días. El «déficit de sueño» nacional significa que nos ocupamos con demasiadas actividades desde el momento en que despertamos hasta que nos metemos en la cama dieciséis, diecisiete o dieciocho agotadoras horas después. Los estadounidenses adultos duermen menos de siete horas diarias por noche, unas dos menos que nuestros bisabuelos cien años atrás. Esto no puede ser beneficioso para el corazón.
¿Cuántas horas duerme usted cada noche? El número mágico, según los expertos es ocho horas. Eso se debe a que cuando a uno se le permite dormir cuanto quiera en un ambiente controlado (como un laboratorio de investigaciones del sueño) tiende a dormir naturalmente ocho horas en un período de veinticuatro.
Es un buen momento para hablar de la apnea del sueño y cómo se relaciona con la salud cardiovascular. La apnea del sueño es una condición grave y potencialmente peligrosa para la vida que se caracteriza por breves interrupciones de la respiración mientras dormimos. Con frecuencia quienes la padecen no están conscientes de que roncan durante la noche, lo que resulta en una súbita interrupción de la respiración por un lapso más o menos largo. Sólo podemos imaginar el daño que ocasionan los bajos niveles de oxígeno al corazón y los pulmones, por no hablar de la preocupación de su cónyuge. Si usted es de los que cuando roncan tumban de la pared las fotos familiares, o su pareja le ha comentado sobre las pausas momentáneas en su respiración, debe buscar atención médica cuanto antes. Puede que para sobrevivir a sus noches necesite una máquina especial de aire a presión.
**Descanse el séptimo día.** Además de dormir lo suficiente, el cuerpo humano necesita un tiempo de descanso cada siete días, una pausa para reponerse. Y esto se logra tomando un receso en la «carrera de ratas» los sábados y domingos. Dios creó la tierra y los cielos en seis días y descansó el séptimo, dándonos un ejemplo y un recordatorio de que necesitamos una tregua en nuestras labores. Todos deberíamos hacer como los atletas de triatlón y otros de alto rendimiento, que se aseguran de descansar un día a la semana.
De lo contrario seremos candidatos seguros a fundirnos.
**Deje que el sol brille.** Puede que usted no vea una clara correlación entre solearse y la salud del corazón, pero déjeme explicarle algo.
Cuando expone su cara y su piel a la luz solar, su piel sintetiza vitamina D a partir de los rayos solares ultravioletas. El cuerpo necesita la vitamina D, que no es en realidad una vitamina, sino una hormona clave que ayuda a regular la salud de más de treinta tejidos y órganos diferentes, entre ellos el corazón. Le recomiendo exponerse al astro rey al menos quince minutos diarios, a fin de incrementar los niveles de vitamina D en su cuerpo.
**Dése gusto con la hidroterapia.** La hidroterapia toma la forma de baños, duchas, lavados y envolturas que emplean agua caliente y fría. Por ejemplo, al levantarme en la mañana me doy una ducha caliente, pero luego abro la llave del agua fría y la dejó correr sobre mí cerca de un minuto. Esto me vigoriza. El agua fría estimula el cuerpo y el aprovechamiento del oxígeno por las células, en tanto que el agua caliente dilata los vasos sanguíneos, mejorando la circulación, y favoreciendo el transporte de más oxígeno al cerebro.
Tomar un baños sauna o de vapor son otras formas de hidroterapia que mejoran la circulación cardiovascular. Le recomiendo especialmente que busque la forma de añadir estas terapias corporales a su horario semanal.
**Por último, mímese a sí mismo con la aromaterapia y la terapia musical.** En la aromaterapia, puede absorber, al frotarlos sobre su piel y sus poros, aceites esenciales extraídos de plantas, flores y especias, o simplemente inhalando sus fragancias. El uso de esos aceites esenciales no reparará milagrosamente arterias coronarias bloqueadas, pero le proporcionará bienestar emocional. Pruebe a frotar en las palmas de sus manos unas gotas de mirto, cilantro, hisopo, gálbano o incienso. Luego ahueque las manos, acerque la boca y la nariz e inhale. Una aspiración profunda revitalizará su espíritu.
Y lo mismo resultará de escuchar música suave, relajante y curativa. Sé cuáles son mis gustos en cuanto a esta terapia: música contemporánea de adoración y alabanza a Dios. Cualquiera sea su favorita, comprobará que escuchar una música capaz de levantar su estado de ánimo puede ayudar a sanar su cuerpo, su alma y su espíritu.
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE:
ACONDICIONE SU CUERPO CON EJERCICIOS Y TERAPIAS CORPORALES **
• Hágase el compromiso, y las citas, para hacer ejercicios durante al menos una hora, tres veces a la semana o más.
• Incorpore a su rutina cotidiana entre cinco y quince minutos del método de forma física funcional
• Dé una breve caminata y compruebe que al final del día se sentirá mucho mejor.
• Haga un esfuerzo consciente para practicar ejercicios de respiración profunda una vez al día.
Llene sus pulmones y retenga el aire durante varios segundos antes de exhalar lentamente.
• Váyase a la cama más temprano, prestando especial atención a cuántas horas duerme antes de medianoche. Esfuércese por dormir cada noche ocho horas. Recuerde que el sueño es, aparte de los nutrientes, lo más importante que puede incorporar a su régimen de salud.
• Finalice su próxima ducha cambiando la temperatura del agua a fresca (o fría) y permaneciendo bajo el chorro durante un minuto.
• El próximo sábado o el domingo, tómese el día de descanso, dedíqueselo al Señor y haga algo entretenido y relajante que no haya hecho en mucho tiempo. Procure en su día de descanso no trabajar, hacer compras ni diligencias. Confíe en que Dios hará más con Sus seis días que usted con siete.
• En su próximo receso laboral, salga y siéntese afuera mirando al sol. Báñese con sus rayos durante diez o quince minutos.
• Incorpore a su vida diaria aceites esenciales aromáticos.
• Ponga música de adoración en su hogar, su automóvil, o su iPod. Concéntrese en el plan de Dios para su vida.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para acondicionar su cuerpo con ejercicios y terapias corporales, por favor pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable_.
[LLAVE # 5
_Reduzca las toxinas en su ambiente_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch01577)
El doctor Dobson, que cumplió setenta y cinco años en la primavera boreal del 2006, tiene algo a favor suyo y de su corazón: vive en un lugar alto.
Colorado Springs, en Colorado, adonde se mudó el doctor al año siguiente de su infarto cardiaco de 1990, se halla en la cordillera frontal de las Montañas Rocosas, a mil ochocientos cuarenta metros sobre el nivel del mar. Vivir en la altura obliga al corazón a un buen ejercicio y habilita al sistema cardiovascular para adaptarse a niveles más bajos de oxígeno. Esto me pone un poco envidioso, porque vivo en un estado donde la «montaña» más alta, Britton Hill, tiene cien metros de altura, y el pico más alto cerca de mi casa es el Harriet Himmel Gilman Theater, en el área comercial de West Palm Beach.
El doctor Dobson cuenta con otro factor ambiental favorable a su corazón: ya no vive en el contaminado sur de California, donde un espeso manto de polución atmosférica suele flotar sobre Los Ángeles. La contaminación del aire es sin duda una de las causas de las enfermedades cardiovasculares, según un estudio realizado por la American Heart Association en relación con los efectos a largo plazo de una exposición crónica a ella. En otro estudio realizado en Atenas, Grecia, se compilaron los valores diarios de los principales contaminantes atmosféricos —humo, dióxido de azufre y emisiones de vehículos de motor— registrados en nueve estaciones, así como información relacionada con la cifra de muertes debidas a enfermedades cardiovasculares. Los investigadores pudieron extrapolar una significativa asociación entre las muertes por problemas cardiacos y varios contaminantes del aire, algo en lo que usted debería pensar si sufre de problemas del corazón.
Mudarse a las Rocosas, por ejemplo, no es una panacea, ya que la contaminación del aire _en interiores_ también puede ser peligrosa para su corazón.
La American Lung Association calcula que pasamos un noventa por ciento de nuestro tiempo en interiores, respirando aire acondicionado reciclado en el verano y aire calentado en el invierno, un aire en el que se arremolinan las partículas tóxicas. Las casas bien aisladas y las puertas y ventanas energéticamente eficientes de hoy día atrapan el aire «usado» lleno de dióxido de carbono, dióxido de nitrógeno y caspa de animales.
Estos contaminantes desencadenan y aceleran un estrechamiento de las arterias carótidas.
Le recomiendo que abra sus puertas y ventanas periódicamente, para refrescar el aire que respira, al margen de si la temperatura exterior es muy alta o muy baja. Unos minutos de aire fresco pueden obrar maravillas. También le recomiendo comprar un filtro de aire de alta calidad que elimine y neutralice las diminutas partículas de polvo, hollín, polen, moho y caspa, suspendidas en el aire. Yo he hecho instalar en nuestro hogar cuatro sistemas de tratamiento de aire de alta calidad, los cuales filtran las impurezas dañinas.
Las sustancias químicas y toxinas peligrosas para la salud cardiovascular están también presentes en nuestros alimentos. Si usted se hiciera un análisis de sangre y orina, los técnicos de laboratorio descubrirían decenas de toxinas en su sangre, incluyendo los bifenilos policlorinados (PCB), dioxinas, furanos, metales microscópicos, ftalatos, compuestos orgánicos volátiles (VOC) y cloro.
Algunas toxinas son solubles en agua, lo que significa que se eliminan rápidamente del organismo y no son dañinas. Desafortunadamente, muchas otras son solubles en grasa, lo que quiere decir que eliminarlas por completo de su sistema puede tomar meses o años.
Algunas de las más conocidas toxinas solubles en grasa son las dioxinas, los ftalatos y el cloro, y cuando no se eliminan permanecen almacenadas en sus tejidos adiposos, y obstruyen sus arterias.
La mejor manera de eliminar las toxinas solubles de su sangre es: beber más agua, lo que ayuda a excretarlas por medio de los riñones. También debe incrementar la cantidad de fibra en su dieta para eliminar toxinas a través de los intestinos; hacer ejercicios y sudar, a fin de eliminar las toxinas a través del sistema linfático; y practicar la respiración profunda para eliminarlas por medio de los pulmones.
Otra manera de reducir la cantidad de toxinas que usted ingiere es consumir carne y leche orgánicas o de reses criadas con pasto. Recuerde que la mayor parte de la carne de res, de aves y de cerdo que se produce a escala comercial actúa como un imán químico para las toxinas del medio ambiente, y por tanto nunca será tan saludable como comer la carne de reses alimentadas con pasto. Además, consumir vegetales orgánicos comprados en las tiendas de productos de salud, en quioscos a la orilla de la carretera y en mercados agropecuarios (siempre que sean cultivados en la localidad y no hayan sido rociados con productos químicos) le expondrá a menos residuos de pesticidas.
Debido a su alto contenido de mercurio, el atún enlatado (ahora también disponible en bolsas) es otro alimento que debe comer en cantidades mínimas, pese a que muchas dietas populares incluyen el atún y las ensaladas como plato frecuente para el almuerzo o la comida. Partículas metálicas de mercurio, plomo y aluminio se encuentran todavía en los tejidos grasos del atún, la aguja y la macarela. No obstante, ahora se encuentra en el mercado un atún enlatado que no sólo tiene bajo contenido de mercurio, sino que tiene alto contenido de ácidos grasos omega-3. Este atún puede consumirse con seguridad muchas veces a la semana, y contiene la misma proporción de grasas omega-3 (como el EPA y DHA) que peces como el salmón y las sardinas. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para más información sobre el atún con bajo contenido de mercurio y alto de omega-3.)
QUÉ BEBER
Ya he mencionado los beneficios para la salud de beber agua, pero en lo que respecta a reducir las toxinas en su ambiente, el agua es especialmente importante, gracias a su capacidad para eliminar del organismo toxinas y otros desechos metabólicos. Quienes padecen enfermedades cardiovasculares tienden a tener una mayor carga metabólica.
Nunca se debe subestimar la importancia de beber suficiente agua: ella es una fuerza vital presente en casi todos los procesos corporales, desde la digestión hasta la circulación sanguínea. Su corazón bombea la sangre con más eficiencia cuando usted está bien hidratado.
La respuesta a la necesidad de hidratación no debe ser cambiar el agua por gaseosas regulares o de dieta, ni bebidas como el café, el té o los jugos de frutas, aun cuando estos últimos pueden ser saludables. Las gaseosas normales contienen demasiado azúcar, y las de dieta, edulcorantes artificiales como el aspartame, acesulfame-K, o sucralosa. Si bien la Administración de Alimentos y Fármacos de EUA ha aprobado el uso de los edulcorantes artificiales en las bebidas (y en los alimentos) estos aditivos químicos pueden ser a largo plazo dañinos para su salud.
Nada supera al agua, un líquido creado por Dios para ser totalmente compatible con su cuerpo. Usted debe beber los proverbiales ocho vasos de agua al día.
Ya sé lo que está pensando: _Jordan, si bebo toda esa agua, nunca me podré alejar más de quince pasos de un sanitario._ Cierto, probablemente tendrá que triplicar sus viajes al baño, pero créame cuando le digo que si de verdad quiere bajar de peso, debe tomar en serio la necesidad de beber suficiente agua. No existe otra forma fisiológica de eliminar las reservas de grasa y las toxinas que se acumulan en su organismo.
Claro que no le recomiendo beber agua directamente del grifo. Casi todas las aguas comunales son tratadas rutinariamente con cloro o cloramina, sustancias químicas que son potentes bactericidas. He instalado en mi casa un sistema de filtración de toda el agua doméstica que elimina el cloro y otras impurezas _antes_ de que el agua entre en nuestras tuberías. Nicki y yo podemos confiar en abrir la llave y disfrutar los beneficios de un agua libre de cloro para beber, cocinar y bañarnos. Y como esta agua no tiene ningún regusto químico, es más apta para beber. (Una inversión más modesta sería comprar filtros más baratos para acoplarlos al grifo del fregadero, o una jarra con filtro de carbón por veinte dólares o menos.)
TOXINAS EN OTROS LUGARES DE SU AMBIENTE
Existen otras toxinas no directamente relacionadas con las enfermedades cardiovasculares, pero que es importante mencionar:
• **Plásticos.** Aunque si estoy lejos de casa bebo agua mineral envasada en botellas de plástico, me parece más seguro beberla en botellas o vasos de vidrio, debido a la presencia, en el primer caso, de dioxinas y ftalatos añadidos en el proceso de fabricación del plástico.
• **Productos de Limpieza para el Hogar.** Hoy en día la mayoría de los productos comerciales de limpieza para el hogar contienen sustancias químicas y disolventes potencialmente dañinos que exponen a las personas a los COV, o compuestos orgánicos volátiles, que pueden causar irritación en los ojos, la nariz y la garganta.
Nicki y yo hemos descubierto que los ingredientes naturales como el vinagre, el jugo de limón y el bicarbonato de sodio son perfectos para mantener nuestro hogar reluciente. En los supermercados y en las tiendas naturistas puede encontrar productos de limpieza naturales que no son abrasivos ni potencialmente peligrosos para su familia.
• **Productos para el cuidado de la piel y del cuerpo.** Las sustancias químicas tóxicas como los disolventes y ftalatos se encuentran en los lápices, acondicionadores y brillos labiales, en los tintes, aerosoles y champúes para el cabello, y en el jabón. Señoras: Cada vez que ustedes frotan sobre sus labios un lápiz labial, su piel absorbe inmediatamente esas toxinas. Como en el caso de los productos de limpieza para el hogar, en los mercados naturistas puede encontrar cosméticos naturales, aunque se venden cada vez más en las farmacias y tiendas de productos de belleza corrientes.
• **Dentífricos.** Los dentífricos comunes llevan una etiqueta con la advertencia de que en caso de tragar el producto accidentalmente se debe llamar al centro local de control de venenos ¿Qué quiere decir eso? Pues que la mayoría de los dentífricos comerciales contienen edulcorantes artificiales, nitrato de potasio, fluoruro de sodio y una buena cantidad de otras largas e impronunciables palabras. Una vez más, procure conseguir una versión natural y que no sea dañina para su salud.
Por último, permítame llevarme un momento a los señores a un lado, para hablar de la salud cardiovascular y... su vida sexual. Si usted ha estado teniendo problemas... usted sabe con qué, debe saber que la American Heart Association reporta una fuerte vinculación entre los problemas de las arterias coronarias y las disfunciones eréctiles.
Todo radica en la presión y la circulación de la sangre, así que ahí se lo dejo, por si andaba buscando algo más que seguir con vida para inspirarse.
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE: REDUZCA
LAS TOXINAS EN SU AMBIENTE **
• Preste atención al grado de contaminación dentro y fuera de su hogar, especialmente si está lidiando con una dolencia cardiovascular.
• Beba la cantidad recomendada de ocho vasos de agua diarios, o tres cuartos de litro por cada veinticinco kilos de peso corporal.
• Use siempre que pueda envases de vidrio en lugar de envases de plástico.
• Mejore la calidad del aire dentro de su vivienda abriendo las ventanas y adquiriendo un sistema de filtración de aire.
• Use en su hogar productos de limpieza naturales.
• Use también productos naturales para el cuidado de la piel, del cuerpo y el cabello, incluyendo los cosméticos y la pasta dental.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para reducir las toxinas en su entorno, por favor, pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable._
[LLAVE # 6
_Evite las emociones mortales_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch01785)
«Eso me tocó el corazón».
Eso Todos hemos escuchado —o musitado— esa frase en uno u otro momento, pero ¿ha pensado alguna vez por qué nos referimos al corazón cuando hablamos de nuestras emociones? O, dicho de otro modo, ¿por qué creemos que las emociones se centran en el corazón?
«Por supuesto que las emociones se centran en el cerebro», dice el escritor Joel Achenbach. «Pero cuando experimentamos una emoción poderosa —temor, enojo, pesar, amor— la adrenalina se derrama en la sangre, lo cual incrementa la presión de esta y acelera el ritmo cardiaco. Así que tiene perfecto sentido creer que el corazón controla las emociones. De otro modo, cuando juramos ante la bandera tendríamos que colocarnos la mano en la frente». No creo que nos gustaría hacer eso. A través de la historia, el corazón ha sido citado por salmistas, bardos y compositores de canciones para describir emociones que son casi indescriptibles.
«Jehová es mi fortaleza y mi escudo; en él confió _mi corazón_ , y fui ayudado, por lo que se gozó _mi corazón_ , y con mi cántico le alabaré», escribió David en el Salmo 28.7 (RVR60, las cursivas son mías).
Un par de miles de años más tarde, Shakespeare escribió: «La esperanza es la raíz de todos los quebrantos»; y no podemos olvidar que uno de los superéxitos de Elton John es «Don't Go Breaking My Heart». Las canciones sobre corazones rotos son una constante de la música pop, pero también nos recuerdan que existe una relación real entre las emociones y el estado del corazón. Mi amigo el doctor Don Colbert, autor del excelente libro _Emociones que matan_ , dice que los investigadores han vinculado de manera directa y científica las emociones mortales con las enfermedades cardiovasculares y la hipertensión, y que emociones como la ansiedad y el temor se han relacionado con palpitaciones cardiacas.
La ira, la acritud, la aprehensión, el nerviosismo, la ansiedad y la alarma _son_ emociones mortales, y cuando usted experimenta una de esas sensaciones —justificada o no— alteran la química de su músculo cardiaco y lo que le rodea. He aquí el cuadro que describe Colbert:
¿Qué sucede durante los períodos de estrés? Se libera adrenalina en el sistema, haciendo que el corazón se acelere y su ritmo sea más fuerte. La adrenalina también hace que las arterias coronarias y el corazón se dilaten en un esfuerzo por enviar más oxígeno y nutrientes al músculo cardiaco. Si las arterias coronarias están llenas de placa, o si las paredes arteriales están engrosadas debido al daño de la alta presión sanguínea, entonces en lugar de dilatarse, las arterias coronarias se contraen... El corazón necesita entonces latir más fuerte y rápido... El resultado final será angina, un infarto, arritmia o la liberación de un coágulo de sangre que bloquea totalmente un vaso sanguíneo, produciendo muerte súbita.
Si alberga en su interior sentimientos de enojo u hostilidad, se debe probablemente a que alguien le humilló o dijo de usted algo con mala intención. Créame, sé que hay palabras muy hirientes, capaces de rompernos el corazón. Pero no deben ser ocasión para bajarse del carro de una dieta sana o consolarse comiendo manjares grasientos o golosinas azucaradas que de seguro complicarán sus problemas cardiovasculares. Debe ser el momento de perdonar a aquellos que le han agriado la vida, han hecho declaraciones que buscaban disminuirle a usted o a sus hijos, o han tratado de perjudicar financieramente a su familia.
Si algunas personas malintencionadas le han lastimado con sus comentarios, estoy seguro de que no soy el primero en instarle a poner el pasado en el espejo retrovisor y seguir adelante. Pero _debe_ hacerlo. Si sigue la receta del Gran Médico para un estilo de vida saludable, confío en que esto le ayudará a lidiar con cualesquiera emociones mortales que estén gravitando en su mente. Por favor, recuerde que por más que lo hayan lastimado, aún es posible perdonar. «Porque si perdonáis a los hombres sus ofensas, os perdonará también a vosotros vuestro Padre celestial », dice Jesús en Mateo 6; «mas si no perdonáis a los hombres sus ofensas, tampoco vuestro Padre os perdonará vuestras ofensas » (Mateo 6.14-15).
Perdone a aquellos que le han lastimado y olvide sus ofensas. Luego, memorice este sabio consejo del rey Salomón: «El corazón alegre constituye buen remedio» (Proverbios 17:22).
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE: EVITE LAS
EMOCIONES MORTALES **
• Comprenda el daño que puede hacerle a su corazón cuando está triste, asustado o estresado por la vida diaria.
• Confíe en Dios cuando enfrente circunstancias que le provoquen preocupación o ansiedad.
• Practique el perdón cada día y perdone a aquellos que le hayan ofendido.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para evitar las emociones mortales, por favor, pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable._
[LLAVE # 7
_Viva una vida de oración ycon propósito_](Bras_ISBN9781418582906_epub_c3_r1.html#Anch01989)
Hace unos años, me dio por coleccionar objetos relacionados con el béisbol. Compré algunas cosas de Barry Bonds, entre ellas una pintura suya, un bate y una pelota firmada por él. También compré un gran cuadro pintado por «A-Rod» —Alex Rodríguez, el tercera base de los Yankees de Nueva York— que ahora ocupa un considerable espacio en mi oficina.
Hay otro yankee entre los grandes que también me agrada: su nombre es Yogi Berra, y fue el receptor de la novena desde 1947 hasta 1963. En sus días como jugador activo, Yogi era conocido como un _catcher_ inteligente y reflexivo, pese a las limitaciones de una educación que sólo llegó hasta el octavo grado.
Lo que hacía especial a Yogi —y famoso en esos días— era su rara capacidad para decir algo gracioso sin inmutarse. Si uno le preguntaba qué hora era, Yogi replicaba: «¿Ahora mismo?» Y si le preguntaban por un popular restaurante llamado Charlie's, la respuesta era un equívoco: «Ya nadie va allí. Siempre está demasiado lleno».
En otra ocasión, la esposa de Yogi, Carmen, le hizo una pregunta seria: «Yogi, tú eres de San Luis, vivimos en Nueva Jersey y juegas béisbol en Nueva York. Si te mueres antes que yo ¿dónde quisieras que te entierre?»
«Mejor me das la sorpresa», fue la respuesta de Yogi.
Por último, refiriéndose a jugar en el Yankee Stadium la posición del jardín izquierdo, donde caen desde temprano las sombras de la noche, Yogi comentaba: «Allí se hace tarde muy temprano».
Temo que se haga tarde muy temprano para muchas víctimas de ataques al corazón, muchas de las cuales andan despreocupadamente por la vida, sólo para ver que de pronto se hace de noche y sumergirse en la eternidad.
Para aquellos que quedamos atrás, el pesar de perder a un ser querido en un deceso repentino y casi instantáneo es una conmoción difícil de sobrellevar. Este es un emotivo tema que ciertamente resuena en mi interior, porque mis dos abuelos sucumbieron a infartos cardiacos, uno antes de cumplir los sesenta, y el otro poco después de cumplirlos.
Cuando yo tenía diecisiete meses perdí a mi primer abuelo, abuelito Al, debido a su tercer infarto en menos de dos años.
El padre de mamá tuvo su primer ataque al corazón mientras estaba de vacaciones en Israel con abuela Rose. Caminaba con un grupo de turistas en dirección al Monte del Templo cuando sintió un dolor en el pecho. Un aneurisma se le reventó en uno de sus ventrículos; de algún modo, sobrevivió. Se quedó hospitalizado durante meses en Israel antes de que los médicos le autorizaran regresar a Estados Unidos.
Abuelito Al era un candidato seguro para una operación de desvío coronario, pero entonces no se realizaban tan rutinariamente como en nuestros días. Los cardiólogos lo descartaron. Meses después de regresar de Tierra Santa, mi abuelo sufrió un segundo infarto al miocardio. También lo sobrevivió, pero por un pelo.
Permaneció postrado con oxígeno hasta que el tercer ataque le dio el golpe mortal. Abuelito Al tenía apenas cincuenta y cinco años.
En cuanto al padre de papá, Joshua Rubin «americanizó» su nombre convirtiéndolo en Jerry, para escapar de la persecución a los judíos. Abuelo Jerry no tuvo problemas para adoptar la dieta tradicional americana, rica en comidas chatarra. Antes de irse a dormir le gustaba tomar abundante helado mientras con la mano derecha vaciaba una bolsa de pretzels salados. Compraba los Fudgsicles por docenas, y cada viernes por la noche saciaba su apetito en la heladería Howard Johnson's, donde pedía un sundae con crema de huevo. También frecuentaba las hamburgueserías, y siempre acompañaba su hamburguesa con grandes y cremosos batidos y papas fritas a la francesa.
Abuelo Jerry era un voluminoso dentista que nunca fue al médico. Con una complexión en forma de manzana y un metro ochenta y tres de estatura, disparaba la báscula mucho más allá de los ciento diez kilos. Aunque nunca hizo ejercicios, era fuerte como un toro, y nunca se enfermaba.
Cuando cerró su exitoso gabinete dental a los sesenta y un años, abuelo Jerry estaba listo para disfrutar de sus años de oro. Él y mi abuela Anne compraron una casa de retiro en el sur de Miami Beach. Pero cuando yo tenía nueve años, mis padres recibieron una de esas ominosas llamadas que sólo parecen hacerse a las dos de la madrugada: Abuelo Jerry había fallecido de un repentino y masivo infarto coronario. Había estado leyendo en la cama hasta que se quedó dormido, sólo para despertar tosiendo y jadeando. Todo terminó pronto, los paramédicos no pudieron hacer nada por él. El médico que realizó la autopsia dijo que rara vez había visto tanta placa acumulada en las arterias de una persona.
Me hubiera gustado disfrutar más tiempo en este mundo con mis abuelos, para llegar a conocerlos mejor, pero contar sus historias me da un renovado sentido de propósito para ayudar a quienes sufren de enfermedades cardiovasculares.
Estas son una horrible aflicción que lleva a quienes la sufren a confrontar su mortalidad. La oración es la herramienta más poderosa que poseemos en la lucha por mantener un corazón sano. La oración reconoce que hay algo... Alguien... más allá de los mortales confines de nuestras vidas. Hablar con nuestro Creador por medio de la oración no debe ser el tratamiento de último recurso, sino el primero. Quizás Dios no conteste nuestras oraciones como esperamos, pero la oración transformará nuestros corazones alineándolos mejor con el de Él.
«La oración debe ser el fundamento de toda vida cristiana», escribió Germaine Copeland, autora de _Prayers That Avail Much_ ( _Oraciones con poder_ ). Su libro contiene más de ciento cincuenta oraciones que contemplan prácticamente cualquier situación posible bajo el sol: orar por nuestros familiares para que reciban a Jesús como Señor y Salvador, sanar lastimaduras emotivas, vencer la depresión o superar el miedo. Me atrae particularmente la oración de la señora Copeland para «olvidar el pasado»:
Señor, presento aquí mi pasado y pongo en la adecuada perspectiva las cosas que ya quedaron atrás. Confío en ti, Señor, con todo mi corazón, y no en mi propio raciocinio. Sin importar mi pasado, miro adelante, a lo que está por venir. Me esfuerzo por llegar al final de la carrera y recibir el premio por el cual me estás llamando al cielo, gracias a lo que Cristo Jesús hizo por mí. En Su nombre te lo pido, amén.
Si se siente atraído hacia Dios en un momento crucial como este, le invito a doblar sus rodillas ahora. ¡La oración es un imperativo!
No subestime el poder de comunicarse con su Padre celestial. Cuando usted ora y abre su corazón a Dios, le permite que le llene con las Escrituras, desde las cuales Él puede hablarle con la paz del momento y transformar simultáneamente su salud.
La posibilidad de que los enfermos por quienes se ha orado experimenten una mejor recuperación ha sido tema de estudios en años recientes. El primer estudio importante, en 1988, comprendía a víctimas de infarto al miocardio, fallo cardiaco y otros problemas cardiovasculares, atendidas en el Hospital General de San Francisco. El cardiólogo Randolph Byrd asignó a ciento noventa y dos pacientes de la Unidad de Cuidados Intensivos al grupo de los que no recibirían oraciones. Ni los pacientes, ni los médicos ni las enfermeras sabían en qué grupo estaba cada quién. Los cristianos que debían orar estaban dispersos por todo el país y sólo recibieron los nombres de pila, los diagnósticos y los pronósticos de los pacientes. En sus dramáticas conclusiones, el doctor Byrd encontró que los pacientes por quienes se había orado necesitaron menos medicamentos y experimentaron menos complicaciones.
Nuestro Creador escuchó nuestras oraciones. Lo cual me recuerda otro dicho que hizo famoso a Yogi Berra: «Nada ha terminado hasta que ha terminado».
**LA RECETA DEL GRAN MÉDICO PARA
UN CORAZÓN SALUDABLE: VIVA UNA
VIDA DE ORACIÓN Y CON PROPÓSITO **
• Ore constantemente.
• Confiese las promesas de Dios al levantarse y antes de retirarse a dormir.
• No importa dónde se encuentre usted en su batalla por su salud, encuentre el propósito de Dios para su vida, y vívalo.
_Actúe_
Si quiere aprender a incorporar a su régimen diario los principios para vivir una vida de oración y con propósito, por favor, pase a la página 71 para consultar el plan de batalla de _La receta del Gran Médico para un corazón saludable._
[PLAN DE BATALLA DE LA RECETA
DEL GRAN MÉDICO PARA UN
CORAZÓN SALUDABLE](Bras_ISBN9781418582906_epub_c3_r1.html#Anch02195)
DÍA 1
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta Mateo 6.9-13.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene Avanzada:_ Para las manos y las uñas, meta los dedos en jabón semilíquido cuatro o cinco veces, y lávese las manos con el jabón durante quince segundos, frotándolo sobre las cutículas y enjuagándose con el agua lo más caliente que pueda soportar. Eche otro poco de jabón semilíquido en las manos para lavarse la cara. Luego, llene el aguamanil o lavamanos con agua tan caliente como pueda, y agregue entre una y tres cucharadas de sal de mesa, y entre uno y tres goteros llenos de una solución mineral a base de yodo. Sumerja la cara en el agua y abra los ojos, parpadeando repetidamente bajo el agua. Mantenga los ojos abiertos bajo el agua durante tres segundos. Después de limpiar sus ojos, vuelva a meter la cara en el agua con la boca cerrada, haciendo burbujas a través de la nariz. Saque la cara del agua y sóplese la nariz con una servilleta sanitaria. Para limpiarse las orejas, utilice agua oxigenada y gotas para los oídos con base mineral. Ponga dos o tres gotas en cada oído y manténgalas ahí por un minuto. Luego sacuda la cabeza para que el líquido salga. Para los dientes aplique en el cepillo dos o tres gotas de dentífrico líquido basado en aceites esenciales. Puede usar esto para cepillarse o añadirlo a su crema dental. Después de los dientes, cepíllese la lengua durante quince segundos. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver los productos de higiene avanzada recomendados.)
_Reducir toxinas:_ Abra hoy las ventanas durante una hora. Utilice jabón natural y productos naturales para el cuidado de la piel y del cuerpo (gel para la ducha, cremas, etc.). Use también productos naturales para el cuidado del cutis, de la dentadura y el cabello (champúes, acondicionadores, gels, mousses y lacas). (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver los productos recomendados.)
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudo. (Para ver los productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide.)
_Terapia Corporal:_ Expóngase durante veinte minutos a la luz directa del sol en algún momento del día, pero guárdese de exponerse demasiado entre las diez de la mañana y las dos de la tarde.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas —de la primera a la tercera— de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Cada vez que se enfrente a una circunstancia adversa que le cause preocupación, como un problema de salud, repita lo siguiente: «Señor, yo confío en ti, a ti te entrego el cuidado de mi persona, y creo que cuidarás de mi [insertar la presente situación] y llenarás de fuerza y de salud mi cuerpo». Confiese lo anterior a lo largo del día cada vez que venga a su mente su problema de salud.
**_Desayuno_**
Prepárese en la licuadora un batido con los siguientes ingredientes:
Una taza de yogur natural o kéfir (es mejor el de leche de cabra); una cucharada de aceite de linaza orgánico; una cucharada de miel de abejas orgánica; una taza de frutas orgánicas (bayas, banana, durazno, piña, etc.) frescas o congeladas; dos cucharadas de polvo proteínico a base de leche de cabra (visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas); una pizca de extracto de vainilla (opcional).
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**_Almuerzo_**
Antes de almorzar, beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas. (Para ver los productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide.)
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos, junto con dos onzas de atún bajo en mercurio y rico en omega-3.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Una manzana con cáscara.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
(Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
Salmón, capturado en su medio, al horno, cocido en agua o a la parrilla.
Brócoli al vapor.
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a nueve cápsulas de aceite de hígado de bacalao rico en omega-3. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
**_Refrigerios_**
Tajadas de manzana con mantequilla de almendras crudas.
Una barra nutritiva antioxidante de bayas basada en alimentos enteros con betaglucanos de fibra soluble de avena. (Para ver productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía GPRx Resource Guide.)
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas
**_Antes de irse a la cama_**
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Terapia corporal:_ Tome un baño tibio durante quince minutos añadiéndole ocho gotas de aceites esenciales bíblicos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Pida al Señor que le recuerde a alguien a quien deba perdonar. Tome una hoja de papel y escriba en la parte superior el nombre de esa persona. Trata de recordar cada acto específico suyo que le haya herido. Escriba lo siguiente: «Perdono a [insertar el nombre de la persona] por [insertar lo que hizo contra usted]». Una vez que haya llenado la hoja, rómpala o quémela, y pida a Dios que le dé la fuerza para perdonar de corazón al ofensor.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la fidelidad de su amor incesante y su misericordia renovada cada mañana. Lea en voz alta Romanos 8.35, 37-39.
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 2
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta el Salmo 91.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudo.
_Terapia corporal:_ Dése una ducha caliente y fría. Después de una ducha normal, alterne sesenta segundos de agua tan caliente como pueda resistir, seguidos por sesenta segundos de agua tan fría como la pueda soportar. Repita el ciclo cuatro veces para un total de ocho minutos, finalizando con agua fría.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas —de la primera a la tercera— de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Siga las instrucciones para la salud emotiva de la mañana del Día 1.
**_Desayuno_**
Dos o tres huevos en cualquier estilo, cocinados con una cucharada de aceite de coco extravirgen. (Para ver las marcas recomendadas visite www.BiblicalHealthInstitute.com y haga clic sobre la guía GPRx Resource Guide.)
Cebollas, champiñones y pimientos salteados.
Una rebanada de pan integral de grano entero germinado o sin levadura con mantequilla de almendras y miel.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
Antes de almorzar, beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas. (Para ver los productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide.)
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos, junto con tres huevos hervidos ricos en omega-3.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Uvas orgánicas.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Pollo asado orgánico.
Vegetales cocidos (zanahorias, cebollas, arvejas, etc.).
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a nueve cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
Almendras crudas y tajadas de manzana.
Una barra nutritiva de proteína con sabor a chocolate basada en alimentos enteros con betaglucanos de fibra soluble de avena.
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas.
**_Antes de irse a la cama_**
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Repita las recomendaciones para salud emotiva del Día 1.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la fidelidad de su amor incesante y su misericordia renovada cada mañana. Lea en voz alta 1 Corintios 13.4-8.
_Terapia corporal:_ Dedique diez minutos a escuchar música relajante antes de retirarse a dormir.
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 3
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta Efesios 6.13-18.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Terapia Corporal:_ Expóngase durante veinte minutos a la luz directa del sol en algún momento del día, pero guárdese de exponerse demasiado entre las diez de la mañana y las dos de la tarde.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas —de la primera a la tercera— de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Siga las instrucciones para la salud emotiva de la mañana del Día 1.
**_Desayuno_**
De cuatro a ocho onzas de yogurt de leche entera orgánica o requesón con frutas (piña, duraznos o bayas), miel de abejas y una pizca de extracto de vainilla.
Un puñado de almendras crudas.
Una taza de té caliente con miel de abejas.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
Antes de almorzar, beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas. (Para ver los productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide.)
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos, junto con tres huevos hervidos ricos en omega-3.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Un pedazo de alguna fruta de la estación.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Steak de carnes rojas (res, bisonte, o venado).
Brócoli al vapor.
Batata al horno con mantequilla.
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a 9 cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
Cuatro onzas de yogurt de leche entera con frutas, miel de abejas y unas cuantas almendras
Una barra nutritiva antioxidante de bayas basada en alimentos enteros con betaglucanos de fibra soluble de avena.
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas.
**_Antes de irse a la cama_**
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Terapia corporal:_ Tome un baño tibio durante quince minutos añadiéndole ocho gotas de aceites esenciales bíblicos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Repita las instrucciones para la salud emotiva del Día 1.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día.
Déle gracias por la fidelidad de su amor incesante y su misericordia renovada cada mañana. Lea en voz alta Filipenses 4.4-8,11-13,19.
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 4
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta Mateo 6.9-13.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas del 1 al 3 de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Siga las instrucciones para salud emotiva de la mañana del Día 1.
_Terapia corporal:_ Dése una ducha caliente y fría. Después de una ducha normal, alterne sesenta segundos de agua tan caliente como pueda resistir, seguidos por sesenta segundos de agua tan fría como la pueda soportar. Repita el ciclo cuatro veces para un total de ocho minutos, finalizando con agua fría.
**_Desayuno_**
Tres huevos brevemente hervidos o cocidos en agua.
Cuatro onzas de cereal integral de grano entero germinado con dos onzas de yogurt de leche entera o leche de cabra. (Para ver productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía GPRx Resource Guide.)
Una taza de té caliente con miel de abejas.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
Antes de almorzar, beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas. (Para ver los productos recomendados visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide.)
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos, junto con tres onzas de atún enlatado.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Un racimo de uvas con semillas.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Pechuga de pollo a la parrilla.
Vegetales al vapor.
Una porción pequeña de cereales integrales (quinoa, amaranto, mijo, o arroz integral) salteados con una cucharada de aceite de coco extravirgen.
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, Sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a 9 cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
Manzana y zanahorias con mantequilla de almendras crudas.
Una barra nutritiva antioxidante de bayas basada en alimentos enteros con betaglucanos de fibra soluble de avena.
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas.
**_Antes de irse a la cama_**
Beba de ocho a doce onzas de agua o té con miel de abejas.
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Repita las instrucciones para la salud emotiva del Día 1.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la fidelidad de su amor incesante y su misericordia renovada cada mañana. Lea en voz alta Romanos 8.35, 37-39.
_Terapia corporal:_ Dedique diez minutos a escuchar música relajante antes de retirarse a dormir _Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 5 (DÍA DE AYUNO PARCIAL)
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta Isaías 58.6-9.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas —de la primera a la tercera— de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Siga las instrucciones para la salud emotiva de la mañana del Día 1.
_Terapia Corporal:_ Expóngase durante veinte minutos a la luz directa del sol en algún momento del día, pero guárdese de exponerse demasiado entre las diez de la mañana y las dos de la tarde.
**_Desayuno_**
No desayune (día de ayuno parcial).
Beba de ocho a doce onzas de agua.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
No almuerce (día de ayuno parcial).
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Sopa de pollo (visite www.BiblicalHealthInstitute.com para hallar la receta).
Vegetales encurtidos. (Visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver las marcas recomendadas.)
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a nueve cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
No meriende (día de ayuno parcial).
Beba ocho onzas de agua.
**_Antes de irse a la cama_**
Beba de ocho a doce onzas de agua o té con miel de abejas.
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Terapia corporal:_ Tome un baño tibio durante quince minutos añadiéndole ocho gotas de aceites esenciales bíblicos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Siga las recomendaciones para perdonar de la noche del Día 1.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito?» «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la lealtad de su amor incesante y su misericordia que se renueva cada mañana. Lea en voz alta Isaías 58.6-9.
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 6 (DÍA DE DESCANSO)
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque este es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta el Salmo 23.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Ejercicio:_ No haga ejercicios formales pues es un día de descanso.
_Terapia corporal: No se dé ninguna, pues es día de descanso._
_Salud emotiva:_ Siga las instrucciones para la salud emocional de la mañana del Día 1.
**_Desayuno_**
Dos o tres huevos preparados en cualquier estilo con una cucharada de aceite de coco extravirgen.
Una toronja o pomelo.
Un puñado de almendras.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
Antes de almorzar, beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas.
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos con dos onzas de atún bajo en mercurio y rico en omega-3.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Una manzana orgánica con cáscara.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Pollo asado orgánico.
Vegetales cocidos (zanahorias, cebollas arvejas, etc.).
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a nueve cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
Un puñado de almendras crudas con tajadas de manzana.
Una barra nutritiva antioxidante de bayas basada en alimentos enteros con betaglucanos de fibra soluble de avena.
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas.
**_Antes de irse a la cama_**
Beba de ocho a doce onzas de agua o té con miel de abejas.
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Siga las recomendaciones para perdonar de la noche del Día 1.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la lealtad de su amor incesante y su misericordia que se renueva cada mañana. Lea en voz alta el Salmo 23.
_Terapia corporal:_ Dedique diez minutos a escuchar música relajante antes de retirarse a dormir.
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 7
**_Al levantarse y durante el día_**
_Oración:_ Dé gracias a Dios porque éste es el día que el Señor ha hecho. Regocíjese y gócese en Él. Déle gracias por el aliento en sus pulmones y la vida de su cuerpo. Pida al Señor que sane su organismo y utilice su experiencia en beneficio de las vidas de otros. Lea en voz alta el Salmo 91.
_Propósito:_ Pida al Señor una oportunidad para añadir significado hoy a la vida de alguien. Esté alerta esperando esa oportunidad. Pida a Dios que le utilice en este día para Su propósito.
_Higiene avanzada:_ Siga las recomendaciones de higiene avanzada para la mañana del Día 1.
_Reducir toxinas:_ Siga las recomendaciones para reducir toxinas de la mañana del Día 1.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese a tragar con un vaso de doce a dieciséis onzas de agua o jugo de vegetales crudos.
_Ejercicio:_ Realice ejercicios del método de forma física funcional durante cinco a quince minutos, o pase igual tiempo sobre un minitrampolín. Finalice con cinco a diez minutos de ejercicios de respiración profunda. (Podrá encontrar las rondas —de la primera a la tercera— de estos ejercicios en www.GreatPhysiciansRx.com.)
_Salud emotiva:_ Siga las instrucciones para la salud emocional de la mañana del Día 1.
_Terapia corporal:_ Expóngase durante veinte minutos a la luz directa del sol en algún momento del día, pero guárdese de exponerse demasiado entre las diez de la mañana y las dos de la tarde.
_Salud emotiva:_ Siga las instrucciones para la salud emocional de la mañana del Día 1.
**_Desayuno_**
Prepárese en la licuadora un batido con los siguientes ingredientes:
Una taza de yogur natural o kéfir (es mejor el de leche de cabra); una cucharada de aceite de linaza orgánico; una cucharada de miel de abejas orgánica; una taza de frutas orgánicas (bayas, banana, durazno, piña, etc.) frescas o congeladas; dos cucharadas de polvo proteínico a base de leche de cabra (visite www.BiblicalHealthInstitute.com y haga clic sobre la guía de recursos GPRx Resource Guide para ver los productos recomendados.); una pizca de extracto de vainilla (opcional).
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Almuerzo_**
Antes de almorzar beba ocho onzas de agua.
Mientras almuerza, beba ocho onzas de agua o té caliente con miel de abejas.
Prepare una ensalada grande con verduras de hojas verdes, aguacate, zanahorias, tomates, col morada, cebolla morada, pimientos rojos, y brotes tiernos, junto con tres onzas de salmón (pescado en su medio) frío, cocido en agua o enlatado.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, Sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
Un pedazo de alguna fruta de estación.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros.
**_Cena_**
Antes de cenar beba ocho onzas de agua.
Mientras cena, beba té caliente con miel de abejas.
Pescado de su elección al horno o a la parrilla.
Brócoli al vapor.
Batata horneada con mantequilla.
Prepare una gran ensalada con vegetales de hojas verdes, aguacate, zanahorias, pepinos, apio, tomates, col morada, pimientos rojos, cebolla morada y brotes tiernos.
Aliño: Utilice aceite de oliva extravirgen, vinagre de sidra de manzana o jugo de limón, sal Celtic Sea, plantas medicinales y especias, o mezcle una cucharada de aceite de oliva extravirgen con otra de algún aliño adquirido en tiendas de productos de salud.
_Suplementos:_ Tome dos cápsulas de multivitaminas basadas en alimentos enteros y una cápsula de fórmula energética antioxidante basada en alimentos enteros, y de una a tres cucharaditas o de tres a nueve cápsulas de aceite de hígado de bacalao rico en omega-3.
**_Refrigerios_**
Tajadas de manzana con mantequilla de ajonjolí crudo (tahini).
Una barra nutritiva antioxidante de bayas basada en alimentos enteros con betaglucanos de fibra soluble de avena.
Beba de ocho a doce onzas de agua o té caliente o frío recién hecho con miel de abejas.
**_Antes de irse a la cama_**
Beba de ocho a doce onzas de agua o té con miel de abejas.
_Ejercicios:_ Salga a caminar o participe en una actividad recreativa o deporte favorito.
_Suplementos:_ Tome una porción de mezcla pulverizada de fibras y superalimentos verdes, o cinco cápsulas de superfórmula verde. Ayúdese con un vaso de doce a dieciséis onzas de agua.
_Higiene avanzada:_ Repita las instrucciones de higiene avanzada para la mañana del Día 1.
_Salud emotiva:_ Pida al Señor que le recuerde a alguien a quien deba perdonar. Tome una hoja de papel y escriba en la parte superior el nombre de esa persona. Trata de recordar cada acto específico suyo que le haya herido. Escriba lo siguiente: «Perdono a [insertar el nombre de la persona] por [insertar lo que hizo contra usted]». Una vez que haya llenado la hoja, rómpala o quémela, y pida a Dios que le dé la fuerza para perdonar de corazón al ofensor.
_Terapia corporal:_ Tome un baño tibio durante quince minutos añadiéndole ocho gotas de aceites esenciales bíblicos.
_Propósito:_ Hágase estas preguntas: «¿He vivido hoy una vida con propósito? » «¿Qué he hecho hoy para enriquecer la vida de mi prójimo?»
Comprométase a vivir mañana un día con propósito.
_Oración:_ Dé gracias a Dios por este día, pidiéndole que le dé un descanso nocturno reparador y un comienzo fresco en el nuevo día. Déle gracias por la fidelidad de su amor incesante y su misericordia renovada cada mañana. Lea en voz alta 1 Corintios 13.4-8
_Hora de dormir:_ Acuéstese a las diez y media de la noche.
DÍA 8 EN ADELANTE
Si usted se está sintiendo mejor, puede repetir tantas veces como lo desee el plan de batalla _La receta del Gran Médico para un corazón saludable_.
Para sugerencias detalladas paso a paso, y planes de nutrición y estilo de vida, visite www.GreatPhysiciansRx.com y súmese a la experiencia «Cuarenta días por la salud» si es que desea continuar gozando +de buena salud, o al plan «Una vida de bienestar» si desea mantener su nuevo nivel de salud.
Estos programas online le proveerán comidas diarias adecuadas a su caso y planes de ejercicios, así como las herramientas para determinar su progreso.
Si ha experimentado resultados positivos con el programa _La receta del Gran Médico para un corazón saludable_ , le insto a hablarles de él a sus conocidos y a que les recomiende este libro y este programa. Usted puede aprender a dirigir un pequeño grupo en su iglesia o en su hogar, visitando: www.BiblicalHealthInstitute.com
Recuerde: No necesita ser médico ni especialista de la salud para ayudar a transformar la vida de alguien a quien aprecie: basta con que tenga la voluntad para hacerlo.
Permítame ofrecer ahora esta plegaria de bendiciones tomada de Números 6.24-26:
_Jehová te bendiga, y te guarde;_
_Jehová haga resplandecer su rostro sobre ti, y tenga de ti misericordia;_
_Jehová alce sobre ti su rostro, y ponga en ti paz._
_En el nombre de Yeshua Ha Mashiach, Jesús, nuestro Mesías. Amén._
**_¿Necesita recetas?_**
Para una lista detallada de más de doscientas recetas (en inglés) deliciosas y saludables contenidas en el plan de la receta del Gran Médico para las comidas, visite:
www.BiblicalHealthInstitute.com
NOTAS
**_Introducción_**
. "Cardiovascular Statistics Updated for 2005", del informe de fin de año de la American Heart Association, 30 diciembre 2004.
. "Heart Disease and Stroke Statistics—2006 Update", de la revista digital _Circulation_ , publicada por la American Heart Association, 11 enero 2006, disponible en www.circulationaha.org.
. Ibid.
. "The Heart Truth for Women", nota de prensa emitida por National Heart, Lung, and Blood Institute, parte de los National Institutes of Health de Estados Unidos. "The Heart Truth" es una campaña nacional de concienciación para las mujeres acerca de las enfermedades cardiovasculares y puede ser vista en línea en www.nhlbi.nih.gov/health/hearttruth/espanol.htm.
. "Marketing an Operation: Coronary Artery Bypass Surgery", ensayo por Thomas A. Preston, MD, asociado con la clínica Mount Rainer y disponible en www.drcranton.com/chelation/cabg1.htm.
. De "Oral EDTA Helps Restore Cardiovascular Function", por Gail Valentine, MD, disponible en la página web de Life Enhancement, www.lifeenhancement. com/article_template.asp?ID=531.
. "Chelation Therapy: AHA Recommendation", encontrado en la página web de la American Heart Association, www.americanheart.org/presenter.jhtml?identifier =4493.
. _Congenital Heart Defects Report_ de la American Heart Association, 8 julio 2005.
**_Llave # 1_**
. El estudio Framingham sobre el corazón, bajo la dirección del National Heart, Lung, and Blood Institute, puede verse en http://www.nhlbi.nih.gov/about/framingham/.
. Alex Berenson, "Lipitor or Generic? Billion-Dollar Battle Looms", _New York Times_ , 15 octubre 2005.
. K. M. Anderson, W. P. Castelli, y D. Levy, "Cholesterol and mortality. 30 years of followup from the Framingham study", _JAMA_ 257 (1987):2176-80.
. Sue Goetinck, "Researchers Cite Protein in Heart Protection", _Dallas Morning News_ , 24 noviembre 2005.
. Revista _FDA Consumer_ , julioagosto 1997, número de publicación (FDA) 97-2313.
. _American Journal of Clinical Nutrition_ 80 (diciembre 2004):1492–9, disponible en www.wholegrainscouncil.org/research.htm.
. "Fiber May Reduce Women's Risk of Heart Disease", publicado 1 julio 1999 en www.cnn.com.
. "Dietary Supplement Fact Sheet: Vitamin B-12", del Office of Dietary Supplement, National Institute of Health Clinical Center, y disponible en www.ods.od.nih.gov/factsheets/vitaminb12.asp#h9.
. "Omega-3 Fatty Acids", artículo disponible en www.wholehealthmd.com.
. "Fats and Cholesterol", encontrado en la página web de Harvard School of Public Health y disponible en www.hsph.harvard.edu/nutritionsource/fats.html.
. James H. Dwyer, PhD, Mohamad Navab, PhD, Kathleen M. Dwyer, PhD, et. al, "Oxygenated Carotenoid Lutein and Progression of Early Atherosclerosis: The Los Angeles Atherosclerosis Study", _Circulation_ , de la American Heart Association, 103 (2001):2922, disponible en www.circ.ahajournals.org/cgi/content/abstract/103/24/2922.
. De "Fruits and Vegetables", publicado por Harvard School of Public Health, 13 diciembre 2004, disponible en www.hsph.harvard.edu/nutritionsource.
. Steven Milloy, reportero de Fox News, "Healthy Food Labeling: Buyer Beware", 18 julio 2003, disponible en www.foxnews.com/story/0,2933,92249,00.html.
. "Nuts Cut Coronary Heart Disease Risk," nota de prensa de 8 mayo 2001 de Penn State University, disponible en www.psu.edu/ur/2001/nuts.html.
. Paul Schulick, _Ginger: Common Spice & Wonder Drug_, 3a ed. (Prescott, AZ: Hohm Press, 1996), p. 36.
. "Loma Linda University Reveals First Study on Correlation between High Water Intake and Lowered Coronary Heart Disease", nota de prensa de 25 abril 2002, disponible en www.llu.edu/news/pr/042502water.html.
. F. Batmanghelidj, MD, _You_ ' _re Not Sick, You_ ' _re Thirsty!_ (Nueva York: Warner Books, 2003), p. 206.
**_Llave # 3_**
. Christine Gorman y Alic Park, "Inflammation Is a Secret Killer: The Surprising Link Between Inflammation and Asthma, Heart Attacks, Cancer, Alzheimer's and Other Diseases", _Time_ , 23 febrero 2004.
. "Heart Disease and Stroke", artículo encontrado en la página web de la American Academy of Periodontology y disponible en www.perio.org/consumer/mbc.heart.htm.
**_Llave # 5_**
. De dos informes de prensa: "Bad Air Causes Heart Disease, Says U.S. Group", por la agencia de noticias Reuters, 3 junio 2004, y "Air Pollution, Even at 'Safe Levels,' Is Bad for the Heart", por la American Heart Association, 11 noviembre 2003.
**_Llave # 6_**
. Joel Achenbach, _Why Things Are: Answers to Every Essential Question in Life_ (Nueva York: Ballantine Books, 1991).
. Don Colbert, MD, _Emociones que matan: Entienda la conexión mente-cuerpoespíritu que puede sanarle o destruirle_ (Nashville: Grupo Nelson, 2006), p. 50.
**_Llave # 7_**
. Germain Copeland, _Prayers That Avail Much_ (Tulsa: Harrison House, 1997), 104-5 [ _Oraciones con poder_ (El Paso, TX: Spanish House, 2000)].
. Hilary E. MacGregor, "Researchers Look at the Power of Prayer", _Los Angeles Times_ , 1 junio 2005.
ACERCA DE LOS AUTORES
**J** **ordan Rubin** ha dedicado su vida a transformar la salud de otros, vida a vida. Es consultor certificado en nutrición, instructor certificado de forma física personal, especialista certificado en nutrición y miembro de la Academia Nacional de Medicina Deportiva.
El señor Rubin, de treinta y un años, es fundador y presidente ejecutivo de Garden of Life, Inc., una compañía dedicada a la salud y el bienestar humanos con sede en West Palm Beach, Florida, que produce suplementos nutricionales basados en alimentos enteros, y productos para el cuidado personal. También es presidente y ejecutivo principal de GPRx, Inc., una compañía de salud y bienestar basada en la Biblia que provee recursos educativos, currículos para grupos pequeños, alimentos funcionales, suplementos nutritivos y servicios de bienestar.
Él y su esposa, Nicki, se casaron en 1999 y son padres de un parvulito, Joshua. Residen en Palm Beach Gardens, Florida.
**El doctor en medicina Joseph D. Brasco** tiene amplios conocimientos y experiencia en las especialidades de medicina interna y gastroenterología; es vicepresidente de Asuntos Médicos de Garden of Life. El doctor Brasco, que estudió medicina en el Colegio Médico de Wisconsin en Milwaukee, Wisconsin, es un profesional certificado por la Junta Estadounidense de Medicina Interna. Además de escribir para varias publicaciones médicas, es también coautor con Jordan Rubin de _Restoring Your Digestive Health_
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,882 |
import socket
import threading, signal
import argparse
from time import sleep,time
# GLOBALS
killEvent = threading.Event()
printLock = threading.Lock()
stopServer = False
stopClient = False
startTime = time()
# Classes
class Server( object ):
def __init__( self, host, port ):
super().__init__()
# Class members
self.host = host
self.port = port
self.clients = []
self.clientId = 0
# Set up socket
self.sock = socket.socket( socket.AF_INET, socket.SOCK_STREAM )
self.sock.setsockopt( socket.SOL_SOCKET, socket.SO_REUSEADDR, 1 )
self.sock.settimeout( 5 )
self.sock.bind( ( self.host, self.port ) )
def run( self ):
tsPrint( f'Server starting.' )
self.sock.listen( 5 )
tsPrint( f'Listening on {self.host}:{self.port}' )
try:
while ( not killEvent.is_set() ):
try:
sock, address = self.sock.accept()
tsPrint( "Client connected" )
except socket.timeout:
# Timeout for sock.accept, just jump back
# to start of while loop
continue
tsPrint( "Creating client thread" )
# If we get to here, we have a client conneciton
newThread = ClientThread( self.clientId, address, sock )
self.clients.append( newThread )
newThread.start()
self.clientId += 1
finally:
self.sock.close()
# Here we've been told to die. Wait for client threads to die
for c in self.clients:
c.join()
class ClientThread( threading.Thread ):
def __init__( self, id, address, sock ):
super().__init__()
self.id = id
self.address = address
self.sock = sock
self.bufSize = 1024
self.socketCloseTimeout = 60
self.sinceLastMsg = 0
self.sock.settimeout( 1 )
tsPrint( f'New client [{self.id}] connected from {address}' )
def run( self ):
try:
while ( not killEvent.is_set() ):
try:
data = self.sock.recv( self.bufSize )
if ( data ):
self.sinceLastMsg = 0
self.handleMessage( data )
else:
tsPrint( "Client disconnected." )
break
except socket.timeout:
# Socket timed out, check if time to close socket,
# otherwise just jump to start of loop
if ( self.sinceLastMsg >= self.socketCloseTimeout ):
tsPrint( 'Client timed out, closing socket.' )
break
else:
self.sinceLastMsg += 1
continue
finally:
self.sock.close()
tsPrint( 'Client thread exiting.' )
def handleMessage( self, msg ):
tsPrint( f'[{self.id}] > {msg}' )
def killHandler( signum, stackFrame ):
print('')
tsPrint( f'Shutting down server.' )
# Signal kill event
killEvent.set()
# Functions
def tsPrint( msg ):
'''
Thread safe print.
'''
printLock.acquire()
print( f'[{time() - startTime:5.2f}] {msg}' )
printLock.release()
def main( args ):
# Retister SIGTERM and SIGINT handler
signal.signal( signal.SIGINT, killHandler )
signal.signal( signal.SIGTERM, killHandler )
s = Server( args.host, args.port )
s.run()
tsPrint( f'Server shut down.' )
return 0
def parseArgs():
parser = argparse.ArgumentParser( description = 'Multithreaded TCP server.' )
# host
parser.add_argument(
'--host'
,action = 'store'
,help = 'Host to listen to.'
,type = str
,default = '127.0.0.1'
)
# Port
parser.add_argument(
'-p', '--port'
,action = 'store'
,help = 'Port to listen on.'
,type = int
,default = 10021
)
return parser.parse_args()
# Entry poing
if __name__ == "__main__":
res = main( parseArgs() )
if ( res != 0 ):
print( f'main() returned with error: {res}' ) | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,215 |
For anyone that intends to move to Greenland, it's probably worth knowing how the hospital system works around here. Two words: Doctor Who.
Firstly, there are no private doctors or medical practices in Greenland. Healthcare is mostly free, paid for and provided by the State.
If you get sick one day, you have to call the hospital. They have a free telephone number that you can call '80 11 11′. You can only call between 08.15 – 08.30.
When you call you will probably end up on a waiting list. The random times I've called when the clock has ticked to 8.15 I've made it to number 70, 90 or even 110 on the waiting list. You are normally guaranteed to wait at least half an hour.
You will then get a time to visit the doctor. In many workplaces, it is generally accepted that you can go to an appointment during work hours, and some workplaces will even pay for the time that you take to visit the doctor because it is so inflexible.
You will then be allocated a random doctor to visit. I've never had the same doctor twice, but then again I haven't had too many doctor consultations. The last time I visited, the doctor told me that he normally told his patients to ring and ask after him if they wanted to see him again, and then he would call him back to arrange a time. The unfortunate thing was that he was leaving to move onto another rotation in another hospital in another country, so I was unlikely to see him again.
Greenland has a shortage of doctors, so they fly them in generally from Denmark. Some of the doctors are quite young and only stay for a short period of time due to their need to undertake different rotations and try different specialisations in their early years. So it is possible to have the same doctor twice, but you have to know how to work the system a bit, and get one who stays a little longer. Generally, though, if you don't ask for a particular doctor, you will be allocated a random one.
Anyway the whole system takes a bit of getting used to for people who move here. The plus side is that you get qualified doctors in a modern facility where the emphasis is on treating the patients. Medicine is free, and that you can talk through your symptoms with the doctor first before confirming an appointment. There is also research about Greenland health issues being conducted as well.
If you live in the settlements it is a different process. There are no hospitals in the settlements but they have some well-equipped health centers with health assistants who are able to help with basic treatments. They use Tele-Medicine technology which is like a virtual doctor machine, that has the ability to transmit crucial data to a larger hospital such as the one in Nuuk. If it is a serious illness or emergency, then you will be transported by helicopter to the main hospital in Nuuk like they did in this emergency operation.
So yes… this is really Doctor Who! For more information about the health system the hospital website is available in Greenlandic and Danish.
What's the health system like where you live? Any quirks?
← Hey you, stop being so polite!
In India, we have the land of wildly divergent medical experiences – from 5 star to below no star.
There is a great difference between the rich and poor in India, right? There was a scandal a few years ago when a politician tried to jump the queue and not go through proper channels. People were not very happy with that special treatment when it was exposed! 🙂 India, what an interesting place to live!
Canada (where I'm from originally) has universal health care… which means a lot of it is 'free' but there are often long waiting periods.
Here in India it is all about the money honey… You can even have an x-ray machine brought to your doorstep in under an hour!
Hi Carissa sorry I'm so late in replying. This is more of a hobby and as the days get darkers I'm sleeping so much these days.
As for India – What social system? There really isn't any kind of effective safety net in India. So you are right – there is a great disparity which also means that medical care is equally tiered to who has / does not have money.
Well, it is interesting because I think the hospital system here is one that people love to hate. But if outsiders criticise it, then many will defend it. It is after all free. I was surprised by some comments I got on Facebook in reaction to the article…. More pride than expected!
I think the doctors I've had have been in general very professional, so all in all I'm happy for the system that seems to work here in Greenland.
From the sounds of it, that is a generous healthcare system. In Australia, if you're Australian then your healthcare is partially subsidised. If memory served me correct, I paid $35 the last time I saw the doctor for a consultation after I showed them my healthcare card. As for medicine, I believe we pay the full price for it and I've never gotten any discounts on them, ranging from cold tablets, panadol that you can get from the supermarket to the medicines that can only be bought with prescriptions.
Waiting time to see a doctor is horribly long here in Australia, for both bulk-billed and private clinics. Call a clinic in the morning and the receptionist will tell you that they booked for most of the day and you can come in an wait for a doctor for an indefinite amount of time.
Hey Mabel, I think you are correct. If we call for a normal appointment we will only really get a time the following week or week after that. If there is an emergency though you need to call another specific number and you are able to visit that same day. I was surprised by the free medicine!! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,053 |
package jsat.utils;
import java.io.Serializable;
import java.util.AbstractSet;
import java.util.Iterator;
/**
* A set for integers that is of a fixed initial size, and can only accept
* integers in the range [0, size). Insertions, removals, and checks are all
* constant time with a fast iterator. <br>
* Null values are not supported
*
* @author Edward Raff
*/
public class IntSetFixedSize extends AbstractSet<Integer> implements Serializable
{
private static final int STOP = -1;
private int nnz = 0;
private int first = -1;
private boolean[] has;
//Use as a linked list
private int[] prev;
private int[] next;
/**
* Creates a new fixed size int set
* @param size the size of the int set
*/
public IntSetFixedSize(int size)
{
has = new boolean[size];
prev = new int[size];
next = new int[size];
first = STOP;
}
@Override
public boolean add(Integer e)
{
return add(e.intValue());
}
/**
* Adds a new integer into the set
* @param e the value to add into the set
* @return {@code true} if the operation modified the set, {@code false}
* otherwise.
*/
public boolean add(int e)
{
if(e < 0 || e >= has.length)
throw new IllegalArgumentException("Input must be in range [0, " + has.length + ") not " + e);
else if(contains(e) )
return false;
else
{
if (nnz == 0)
{
first = e;
next[e] = prev[e] = STOP;
}
else
{
prev[first] = e;
next[e] = first;
prev[e] = STOP;
first = e;
}
nnz++;
return has[e] = true;
}
}
@Override
public boolean remove(Object o)
{
if(o instanceof Integer)
return remove_int((Integer)o);
return super.remove(o);
}
/**
* Removes the specified integer from the set
* @param val the value to remove
* @return {@code true} if the set was modified by this operation,
* {@code false} if it was not.
*/
public boolean remove(int val)
{
return remove_int(val);
}
@Override
public boolean contains(Object o)
{
if(o instanceof Integer)
{
int val = (Integer)o;
return contains(val);
}
else
return false;
}
/**
* Checks if the given value is contained in this set
* @param val the value to check for
* @return {@code true} if the value is in the set, {@code false} otherwise.
*/
public boolean contains(int val)
{
if(val < 0 || val >= has.length)
return false;
return has[val];
}
private boolean remove_int(int index)
{
if (contains(index))
{
if (first == index)
first = next[index];
else
next[prev[index]] = next[index];
if (next[index] != STOP)
prev[next[index]] = prev[index];
next[index] = STOP;
has[index] = false;
nnz--;
return true;
}
else
return false;
}
@Override
public Iterator<Integer> iterator()
{
final Iterator<Integer> iterator = new Iterator<Integer>()
{
int prev = STOP;
int runner = first;
@Override
public boolean hasNext()
{
return runner != STOP;
}
@Override
public Integer next()
{
prev = runner;
runner = next[runner];
return prev;
}
@Override
public void remove()
{
remove_int(prev);
}
};
return iterator;
}
@Override
public int size()
{
return nnz;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,532 |
If you are a business owner you understand how important presentation is. Whether its brochures, business cards, or a sign for your company, you want it to draw the eyes attention in a pleasing way. To not only show off who you are as a business but also your information for them to call or come in and check things out. So here are some of the top tips for success when using your Queens brochure printing company.
Just like you want to build relationships with your clients, you also want to build a relationship with your printer. You want a good report so when you need things done and you are trying to get a job done right, you know you can trust your printer to handle it. You both have plenty of things on your plate, and it helps everything go a lot smoother when you have good communication and understanding with one another.
When going into a printing project you always want to be sure to know what you want and how you want it. Be sure to have all of these details hashed out before starting your project because your printer is not a mind reader, they can only make what you want if you tell them what you want. They are always able to give you pointers and let you know what works best usually for different projects, but in the end, it is about your vision. Here are a few details to make sure you have worked out before getting started. The deadline for the project, whether its the timeline for when you need it done or if its being handled for a client and they have a specific date they need it by. You also need to be aware of your budget for this project, how much are you able to spend. Make sure to let your printer know what the project is for so they can properly help you to make sure its done best for what you need. Also, be sure to know the quantities of what you need printed. Know your format! What kind of paper do you want? you can ask for paper selections. You should also know if you want a matte or gloss finish, and if you want any specialty selections such as a fold on you business cards. Lastly, make sure to proof you project, you can always ask for a PDF or some kind of attachment in an email to see before hand what the end product will look like before it is sent to print. Make sure all of your information is spelled correctly, your business and personal name, your address, etc. Presentation is everything.
Queens Brochure Printing | Last But Not Least!
It is finally time to print! Be sure upon receiving your finished project to check it out and make sure everything has come out just as you and your printer discussed! Double check all of the things we went over before and be sure it is to your liking. Now that you have checked your order, be sure to give your printer a call if there is something not quite right, but especially if they did a good job! If you did your research and you are dealing with a good Queens Brochure Printing company, they will help you when something goes wrong, and they always want to hear when they did a good job on your project for you! So letting them know will help build a stronger relationship with your printing company. If you follow these tips you will have the best printing experience possible! | {
"redpajama_set_name": "RedPajamaC4"
} | 88 |
Q: Смена вида деятельности, нужен совет Добрый день!
Долго думал, как сформулировать вопрос, напишу коротко как смогу. Прошу сильно не пинать, на других форумах тема наверняка не раз поднималась, но аналогичных вопросов на ХэшКоде я еще не видел.
Коротко о том, чем я занимался и что умею.
В студенчестве почти все свободное уделял программированию - C, C++, WinAPI, asm, Delphi - мне это было жутко интересно, т.к. преподавали у нас один Pascal.
Слово Web меня пугало, интернета у меня тогда не было, поэтому ограничивался только Desktop программированием, пользовался материалами, которые изредка скачивал у друзей, у которых был интернет (тогда еще dial-up).
Теперь, проработав несколько лет, увидел, что, надо постоянно развиваться, изучать новые технологии и при этом еще успевать программировать. Это меня еще дальше оттолкнуло от Web программирования (мне всегда казалось, что там много шаманства). Тем не менее, пришлось изучать и новые технологии и научиться делать не очень сложные ASP.NET приложения.
Везде, где работал, занимал ведущие места, есть хорошее знание математики.
К чему это все. Профессия программиста меня начала утомлять, день проходит "работа - дом - покушать - спать". От этого и здоровье страдает и личная жизнь.
Такими темпами далеко не пойдешь.
Сейчас у меня есть достойная работа и там не надо заниматься изучением каждой новомодной плюшки и Web'а, проект типа какого-нибудь архиватора - нашел то, чем больше всего нравится заниматься.
Зарплата моя, по меркам города (живу в провинции), хоть и высокая, но жилье купить точно не хватит.
Тем временем.
Есть знакомый, лет на 5 младше меня. Он занимается обычной разработкой сайтов. Дело может и немного скучноватое, но он уже открыл свой офис, набрал людей, купил жилье, ездит в дорогом автомобиле. Да, у него конечно есть компьютер (у меня только со 2 курса) и интернет (у меня года 3 назад) уже со школы, но программировать он толком не умеет, а сайты на CMS штампует.
В связи с чем возник вопрос, может мне тоже пора начать работать на себя, уйти от разработчика?
Надеюсь у меня уже есть достаточно опыта, что бы организовать работу небольшой IT команды, настроить рабочие места, софт и т.д. Меня это привлекает, начать что-то свое, занять свою нишу. Начинать какой-нибудь стартап проект не хочется, больше привлекает занятие типа разработки того же сайта, где уже все пройдено (есть множество CMS, шаблонов, для нестандартных случаев наверное программируются не очень сложно), одна только проблема - найти клиентов.
*
*Правильно ли я рассуждаю или это все мое узконаправленное мировоззрение?
*У кого какой опыт, что посоветуете?
*Отдельно хотелось бы узнать мнение тех, у кого есть свой опыт.
UPD
Некоторые пояснения, вопросы можно немного переформулировать:
- Какова на самом деле реальность?
- Не хотелось бы не повторять чужих ошибок, у кого есть опыт начала чего-то своего, кто чем занимается?
A: Если вы задаете такой вопрос, значит не уверены. Любой бизнес, будь это торговля шаурмой или строительство самолетов - это риск. Если у вас нет делового напора, наглости, то это не для вас. Ваш знакомый, я уверен, ни у кого не спрашивал, начать ему работать на себя или нет.
A: Ваши рассуждения разумны, но вы забыли про, наверно, самое важное - конкуренцию. Сейчас есть очень много людей, которые думают так же, открывают свое дело и прогорают, потому что не могут донести до потребителей чем они лучше чем еще несколько десятков фирм из бизнес-центра напротив. Ответьте сами себе на этот вопрос, если можете, и вы сами поймете, стоит ли начинать.
A: не буду пытаться дать Вам ответ на поставленный вопрос, но немного раскрою пелену происходящего в сфере ИТ на данный момент ( возможно с большими погрешностями +-inf):
в настоящий момент отчетливо выделяются 2 направления по "легкой" монетизации труда программистов:
- игровая индустрия: буть то консоли, PS, мобилки или браузерные игрушки
- и собственно Web.
про 1ю сказать толком не могу ни чего т.к ни разу себя не применял для игростроя, да и одному там ловить нечего...
А вот про web говорить особо и не надо: огромный спрос на изготовление разномастных сайтов дает каждому адекватному фрилансеру место под солнцем (и не только фрилансеру)...
что-то знаете, что-то умеете, есть портфолио в данном направлениее - смело выкладывайте в сеть, первый заказ не заставит себя ждать.
чем специфичнее ваше направление и выше спрос на него тем дороже будет оплачиватся Ваш труд.
если соотношение цена/качество устраивает заказчика с большой вероятностью он обратится к Вам снова. При удачных раскладах через доволько короткий срок Вы обрастете своей стабильной клиентской базой и таким количеством заказов, что в одиночку этот объем вывезти будет тяжело - тут уже начинается предпринимательская деятельность: нанимаете других фрилансеров по более низкой ставке, контролируете их, маржу кладете в карман. при должном настырстве и умении себя продавать за пару тройку лет с нуля можно выйти на уровень хорошей веб студии и снимать сливки с труда окружающих.
P.S: из личного опыта - за 2 года на фрилансе поднялся с 45т.р/мес в офисе до $45/час работая в свое удовольствие. открывать офис не собираюсь. | {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,937 |
\section{Introduction}
There has been a lot of recent interest in the study of $pp\to\gamma\gamma+{}$jets processes as
an important background to $pp\to H \to \gamma\gamma$ which is one of the cleanest decay
channels for studies of the Higgs properties \cite{Aad:2012tfa,Chatrchyan:2012ufa}. These processes also have
importance outside the realm of Higgs physics testing our ability to model isolated photon
radiation in association with strong interactions, see for example Refs.~\cite{Chatrchyan:2013oda,Aad:2013gaa}
for recent experimental studies of the closely related process of photon production in association with hard
jets. From a theoretical point of view di-photon production is
under good control with corrections known up to NNLO in QCD~%
\cite{Catani:2011qz}. NLO QCD corrections to $pp\to\gamma\gamma+j$ have been
available for some time~\cite{DelDuca:2003uz} and have recently been
re-explored~\cite{Gehrmann:2013aga} to investigate the impact of using different photon isolation criteria comparing the smooth
cone isolation~\cite{Frixione:1998jh} with the fixed cone isolation favoured
in experimental studies. The computation of $pp\to\gamma\gamma+2j$ including NLO QCD corrections has been
completed quite recently~\cite{Gehrmann:2013bga,Bern:2013bha}.
In this paper we extend the available range of predictions for $pp\to\gamma\gamma+{}$jets
to include up to three hard jets with NLO accuracy in QCD. The results show a
significant reduction in the uncertainty of the theoretical predictions and
have noticeable corrections to the shapes of the distributions when going from LO to NLO. As well as
providing new phenomenological studies relevant for the current measurements
at ATLAS and CMS we also present new developments to the {\sc NJet}{}~C++ code
enabling more efficient computations of high multiplicity processes at NLO.
Modern methods for scattering amplitude computations based on unitarity~\cite{Bern:1994zx},
generalized unitarity~\cite{Britto:2004nc} and integrand
reduction~\cite{Ossola:2006us} have opened up the possibility of performing precision
phenomenological studies with multi-particle final states at colliders. The current
state-of-the-art NLO QCD processes include
$pp\to W/Z+4j$~\cite{Berger:2010zx,Ita:2011wn}, $pp\to 4j$~\cite{Bern:2011ep,Badger:2012pf}, $pp\to W+5j$~\cite{Bern:2013gka}, $pp\to 5j$~\cite{Badger:2013yda},
all of which have been obtained using on-shell
methods. A wealth of $2\to 4$ and general lower multiplicity processes are now
becoming available in an increasing number automated codes \cite{Berger:2008sj,Hirschi:2011pa,Bevilacqua:2011xh,Cullen:2011ac,Cascioli:2011va,Actis:2012qn}.
This article is organized as follows: we begin by outlining our computational
set-up in Section~\ref{sec:setup}, where for completeness we review the well known
decomposition of next-to-leading order differential cross sections and describe
the interface of {\sc NJet}{} with the Sherpa Monte-Carlo, which we used for the computation of the unresolved real
radiation contributions and phase-space integration. In Section~\ref{sec:results} we provide results for the LHC
at centre-of-mass energy of $8$~TeV for both $pp\to\gamma\gamma+2j$ and $pp\to\gamma\gamma+3j$.
We present differential distributions for some important observables, particularly those
used in Higgs productions studies with vector-boson fusion phase space in the case of $pp\to\gamma\gamma+2j$.
We present a study of the dependence on the renormalization scale of the NLO predictions and investigate the
uncertainty due to the choice of Parton Distribution Functions (PDFs) on total cross sections and distributions.
In Section~\ref{sec:conclusions} we present our conclusions.
\section{Computational set-up \label{sec:setup}}
The computation is performed in the five-flavour scheme with massless b-quark included in the
initial state. The basic partonic sub-processes considered are:
\begin{align*}
&0 \to \gamma \gamma q\bar{q} gg &
&0 \to \gamma \gamma q\bar{q} q'\bar{q}'
\end{align*}
for $pp\to \gamma\gamma+2j$ and
\begin{align*}
&0 \to \gamma \gamma q\bar{q} ggg &
&0 \to \gamma \gamma q\bar{q} q'\bar{q}'g,
\end{align*}
for $pp\to \gamma\gamma+3j$, from which all relevant channels can be derived using crossing
symmetries. Channels with like-flavour fermions are obtained from the above using the appropriate
(anti-)symmetrization relations.
We do not include loop-induced and formally higher order sub-processes
$0 \to \gamma\gamma + 4g$ and $0 \to \gamma\gamma + 5g$.
We can schematically write down the NLO partonic cross section as a sum of four finite contributions
which can be integrated separately over their respective phase spaces,
\begin{equation}
d\sigma_n^{NLO} =
\int_n d\sigma_n^{B} +
\int_n d\sigma_n^{V} +
\int_n d\sigma_n^{I} +
\int_{n+1}d\sigma_{n+1}^{RS},
\label{eq:NLOdecomp}
\end{equation}
where $d\sigma_n^{B}$ denotes the leading order contribution, $d\sigma_N^{I}$ contains the integrated
dipole subtraction terms, including factorization contributions from initial state singularities, $d\sigma_n^V$
the one-loop virtual corrections and $d\sigma_{n+1}^{RS}$ the infra-red finite contributions from real-radiation
and dipole subtraction terms. This hard partonic cross section is then convoluted with the parton distribution
functions to obtain the cross sections for hadronic collisions.
The computation of the Born and real-emission matrix elements is performed using the colour dressed recursive
Berends-Giele formulation~\cite{Berends:1987me} implemented in the Comix amplitude generator~\cite{Gleisberg:2008fv}.
The subtraction of infra-red singularities is performed using the Catani-Seymour~\cite{Catani:1996jh} dipole method.
The evaluation of these contributions is performed using the Sherpa~\cite{Gleisberg:2007md,Gleisberg:2008ta} package, which is also used
for the determination and organization of the partonic subprocesses and the integration over the phase space.
The one-loop virtual amplitudes are provided using the automated generalized unitarity framework implemented
in the latest version of the {\sc NJet}{}~library\footnote{The code {\sc NJet 2.0}{} is available at \url{https://bitbucket.org/njet/njet/downloads}.}.
{\sc NJet 2.0}~is an updated code based on {\sc NJet}~\cite{Badger:2012pg} and
{\sc NGluon}~\cite{Badger:2010nx}. The primitive kinematic objects are constructed
using a numerical generalized unitarity algorithm~\cite{Bern:1994zx,Britto:2004nc,Ossola:2006us,Ellis:2007br,Forde:2007mi,Giele:2008ve,Berger:2008sj,Badger:2008cm}
with Berends-Giele recursion
relations for the tree-level input~\cite{Berends:1987me}. The extended code can compute arbitrary
primitive amplitudes for vector bosons (W, Z and $\gamma$) and massless QCD
partons. Full colour sums are implemented for pure QCD with up to five jets,
vector bosons with up to five jets and di-photon production with up to four
jets. We interface this code to the Sherpa Monte-Carlo via the updated Binoth
Les Houches Accord~\cite{Binoth:2010xt,Alioli:2013nda} to obtain the virtual events. In addition to the standard
internal checks we have managed to check individual phase-space points for the
processes used in this paper against those obtained with {\sc
GoSam}~\cite{Cullen:2011ac} and {\sc MadLoop}~\cite{Hirschi:2011pa}. For both
processes we have neglected the small effect of top quark loops in the virtual
amplitudes. In the case of $pp\to\gamma\gamma+3j$ we also neglect the
contribution from vector loops where the photons couple directly to a virtual
fermion loop. These corrections have been included in $pp\to\gamma\gamma+2j$
contributing less than 0.5\% of the total cross section, therefore
they are expected to be negligible for $pp\to\gamma\gamma+3j$.
For the current calculation we made use of some new features to optimise the
computation time needed for the virtual corrections. Firstly we use a C++
library Vc~\cite{Kretz_Lindenstruth_2012} to utilize vector capabilities of
modern CPUs and gain a factor of $\sim2$ in the computation speed. We also
separate leading and sub-leading contributions in colour such that the simpler,
dominant contributions can be sampled more often than the sub-leading terms.
The definition of our leading terms include all multi-quark processes in the
large $N_c$ limit and processes with two or more gluons in the final state
using the de-symmetrized form of the colour sum that efficiently exploits the
Bose symmetry of the phase space~\cite{Ellis:2009zw,Badger:2012pg}. The
de-symmetrized sums give full colour information and are faster than leading
colour when we have many final state gluons. In Figure~\ref{fig:colour} we
show virtual corrections to the 3rd leading jet transverse momentum in $pp\to
\gamma\gamma+3j$. The corrections from the sub-leading part are around $10\%$
on average with a slight rise to around $20\%$ at large $p_T$. In the case of
$pp\to \gamma\gamma+3j$ the virtual cross sections are about $1/3$ of the size
of the total cross section. For the current implementation of $pp\to
\gamma\gamma+3j$, the leading virtual events are generated approximately $7$
times faster than the sub-leading events.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.65\textwidth]{{plotsAA3j/plot_AA3j_colour}.pdf}
\end{center}
\caption{Full colour and leading approximation (as explained in the text)
for the virtual corrections to the transverse momentum of the 3rd jet in $pp\to \gamma\gamma+3j$.}
\label{fig:colour}
\end{figure}
In order to maximise the phenomenological predictions that we can extract from
processes with such complicated final states, we make use of the ROOT Ntuple
format provided by Sherpa~\cite{Binoth:2010ra}. During the course of event
generation, additional weights from the poles in the loop process and from the
subtraction terms are stored along with a full list of kinematic variables and
couplings. This allows the events to be re-weighted to different scale and PDF
choices during analysis. Even within this approach a full
study of PDF uncertainties and scale variations can be computationally
intensive. During this work we have developed an interface to the
APPLgrid library~\cite{Carli:2010rw} to allow for specific observables to be
efficiently re-weighted changing the scales and the specific PDF set used.
\section{Numerical results \label{sec:results}}
All the results presented in this section are for $pp$ collisions with a centre-of-mass energy of
8~TeV. We consider the following kinematic cuts on the external momenta, which are inspired by
typical experimental cuts used in the analyses at LHC
\begin{align*}
&p_{T,j} > 30 \,{\rm GeV} && |\eta_j|\leq 4.7 && \\
&p_{T,\gamma_1} > 40 \,{\rm GeV} && p_{T,\gamma_2} > 25 \,{\rm GeV} && |\eta_\gamma|\leq 2.5 \\
&R_{\gamma,j} = 0.5 && R_{\gamma,\gamma} = 0.45
\end{align*}
where the photon transverse momenta have been ordered by size. The jets are defined using the
anti-$k_T$ algorithm \cite{Cacciari:2008gp} with cone size $R=0.5$ as implemented in {\sc FastJet}~\cite{Cacciari:2011ma}.
Photons are selected using the Frixione smooth cone isolation criterion~\cite{Frixione:1998jh}. A photon
is considered isolated if the total hadronic energy inside all cones of radius $r_\gamma < R$
\begin{equation}
E_{\text{hadronic}}(r_\gamma) \leq \epsilon \,p_{T,\gamma} \left( \frac{1-\cos{r_\gamma}}{1-\cos{R}} \right)^n
\label{eq:frixionecone}
\end{equation}
with $\epsilon=0.05$, $R=0.4$ and $n=1$. We use the NLO CT10 PDF set~\cite{Lai:2010vv} for our central predictions
with the strong coupling running from $\alpha_s(M_Z) = 0.118$, and the electromagnetic coupling fixed
at $\alpha=1/137.036$. In particular we use the same (NLO) PDF set and definition of the strong coupling constant
both for LO and NLO predictions. Using a NLO PDF set for the LO computation includes higher order terms that go
beyond a fixed order prediction, nevertheless such a set-up allows us
to separate NLO effects coming from the running of the strong coupling and from
PDFs and to highlight the impact of corrections coming from the NLO matrix
elements.
We choose a dynamical value for the factorization and renormalization scales which are kept equal,
$\mu_R = \mu_F$, when performing scale variations. We have investigated the dependence on a number of different
functional forms which we will denote as:
\begin{align}
\widehat{H}_T &= p_{T,\gamma_1} + p_{T,\gamma_{2}} + \sum_{i\in\text{partons}} p_{T,i} \\
\widehat{H}_T' &= m_{\gamma\gamma} + \sum_{i\in\text{partons}} p_{T,i} \\
\widehat{\Sigma}^2 &= m_{\gamma\gamma}^2 + \sum_{i\in\text{partons}} p_{T,i}^2 \\
H_T' &= m_{\gamma\gamma} + \sum_{i\in\text{jets}} p_{T,i} \\
\Sigma^2 &= m_{\gamma\gamma}^2 + \sum_{i\in\text{jets}} p_{T,i}^2
\label{eq:scaledefs}
\end{align}
where $m_{\gamma\gamma} = \sqrt{p_{T,\gamma_1}^2+p_{T,\gamma_2}^2}$. The quantities $H_T'$ and
$\Sigma^2$ are constructed after the clustering of final state partons into jets. Notice that partonic and jet-level
scales will only differ at NLO, where the additional unresolved radiation enters in the clustering algorithm.
\subsection{Results for $pp\to \gamma\gamma+2j$}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{plotsAA2j/AA2j_scalevar.pdf}
\end{center}
\caption{Scale variations on the total cross section for $pp\to \gamma\gamma+2j$ for a variety
of dynamical scales. Dashed lines are LO accuracy whereas solid lines are NLO accuracy. The vertical
line at $x=1$ corresponds to the central scale whereas the lines at $x=0.5$ and $x=2$ show the bounds
of the scale variation region.}
\label{fig:AA2j_scalevar}
\end{figure}
We first consider the production of a photon pair in association with two jets. In this case
we compare our predictions with the recent results of reference~\cite{Gehrmann:2013bga}
and present additional studies of PDF variations and dynamical scale choices.
For the latter we rely on the possibility to substantially reduce the computational cost
by the use of our APPLgrid set-up.
In Figure~\ref{fig:AA2j_scalevar} we show the dependence of the total inclusive cross section
upon variation of the renormalization and factorization scales with $\mu_R=\mu_F$. We consider the five different
dynamical scales defined in Eq.~\eqref{eq:scaledefs}. Though the scale choices are closely related to each other
we find significant deviations for the value of the total cross section. Nevertheless the NLO
predictions show a significant reduction in the dependence on the scale variation compared to the
LO ones. Taking the envelope of all the scales considered we see that the LO predictions vary in the interval
$1.64-3.04$ whereas NLO ones lie within $2.46-3.58$ when the scales are varied over the range
$x\in[0.5,2]$ around the central choice. This represents a reduction in the scale variation uncertainty
from $\sim 30\%$ at LO to $\sim 20\%$ at NLO.
We notice that choosing $\mu_R=\sqrt{\widehat{\Sigma}^2}$ as a scale leads to significantly larger
predictions for the total cross section than with the other scales considered here. Moreover the scale variation
profile closely resembles the one of a leading order prediction. We therefore consider this choice of scale
disfavoured with respect to the others, which provide results that are in better agreement with each other.
In general, we find that the $H_T$-based scales lead to a broader profile of scale variations and on average
favour harder events than the $\Sigma$-based scales. On the other hand we find the shapes of the distributions
to be quite stable with respect to the scale choice.
The results for the total cross section and distributions using $\mu_R=\sqrt{\Sigma^2}/2$ are
in good agreement with those obtained previously by Gehrmann, Greiner and Heinrich~\cite{Gehrmann:2013bga}.
In the following we opt for the scale $\widehat{H}_T'$ for computing our predictions for the total cross section and
the differential distributions presented here. This scale has been widely used in studies of $W+{}$jets (see for example~\cite{Bern:2013gka}).
The values for the total cross sections at both LO and NLO computed our default choices for scale, PDF set and physical
parameters are found to be:
\begin{align}
\sigma_{\gamma\gamma+2j}^{LO}(\widehat{H}_T'/2) &= 2.046(0.002)^{+0.534}_{-0.396}\,{\rm pb} &
\sigma_{\gamma\gamma+2j}^{NLO}(\widehat{H}_T'/2) &= 2.691(0.007)^{+0.167}_{-0.225}\,{\rm pb}
\label{eq:AA2jinclXS}
\end{align}
where the sub-scripts(super-scripts) show the maximum deviation from the central value in the negative(positive) direction
over the range $x\in[0.5,2]$ for $\mu_R = x \widehat{H}_T'/2$. Monte-Carlo integration errors are shown in brackets.
Figure \ref{fig:aa2j_jet_pt} shows the differential distributions for the ordered jet transverse momenta.
The results for the scale $\widehat{H}_T'/2$ are consistent with those obtained at the scale $\sqrt{\Sigma^2}$
with a significant reduction in scale variation from around $20\%$ at LO to $\sim 10\%$ at NLO in both cases.
The $K$-factor is fairly constant at around $1.1$ for $p_T$ higher than 200 GeV rising to $1.4$ as we approach the $p_T$ cut.
This larger $K$-factor in the low $p_T$ region could be the indication of the presence of large logarithms beyond fixed order NLO.
The di-photon invariant mass $m_{\gamma\gamma}$ and the di-photon rapidity distributions $\eta_{\gamma\gamma}$ are shown in
Figure~\ref{fig:aa2j_photonpair}. They receive slightly larger NLO corrections with respect to the jet transverse momenta with
the $K$-factor for the former ranging from $1.2$ in the large invariant mass region to $1.7$ for lower invariant masses, while
for the latter NLO correction induce a roughly flat $K$-factor of~$1.3$.
Figure \ref{fig:aa2j_vbf} shows four distributions of angular quantities that are usually used in analyses of
Higgs production in vector boson fusion (VBF), where additional cuts are imposed in order to reduce QCD
backgrounds in $pp\to H (\to\gamma\gamma)+2j$ studies.
Owing to the increased phase-space available to the real radiation at NLO we see large deviations
from the shapes of the leading order distributions. These features have been pointed out for the jet-pair
azimuthal angle $\Delta\phi_{j_1j_2}$ and for the separation of the leading-photon/leading jet,
\begin{align}
R_{\gamma_1 j_1} = \sqrt{\Delta y_{\gamma_1 j_1}^2 + \Delta\phi_{\gamma_1 j_1}^2},
\label{eq:R11}
\end{align}
in Ref.~\cite{Gehrmann:2013bga}. We also see increasing deviations for large rapidities of the jet pair $\eta_{j_1 j_2}$
and even more so in the relative rapidity of the diphoton-dijet system,
\begin{align}
y_{\gamma\gamma jj}^* = y_{\gamma\gamma} - (y_{j_1}+y_{j_2})/2.
\label{eq:ystar}
\end{align}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_jet_pT_1}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_jet_pT_2}.pdf}
\end{center}
\caption{Differential distributions for jet transverse momenta. The lower plot shows the ratio
of NLO to LO including the scale variation bands estimated over the range of $x\in[0.5,2]$.}
\label{fig:aa2j_jet_pt}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_photon_mass}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_photon_eta}.pdf}
\end{center}
\caption{Differential distributions for di-photon invariant mass and rapidity $pp\to\gamma\gamma+2j$.}
\label{fig:aa2j_photonpair}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_photon_jet_R11}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_jet_jet_phi12}.pdf} \\
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_jet_jet_eta12}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA2j/plot_p36_e18_s0.3_maahthat_diphoton_dijet_ystar}.pdf}
\end{center}
\caption{Differential distributions for the angular observables $R_{\gamma_1 j_1}$ (see Eq.~\eqref{eq:R11}),
$\Delta\phi_{j_1j_2}$, $\eta_{j_1 j_2}$ and $y_{\gamma\gamma jj}^*$ (see Eq.~\eqref{eq:ystar}) in $pp\to\gamma\gamma+2j$.}
\label{fig:aa2j_vbf}
\end{figure}
Using the APPLgrid set-up described in the previous section, we have also performed a study of PDF uncertainties on
$pp\to\gamma\gamma+2j$, concentrating on the total cross section and the invariant mass distribution of the photon
pair. In Table \ref{tab:aa2jXSpdf} we show the central value and PDF uncertainties for the total cross section evaluated
at the central scale ($\mu_R=\widehat{H}_T'/2$) using four different NLO PDF sets: CT10~\cite{Lai:2010vv},
MSTW2008~\cite{Martin:2009iq}, ABM11~\cite{Alekhin:2012ig} and NNPDF2.3~\cite{Ball:2012cx}. All PDF sets are compared
using the same value of $\alpha_s(M_Z) = 0.118$ in order to disentangle PDF and strong coupling constant uncertainties.
PDF uncertainties are considerably smaller than the theoretical uncertainty estimated from scale variations and range from
1\% for MSTW and NNPDF to 3.5\% for ABM. We note, however that the ABM11 uncertainty includes errors associated with
$\alpha_s$ variations. In general, we find that central predictions from different PDF sets differ by amounts which are larger
than the nominal PDF uncertainty of each set.
\begin{table}
\centering
\renewcommand\arraystretch{1.3}
\begin{tabular}{lccc}
\hline
PDF set & $\sigma_{\gamma\gamma+2j}^{NLO}(\widehat{H}_T'/2)$
& $\delta\sigma_{\gamma\gamma+2j}^{NLO,PDF+}(\widehat{H}_T'/2)$
& $\delta\sigma_{\gamma\gamma+2j}^{NLO,PDF-}(\widehat{H}_T'/2)$ \\
\hline
CT10nlo & $2.69102$ & $+0.0357456$ & $-0.042148$ \\
NNPDF2.3 & $2.77285$ & $+0.0167702$ & $-0.016770$ \\
MSTW2008 & $2.71578$ & $+0.0184072$ & $-0.016373$ \\
ABM11 & $2.73791$ & $+0.0659662$ & $-0.065966$ \\
\hline
\end{tabular}
\caption{The total cross section for $pp\to \gamma\gamma+2j$ computed with different PDF sets at $\alpha_s(M_Z)=0.118$.
The PDF uncertainties are computed from the relevant PDF error sets.
}
\label{tab:aa2jXSpdf}
\end{table}
In Figure \ref{fig:AA2jmaadistPDF} we show the distributions for the invariant mass of the photon pair ($m_{\gamma\gamma}$)
computed at the scale $\mu_R=\widehat{H}_T'/2$ and the four PDF sets considered before at $\alpha_s(M_Z) = 0.118$.
The upper (log scale) plot shows the absolute predictions for all sets, which are indeed extremely
close to each other. In the lower plot, where we show the ratio of each set to the central value
computed using CT10, we do however notice discrepancies in the predictions of the order of a few percent.
These differences are especially pronounced at high $m_{\gamma\gamma}$ where the spread between them reaches $\sim 6\%$,
a value larger than PDF uncertainties coming from individual sets (represented by the shaded regions). We also note that, while
the differences between CT10, MSTW08 and NNPDF2.3 are mostly due to an overall normalization factor, the ABM11 predictions
stand out from the others also in the shape being higher for low values of $m_{\gamma\gamma}$ and dropping below the other
predictions for higher values of the invariant mass.
\begin{figure}[h]
\begin{center}
\includegraphics{{plotsAA2j/AA2j_plot_pdfvar_phtmass}.pdf}
\end{center}
\caption{The $m_{\gamma\gamma}$ distributions for $pp\to \gamma\gamma+2j$ for the four PDF sets described in the text
at $\alpha_s{M_Z} = 0.118$. The lower plot shows the ratio of each set to CT10 with the shaded region representing
the PDF uncertainty.}
\label{fig:AA2jmaadistPDF}
\end{figure}
\subsection{Results for $pp\to \gamma\gamma+3j$}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{plotsAA3j/AA3j_scalevar.pdf}
\end{center}
\caption{The result of scale variations on the total inclusive cross section for
$pp\to\gamma\gamma+3j$. LO~curves are represented with dashed lines while NLO
curves are represented with solid lines.}
\label{fig:AA3j_scalevar}
\end{figure}
We now consider the production of a photon pair in association with three jets.
As in the previous section we studied the dependence
of the total cross section upon variation of renormalization and factorization
scales with the choices of dynamical scales defined in Eq.~\eqref{eq:scaledefs}.
The results in Figure~\ref{fig:AA3j_scalevar} show reasonable
differences between quantities based on jets versus quantities based on
partons. Overall we find a significant improvement in the uncertainty
estimated from scale variations when going from LO to NLO. The envelope of
predictions from all scale choices varied over the range $x\in[0.5,2]$ is
around $0.67-0.99$~pb at NLO compared to $0.46-1.28$~pb at LO. This represents a
decrease in variation from $\sim 50\%$ at LO to $\sim 20\%$ at NLO. As in the two
jet case the scales based on $\Sigma^2$ give generally larger predictions than
those based on $H_T$. Other than the overall normalization, we find that all
scales give very similar predictions for shapes of the distributions.
Comparing Figure~\ref{fig:AA3j_scalevar} with Figure~\ref{fig:AA2j_scalevar} we see that
the peak in the NLO curve for $\Sigma^2$ has moved further to the right than the
$H_T$ scales which may suggest that a range of $x\in[1,4]$ would be more appropriate
here. Since we would like to make predictions for jet ratios we need to have
as consistent description of $\gamma\gamma+3j$ and $\gamma\gamma+2j$ as
possible and therefore we prefer the $H_T$ scales. In the following we choose to adopt
the central scale of $\widehat{H}_T'/2$ for the total rates and distributions, though theoretical
uncertainties are likely underestimated by the simple scale variations following the discussion above.
For the total cross sections at LO and NLO we find,
\begin{align}
\sigma_{\gamma\gamma+3j}^{LO}(\widehat{H}_T'/2) &= 0.643(0.003)^{+0.278}_{-0.180}\,{\rm pb} &
\sigma_{\gamma\gamma+3j}^{NLO}(\widehat{H}_T'/2) &= 0.785(0.010)^{+0.027}_{-0.085}\,{\rm pb}
\label{eq:AA3jinclXS}
\end{align}
where the sub-scripts(super-scripts) show the maximum deviation from the
central value in the negative(positive) direction over the range $x\in[0.5,2]$
for $\mu_R = x \widehat{H}_T'/2$ and Monte-Carlo integration errors are shown in
brackets.
The distributions for the jets transverse momenta are shown in Figure~\ref{fig:AA3j_jpt}. The
$K$-factor is quite flat with a value between $1.0$ and $1.2$ except in the
low $p_T$ where it rises to around $1.4$ for the leading jet $p_T$. Again, this
may suggest the presence of large logarithms in the missing higher order
contributions in this region. The distribution of the 3rd jet shown in Figure~\ref{fig:AA3j_jpt3} has a noticeably
flatter $K$-factor than the ones for the two leading jets.
The di-photon invariant mass distribution in Figure~\ref{fig:photon_mass} receives significant corrections to
the shape at NLO with the $K$-factors increasing from around $1.0$ at low $m_{\gamma\gamma}$ to $1.4$ at
large values of the photon pair invariant mass. In Figure~\ref{fig:extra_dist} we show the leading jet/leading
photon separation $R_{\gamma_1 j_1}$ and the azimuthal separation of the two leading jets $\Delta\phi_{j_1 j_2}$.
Both quantities receive large NLO corrections for small values of the observable, though notably not as large as
the corresponding distributions in $pp\to\gamma\gamma+2j$ where there is a more substantial increase in the
available phase-space at NLO.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_jet_pT_1}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_jet_pT_2}.pdf}
\end{center}
\caption{Differential cross section as a function of $p_T$ of the 1st and 2nd leading jets.}
\label{fig:AA3j_jpt}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfloat[]{\label{fig:AA3j_jpt3}%
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_jet_pT_3}.pdf}}
\subfloat[]{\label{fig:photon_mass}%
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_photon_mass}.pdf}}
\end{center}
\caption{Differential cross section as a function of $p_T$ of the 3rd leading jet and
di-photon invariant mass in $pp\to \gamma\gamma+3j$.}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_photon_jet_R11}.pdf}
\includegraphics[width=0.45\textwidth]{{plotsAA3j/plot_q20.6_l20_sm0.2_maahthat_jet_jet_phi12}.pdf}
\end{center}
\caption{Photon/jet separation and di-jet azimuthal angle distributions in $pp\to \gamma\gamma+3j$.}
\label{fig:extra_dist}
\end{figure}
The PDF analysis was performed with the same set-up as for $pp\to \gamma\gamma+2j$ with the
four PDF sets compared at the same value of $\alpha_s(M_Z)=0.118$. Table~\ref{tab:aa3jXSpdf}
shows the results for the total cross section at the central scale $\mu_R = \widehat{H}_T'/2$.
Again we see that PDF uncertainties for all sets are noticeably smaller than theoretical uncertainties
estimated from scale variations and that central predictions from different sets differ by more than the
nominal uncertainty obtained with each set.
Figure~\ref{fig:AA3jmaadistPDF} shows the comparison of the $m_{\gamma\gamma}$ distribution
for the different PDF~sets. The deviation between ABM11 and NNPDF, which again give the lower and higher limit
of the predictions from different sets, is somewhat larger than that seen in $pp\to \gamma\gamma+2j$,
though the results are consistent within the theoretical uncertainties determined by scale variations.
\begin{table}
\centering
\renewcommand\arraystretch{1.3}
\begin{tabular}{lccc}
\hline
PDF set & $\sigma_{\gamma\gamma+3j}^{NLO}(\widehat{H}_T'/2)$
& $\delta\sigma_{\gamma\gamma+3j}^{NLO,PDF+}(\widehat{H}_T'/2)$
& $\delta\sigma_{\gamma\gamma+3j}^{NLO,PDF-}(\widehat{H}_T'/2)$ \\
\hline
CT10nlo & $0.746696$ & $+0.0123788$ & $-0.0133826$ \\
NNPDF2.3 & $0.773112$ & $+0.0056425$ & $-0.0056425$ \\
MSTW2008 & $0.752756$ & $+0.0068782$ & $-0.0050721$ \\
ABM11 & $0.731019$ & $+0.0241568$ & $-0.0241568$ \\
\hline
\end{tabular}
\caption{The total cross section for $pp\to \gamma\gamma+3j$ computed with different
PDF sets at $\alpha_s(M_Z)=0.118$. The PDF uncertainties are computed from the relevant PDF error sets.
}
\label{tab:aa3jXSpdf}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics{{plotsAA3j/AA3j_plot_pdfvar_phtmass}.pdf}
\end{center}
\caption{The $m_{\gamma\gamma}$ distributions for $pp\to \gamma\gamma+3j$ for the four PDF sets described in the text
at $\alpha_s{M_Z} = 0.118$. The lower plot contains the ratio of each set to CT10 with the shaded region representing
the PDF uncertainties.}
\label{fig:AA3jmaadistPDF}
\end{figure}
\subsection{The three-to-two jet ratio}
In the context of multi-jet production studies it is interesting to look at the ratio of $pp\to \gamma\gamma+3j$
over $pp\to \gamma\gamma+2j$ which we will denote as $R_{3/2}$.
Due to the cancellation of many uncertainties both theoretical and experimental, ratios such as this one are prime
observables for the determination of physical parameters (like $\alpha_s$).
In the case of the production of di-photon in association with jets, we find $R_{3/2}$ to be:
\begin{align}
R_{3/2}^{LO}(\mu_R=\widehat{H}_T'/2) &= 0.314(0.002) &
R_{3/2}^{NLO}(\mu_R=\widehat{H}_T'/2) &= 0.276(0.004)
\label{eq:3to2ratio}
\end{align}
where the numbers in brackets refer to Monte-Carlo errors. We have checked that all scale choices
are in much better agreement for $R_{3/2}$ with the range of predictions lying within $\sim 8\%$
of this value. In Figure \ref{fig:jetratio_pT_1} we show the differential ratio with the $p_T$ of the
leading jet. The NLO corrections become more important for $p_T>100$ GeV and reaching
about $15\%$ at high $p_T$.
\begin{figure}[h]
\begin{center}
\includegraphics{{plotsAA3j/plot_maahthat_jetratio_pT_1}.pdf}
\end{center}
\caption{The ratio of of $pp\to \gamma\gamma+3j$ over $pp\to \gamma\gamma+2j$ as a function of leading jet~$p_T$.}
\label{fig:jetratio_pT_1}
\end{figure}
\section{Conclusions \label{sec:conclusions}}
In this paper we have presented a study of di-photon production in association with jets.
The first calculation of the full NLO QCD corrections to the process $pp\to\gamma\gamma+3j$ is presented and discussed,
together with a variety of results for $pp\to\gamma\gamma+2j$. We find that the inclusion of NLO QCD corrections leads
to a significant reduction of theoretical uncertainties both on total cross sections and distributions.
We have studied distributions for a number of observables, of particular interest are those relevant for Higgs production in VBF
analyses which are used when modelling $\gamma\gamma+{}$jets as a background to $pp\to H+\text{jets}\to \gamma\gamma+{}$jets.
The present study is based on the use of the Frixione smooth cone isolation criterion to define the final state photons, which
provides a theoretically clean way of suppressing the fragmentation component in direct photon production. It would be
also interesting to consider alternative isolation criteria, like the fixed cone isolation, which require the inclusion of fragmentation
functions, and were shown to have a significant effect in lower multiplicity processes \cite{Gehrmann:2013aga}.
Moreover the majority of experimental analyses involving direct photon production (both in association with jets or not) rely
on the cone isolation for photon identification.
Nonetheless, we hope that the results presented here will be of use in future experimental analyses and look forward
to direct comparisons with the LHC data.
\acknowledgments{%
We would like to thank Peter Uwer, Benedikt Biedermann, Thomas Gehrmann, Gudrun Heinrich and
Nicolas Greiner and Nicolas Chanon for helpful discussions. We are grateful to the Humboldt-Universit\"{a}t zu Berlin
for providing computing resources. This work has been supported by the Alexander von Humboldt Foundation,
in the framework of the Sofja Kovaleskaja Award 2010, endowed by the German Federal Ministry of Education and Research.
}
\providecommand{\href}[2]{#2}\begingroup\raggedright | {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,618 |
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Kulturarv er et begreb, der bruges som betegnelse for kulturelle elementer, som anses for at være med til at 1) minde os om fortiden og 2) danne en fælles erindring om den. Erindringen kan være fælles for hele verden eller en mindre gruppe som en nation eller et folkeslag.
Materiel og immateriel
Der kan skelnes mellem materiel og immateriel kulturarv. Den materielle udgøres af fx monumenter, mens den immaterielle er fx skikke.
UNESCOs verdenskulturarvsliste
I 1972 vedtog UNESCO en konvention til beskyttelse af verdensarv, der omfattede den materielle kulturarv (Danmark ratificerede konventionen i 1979), og i 2003 blev en lignende konvention vedtaget for den immaterielle kulturarv (Danmark ratificerede konventionen i 2009).
Definition
I en dansk sammenhæng dukker begrebet Kulturarv for alvor op i 1980-erne. Forud for brugen af dette begreb gik brugen af begreberne "identitet" og "kollektiv hukommelse". Fra 1980-erne ekspanderede begrebet kulturarv stærkt i den kulturpolitiske og institutionspolitiske debat, hvor der nu tales om "den danske kulturarv" eller "Danmarks kulturarv", især med fokus på museers og bibliotekers forvaltning af kulturarven. Kulturministeriet introducerede begrebet kulturarv på en konference i 1999 i en tale af daværende kulturminister Elsebeth Gerner Nielsen. I forbindelse med, at Kulturministeriet i 2003 udarbejdede en rapport om bevaring af Kulturarven, fastlagde ministeriet, at kulturarvsinstitutionerne i Danmark er:
på arkivområdet: Statens Arkiver,
på biblioteksområdet: Det Kgl. Bibliotek (tidligere Det Kongelige Bibliotek og Statsbiblioteket),
på museumsområdet: Statens Museum for Kunst, Nationalmuseet, Statens Naturhistoriske Museum og Det danske Filmmuseum.
I det seneste årti er kulturarvsbegrebet blevet udvidet til at omfatte både den kanoniserede (fin)kulturelle arv og den fødte kulturarv. Kulturarvsbegrebet er også udvidet til det digitale område, idet Det Kgl. Bibliotek via Netarkivet indsamler den digitale kulturarv fra den danske del af Internettet.
Eksterne henvisninger
Dansk Kulturarv – Hjemmeside som formidler Kulturarv fra Danmarks Radio, Det Danske Filminstitut, Det Kgl. Bibliotek, Nationalmuseet, Statens Arkiver, Statens Museum for Kunst & Slots- og Kulturstyrelsen.
Kulturarven – definition af kulturarv fra Slots- og Kulturstyrelsen
UNESCO – immateriel kulturarv – artikel fra Undervisningsministeriet
Cultural Heritage – UNESCOs engelske hjemmeside om kulturarv
Noter | {
"redpajama_set_name": "RedPajamaWikipedia"
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11 Twisted Ray Garton Books That Will Give You Nightmares
These stories will haunt you long after you've turned the last page.
Promoted by Open Road Media | By Dave Adams | Published Aug 13, 2019
Sex-crazed vampires. Demonic heavy metal rockers. Man-eating arachnids. No matter what form it takes, the evil that lurks within the pages of a Ray Garton novel wants one thing and one thing only—to scare the bejesus out of you. Godfather of "splatterpunk" and Bram Stoker Award finalist for his novel Live Girls, Garton serves up an endless supply of chills, thrills, and gore in these 11 twisted books, each of which features unique, relatable characters facing off against the most unspeakable of terrors. But be warned—once you've let this Grand Master of Horror into your head, you won't be able to get him out.
The Loveliest Dead
By Ray Garton
When she inherits her estranged father's house in Eureka, California, Jenna Kellar thanks God for small miracles. Her youngest son, Josh, has just died, and she, her husband, David, and their surviving son, Miles, need a fresh start. But when they arrive at the ramshackle house, the Kellars discover that it's already occupied—by ghosts. Spooky children play on the backyard swing set. An ominous figure watches over Miles as he sleeps. When Jenna sees Josh among the spirits, she sets out to communicate with him. But does the door she's about to open lead to heaven—or hell?
Related: 13 People Share the Scariest Books They've Ever Read
The Girl in the Basement
15-year-old Ryan Ketterling knows there's a lot to like about his new foster home. The Prestons genuinely care about their foster kids, one of whom is a cute girl named Lyssa. She and Ryan have a budding romance, but it's another girl in the Preston household that he can't get off his mind. Maddy lives in the basement. She's only nine years old, but she talks in the deep, gravelly voice of an adult. And she knows things—things about Ryan that no one else knows. And things about the future. Ryan thinks he can help Maddy, but he has no idea what kind of trouble she's really in.
'Nids
It's Saturday night in Hope Valley and Lovers' Lookout is full. Three parked cars, six teenagers hoping to make tonight unforgettable. And it will be—but not for the reasons they think. An accident at the BioGenTech research lab on the outskirts of town has just released a genetically-modified monster into the wild. It's huge, mean, and hungry—and it's headed straight for Lovers' Lookout on eight hairy legs. Once the beast gets its first taste of human flesh, it won't stop until the entire town is destroyed.
Crucifax
When heavy metal rocker Mace arrives in California's San Fernando Valley, his siren song of sex, drugs, and rock 'n' roll is music to Mallory Carr's ears. But what's up with the strange, ratlike creatures that follow Mace wherever he goes? And the axe blade on the crucifix he wears around his neck? By the time Jeff Carr realizes how far his big sister and her friends have fallen under the stranger's spell, it's too late—Mace has evil plans for his young groupies, and he'll crush anyone and anything that gets in his way.
Related: 10 Scary Podcasts for Fans of the Paranormal
Want more horror books? Sign up for The Lineup's newsletter, and get our freakiest recommendations delivered straight to your inbox.
Dumped by his girlfriend and passed over for a promotion, Davey Owen wanders through a Times Square doorway marked "LIVE GIRLS." Inside, a beautiful woman gives him the greatest pleasure of his life—and the bite marks to remember her by. But in the following days, Davey starts to feel very strange. And when he learns that a fugitive killer visited the same peep show, he begins to suspect that the women of LIVE GIRLS are not who they claim to be. But who—or what—are they? To find out, he'll have to face off against mankind's bloodthirsty enemy in this Bram Stoker Award-nominated horror thriller.
Related: Bram Stoker's Descendant Gives Dracula a Creepy Origin Story
After battling a coven of Times Square vampires, Davey Owen escaped to Los Angeles with his true love, Casey Thorne. But the undead never forget—and never forgive. When the Midnight Club vampires come calling on Davey and Casey, the duo must unleash the otherworldly powers they've kept locked within themselves for years. Meanwhile, a world-renowned horror writer is on a mission to prove that vampires exist, and the trail leads straight to L.A. Can Davey and Casey keep their deepest, darkest secret safe while saving the world?
Related: 17 Romantic Horror Movies for Your Next Night In
Darklings
What drives murderers and madmen to commit their vile acts? In this twisted horror thriller, the answer is as simple—and as terrifying—as a parasite. When Darklings slither from the body of a serial killer dying in a hospital emergency room, they seek a new host. Soon four innocent people have become living, breathing embodiments of evil, their minds ravaged by the sinister organism. Where did it come from? What does it want? To find the answers, a small town must race against the clock to stop a monster that multiplies as fast as it kills.
Lot Lizards
Long-haul trucker Bill Ketter is lonely, bored, and certain that his wife has finally left him for good. So when a "lot lizard"—a prostitute who plies her trade at truck stops—knocks on his cab door, he opens it. But it's not Bill's money the girl is after—it's his blood. Before he knows it, Bill is one of the undead. But that doesn't mean he won't try to stop a trailer full of vampires from devouring dozens of snowbound travelers, including his estranged wife and teenage son. To do so, he'll have to battle an evil as powerful as it is ancient.
Related: Vampires, Witches, and More: The 9 Best Anne Rice Books
The New Neighbor
A demonic force has taken root on Deerfield Avenue. It feeds on the souls of young and old alike, turning them into husks of their former selves. It arrived under the cover of darkness, occupying the house across the street from the Pritchard family. Its name? Lorelle Dupree. One by one, the men and women of this suburban neighborhood fall victim to her Dupree's sinister charms, until a preacher sees her for what she truly is: A succubus. But is it too late to stop her evil plan?
Dark Channel
To some, the Universal Enlightened Alliance is a community built on peace, self-affirmation, and the wisdom of its leader, Hester Thorne, and Orrin, the centuries-old entity she channels. To others, Hester is a con artist dressed in New Age clothing. But when a reporter investigating the Alliance goes missing and a wife loses her husband and two children to the collective, a much more chilling reality begins to emerge. The Alliance is a cult—and its leaders are about to unleash an unspeakable horror on humankind.
Methods of Madness
Garton's first story collection, originally published in 1990, is a demented smorgasbord of sinister delights. In "Fat," an overweight man takes revenge on the woman who rejected him. "Something Kinky" is the chilling chronicle of a husband who strays outside his marriage and pays the ultimate price. In the novella "Dr. Krusadian's Method," a radical new therapy is applied to child abusers, with gruesome results. Dark, gory, titillating, and fiendishly clever, these short tales pack a terrifying punch.
Related: 9 Horror Short Story Collections to Keep You up at Night
This post is sponsored by Open Road Media. Thank you for supporting our partners, who make it possible for The Lineup to celebrate the horror stories you love. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,351 |
\section{Introduction}
Given a positive number $\lambda$ and a
time-homogeneous L\'{e}vy process $L$, the Ornstein-Uhlenbeck (OU) process driven by $L$ is defined by
\begin{equation}
Y_{t}=e^{-\lambda t}Y_{0}+e^{-\lambda t}\int_{0}^{\lambda
t}e^{s}dL_{s}, \label{OU2}
\end{equation}
where $Y_{0}$ is assumed to be independent of $\{L_{t}\}_{t\geq
0}$. Following the terminology introduced by Barndorff-Nielsen and
Shephard in \cite{BNS1}, we shall call $L$ the background driving
L\'{e}vy process (BDLP). It is easy to see that (\ref{OU2}) is the
unique strong solution of the stochastic differential equation
\begin{eqnarray}
dY_{t}&=&-\lambda Y_{t}dt+dL_{\lambda t}.\label{OU1}
\end{eqnarray}
Under some regularity conditions on the L\'{e}vy measure of $L$ and
if $\lambda>0$, $Y$ admits a unique invariant distribution $F_{Y}$.
Owing to the scaling of the time index of $L$ in (\ref{OU1}) by
$\lambda$ (i.e. the term $L_{\lambda t}$), $F_{Y}$ is independent of
$\lambda$.
Let us suppose now that we have discrete-time observations
$Y_{0},Y_{h},\cdots$ $,Y_{(n-1)h}$ with $h>0$ from $\lbrace
Y_{t}\rbrace_{t\geq 0}$ as it is defined by (\ref{OU2}). The
objective here is to estimate the parameters of the model using
these discrete-time observations. In particular, we are interested
in estimating $\lambda$ and moments of $L_{1}$. We derive strongly consistent method of moments
estimators and prove that they are asymptotically normal. In this paper, we
consider $\lambda$ and only the first two moments of $L_{1}$, i.e.
$\mathbb{E}L_{1}$ and $\mathbb{E}{L_{1}^{2}}$. However, the
methodology can be extended to higher moments as well (see Remark \ref{Remark32}). Similar
methods of moments estimators have been used elsewhere as well (see
Valdivieso et al \cite{Valdivieso} for example), but without theoretical
justification. The proof of their consistency and asymptotic
normality is presented, according to the best knowledge of the
author, for the first time in the present work.
Our motivation for studying this problem comes from continuous
stochastic volatility models in financial mathematics.
Barndorff-Nielsen and Shephard (in \cite{BNS1}; see also
\cite{BNS2}) model stock price as a geometric Brownian motion and the
diffusion coefficient of this motion as an OU process that is driven
by a subordinator (a L\'{e}vy process that is nonnegative and
nondecreasing). Other continuous stochastic volatility models can be
found in Kl\"{u}ppelberg et al \cite{KLM1} and in Shephard \cite{Shephard}. Some papers that
consider statistical inference of these models are
Barndorff-Nielsen and Shephard \cite{BNS1}, Brockwell et al \cite{BrockwellDavisYang}, Haug et al \cite{HKLM}, Jongbloed et al \cite{JMV}.
The paper is organized as follows. Section 2 presents several known
results concerning OU processes and L\'{e}vy processes. In section
3 we consider strongly consistent estimators of the first two
moments of $L_{1}$ and of $\lambda$ and we provide a methodology to
express any moment of the stationary distribution of $\{Y\}_{t\geq
0}$ in terms of the moments of $L_{1}$. In section 4, we prove that
these estimators are asymptotically normal. Section 5 discusses
modeling issues and simulation techniques and presents simulation
results for gamma OU process and inverse Gaussian OU process. In
section 6 we fit the model to real log(VIX) data and we argue that an OU model is a good candidate for modeling log(VIX). Finally, section 7 contains a summary and a discussion on future work.
We would like to mention here, that after completion of this work,
the author learned about the results in Jongbloed et al \cite{JMV}. In \cite{JMV},
the authors assume that $L$ is a subordinator. Let $F_{L}$ denote the L\'{e}vy measure of $L$, $Y$
the unique stationary solution to (\ref{OU2}) (which exists if $\int_{x>1}\log(x)F_{L}(dx)<\infty$ for example) and $F_{Y}$ its probability law.
The characteristic function of $Y$ is given by
\begin{displaymath}
\phi_{F_{Y}}(t):=\int e^{itx}F_{Y}(dx)=\exp(\int_{0}^{\infty}[e^{itx}-1]\frac{\kappa(x)}{x}dx),
\end{displaymath}
where $\kappa(x)=F_{L}(x,\infty)$. Hence, the stationary
distribution $F_{Y}$, of the OU process $Y$, is being determined
by the canonical function $\kappa(x)$. In \cite{JMV}, the authors
develop a nonparametric inference procedure for $\lambda$ and for
the canonical function $\kappa(x)$. The results in the present
paper complement the results of \cite{JMV}.
\section{Assumptions and Preliminary Results}
Consider a probability space $(\Omega, \mathfrak{F}, P)$ equipped
with a filtration $\mathfrak{F}_{t}$.
\begin{defi}
A one dimensional $\mathfrak{F}_{t}$ adapted
L\'{e}vy process is usually denoted by
$L_{t}=L_{t}(\omega)$, $t\geq 0$, $\omega\in \Omega$ and is a
stochastic process that satisfies the following:
\begin{enumerate}
\item{$L_{t}\in \mathfrak{F}_{t}$ for all $t\geq 0$.}
\item{$L_{0}=0$ a.s.}
\item{$L_{t}-L_{s}$ is independent of $\mathfrak{F}_{s}$ and has the same distribution as $L_{t-s}$.}
\item{It is a process continuous in probability.}
\end{enumerate}
\end{defi}
We assume that we are working with a c\`{a}dl\`{a}g L\'{e}vy
process (i.e. it is right continuous with left limits). It is well known that every L\'{e}vy process has such a modification.
Furthermore, if $F_{L}$ denotes the L\'{e}vy measure of $L_{1}$, we
will assume that there exist a constant $M>0$ such that
\begin{equation}
\int_{|x|>1}e^{v x}F_{L}(dx)<\infty, \hspace{0.5 cm} \textrm{ for
every } |v|\leq M. \label{MomentCondition}
\end{equation}
Condition (\ref{MomentCondition}) gaurantees that the moment
generating function $v\rightarrow \mathbb{E}e^{vL_{1}}$ exists at
least for $|v|\leq M$ (see Wolfe \cite{Wolfe} and Eberlein and
Raible \cite{EberleinRaible}).
We shall write
\begin{enumerate}
\item{$\mathbb{E}L_{1}=\mu$.}
\item{$\textrm{Var}(L_{1})=\sigma^{2}$.}
\end{enumerate}
Moreover, we shall assume that $Y_{0}$ is independent of $\{L_{t}\}_{t\geq 0}$ and that
\begin{equation}
Y_{0} \stackrel{\mathfrak{D}}{=} \int_{0}^{\infty}e^{-s}dL_{s}.\label{DistributionCondition}
\end{equation}
The integral on the right hand side of (\ref{DistributionCondition}) is well defined (see Sato \cite{Sato} for example).
The following proposition, which is a reformulation of Propositions 1 and 2 in Brockwell \cite{Brockwell}, characterizes the stationarity of the OU process $\{Y_{t}\}_{t\geq 0}$.
\begin{prop}
If $Y_{0}$ is independent of $\{L_{t}\}_{t\geq 0}$ and $EL_{1}^{2}<\infty$ then
$\{Y_{t}\}_{t\geq 0}$ is weakly stationary if and only if $\lambda>0$ and $Y_{0}$ has the
same mean and variance as $\int_{0}^{\infty}e^{-s}dL_{s}$. If in addition
$Y_{0}$ has the same distribution as
$\int_{0}^{\infty}e^{-s}dL_{s}$, then $\{Y_{t}\}_{t\geq 0}$ is strictly
stationary and vice-versa.\label{Proposition22}
\end{prop}
In Masuda \cite{Masuda} now, the author proves, under mild regularity conditions, that the OU process $Y$ is strong Feller, its probability law has a smooth transition density, is ergodic and exponentially $\beta-$mixing (strong mixing). Before mentioning the results of \cite{Masuda} that we will use in the present paper, let us recall the definitions of a self-decomposable law on $\mathbb{R}$ and of $\beta-$mixing.
\begin{defi}
Let $\lambda$ be a positive number. Then, an infinitely divisible
distribution $F_{Y}$ is called $\lambda-$self-decomposable, if there
exists a random variable $X=X_{t,\lambda}$, such that, for each
$t\in\mathbb{R}_{+}$
\begin{displaymath}
\phi_{F_{Y}}(u)=\phi_{F_{Y}}(e^{-\lambda t}u)\phi_{F_{X}}(u), u\in \mathbb{R},
\end{displaymath}
where $\phi_{F_{Y}}(u)$ and $\phi_{F_{X}}(u)$ are the characteristic
functions corresponding to $F_{Y}$ and $F_{X}$ respectively. For the
sake of notational convenience we will just say that $F_{Y}$ is
called self-decomposable.
\end{defi}
If $\int_{|x|>1}\log(|x|)F_{L}(dx)<\infty$, then the class of all possible invariant distributions of $Y$ forms the class of all self-decomposable distributions $F_{Y}$ (see Sato \cite{Sato}). In particular, the latter is implied by (\ref{MomentCondition}).
\begin{defi}
For a stationary process $Y=\{Y_{t}\}_{t \geq 0}$ define the $\sigma$-algebras
$\mathfrak{F}_{1}=\mathfrak{F}_{(0,u)}=\sigma(\{Y_{v}\},0\leq v<u)$
and
$\mathfrak{F}_{2}=\mathfrak{F}_{[u+t,\infty)}=\sigma(\{Y_{v}\},v\geq
u+t)$. Then
\begin{enumerate}
\item{$Y$ is called $\beta-$mixing (or strong mixing) if: $$\beta(t)=\sup_{A\in\mathfrak{F}_{1},B\in\mathfrak{F}_{2}}|P(A\cap B)-P(A)P(B)|\rightarrow 0 \textrm{ as } t\rightarrow \infty.$$}
\item{$Y$ is called $\beta-$mixing with exponential rate if for some $k>0$ and $a>0$: $$\beta(t)\leq k e^{-at} \textrm{ for } t\geq 0.$$}
\end{enumerate}
\end{defi}
The following theorem is Theorem 4.3 in Masuda \cite{Masuda} and discusses the mixing properties of $\{Y_{t}\}_{t\geq 0}$.
\begin{thm}
Let $\lambda>0$ and $\{Y_{t}\}_{t\geq 0}$ be the strictly stationary OU process given by (\ref{OU2})
with self-decomposable marginal distribution $F_{Y}$. If we have that
$$\int_{\mathbb{R}}|x|^{p}F_{Y}dx<\infty$$
for some $p>0$, then there exists a constant $a>0$ such that $\beta(t)=O(e^{-at})$ as $t\rightarrow \infty$. In particular, $Y$ is ergodic. \label{Theorem23}
\end{thm}
\section{Method of Moments Estimation}
We aim at estimation of the model parameters $\theta_{0}=(\mu,\sigma^{2},\lambda)$ from a sample of equally spaced observations from (\ref{OU2}) by matching moments and empirical autocorrelation function to their theoretical counterparts.
Proposition \ref{Proposition31} below relates the theoretical moments of $L_{1}$ with the theoretical
moments of the stationary distribution $F_{Y}$ of $\{Y_{t}\}$.
\begin{prop}
Suppose that $\lbrace L_{t}\rbrace_{t\geq 0}$ is a L\'{e}vy process
such that $\mathbb{E}L_{1}=\mu<\infty$,
$\textrm{Var}{L_{1}}=\sigma^{2}<\infty$ and that
(\ref{MomentCondition}) holds. Let $M$ be the largest constant
satisfying (\ref{MomentCondition}) and assume that $\lambda<M$.
Then, the following are true
\begin{enumerate}
\item{$\mathbb{E}Y_{0}=\mu$}
\item{$\textrm{Var}{Y_{0}}=\frac{\sigma^{2}}{2}$}
\end{enumerate}\label{Proposition31}
\end{prop}
\begin{proof}
Let $\gamma(v)$ be the cumulant function of $L_{1}$, i.e.
\begin{equation}
\gamma(v)=\ln \mathbb{E}e^{vL_{1}}\label{CumulantOfL1}
\end{equation}
By the L\'{e}vy- Khinchine representation Theorem we get that $\gamma(v)$ has the form
\begin{equation}
\gamma(v)=b v+\frac{c}{2} v^{2}+\int_{\mathbb{R}}(e^{vx}-1-vx)F_{L}(dx), \label{CumulantOfL2}
\end{equation}
which is valid for $|v|\leq M$. Moreover, $\gamma$ is continuously
differentiable (see Lukacs \cite{Lukacs}).
Using the assumptions $\mathbb{E}L_{1}=\mu$ and $\textrm{Var}{L_{1}}=\sigma^{2}$ and relations (\ref{CumulantOfL1}) and (\ref{CumulantOfL2}), it is easy to see that $b=\mu$ and $c=\sigma^{2}-\int_{\mathbb{R}}x^{2}F_{L}(dx)$.
In order to calculate $\mathbb{E}Y_{0}$ and $\textrm{Var}{Y_{0}}$ we use the following formula:
\begin{equation}
\mathbb{E}e^{\int_{0}^{\infty}\lambda e^{-s}dL_{s}}=e^{\int_{0}^{\infty}\gamma(\lambda e^{-s})ds},\label{CumulantOfL3}
\end{equation}
which is valid since $\lambda<M$ (see Lemma 3.1 of Eberlein and Raible \cite{EberleinRaible}).
Recall now that we have assumed $Y_{0}=\int_{0}^{\infty}e^{-s}dL_{s}$ in distribution. The latter and (\ref{CumulantOfL3}) imply that:
\begin{eqnarray}
\mathbb{E}Y_{0}&=&\frac{d}{d\lambda}\mathbb{E}e^{\int_{0}^{\infty}\lambda e^{-s}dL_{s}}|_{\lambda=0}=\nonumber\\
&=& \frac{d}{d\lambda}e^{\int_{0}^{\infty}\gamma(\lambda e^{-s})ds}|_{\lambda=0}=\nonumber\\
&=& \mu
\end{eqnarray}
In a similar way we get that $\mathbb{E}Y_{0}^{2}=\frac{\sigma^{2}}{2}+\mu^{2}$. This concludes the proof of the proposition.
\end{proof}
\begin{rem}
We would like to note here, that the proof of Proposition \ref{Proposition31} can be used for the calculation of higher moments of $Y_{0}$.\label{Remark32}
\end{rem}
It follows directly by (\ref{OU2}) that the theoretical autocovariance and autocorrelation function of $Y_{t}$ are given by the formulas
\begin{enumerate}
\item{autocovariance: $\gamma(h)=cov(Y_{t+h},Y_{t})=\frac{\sigma^2}{2}e^{-\lambda h}$, for $h\in \mathbb{N}_{0}$.}
\item{autocorrelation: $\rho(h)=corr(Y_{t+h},Y_{t})=e^{-\lambda h}$, for $h\in \mathbb{N}_{0}$.}\label{TheoreticalAutoCorrelationOfY}
\end{enumerate}
On the other hand, the empirical moments, autocorrelation and
autocovariance function are given by the formulas below. Let $d\geq
0$ be fixed. Then, we have:
\begin{enumerate}
\item{Sample mean: $\bar{Y}_{\cdot}=\frac{1}{n}\sum_{i=1}^{n} Y_{i}$.}
\item{Sample variance: $\frac{1}{n}\sum_{i=1}^{n} (Y_{i}-\bar{Y}_{\cdot})^{2}$.}
\item{Sample autocovariance: $\hat{\gamma}_{n}=(\hat{\gamma}_{n}(0),\hat{\gamma}_{n}(1),\cdots,\hat{\gamma}_{n}(d))^{T}$ where for $h\in \lbrace 0,\cdots, d\rbrace$ we define $\hat{\gamma}_{n}(h)=\frac{1}{n}\sum_{i=1}^{n-h} (Y_{i+h}-\bar{Y}_{\cdot})(Y_{i}-\bar{Y}_{\cdot})$.}
\item{Sample autocorrelation: $\hat{\rho}_{n}=(\hat{\rho}_{n}(0),\hat{\rho}_{n}(1),\cdots,\hat{\rho}_{n}(d))^{T}$ where for $h\in \lbrace 0,\cdots, d\rbrace$ we define $\hat{\rho}_{n}(h)=\frac{\hat{\gamma}_{n}(h)}{\hat{\gamma}_{n}(0)}$.}\label{EmpiricalAutoCorrelationOfY}
\end{enumerate}
We have the following Theorem:
\begin{thm}
Let $\mu,\sigma^{2}, \gamma(\cdot), \hat{\gamma}_{n}(\cdot),\rho(\cdot)$ and $\hat{\rho}_{n}(\cdot)$ be defined as above. Then, the following statements are true
\begin{enumerate}
\item{$\bar{Y}_{\cdot}\stackrel{n\rightarrow\infty}{\longrightarrow} \mu$ almost surely}
\item{$\frac{1}{n}\sum_{i=1}^{n} (Y_{i}-\bar{Y}_{\cdot})^{2}\stackrel{n\rightarrow\infty}{\longrightarrow} \frac{\sigma^{2}}{2}$ almost surely}
\item{$(\hat{\gamma}_{n}(1),\cdots,\hat{\gamma}_{n}(d))\stackrel{n\rightarrow\infty}{\longrightarrow}(\gamma(1),\cdots,\gamma(d))$ almost surely}
\item{$(\hat{\rho}_{n}(1),\cdots,\hat{\rho}_{n}(d))\stackrel{n\rightarrow\infty}{\longrightarrow}(\rho(1),\cdots,\rho(d))$ almost surely}
\end{enumerate}\label{Theorem32}
\end{thm}
\begin{proof}
Due to our assumptions, the process $\{Y_{t}\}_{t\geq 0}$ is
strictly stationary. Moreover by Theorem \ref{Theorem23} it is also
$\beta-$mixing with exponential decaying rate. These two results
imply ergodicity of $\{Y_{t}\}_{t\geq 0}$. The latter together with
strict stationarity imply that empirical moments and sample
autocovariance functions are strongly consistent estimators of the
corresponding theoretical quantities Billingsley \cite{Billingsley}.
Then, the statement of the Theorem follows.
\end{proof}
For the mean reverting parameter $\lambda$ we have the following Lemma.
\begin{lem}
Let $K$ be a compact subset of $\mathbb{R}_{+}$
such that the true value of $\lambda$, say $\lambda_{o}$, belongs to
$K$ and let $\hat{\lambda}_{n}=\textrm{argmin}_{\lambda \in
K}\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda h})^{2}$. Then $\hat{\lambda}_{n}$ exists, is locally unique and
\begin{equation}
\hat{\lambda}_{n}\stackrel{n\rightarrow\infty}{\longrightarrow} \lambda_{o} \textrm{
almost surely}.
\end{equation}\label{Lemma33}
\end{lem}
\begin{proof}
Consider the functions $\Delta_{n}(\lambda)=\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-\rho_{\lambda}(h))^{2}$ and $\Delta_{0}(\lambda)=\sum_{h=1}^{d}(\rho_{\lambda_{o}}(h)-\rho_{\lambda}(h))^{2}$
where $\rho_{\lambda}(h)=e^{-\lambda h}$. Theorem \ref{Theorem32} implies that for all $\lambda\in K$:
\begin{displaymath}
\Delta_{n}(\lambda)\stackrel{n\rightarrow\infty}{\longrightarrow}\Delta_{0}(\lambda) \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
By Theorem II.1 in Andresen and Grill \cite{AndresenGrill} we have
\begin{displaymath}
\sup_{\lambda\in K}|\Delta_{n}(\lambda)-\Delta_{0}(\lambda)|\stackrel{n\rightarrow\infty}{\longrightarrow}0 \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
Observe now that $\Delta_{0}(\lambda)$ is a sum of nonnegative
terms. It becomes zero if and only if $\lambda=\lambda_{0}$. Hence,
$\Delta_{0}(\lambda)$ has a unique minimum at $\lambda=\lambda_{0}$
which is equal to zero. We get
\begin{displaymath}
\Delta_{n}(\lambda_{0})\stackrel{n\rightarrow\infty}{\longrightarrow}0 \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
Furthermore, for $n$ finite we have that $0\leq \Delta_{n}(\hat{\lambda}_{n})\leq \Delta_{n}(\lambda_{0})$. Therefore we get
\begin{displaymath}
\Delta_{n}(\hat{\lambda}_{n})\stackrel{n\rightarrow\infty}{\longrightarrow}0 \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
Moreover, we have
\begin{eqnarray}
|\Delta_{n}(\hat{\lambda}_{n})-\Delta_{0}(\hat{\lambda}_{n})|&=&
|\sum_{h=1}^{d}[\hat{\rho}_{n}^{2}(h)-\rho_{\lambda_{0}}^{2}(h)+2\rho_{\hat{\lambda}_{n}}(h)
(\rho_{\lambda_{0}}(h)-\hat{\rho}_{n}(h))]|\leq\nonumber\\
&\leq&\sum_{h=1}^{d}[|\hat{\rho}_{n}(h)|+|\rho_{\lambda_{0}}(h)|+2|\rho_{\hat{\lambda}_{n}}(h)|]|\rho_{\lambda_{0}}(h)-\hat{\rho}_{n}(h)|\leq\nonumber\\
&\leq& 4\sum_{h=1}^{d}|\rho_{\lambda_{0}}(h)-\hat{\rho}_{n}(h)|\stackrel{n\rightarrow\infty}{\longrightarrow} 0 \hspace{0.2cm} \textrm{almost surely.}\nonumber
\end{eqnarray}
Here, we used the relation $|\hat{\rho}_{n}(h)|\leq 1$ which follows
immediately from Cauchy-Schwarz inequality. The above imply that
\begin{displaymath}
\Delta_{0}(\hat{\lambda}_{n})\stackrel{n\rightarrow\infty}{\longrightarrow}0 \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
But, $\lambda_{0}$ is the unique minimum of $\Delta_{0}(\lambda)$ and it satisfies $\Delta_{0}(\lambda_{0})=0$. Thus,
\begin{displaymath}
\Delta_{0}(\hat{\lambda}_{n})\stackrel{n\rightarrow\infty}{\longrightarrow}\Delta_{0}(\lambda_{0})=0 \hspace{0.2cm} \textrm{almost surely}.
\end{displaymath}
Hence, we easily conclude (Corollary II.2 in \cite{AndresenGrill}) that $\hat{\lambda}_{n}$ is locally uniquely determined and that
\begin{displaymath}
\hat{\lambda}_{n}\stackrel{n\rightarrow\infty}{\longrightarrow} \lambda_{o} \textrm{
almost surely}.
\end{displaymath}
\end{proof}
\begin{rem}
Theorem \ref{Theorem32} and Lemma \ref{Lemma33} give us two strongly
consistent estimators for $\lambda$. The first one is
$\hat{\lambda}_{1,n}=-\log(\hat{\rho}_{n}(1))$ and the second one is
$\hat{\lambda}_{2,n}=\textrm{argmin}_{\lambda}\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda
h})^{2}$. One could use, for example, $\hat{\lambda}_{1,n}$ as an
initial value to an algorithm that calculates $\hat{\lambda}_{2,n}$.
\end{rem}
Summarizing, we have that $\hat{\mu}_{n}$, $\hat{\sigma}^{2}_{n}$
and $\hat{\lambda}_{1,n}, \hat{\lambda}_{2,n}$ are strongly
consistent estimators of $\mu,\sigma^{2}$ and $\lambda$
respectively, where:
\begin{eqnarray}
\hat{\mu}_{n}&=&\frac{1}{n}\sum_{i=1}^{n} Y_{i}\nonumber\\
\hat{\sigma}^{2}_{n}&=&2\frac{1}{n}\sum_{i=1}^{n} (Y_{i}-\hat{\mu}_{n})^{2} \label{StrongConsistentEstimators}\\
\hat{\lambda}_{1,n}&=&-\log(\hat{\rho}_{n}(1))\nonumber\\
\hat{\lambda}_{2,n}&=&\textrm{argmin}_{\lambda}\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda h})^{2}.\nonumber
\end{eqnarray}
\begin{rem}
For a stationary model, the parameter $\lambda$ has to be positive.
However, if we compute $\hat{\lambda}_{2,n}$ as the unrestricted
minimum $\hat{\lambda}_{2,n}=\textrm{argmin}_{\lambda \in
\mathbb{R}_{+}}\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda h})^{2}$
we may end up with a negative estimator $\hat{\lambda}_{n}$. In this
case, we define the estimator of $\lambda$ to be zero and we take
this as an indication that the data is not stationary.
\end{rem}
\section{Asymptotic Properties of the Moment Estimators}
In this section we prove that the estimators defined by (\ref{StrongConsistentEstimators}) are asymptotically normal.
If $\beta$ is a vector, then we define by $\beta^{T}$ its transpose.
We begin with the following central limit theorem.
\begin{thm}
Let us assume that there exists a $\delta>0$ such that $\mathbb{E}Y_{0}^{4+\delta}<\infty$.
Define
\begin{eqnarray}
\hat{\psi}_{n}&=&(\hat{\mu}_{n},\hat{\gamma}_{n}(0),\hat{\gamma}_{n}(1),\cdots,\hat{\gamma}_{n}(d))^{T}\nonumber\\
\psi_{o}&=&(\mu,\gamma(0),\gamma(1),\cdots,\gamma(d))^{T}\nonumber\\
\Sigma&=&[\sigma_{k,l}]_{k,l=1}^{d+2}\textrm{ with elements
}\nonumber\\
&
&\sigma_{k,l}=cov(Z_{1}^{k},Z_{1}^{l})+2\sum_{i=1}^{\infty}cov(Z_{1}^{k},Z_{i+1}^{l})\textrm{ where }\nonumber\\
Z_{i}&=&(Y_{i},(Y_{i}-\mu)^{2},
(Y_{i+1}-\mu)(Y_{i}-\mu),\cdots,(Y_{i+d}-\mu)(Y_{i}-\mu))^{T}\nonumber
\end{eqnarray}
Then, the following holds:
\begin{equation}
\sqrt
n(\hat{\psi}_{n}-\psi_{o})\stackrel{\mathfrak{D}}{\longrightarrow} N(0, \Sigma)\label{CLT1}
\end{equation}
where $N(0,\Sigma)$ is the multivariate normal distribution with mean $0$ and variance-covariance matrix $\Sigma$.\label{Theorem41}
\end{thm}
\begin{proof} The proof of this theorem is similar to the proof of Proposition 3.7 of Haug et al \cite{HKLM}. Let us define
\begin{enumerate}
\item{$\gamma_{n}^{*}(h)=\frac{1}{n}\sum_{i=1}^{n} (Y_{i+h}-\mu)(Y_{i}-\mu), h\in\lbrace 0,\cdots, d\rbrace.$}
\item{$\gamma_{n}^{*}=(\gamma_{n}^{*}(0),\cdots,\gamma_{n}^{*}(d))^{T}$.}
\end{enumerate}
We first prove that (\ref{CLT1}) is true with $\hat{\psi}_{n}^{*}=(\hat{\mu}_{n},\hat{\gamma}_{n}^{*}(0),\hat{\gamma}_{n}^{*}(1),\cdots,\hat{\gamma}_{n}^{*}(d))^{T}$ in place of $\hat{\psi}_{n}$.
By the well known Cramer-Wold device, it is sufficient to prove that for every $\beta\in \mathbb{R}^{d+2}$ such that $\beta^{T}\Sigma\beta>0$ we have
\begin{equation}
\sqrt{n}(\frac{1}{n}\sum_{i=1}^{n}\beta^{T}Z_{i}-\beta^{T}\psi_{0})\stackrel{\mathfrak{D}}{\longrightarrow}N(0, \beta^{T}\Sigma\beta).\label{CLT2}
\end{equation}
It is well known (see \cite{Billingsley} for example) that strong mixing and the corresponding decaying rate are preserved under linear transformations. Thus, the sequence $\lbrace\beta^{T}Z_{i}\rbrace$ is strong mixing with exponential decaying rate. Since, by assumption $\mathbb{E}|Z_{1}|^{2+\epsilon}$ for some $\epsilon>0$, the central limit theorem for strong mixing processes is applicable (Theorem 7.3.1 in Ethier and Kurtz \cite{EK}). Hence, we have as $n\rightarrow \infty$ that
\begin{equation}
\sqrt{n}(\frac{1}{n}\sum_{i=1}^{n}\beta^{T}Z_{i}-\beta^{T}\psi_{0})\stackrel{\mathfrak{D}}{\longrightarrow} N(0, \tilde{\sigma}^{2}).\label{CLT3}
\end{equation}
But, we easily see that $\tilde{\sigma}^{2}=var(\beta^{T}Z_{1})+2\sum_{i=1}^{\infty}cov(\beta^{T}Z_{1},\beta^{T}Z_{i+1})=\beta^{T}\Sigma\beta$. So (\ref{CLT2}) holds.
Now recall that by Theorem \ref{Theorem32} we have
\begin{equation}
\hat{\psi}_{n}\stackrel{n\rightarrow\infty}{\longrightarrow}\psi_{0}\hspace{0.2cm} \textrm{ almost surely}.
\end{equation}
Following the proof of proposition 7.3.4 of Brockwell and Davis \cite{BrockwellDavis} we get
\begin{equation}
\sqrt{n}(\frac{1}{n}\sum_{i=1}^{n}\beta^{T}Z_{i}-\beta^{T}\hat{\psi}_{n})\stackrel{n\rightarrow\infty}{\longrightarrow} 0 \hspace{0.2cm} \textrm{in probability}.\label{CLT4}
\end{equation}
Therefore, $\hat{\psi}_{n}$ has the same asymptotic behavior as $\hat{\psi}_{n}^{*}$. The latter and (\ref{CLT2}) imply the Theorem.
\end{proof}
\begin{cor}
Let the conditions of Theorem \ref{Theorem41} hold. Then we have
\begin{displaymath}
\sqrt{n}(\hat{\rho}_{n}-\rho)\stackrel{\mathfrak{D}}{\longrightarrow} N(0, \Sigma_{\rho}).\label{CLT5}
\end{displaymath}
\end{cor}
\begin{proof}
It follows directly by Theorem \ref{Theorem41} and delta method (Theorem 3.1 in A.W.van der Vaart \cite{VanDerVaart}).
\end{proof}
Finally, we prove central limit theorem for $\hat{\theta}_{n}=(\hat{\mu}_{n},\hat{\sigma}^{2}_{n},\hat{\lambda}_{2,n})^{T}$. Let us denote $\sigma^{2}_{Y}=\gamma(0)=\textrm{Var}(Y_{0})=\frac{\sigma^{2}}{2}$ and define the following mappings.
\begin{equation}
G:\mathbb{R}\times [0,\infty)^{2}\longrightarrow \mathbb{R}\times
[0,\infty)^{2}: G(\mu,\sigma^{2}_{Y},\lambda)=
\begin{cases}
(\mu,2\sigma^{2}_{Y},\lambda), & \lambda>0 \cr
(\mu,2\sigma^{2}_{Y},0), & \lambda\leq 0. \cr
\end{cases}
\end{equation}
\begin{equation}
F:\mathbb{R}^{d+1}_{+}\longrightarrow \mathbb{R}_{+}:
F(\hat{\rho})=\textrm{argmin}_{\lambda}\sum_{h=0}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda h})^{2}=\hat{\lambda}_{2,n}
\end{equation}
and $H$ as follows:
\begin{equation}
H:\mathbb{R}^{d+2}\longrightarrow \mathbb{R}\times [0,\infty)^{2}:
H(\mu,\gamma^{T})=G(\mu,\sigma^{2}_{Y},F(\rho)),
\end{equation}
where $\rho(h)=\frac{\gamma(h)}{\gamma(0)}$ for $h=0,\cdots,d$.
\begin{thm}
Let the conditions of Theorem \ref{Theorem41} be satisfied. Let us define:
\begin{eqnarray}
\hat{\theta}_{n}&=&(\hat{\mu}_{n},\hat{\sigma}^{2}_{n},\hat{\lambda}_{2,n})^{T}\nonumber\\
\theta_{o}&=&(\mu,\sigma^{2},\lambda)^{T}.\nonumber
\end{eqnarray}
Then the following holds:
\begin{equation}
\sqrt
n(\hat{\theta}_{n}-\theta_{o})\stackrel{\mathfrak{D}}{\longrightarrow}[\frac{\partial
H(\mu,\gamma^{T})}{\partial
(\mu,\gamma^{T})}] N(0, \Sigma)
\end{equation}
\end{thm}
\begin{proof}
It follows directly by Theorem \ref{Theorem41} and delta method applied to the differentiable map $H$.
\end{proof}
\section{Modeling and Simulation}
In this section we discuss modeling issues of L\'{e}vy driven OU
processes and present some simulation results for a gamma OU process
and an inverse Gaussian OU process. We use the simulated
data to test the performance of our estimators. However, first we mention the necessary ingredients for the simulation process.
A very important ingredient in modeling of L\'{e}vy driven OU processes is the connection between the L\'{e}vy density of the stationary distribution of $Y$ to the L\'{e}vy density of the probability law of $L_{1}$. In particular we have the following proposition.
\begin{prop} Assume that the L\'{e}vy density of $Y$, $\nu_{Y}(x)$, is differentiable and denote the L\'{e}vy density of the probability law of $L_{1}$ by $\nu_{L}(x)$. Then the following relation holds.
\begin{equation}
\nu_{L}(x)=-\nu_{Y}(x)-x\nu'_{Y}(x).\label{RelationForLevyDensities}
\end{equation}\label{Proposition51}
\end{prop}
\begin{proof}
It follows directly by the fact that the stationary solution, $Y$, to $(\ref{OU1})$ satisfies
\begin{displaymath}
Y\stackrel{\mathfrak{D}}{=}\int_{0}^{\infty}e^{-\lambda s}dL(\lambda
s).
\end{displaymath}
See \cite{BNS1} and \cite{BNS2} for more details.
\end{proof}
Hence, given $\nu_{L}(x)$ we can find $\nu_{Y}(x)$ and vice-versa. One can specify the law of the one dimensional marginal distribution of the OU process $Y$ and work out the density of the BDLP, $L_{1}$. One can also go the other way and model through the BDLP. Of course, there are constraints on valid BDLP's which must be satisfied. In particular, if
\begin{displaymath}
\int_{\mathbb{R}}\min\{1,x^{2}\}\nu_{L}(x)dx
\end{displaymath}
then $\nu_{L}(x)$ is the density of a L\'{e}vy jump process $L$ and there exists an OU process $Y$ such that $L$ is the BDLP of $Y$.
A very good survey on the relation between several distributions of $Y$ and $L$ is Barndorff-Nielsen and Shephard \cite{BNS2} (see also Schoutens \cite{Schoutens}).
Another important ingredient in simulations is the infinite series representation of L\'{e}vy integrals (see Rosinski \cite{Rosinski}). For simplicity, we restrict attention to L\'{e}vy processes, $L$, that are subordinators, i.e. they are nonnegative and nondecreasing. It is easy to see that subordinators have no Gaussian component, nonnegative drift and a L\'{e}vy measure that is zero on the negative half-line. If $Y$ models stochastic volatility then it has to be positive and such a choice of the BDLP guarantees that.
Let us denote by $\Gamma_{L}^{+}$ the tail mass function of $\nu_{L}$, i.e.
\begin{equation}
\Gamma_{L}^{+}(x)=\int_{x}^{\infty}\nu_{L}(y)dy\label{ITMF1}
\end{equation}
and by $\Gamma_{L}^{-1}$ the generalized inverse function of $\Gamma_{L}^{+}$, i.e.
\begin{equation}
\Gamma_{L}^{-1}(x)=\inf\lbrace y>0:\Gamma_{L}^{+}(y)\leq x\rbrace \label{ITMF2}.
\end{equation}
In order to simulate from (\ref{OU2}) we need to be able to simulate from $e^{-\lambda t}\int_{0}^{\lambda
t}e^{s}dL_{s}$. The key result here is the following infinite series representation of this type of integrals (Rosinski \cite{Rosinski}):
\begin{prop}
Consider a subordinator $L$ with positive increments. Let $f$ be a positive and integrable function on $[0,T]$. Then
\begin{equation}
\int_{0}^{T}f(s)dL_{s}=\sum_{i=1}^{\infty}\Gamma_{L}^{-1}(\alpha_{i}/T)f(T r_{i}),\label{SeriesRepresentation}
\end{equation}
where the equality is understood in distributional sense, $\lbrace \alpha_{i}\rbrace$ and $\lbrace r_{i}\rbrace$ are two independent sequences of random variables such that $r_{i}$ are independent copies of a uniform random variable in $[0,1]$ and $\lbrace\alpha_{i}\rbrace$ is a strictly increasing sequence of arrival times of a Poisson process with intensity $1$.
\end{prop}
\begin{rem}
We note here that the convergence of the series (\ref{SeriesRepresentation}) is often quite slow.\label{Remark53}
\end{rem}
Using (\ref{SeriesRepresentation}) we can then simulate a L\'{e}vy driven OU process. In particular, if $\Delta$ denotes the time step, we will use the identity
\begin{eqnarray}
Y_{t+\Delta}&=&e^{-\lambda \Delta}(Y_{t}+e^{-\lambda t}\int_{ t}^{
t+\Delta}e^{\lambda s}dL_{\lambda s})\nonumber\\
&=&e^{-\lambda \Delta}(Y_{t}+\int_{0}^{
\Delta}e^{\lambda s}dL_{\lambda s}). \label{OUsimulation}
\end{eqnarray}
Let us demonstrate the validity of our estimators modeling through
the BDLP. We consider two cases: $(a)$ when $Y_{0}\sim
\textrm{Gamma}(a,b)$ and $(b)$ when $Y_{0}\sim \textrm{IG}(a,b)$,
where IG stands for inverse Gaussian.
Regarding the $\lambda$ parameter, we recall our estimators:
$\hat{\lambda}_{1,n}=-\frac{\log(\hat{\rho}_{n}(1))}{\Delta}$ and
$\hat{\lambda}_{2,n}=\textrm{argmin}_{\lambda}\sum_{h=1}^{d}(\hat{\rho}_{n}(h)-e^{-\lambda
h \Delta})^{2}$. In (\ref{StrongConsistentEstimators}) we defined
$\hat{\lambda}_{1,n}$ and $\hat{\lambda}_{2,n}$ for $\Delta=1$, but
of course one can generalize them to any $\Delta>0$.
\subsection{Gamma OU model}
Assume that the driving L\'{e}vy process $L$ is a compound Poisson process and in particular, that $L_{t}=\sum_{n=1}^{N_{t}}x_{n}$ where $N_{t}$ is Poisson with intensity parameter $a$ and $x_{n}$ are independent identically distributed $\textrm{Gamma}(1,b)$ random variables. Using (\ref{RelationForLevyDensities}) we get that $Y_{0}\sim \textrm{Gamma}(a,b)$. It is known (see \cite{BNS1}) that in this case
\begin{displaymath}
\Gamma_{L}^{-1}(x)=\max\lbrace 0,-\frac{1}{b}\log(\frac{x}{a})\rbrace.
\end{displaymath}
Using this and equations (\ref{SeriesRepresentation}) and (\ref{OUsimulation}) we can easily simulate from a $\textrm{Gamma}(a,b)$-OU process. We also need to know how the parameters $\mu$ and $\sigma^2$ relate to $a$ and $b$. Since $\mathbb{E}L_{1}=\mu$ and $\textrm{Var}(L_{1})=\sigma^{2}$ implies $\mathbb{E}Y_{0}=\mu$ and $\textrm{Var}(Y_{0})=\frac{\sigma^{2}}{2}$, we have that $a=2\frac{\mu^{2}}{\sigma^{2}}$ and $b=2\frac{\mu}{\sigma^{2}}$.
\vspace{0.1cm}
We simulated 100 independent paths of a gamma OU process of 1000
observations each, with time step $\Delta=0.1$, using
(\ref{OUsimulation}). We chose $\mu=2$, $\sigma^{2}=0.25$ and in
order to capture possible different behaviors of the intensity
parameter we chose two different values for $\lambda,$ $0.5$ and
$5$.
\vspace{0.1cm}
Tables I and II, summarize the results for $\theta_{0}=(2,0.25,0.5)$
and for $\theta_{0}=(2,0.25,5)$ respectively.
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
True Values & Est. Values & Sample Std. Error & Comments \\
\hline $\mu=2$ & $1.995458$ & $0.0702198$ & - \\
\hline $\sigma^{2}=0.25$ & $0.2350207$ & $ 0.05352894$ & - \\
\hline $\lambda=0.5$ & $0.566116$ & $0.1126439$ & $\hat{\lambda}_{n}=\hat{\lambda}_{1,n}$ \\
\hline $\lambda=0.5$ & $0.5879571$ & $ 0.1441501$ & $\hat{\lambda}_{n}=\hat{\lambda}_{2,n}$\\
\hline
\end{tabular}
\end{center}
\caption{$\theta_{0}=(2,0.25,0.5)$}
\end{table}
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
True Values & Est. Values & Sample Std. Error & Comments \\
\hline $\mu=2$ & $2.003799$ & $0.02094129$ & - \\
\hline $\sigma^{2}=0.25$ & $ 0.2473567$ & $ 0.01608991$ & - \\
\hline $\lambda=5$ & $5.12962$ & $0.4463517$ & $\hat{\lambda}_{n}=\hat{\lambda}_{1,n}$ \\
\hline $\lambda=5$ & $5.186585$ & $0.5898125$ & $\hat{\lambda}_{n}=\hat{\lambda}_{2,n}$ \\
\hline
\end{tabular}
\end{center}
\caption{$\theta_{0}=(2,0.25,5)$}
\end{table}
\newpage
\subsection{Inverse Gaussian OU model}
It is well known that if $Y_{0}\sim \textrm{IG}(a,b)$, then the
L\'{e}vy density of $Y$ is
\begin{equation}
\nu_{Y}(x)=\frac{1}{\sqrt{2\pi}}ax^{-3/2}e^{-\frac{1}{2}b^{2}x}.
\end{equation}
Consider now the Lambert-W function, $L_{w}(\cdot)$, which satisfies $L_{w}(x)e^{L_{w}(x)} =x$. As it is also shown in Gander and Stephens \cite{GanderStephens}, equations (\ref{RelationForLevyDensities}), (\ref{ITMF1}) and (\ref{ITMF2}) imply that the inverse tail mass function of the BDLP of an IG(a,b)-OU process is given by
\begin{equation}
\Gamma_{L}^{-1}(x)=\frac{1}{b^2}L_{w}(\frac{a^{2}b^{2}}{2\pi x^{2}})\label{ITMF3}.
\end{equation}
Using the latter and equations (\ref{SeriesRepresentation}) and (\ref{OUsimulation}) we can easily simulate an IG(a,b)-OU process. We also need to know how the parameters $\mu$ and $\sigma^2$ relate to $a$ and $b$. Since $EL_{1}=\mu$ and $Var(L_{1})=\sigma^{2}$ implies $EY_{0}=\mu$ and $Var(Y_{0})=\frac{\sigma^{2}}{2}$, we have that $a=\mu\sqrt{\frac{2\mu}{\sigma^{2}}}$ and $b=\sqrt{\frac{2\mu}{\sigma^{2}}}$.
\vspace{0.1cm}
We simulated 100 independent paths of an IG-OU process of 1000
observations each with time step $\Delta=0.1$, using
(\ref{OUsimulation}). As before, we chose $\mu=2$, $\sigma^{2}=0.25$
and two different values for $\lambda,$ $0.5$ and $5$.
Tables III and IV, summarize the results for $\theta_{0}=(2,0.25,0.5)$
and for $\theta_{0}=(2,0.25,5)$ respectively.
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
True Values & Est. Values & Sample Std. Error & Comments \\
\hline $\mu=2$ & $ 1.986862 $ & $ 0.06476202 $ & - \\
\hline $\sigma^{2}=0.25$ & $ 0.2331244 $ & $ 0.05235387 $ & - \\
\hline $\lambda=0.5$ & $ 0.5581237$ & $ 0.1128397$ & $\hat{\lambda}_{n}=\hat{\lambda}_{1,n}$ \\
\hline $\lambda=0.5$ & $ 0.6050457$ & $ 0.1376689$ & $\hat{\lambda}_{n}=\hat{\lambda}_{2,n}$\\
\hline
\end{tabular}
\end{center}
\caption{$\theta_{0}=(2,0.25,0.5)$}
\end{table}
\vspace{0.1cm}
\begin{table}[!h]
\begin{center}
\caption{$\theta_{0}=(2,0.25,5)$}
\begin{tabular}{|c|c|c|c|c|}
\hline
True Values & Est. Values & Sample Std. Error & Comments \\
\hline $\mu=2$ & $ 1.955288$ & $ 0.03107831$ & - \\
\hline $\sigma^{2}=0.25$ & $ 0.2452349$ & $ 0.01750871$ & - \\
\hline $\lambda=5$ & $5.05211$ & $ 0.4262788$ & $\hat{\lambda}_{n}=\hat{\lambda}_{1,n}$ \\
\hline $\lambda=5$ & $ 5.158421$ & $ 0.659508$ & $\hat{\lambda}_{n}=\hat{\lambda}_{2,n}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\newpage
\section{Real Data Analysis}
In 1993, the Chicago Board Options Exchange (CBOE) introduced the
CBOE volatility index, VIX, and it quickly became a popular measure
for stock market volatility. In 2003, the VIX methodology was
updated (see www.cboe.com for more details on the old and new VIX
methodology). VIX measures the implied volatility of S\&P $500$
index options and it provides a minute-by-minute snapshot of the
markets expectancy of volatility over the next $30$ calendar days.
We fitted the gamma OU model and the IG OU model to daily log
opening values of the VIX for the year 2004 (VIX values are
calculated using the new methodology). The data are taken from
www.cboe.com.
We use the values from $1/2/2004$ till $9/30/2004$ for the
calibration of the model (in total $189$ data points) and the values
from $10/1/2004$ till $11/30/2004$ (in total $41$ data points) for
testing the model.
Table V summarizes the estimators, given by (\ref{StrongConsistentEstimators}), of the parameters of the model. We used $\hat{\lambda}_{2,n}$ to estimate $\lambda$.
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
parameter & $\mu$& $\sigma^2$ & $\lambda$ \\
\hline estimated value & $2.781769$ & $0.01919740$ & $ 0.1767250$\\
\hline
\end{tabular}
\end{center}
\caption{Estimated values for the parameters.}
\end{table}
\vspace{0.2cm}
In Figure 1, we see the first 10 lags of the
empirical autocorrelation function of the $\log(\textrm{VIX})$ for
$1/2/2004$ till $9/30/2004$ versus the theoretical autocorrelation
function of the OU model with $\lambda=0.1767250$, i.e.
$\rho(h)=e^{-0.1767250 h}$.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=11 cm, height=4.5 cm]{figure1}
\caption{ True time series acf versus the model implied acf with
$\lambda=0.1767250$.}
\end{center}
\end{figure}
As we saw before, the autocorrelation function of an OU model is
exponentially decreasing, i.e. it has the form $e^{-\lambda h}$.
Figure 1 shows that $e^{-0.1767250 h}$ approximates sufficiently
well the empirical autocorrelation function of $\log(\textrm{VIX})$
for $1/2/2004$ till $9/30/2004$, which is also exponentially
decreasing. Hence, we conclude that an OU model is a good candidate
for describing this data set.
To investigate the model fit, we performed a Ljung-Box test for the
squared residuals. We used the estimated values from Table V and since our data is daily
opening values we chose $\Delta=1$ for the time step. The test statistic used 10 lags of the empirical
autocorrelation function. The gamma OU model performed better than
the the IG-OU model. The null hypothesis was not rejected at the
0.05 level and the $p-$value was quite high, $0.6165$. In Figure 2
we see the empirical autocorrelation function of the residuals of $\log(\textrm{VIX})$ and in Figure 3 we see the actual residuals of the gamma OU model.
In Figure 4 we see in one figure: the actual time series from
$10/1/2004$ till $11/30/2004$, the one step ahead predicted time
series and $95\%$ bootstrap upper and lower confidence bounds of the
one step ahead predicted time series. In order to create the one
step ahead predicted time series we averaged over $50$ paths. We observe that the real time series (solid line) is most of the
time within the 95\% bootstrap upper and lower confidence bounds of
the one step ahead predicted time series (dotted lines), with very
few exceptions.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=11 cm, height=4.0 cm]{figure2}
\caption{ Empirical acf for the residuals of $\log(\textrm{VIX})$.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=11 cm, height=4.0 cm]{figure3}
\caption{The actual residuals.}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=11 cm, height=4 cm]{figure4}
\caption{Actual time series versus one step ahead predicted time series.}
\end{center}
\end{figure}
\newpage
\section{Discussion And Future Work}
In this paper, we consider an Ornstein-Uhlenbeck process driven by a
general L\'{e}vy process. We derive strong consistent estimators for
the parameters of the model and we prove that they are
asymptotically normal. Using simulated data, we show that the
estimators perform well at least for a gamma OU model and for an IG-OU model. Lastly we fit
the model to real data and we see that a L\'{e}vy driven OU model is a good
candidate for describing $\log(\textrm{VIX})$.
There are some interesting extensions to the model
studied in this paper. One such extension is a coupled two
dimensional OU process driven by a two dimensional L\'{e}vy process.
This model is important for financial applications, since it could
be used to model log of the price and stochastic volatility
simultaneously. An interesting question for financial
applications is option pricing in these type of models (see
Nicolato and Venardos \cite{Nicolato} for some recent related results). These questions
will be addressed in future work.
\section*{Acknowledgements}
I would like to thank Professor D. Madan, Professor M. Fu, Professor E. Slud and Professor G. Skoulakis
for helpful discussions. Moreover, I would like to thank my
colleague Ziliang Li for his initial involvement in this project,
for bringing to my attention \cite{JMV}, for providing me with the
code of the Lambert-W function and for helpful suggestions.
| {
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{"url":"https:\/\/www.encyclopediaofmath.org\/index.php?title=De_Moivre_formula&oldid=29591","text":"# De Moivre formula\n\nThe formula expressing the rule for raising a complex number, expressed in trigonometric form\n\nto an -th power. According to de Moivre's formula the modulus of the complex number is raised to that power and the argument is multiplied by the exponent:\n\nThe formula was found by A. de Moivre (1707), its modern notation was suggested by L. Euler (1748).\n\nDe Moivre's formula can be used to express and in powers of and :\n\nInversion of de Moivre's formula leads to a formula for extracting roots of a complex number:\n\nwhich is also sometimes called de Moivre's formula.","date":"2019-11-11 22:22:46","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\": 0.9899738430976868, \"perplexity\": 378.02680498652967}, \"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-2019-47\/segments\/1573496664439.7\/warc\/CC-MAIN-20191111214811-20191112002811-00129.warc.gz\"}"} | null | null |
Q: Run a jar with a given process name I need to run couple of java jars in the fedora. When I execute the java -jar application.jar for each jar, it will run with same process name("java"). Lets say I need to kill these processes at a totally different time and from a different location.
It would be easier to do these, if I can run each jar with a customized process name.Can somebody tell me that something like java -jar application.jar -process_name exists?
Or is there any way to run Subprocess.Popen() with a given process name?
Because I am using python to run these jars.
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/en-academic.com\/dic.nsf\/enwiki\/721698","text":"# Inventory turns\n\nInventory turns\n\nIn business management, inventory turns often referred to as stockturn,stock turns, turns, and stock turnover.\n\nThis measures the number of times invested in goods to be sold or used over in a year.\n\n$mbox\\left\\{Inventory Turns\\right\\} = frac\\left\\{Cost of Goods Sold \\left(over a given period\\right)\\right\\}\\left\\{Average Inventory\\left(for the period\\right)\\right\\}$\n\nAn item whose inventory is sold (turns over) once a year has higher holding cost than one that turns over twice, or three times, or more in that time. The purpose of increasing inventory turns is to reduce inventory for three reasons.\n\n*Increasing inventory turns reduces holding cost. The organization spends less money on rent, utilities, insurance, theft and other costs of maintaining a stock of good to be sold.\n\n*Reducing holding cost increases net income and profitability as long as the revenue from selling the item remains constant.\n\n*Items that turn over more quickly increase responsiveness to changes in customer requirements while allowing the replacement of obsolete items. Which is a major concern in fashion industries.\n\nHowever high turns may indicate that the inventory is too low. This often can result in stock shortages.\n\nNote\n\nSome computer programs measure the stock turns of an item using the actual number sold.\n\n$mbox\\left\\{Inventory Turns\\right\\} = frac\\left\\{Number of Units Sold \\left(over a given period\\right)\\right\\}\\left\\{Average Number of Units \\left(For the period\\right)\\right\\}$\n\nIn practice this tends to be confusing and can give misleading results if averaged out over a department.\n\nAlso sometimes we measure stock turn rates based on annual sales at retail divided by average inventory at retail. This measurement, is sometimes available in computer systems, is based on the \"value\" that your customer perceives product (actual selling price which may include markdowns) and the \"value\" of your inventory. These computer systems can devalue inventory as markdowns occur even before they are sold. One valid reason for using retail for these calculations is that if a \\$100 retail-priced item (that was \\$50 at cost) is now only worth \\$80, then the retailer would find it difficult to pay \\$50 for an item only worth \\$80 to customers.\n\nRetail-calculated stock turns rates tend to be lower than those calculated at cost.\n\nThe important issue is that any organization should be consistent in the formula that it uses.\n\nee also\n\n*Cost accounting\n*Throughput accounting\n\nWikimedia Foundation. 2010.\n\n### Look at other dictionaries:\n\n\u2022 inventory turns \u2014 \u00a0How many times inventory is sold and replaced per year. \u00a0\u25ba \u201cOverburdened inventories, resulting in poor inventory turns and on time performance that is far less than desirable, are serious problems for many companies.\u201d (Production & Inventory\u2026 \u2026 \u00a0 American business jargon\n\n\u2022 Inventory \u2014 means a list compiled for some formal purpose, such as the details of an estate going to probate, or the contents of a house let furnished. This remains the prime meaning in British English.[1] In the USA and Canada the term has developed from a\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Inventory turnover \u2014 The ratio of annual sales to average inventory which measures the speed that inventory is produced and sold. Low turnover is an unhealthy sign, indicating excess stocks and\/or poor sales. The New York Times Financial Glossary * * * inventory\u2026 \u2026 \u00a0 Financial and business terms\n\n\u2022 turns \u2014 turnover or turns Terms used to describe the number of operating cycles in a defined period of time or the length of each specific operating cycle. Typical turnover cycles are: the rate at which accounts receivable converts to cash, the rate at\u2026 \u2026 \u00a0 Financial and business terms\n\n\u2022 inventory turnover \u2014 The ratio of annual sales to average inventory, which measures the speed at which inventory is produced and sold. Low turnover is an unhealthy sign, indicating excess stocks and\/or poor sales. Bloomberg Financial Dictionary * * * inventory\u2026 \u2026 \u00a0 Financial and business terms\n\n\u2022 Belbin Team Inventory \u2014 The Belbin Team Inventory, also called the Belbin Self Perception Inventory or the Belbin Team Role Inventory, is a test used to gain insight into an individual s behavioural type. It was developed by Dr. Meredith Belbin after studying numerous\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 GMROII \u2014 Gross Margin Return on Inventory Investment (GMROII) is a ratio in microeconomics that describes a seller s income on every dollar spent on inventory. It is one way to determine how valuable the seller s inventory is, and describes the\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Theory of constraints \u2014 Part of a series of articles on Industry Manufacturing methods Batch production \u2022 Job production Continuous production Improvement method \u2026 \u00a0 Wikipedia\n\n\u2022 turnover \u2014 or turns Terms used to describe the number of operating cycles in a defined period of time or the length of each specific operating cycle. Typical turnover cycles are: the rate at which accounts receivable converts to cash, the rate at which\u2026 \u2026 \u00a0 Financial and business terms\n\n\u2022 Stock management \u2014 is the function of understanding the stock mix of a company and the different demands on that stock. The demands are influenced by both external and internal factors and are balanced by the creation of Purchase order requests to keep supplies at\u2026 \u2026 \u00a0 Wikipedia","date":"2021-06-18 09:46:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.25378116965293884, \"perplexity\": 4288.167346243388}, \"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-2021-25\/segments\/1623487635920.39\/warc\/CC-MAIN-20210618073932-20210618103932-00182.warc.gz\"}"} | null | null |
Q: How does captive.apple.com force the captive login page when other websites don't? Lots of hotel, coffee shop, etc. networks require accepting a ToS via a captive page. Sometimes when I join these networks I can't get that captive page to show by just going to regular websites - the requests just time out. But I can always get it to show by going to captive.apple.com. What does this do differently that other websites aren't doing to force the captive page?
A: This discussion suggests that captive pages are triggered by an http request but not an https request -- or maybe not an http request that the destination website redirects to https. I'm not sure if it's the http-to-https redirect or the mere fact of using https that's the problem. You could test that by entering a URL including the https scheme and seeing if that triggers the captive page.
Anyway, I thought it was odd that whenever the captive.apple.com/hotspot-detect.html URL was given, it was with the http scheme, but it makes sense after reading this.
| {
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\section{Introduction}
Rainfall plays an important role in the formation of fauna and flora of natural life.
An accurate prediction of a rainfall event can not only support causal usages but also provide early warnings of floods or traffic accidents\cite{pham2020development}.
Despite the availability of advanced technology and sufficient amount of weather data, the prediction of rainfall event is extremely complex \cite{manandhar2016gps}. The changing climatic conditions and the increasing greenhouse emissions have made it difficult for humans to properly understand the weather~\cite{fathima2019chaotic}. In order to develop efficient rainfall prediction systems, firstly a thorough study of various meteorological parameters should be made in the context of rainfall occurrence.
\subsection{Related Work}
There is an extensive research carried out in the area of weather forecasting.
In \cite{yen2019application}, Principal Component Analysis (PCA) is used to analyze the hourly meteorological data of southern Taiwan. The results showed that the most dominant factors controlling the rainfall are the air pressure and humidity. The authors in \cite{manandhar2019data} proposes a systematic approach to analyze different ground based weather parameters from year 2012-2015 of Nanyang Technological University weather station. Features such as Precipitable Water Vapor (PWV) and solar radiation stand out for rainfall prediction. Oswal \textit{et al.}~\cite{oswal2019predicting} adopted a pairwise correlation matrix to understand interactions between parameters.
A correlation-based feature selection technique to assemble an effective subset to develop precipitation prediction model is also presented in \cite{manandhar2018systematic}.
\subsection{Contributions of the paper}
The main contributions of this paper include:
\begin{itemize}
\item we present a methodical approach to analyze and assess the impact of different meteorological parameters on the occurrence of rainfall;
\item we also share the source-code of our methodology in the spirit of reproducible research\footnote{The code related to this paper is available here: \url{https://github.com/Sammyy092/Impact-of-meteorological-parameters-on-rainfall}.}.
\end{itemize}
\section{Results \& Discussions}
\subsection{Dataset}
The rainfall data for our research is collected from National Oceanic and Atmospheric Administration (NOAA) Climate Data Online service (CDO\footnote{https://www.ncdc.noaa.gov/cdo-web/}).
The downloaded monthly data from Jan 2015-Dec 2020 is recorded from the weather stations located in Alpena Regional Airport, Michigan, U.S.
The meteorological parameters utilized in our work are: precipitation ($PRCP$), average wind speed ($AWND$), direction of fastest 2-minute wind ($WDF2$), direction of fastest 5-second wind ($WDF5$), fastest 2-minute wind speed ($WSF2$), fastest 5-second wind speed ($WSF5$), average temperature ($TAVG$), minimum temperature ($TMIN$), maximum temperature ($TMAX$). These features were selected because they are interdependent and influence the rainfall. Feature $PRCP$ is rainfall measured in inches (in inches) using a rain gauge, and we consider this feature as the rainfall indicator. In this way we will compare other parameters with $PRCP$ in order to find the relation and influence of each parameter for $PRCP$.
\subsection{Correlation Analysis}
In this section we describe the correlation of different features with respect to $PRCP$. The cross-correlation values among the features were computed by calculating the correlation coefficients of feature matrix $X$, having dimension $m \times n$, denoted as: $X=[v_1, v_2,\ldots, v_n]$, where $v_1, v_2,\ldots, v_n$ are the vectors of $n$ number of meteorological features. Each vector is of length $m$ indicating a weather recording at a particular time. The correlation values of all the features are shown in figure. We observe from Fig.~\ref{fig:corr_mat} that
there is a significant correlation between $PRCP$ and the features related to data type wind (i.e. $WSF5$, $WSF2$, $WDF5$, $WDF2$, $AWND$). $PRCP$ has a positive correlation with $AWND$, $WSF2$ and $WSF5$ of value (0.25, 0.24 and 0.26).The reason of the positive correlation between the wind and rain is that the winds carry an amount of moisture in it which can highly affect the amount of precipitation in an area. Faster winds and precipitation are strongly correlated in nature where faster winds cause rain, showing major significance on daily rainfall variability.
Furthermore, using the annual data we found a negative correlation between rainfall and temperature (\textit{i.e.}, as temperature increases, rainfall drops). We can see a value of $-0.167$ when $TMAX$ is related with $PRCP$. The amount of precipitation gets lower with higher temperatures and vice versa. Hence, a strong negative correlation occurs on land, as temperature favor more dry conditions and less evaporative cooling.
\begin{figure}[htb]
\centering
\includegraphics[width=.45\textwidth]{corr.png}
\caption{We compute the correlation values of the different meteorological parameters and precipitation.}
\label{fig:corr_mat}
\end{figure}
\vspace{-0.3cm}
\subsection{Feature Importance}
This section describes the importance of each input meteorological parameter when predicting the $PRCP$. To accomplish the task, we have used the least absolute shrinkage and selection operator (LASSO)~\cite{li2005lasso}.
The LASSO is the penalized least squares regression case with $\ell_1$-penalty function which is estimated as:
\[L=\sum_{j}^{}(y_j-\sum_{k}^{}\beta _{k}v_{jk})^2+\lambda \sum_{k}^{} \left \|\beta_{k} \right \|_{1}\]
where $v_{jk}$ denotes the $k$th meteorological feature in the $jth$ datum, $y_{j}$ is the response feature value in this datum and $\beta_{k}$ represents the regression coefficient of the $k$th feature. Because of
$\ell_1$ function $\sum_{k}$ $\left \| \beta _{k} \right \|_{1}$, the LASSO feature selection technique typically produces estimates in which some of the coefficients are set exactly to zero thereby performing
feature selection.
Figure~\ref{fig:us_data} shows the feature importance values of each meteorological variable in terms of precipitation prediction. We can observe from the figure that features $TMIN$, $WSF2$, $WSF5$ have the highest values. The weather parameters $TMAX$ and $AWND$ posses the lowest scores meaning that they have a low influence in detecting the rainfall. Furthermore, all the features were selected and given values during the feature selection process, it means that none of the features have a non-zero coefficient after the shrinking process.
\begin{figure}[htb]
\centering
\includegraphics[width=.43\textwidth ]{features_USA.png}
\caption{We compute the scores of different features in predicting rainfall.}
\label{fig:us_data}
\end{figure}
\vspace{-0.4cm}
\section{Conclusion \& Future Work}
An accurate prediction of rainfall is not an easy task due to the changing behavior of climate. Therefore, we have presented a methodical analysis of different meteorological parameters associated with the occurrence of rainfall in order to understand the importance of each parameter for rainfall. We obtained
that
wind is strongly correlated with rainfall indicating that strong winds can highly influence the occurrence of rainfall.
We also obtained that wind speed and minimum temperature are
the most important features in predicting the rainfall. In future we plan to use this reduced subset of features for accurate estimation of the time of occurrence of rainfall.
\bibliographystyle{IEEEtran.bst}
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This week is all about finding your groove and familiarising yourself with the main movements you'll do during the next four weeks.
Find a local park or oval and measure out approximately 100 metres. Place a cone or marker at each end along with a mat or towel. The session requires you to surge – think 70 per cent of your maximum speed – from one end to the other. At one end, stop and do 10 push-ups. At the other, do 10 sit-ups. Continue to run back and forth, augmenting sprints with bodyweight moves until you've clocked your nominated sets and reps (see options based on fitness level). Keep it moving and don't be tempted to down tools during a set; that's what the 90-second breaks between sets are for! | {
"redpajama_set_name": "RedPajamaC4"
} | 8,001 |
\section{Introduction}\label{sec1}
It has been known for a long time that the dynamics of open quantum systems subject to external driving forces and coupled to environmental modes (`heat bath') can be described by master equations \cite{Weiss2007,Breuer2006,Gardiner2004}.
For a Markovian bath, the memory time of the bath represents the shortest time scale of the problem. The master equation is then of
Lindblad type \cite{Lindblad1976,Lindblad1983}, where a Hamiltonian describes the coherent time evolution of the system's density matrix and a Lindbladian captures the dissipative dynamics. (We here use `Lindbladian' for the dissipator terms in the master equations below.) The Lindblad equation is the most general Markovian master equation which preserves the trace and positive semi-definiteness of the density matrix.
A major development over the past two decades has come from the realization that driven dissipative (DD) quantum systems can be stabilized in a pure quantum state by appropriate engineering of the driving fields and of the coupling to the dissipative environment \cite{Plenio1999,Beige2000,Plenio2002,Diehl2008,Kraus2008,Diehl2010,Diehl2011,Bardyn2013,Zanardi2014,Albert2014,Jacobs2014,Albert2016,Goldman2016,Wiseman2010}.
Such states are eigenstates of the corresponding Lindbladian with zero eigenvalue, i.e., the operation of the Lindbladian leaves them inert. We therefore will refer to these DD stabilized states
as \emph{dark states} in what follows. Rather than viewing the coupling to a dissipative environment as foe (e.g., leading to decoherence of quantum states and undermining the utilization of similar platforms for quantum information processing), the combined effect of drive and dissipation can thus be harnessed to engineer quantum-coherent pure states.
Going beyond dark states, the stabilization of a \emph{dark space} \cite{Iemini2015,Iemini2016,Santos2020} --- a manifold spanned by multiple degenerate dark states ---
raises the prospects of employing such systems as viable platform for quantum information processing.
Reference~\cite{Touzard2018} reports on recent experimental results in this direction.
Using trapped ions or superconducting qubits, the above ideas have already allowed for first qubit stabilization experiments \cite{Geerlings2013,Lu2017,Touzard2018}, for
the implementation of quantum simulators \cite{Barreiro2011,Schindler2013}, and for the generation of selected highly entangled multi-particle states \cite{Shankar2013,Leghtas2013,Reiter2016,Liu2016}. Systems composed of many coupled qubits stabilized by DD mechanisms could eventually result in
universal quantum computation platforms \cite{Verstraete2009,Fujii2014}, where fault tolerance is the consequence of
autonomous error correction \cite{Terhal2015} due to the engineered dissipative environment, without the need for active feedback \cite{Wiseman2010,Kerckhoff2010,Murch2012,Kapit2015,Kapit2016}.
Recent experimental progress on autonomous error correction in DD qubit systems has been described in Refs.~\cite{Leghtas2013,Liu2016,Reiter2017,Puri2019}.
At present, reported fidelities in DD qubit setups (which by construction are stable in time) are typically below 90$\%$ for state stabilization, with significantly lower fidelities
for single- or two-qubit gate operations.
Another important and at first glance unrelated development towards the (so far elusive) goal of fault-tolerant universal quantum computation comes from the field of topological quantum computation \cite{Nayak2008}.
By using topological quasiparticles \cite{Wen2017} for encoding and processing quantum information, the latter is nonlocally distributed in space
and thereby protected against local environmental fluctuations. In general terms,
for practically useful and scalable DD systems with multiple degenerate dark states, the coupling to the environment has to be carefully engineered such that it is blind to all system operators acting within the targeted dark space manifold \cite{Facchi2000}.
It will thus be imperative to avoid residual (uncontrolled and unwanted) noise sources.
In that regard, platforms harboring topological quasiparticles may offer a key advantage since they should come with a strongly
reduced intrinsic sensitivity to residual environmental fluctuations as compared to conventional systems.
The simplest candidate for topological quasiparticles is given by
Majorana bound states (MBSs), which are localized zero-energy states in topological superconductors. For Majorana reviews, see Refs.~\cite{Alicea2012,Leijnse2012,Beenakker2013,Sarma2015,Aguado2017,Lutchyn2018,Zhang2019a}.
Topological codes relying on MBSs have so far been discussed in the context of active error correction \cite{Alicea2011,Terhal2012,Hyart2013,Vijay2015,Aasen2016,Landau2016,Plugge2016,Plugge2017,Karzig2017,Litinski2017,Wille2019},
where periodically repeated stabilizer measurements are needed for fault tolerance.
It remains an important challenge to devise feasible and scalable Majorana platforms exploiting passive error correction strategies,
where DD mechanisms serve to continuously measure the system in a way that the desired highly entangled many-body quantum state becomes stabilized automatically, see, e.g., Ref.~\cite{Herold2017}.
While this ambitious goal is beyond the scope of our work, we here analyze related questions for DD systems with up to eight MBSs.
For a mesoscopic floating (not grounded) topological superconductor harboring four MBSs, strong charging effects
\cite{Fu2010} imply that the ground state is doubly degenerate under Coulomb valley conditions (see Sec.~\ref{sec2a} for details).
Such a superconducting island is therefore a good candidate for a topologically protected Majorana qubit,
named Majorana box qubit \cite{Plugge2017} or tetron \cite{Karzig2017}. Thanks to the nonlocal Majorana encoding of quantum information, such a qubit allows
for unique addressability options via electron cotunneling when quantum dots (QDs) or normal leads are attached to the island by tunneling contacts, see also Refs.~\cite{Gau2018,Munk2019}.
Majorana qubits have not yet been experimentally realized. However, the recent emergence of new Majorana platforms (see, e.g.,
Refs.~\cite{Liu2018b,Zhang2018b,Wang2018b,Sajadi2018,Ghatak2018,Murani2019}) in addition to the semiconductor nanowire platform mainly explored
so far \cite{Lutchyn2018,Zhang2019a} indicates that they may be available in the foreseeable future. We note that alternative Majorana qubit designs have been put forward, e.g., in Refs.~\cite{Terhal2012,Hyart2013,Aasen2016}. Many of the ideas discussed below can be adapted to those setups as well.
\subsection{Motivation and goals of this work}
We here show that once available, Majorana box devices yield highly attractive platforms for implementing DD protocols
aimed at the realization of dark states and/or dark spaces. The driving field is applied to the tunnel link connecting a pair of QDs, and
dissipation is due to environmental electromagnetic noise.
To the best of our knowledge, apart from a distantly related proposal for the
DD stabilization of Majorana-based quantum memories \cite{Bardyn2016},
no studies of DD Majorana systems have appeared in the literature so far.
We note that the DD engineering of MBSs in cold-atom based Kitaev chains \cite{Diehl2011,Bardyn2013,Goldman2016} differs from our ideas: We consider topological superconductors harboring native MBSs, and then subject the resulting Majorana systems to DD stabilization
and manipulation protocols targeting dark states and/or dark spaces. Our unique platform enables us to employ QDs as external knobs to be used not only for state engineering but also for state manipulation.
Our motivation for designing and studying novel DD stabilization and manipulation schemes using Majorana platforms rests on several arguments and expectations:
\begin{enumerate}
\item
Since uncontrolled environmental effects are largely suppressed by topological protection mechanisms, one may reach higher
fidelities than those reported so far for DD dark state or dark space implementions using conventional (topologically trivial) platforms. This point should be especially important for high-dimensional dark spaces, where residual noise effects could
break the degeneracy of the dark states spanning the dark space manifold \cite{Facchi2000}. Such spaces are highly attractive candidates for implementing fault-tolerant quantum computing platforms. These topological protection elements are especially important for platforms where the Lindblad spectrum is not gapped.
\item It is known that for large-scale Majorana surface codes, where active feedback is needed for code stabilization,
the fault-tolerance error threshold is much more benign than for conventional bosonic surface codes, see Refs.~\cite{Vijay2015,Plugge2016,Fowler2012} and references therein.
In particular, in Majorana surface codes no ancilla qubits are needed for stabilizer readout at all.
We expect that our dark space constructions using MBS systems can allow for similar fault tolerance advantages over conventional dark space realizations.
However, more work is needed to reach a quantitative conclusion on this point.
\item The DD stabilization and manipulation of Majorana-based dark states or dark spaces offers several practical
advantages. In particular, the robustness of such states as quantified by the dissipative gap is expected to be superior to
quantum states that are encoded without DD mechanisms in native Majorana devices, see Sec.~\ref{sec3}.
Moreover, a small overlap between MBSs is often tolerable, without causing dephasing of dark states, cf.~Sec.~\ref{sec3e}.
\item When steering a state into the dark space or manipulating a state within the dark space, one may need to maximize its purity, having in mind quantum information manipulation protocols. For this purpose, we may adiabatically switch on a suitable perturbation either to the Lindbladian dissipator or to the accompanying Hamiltonian, thereby breaking the degeneracy of
the dark space. In this manner, one can revert to a specific pure dark state, manipulate this state, and subsequently adiabatically switch off this perturbation again.
The DD Majorana platforms discussed below offer convenient tools to switch on and off such degeneracy-breaking perturbations.
\end{enumerate}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.9\columnwidth]{f1}
\end{centering}
\caption{Schematic sketch of a driven dissipative Majorana box setup. The superconducting island harbors four Majorana operators $\gamma_\nu$, three of which are tunnel-coupled to two single-level quantum dots (QDs, in blue).
The Majoranas could be realized as end states of two parallel topological superconductor nanowires (green) which are electrically connected by a superconducting bridge (orange) \cite{Plugge2017}.
The tunnel links connecting QDs to MBSs are shown as dashed lines. The phases $\beta_j$ in Eq.~\eqref{betadef} are also indicated. Due to the large box charging energy,
transport between different QDs through the Majorana island proceeds only via cotunneling processes.
These cotunneling processes can be inelastic, involving the emission or absorption of photons from the dissipative electromagnetic environment.
In addition, a driving field can pump electrons via a tunnel link between the QDs (solid line).
}
\label{fig1}
\end{figure}
The dynamics of the Majorana degrees of freedom in a device such as the one depicted in Fig.~\ref{fig1} will here be discussed on several conceptual levels. We show that our DD protocols indeed give rise to master equations of Lindblad type. These equations contain both a Hamiltonian (governing the unitary part of the time evolution) and a Lindbladian (causing dissipative dynamics). By choosing suitable parameter values as discussed in Sec.~\ref{sec3}, we demonstrate that an arbitrary dark state can be stabilized. In more complex two-box devices, see Sec.~\ref{sec4}, the Lindbladian can be engineered to support a multi-dimensional dark space.
As a generic initial state is driven towards the dark space, we show (see also Ref.~\cite{ourprl}) how to optimize the purity, the fidelity (i.e., the overlap of the state with the target dark space), and the speed of approach.
A major benefit of applying DD strategies to a topologically nontrivial system comes from the insight that
one can here implement unidirectional cotunneling processes in an elementary and practically useful manner. Using Majorana boxes which are tunnel-coupled to two quantum dots,
we show that the combination of driving fields, energy relaxation, and the tunability of tunneling amplitudes allows for the controlled design of
directed cotunneling processes. The latter directly determine the important jump operators in the Lindblad equation.
In our accompanying short paper \cite{ourprl}, we provide a summary of our key ideas and apply them to show that in a two-box setup, one can stabilize and manipulate
`dark qubit' states. In effect, the topologically protected native Majorana qubit discussed in Refs.~\cite{Plugge2017,Karzig2017} (which exists in a single box) is thereby stabilized by adding another protection layer due to DD mechanisms (at the prize of adding a second box).
\subsection{Overview}
In order to guide the focused reader through this long article, we here provide a short overview summarizing the content of the subsequent sections.
In addition, Table~\ref{table1} summarizes the key symbols and notations used throughout this paper.
\begin{itemize}
\item In Sec.~\ref{sec2}, we introduce the theoretical concepts and physical ingredients needed for the
DD stabilization and manipulation of dark states using a single Majorana box, see Fig.~\ref{fig1}, and we derive the dynamical equations. Our model is introduced in Sec.~\ref{sec2a},
where the dissipation arises from environmental electromagnetic fluctuations and the drive is applied to a pair of QDs.
We subsequently derive the Lindblad equation \cite{Weiss2007,Breuer2006,Gardiner2004,Lindblad1976,Lindblad1983} governing the time evolution of the combined QD-Majorana system in Sec.~\ref{sec2b},
where we also present numerical results for the dynamics obtained from this Lindblad master equation.
Remarkably, up to initial transient behaviors, one can describe the dynamics in the Majorana sector in terms of a reduced
Lindblad equation, where the QD degrees of freedom have been traced out. We describe this step in
Sec.~\ref{sec2c}, along with a discussion of the conditions under which this reduced Lindblad equation applies.
All of our subsequent results are obtained by employing this reduced Lindblad equation.
\item In Sec.~\ref{sec3} we then describe dark state stabilization protocols for the single-box device in Fig.~\ref{fig1}.
We begin in Sec.~\ref{sec3a} with the case of Pauli operator eigenstates, followed by the stabilization of the so-called magic state in Sec.~\ref{sec3b}.
In Sec.~\ref{sec3c}, the role of increasing temperature on our stabilization protocols is examined.
Interestingly, as shown in Sec.~\ref{sec3d}, we find that for certain parameter settings, dark states can be stabilized even in the absence of any drive.
However, the field-free stabilization is limited to very special conditions and is also characterized by rather small dissipative gaps.
In practical implementations, it will thus be preferable to employ a driving field.
Finally, in Sec.~\ref{sec3e}, we discuss additional points, e.g., concerning the role of Majorana state overlaps or how to perform a parity readout of the stabilized states.
\item In Sec.~\ref{sec4}, we turn to a setup with two coupled boxes and present our DD stabilization and manipulation protocols for quantum states that belong to a dark space manifold. The Lindblad equation for this setting is derived in Sec.~\ref{sec4a}. We explain how one can engineer a degenerate dark space in Sec.~\ref{sec4b}.
This topic is the main focus of Ref.~\cite{ourprl}, and the discussion is therefore kept rather short here. Finally, in Sec.~\ref{sec4c},
we show how to stabilize Bell states in the two-box setting.
\item
The paper concludes with a summary and an outlook in Sec.~\ref{secConc}.
\end{itemize}
Technical details and additional information can be found in three Appendices. Let us also remark that we often use units with $\hbar=k_B=1$.
\begin{table*}[t]
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{c|l|c} \hline\hline
Symbol & Meaning & First appearance\\ \hline \hline
\emph{Model parameters:} & & \\ \hline
$A$ & drive amplitude & \eqref{Hdriv}\\ \hline
$\alpha$ & dimensionless system-bath coupling for Ohmic bath & \eqref{alphadef}\\ \hline
$\beta_j$ & phases of the tunnel couplings $\lambda_{j\nu}$ & \eqref{betadef} \\ \hline
$E_C$ & charging energy of the Majorana box & \eqref{Hbox} \\ \hline
$\epsilon_{j}$ & level energy of the respective quantum dot & \eqref{HDots}\\ \hline
$g_0$ & cotunneling scale for single-box setup, $g_0=t_0^2/E_C$ & \eqref{Heff}\\ \hline
$\tilde g_0^{}$ & cotunneling scale for double-box setup & \eqref{tildeg0}\\ \hline
$\lambda_{j\nu}$ & tunnel coupling between QD fermion $d_j$ and Majorana operator $\gamma_{\nu}$ & \eqref{hatlambda}\\
& (`state design parameters') & \\ \hline
$M$ & number of MBSs on Majorana box & \eqref{Wopdef} \\ \hline
$\omega_0$ & drive frequency & \eqref{Hdriv} \\ \hline
$\omega_c$ & cut-off frequency for Ohmic bath & \eqref{spectraldensity}\\ \hline
$T$ & temperature & \eqref{condit} \\ \hline
$t_0$ & overall scale of tunnel couplings between QDs and Majorana box & \eqref{Htun} \\ \hline
$t_{LR}$ & tunnel coupling connecting both Majorana boxes, see Sec.~\ref{sec4} & \eqref{LRCoup} \\ \hline \hline
\emph{Dynamical quantities:}& & \\ \hline
$D$ & dark space dimension & Sec.~\ref{sec3e4}\\ \hline
$\Delta_{z,x,y,m}$ & dissipative gap for the respective dark state & e.g., see \eqref{deltaz}\\ \hline
$h_{\pm}, \tilde h^{}_\pm$ & Lamb shift parameters for full and reduced Lindblad eq., respectively & \eqref{rate2}, \eqref{mod22}\\ \hline
$J_\pm, \Gamma_{\pm}, H_{\rm L}$ & jump operators, transition rates, and Hamiltonian for full Lindblad eq. & \eqref{jumpops}, \eqref{rate2}, \eqref{jzdef}\\ \hline
$\tilde J^{}_\pm, \tilde\Gamma^{}_{\pm}, \tilde H^{}_{\rm L}$ & jump operators, transition rates, and Hamiltonian for reduced Lindblad eq. & \eqref{jumpN}, \eqref{mod1}, \eqref{mod22}\\ \hline
$K_{j=1,\ldots,6}, \tilde \Gamma^{}_{j}$ & jump operators and transition rates for two-box setup & \eqref{jumpK}, \eqref{trans22}\\ \hline
$p$ & occupation probability of high-lying QD & \eqref{ssform}\\ \hline
$\rho(t)$ & reduced density matrix for combined QD-Majorana system & \eqref{UnwantedTerms}\\ \hline
$\rho_{\rm M}(t)$ & reduced density matrix for the Majorana sector & \eqref{factorize} \\ \hline
$(\tau_x,\tau_y,\tau_z)$ & Pauli operators for QD pair in single-occupancy regime $N_{\rm d}=1$ & \eqref{taudef} \\ \hline
$\theta_{j\nu},\theta$ & fluctuating electromagnetic phases & \eqref{hatlambda}, \eqref{phasedef}\\ \hline
$\hat W_{jk}, \hat W_{x,y,z}$ & fluctuating cotunneling operators & \eqref{Wopdef}, \eqref{wdef1} \\ \hline
$W_{jk}, W_{x,y,z}$ & cotunneling operators for $\theta_{j,\nu}=0$ & \eqref{wdef2}\\ \hline
$(X,Y,Z)$ & Pauli operators of Majorana box & \eqref{PauliOp} \\ \hline
\hline
\end{tabular}
\caption{List of important symbols.
\label{table1}}
\end{center}
\end{table*}
\section{Driven dissipative Majorana dynamics}\label{sec2}
We start this section by discussing the Majorana box \cite{Plugge2017,Karzig2017}.
Our DD model as well as the physical assumptions behind it are explained in Sec.~\ref{sec2a}.
We then derive the Lindblad master equation governing the dynamics of the reduced density matrix of the Majorana sector.
To that end, we first trace over the environmental degrees of freedom in Sec.~\ref{sec2b}, and then over the QD fermions in Sec.~\ref{sec2c}.
\subsection{Model and low-energy theory} \label{sec2a}
In this subsection, we introduce the model for the DD Majorana setup illustrated in Fig.~\ref{fig1}. We also
outline the hardware ingredients needed for implementing our dark state stabilization and manipulation protocols.
For concreteness, we refer to a possible realization using proximitized semiconductor nanowires \cite{Plugge2017,Karzig2017}.
In addition, we describe the effective low-energy Hamiltonian obtained
after the high-energy charge states on the Majorana island are projected away.
\subsubsection{Majorana box}
Consider the setup depicted in Fig.~\ref{fig1}, where a floating topological superconductor island harbors $M$ zero-energy MBSs. For this case we have $M=4$, but for generality, we shall allow for general (even) values of $M$.
The MBSs correspond to the Majorana operators $\gamma^{}_\nu=\gamma_\nu^\dagger$, with anticommutator
$\{\gamma_\nu,\gamma_{\nu'}\}=2\delta_{\nu\nu'}$ and $\nu=1,\ldots,M$. As indicated in Fig.~\ref{fig1}, they could be realized as end states of two parallel InAs/Al nanowires \cite{Lutchyn2018}. We consider class-$D$ topological superconductor wires, where time reversal symmetry is broken by
a magnetic field \cite{Alicea2012}.
Both nanowires are electrically connected by a superconducting bridge such that the entire island has a common charging energy, $E_C=e^2/(2C)$, with
typical values of the order $E_C\approx 1$~meV \cite{Lutchyn2018}.
The isolated island (`box') has the Hamiltonian (we work in the Schr\"odinger picture for now)
\begin{equation} \label{Hbox}
H_{\rm box}=E_C( \hat N-N_g)^2.
\end{equation}
The operator $\hat N$ refers to the total electron number on the box, and $N_g$ is a tunable backgate parameter.
In Eq.~\eqref{Hbox} we have neglected
hybridization energies resulting from a finite overlap between different MBS pairs. These energy scales are
exponentially small in the respective MBS-MBS distance. As will be discussed in Sec.~\ref{sec3e}, a small hybridization between MBSs
is often tolerable for DD-generated dark states or dark spaces. For the native Majorana qubit, such effects cause dephasing.
Our theory requires several conditions to be satisfied.
First, we assume that our DD protocols only involve energy scales well below both $E_C$ and the superconducting (proximity) gap $\Delta$.
This assumption implies that the ambient temperature satisfies $T\ll {\rm min}\{E_C,\Delta\}$, which typically requires temperatures below $100$~mK in semiconductor-based
Majorana platforms \cite{Lutchyn2018}.
We can then neglect the effects of above-gap continuum quasiparticles, as has tacitly been assumed in Eq.~\eqref{Hbox}, which otherwise constitute an intrinsic source of dissipation in the Majorana sector.
In practice, one also needs to ensure that accidental low-energy Andreev states are not accessible, see Ref.~\cite{Manousakis2019} for a recent discussion.
Second, we consider Coulomb valley conditions \cite{Nazarov,AltlandBook}, i.e., $N_g$ is tuned close to an integer value and the box is only weakly coupled to the QDs in Fig.~\ref{fig1}. In that case, $H_{\rm box}$ leads to charge quantization, which dictates the fermion number parity of the island. At temperatures well below the superconducting gap,
only the Majorana sector of the full Hilbert space of the box has to be kept \cite{Fu2010}. For $M=4$,
we arrive at a parity constraint in the Majorana sector, $\gamma_1\gamma_2\gamma_3\gamma_4=\pm 1$ \cite{Beri2012}, and
the lowest-energy island state is then doubly degenerate.
The corresponding Pauli operators associated with the resulting Majorana qubit
are represented by Majorana bilinears \cite{Beri2012,Landau2016,Plugge2016},
\begin{equation}\label{PauliOp}
X=i\gamma_1\gamma_3,\quad Y=i\gamma_3\gamma_2,\quad Z=i\gamma_1\gamma_2.
\end{equation}
The fact that Pauli operators correspond to spatially separated pairs of Majorana operators allows for unusually versatile qubit access options.
The qubit is encoded either in the even parity sector, i.e.,
by using the two degenerate states with fermionic occupation number $N_m=0$ and $N_m=2$ in the Majorana sector (with one Cooper pair less for the $N_m=2$ state),
or in the odd parity sector, where both states have $N_m=1$.
\subsubsection{Quantum dots}
We next turn to the Hamiltonian describing the two QDs, $H_{\rm d}$, in Fig.~\ref{fig1}.
We start from a general single-dot Hamiltonian, $H_{\rm QD}=\sum_{\alpha} h_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}+\epsilon_C\left(\hat n-n_g\right)^2$,
where $\alpha$ labels electron spin and orbital degrees of freedom, $d_{\alpha}$ are fermion operators with $\hat n=\sum_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}$, $h_{\alpha}$ describes a single-particle energy,
and $\epsilon_C$ is the (large) dot charging energy \cite{Karzig2017,Flensberg2011,Nazarov,AltlandBook}. On low energy scales,
the dot can then effectively be described by a single spinless fermion level.
Denoting the corresponding level energy by $\epsilon_j$ for QD $j=1,2$, one arrives at
\begin{equation}\label{HDots}
H_{\rm d}=\sum_{j=1,2} \epsilon_j d_j^{\dagger} d^{}_j,
\end{equation}
see Ref.~\cite{Karzig2017} for details.
The energies $\epsilon_j$ can be controlled by variation of the gate voltage parameter $n_g$. Without loss
of generality, we take $\epsilon_2>\epsilon_1$ throughout, where both energies should
satisfy $|\epsilon_j|\ll {\rm min}\{E_C,\Delta\}$.
In addition, we employ a time-dependent electromagnetic driving field which can pump single electrons between
the two QDs via a tunnel link.
To that end, a suitable AC voltage can be applied to a gate electrode located near this link.
The respective Hamiltonian contribution is given by \cite{Platero2004}
\begin{equation}\label{Hdriv}
H_{{\rm drive}}(t) = w(t) d_1^{\dagger} d_2^{}+{\rm h.c.},\quad
w(t) = t_{12}+2A \cos\left(\omega_0 t\right),
\end{equation}
where $\omega_0$ denotes the drive frequency and $A$ the drive amplitude. In what follows, we assume that the static contribution vanishes, $t_{12}=0$,
because a small coupling $t_{12}\ne 0$ will not affect the dissipator in the Lindblad equation, see Eq.~\eqref{GeneralLindblad} below,
and thus does not change the physics in a qualitative manner.
In this work, we consider the Coulomb valley regime where the total charge on the box is fixed by the charging term in Eq.~\eqref{Hbox} on time scales $\delta t>1/E_C$ \cite{Romito2014}.
The total particle number on the QDs, $N_{\rm d}=\sum_j d_j^\dagger d_j^{}$, is therefore also conserved on such time scales. For even $N_{\rm d}\in\{0,2\}$, the inter-QD dynamics is effectively frozen out. We here mainly focus on the case $N_{\rm d}=1$, where the pair of QDs forms a spin-1/2 degree of freedom corresponding to Pauli operators
$\tau_{x,y,z}$ with $\tau_\pm=(\tau_x\pm i \tau_y)/2$,
\begin{equation}
\tau_+ = \tau_-^\dagger = d_1^{\dagger}d_2^{}, \quad
\tau_z = d_1^{\dagger}d_1^{} - d_2^\dagger d_2^{} = 2\tau_+\tau_--1.\label{taudef}
\end{equation}
We next turn to the tunnel couplings connecting the QDs to the island.
\subsubsection{Tunnel couplings and electromagnetic environment}
In the above parameter regime, tunneling to the box has to proceed via MBSs since no other low-energy island states are available.
Such processes can be inelastic due to the coupling to a bosonic environment. We here consider the case of a dissipative electromagnetic
environment, which can be modeled by including fluctuating phases $\theta_{j\nu}$ in the tunneling matrix elements \cite{Nazarov,Devoret1990,Girvin1990},
\begin{equation}\label{hatlambda}
\hat\lambda_{j\nu}=\lambda_{j\nu}e^{i\theta_{j\nu}},
\end{equation}
with dimensionless complex-valued parameters $\lambda_{j\nu}$ subject to ${\rm max}\{|\lambda_{j\nu}|\}=1$.
Here $\lambda_{j\nu}$ determines the transparency of the tunnel link between the QD fermion $d_j$ and the Majorana state $\gamma_\nu$ in the absence of electromagnetic noise
\cite{Zazunov2016}.
The parameters $\lambda_{j\nu}$ play an important role in our DD scheme below. Both their amplitude as well as their phase can be tuned by varying the voltage on
a local gate near the tunnel contact in question, see Ref.~\cite{Lutchyn2018} and references therein.
With the overall hybridization energy $t_0$ characterizing
the QD-MBS couplings, the tunneling Hamiltonian is then given by \cite{Nazarov,Devoret1990,Girvin1990}
\begin{equation}\label{Htun}
H_{{\rm tun}} =t_0 e^{-i\hat\phi}\sum_{j,\nu} \hat \lambda_{j\nu} d^{\dagger}_j \gamma_\nu + {\rm h.c.}
\end{equation}
The phase operator $\hat\phi$ of the island has the commutator $[\hat N,\hat\phi]=-i$ with the number operator $\hat N$ in Eq.~\eqref{Hbox}.
The $e^{i\hat\phi}$ ($e^{-i\hat \phi}$) factor in Eq.~\eqref{Htun} thus ensures that an electron charge is added to (subtracted from) the island in a tunneling process.
It is well known that the electromagnetic potential fluctuations predominantly couple to the phase of the wave function \cite{Devoret1990,Girvin1990}. This fact is
expressed by the appearance of the fluctuating tunnel couplings $\hat\lambda_{j\nu}$, see
Eq.~\eqref{hatlambda}, in the tunneling Hamiltonian \eqref{Htun}.
For concreteness, we assume that the electromagnetic environment can be modeled by a single bosonic bath, see also Ref.~\cite{Munk2019}.
Representing the bath by an infinite set of harmonic oscillators \cite{Weiss2007,Breuer2006}, the environmental Hamiltonian is
$H_{\rm env}=\sum_m E_m b_m^{\dagger} b_m^{}$, with the energy $E_m>0$ of the photon mode
described by the boson annihilation operator $b_m$. In practice, the relevant bath energies $E_m$ are strongly suppressed above a cutoff frequency
$\omega_c$. With dimensionless real-valued couplings $g_{j\nu,m}$, the stochastic phase operators $\theta_{j\nu}$ are written as
\begin{equation}\label{phasedef}
\theta_{j\nu}=\sum_{m} g_{j\nu,m} \left(b^{}_m+b_m^\dagger\right).
\end{equation}
Clearly, they commute with each other, $[\theta_{j\nu},\theta_{j'\nu'}]=0$.
\subsubsection{Low-energy theory}\label{spsec1}
We are interested in the parameter regime defined by the conditions
\begin{equation}\label{condit}
{\rm max}\{T,A,t_0,\omega_0,\omega_c,|\epsilon_j|\}\ll {\rm min}\{E_C,\Delta\}.
\end{equation}
The parameters on the left side of Eq.~\eqref{condit} affect the dissipative transition rates in the Lindblad equation \eqref{GeneralLindblad} below. These rates in turn
set the time scale on which dark states are approached.
We will adopt a concise description, whereby for engineering a stabilization protocol targeting a specific dark state, it suffices to adjust the complex-valued tunnel link parameters $\lambda_{j\nu}$,
see Sec.~\ref{sec3}. In practice, those \emph{state design parameters} can be changed via gate voltages.
We also note that under the conditions in Eq.~(\ref{condit}), boson-assisted processes can neither excite above-gap quasi-particles nor higher-energy charge states on the island.
The full Hamiltonian can then be projected onto the doubly degenerate ground-state space of the box, $H(t)\to H_{\rm eff}(t)$.
Using a Schrieffer-Wolff transformation to implement this projection, and noting that $H_{\rm box}$ then reduces to an irrelevant constant energy shift,
we arrive at the effective low-energy Hamiltonian
\begin{equation}\label{heff0}
H_{\rm eff}(t)=H_{\rm d}+ H_{\rm env} + H_{\rm drive}(t)+H_{\rm cot},
\end{equation}
with the drive term in Eq.~\eqref{Hdriv} and the cotunneling contribution
\begin{equation}
H_{\rm cot} = g_0\sum_{j,k=1,2}\hat W_{jk}
\left( 2 d_j^\dagger d_k^{}-\delta_{jk} \right), \quad g_0\equiv \frac{t_0^2}{E_C}.\label{Heff}
\end{equation}
We here use the operators
\begin{equation}\label{Wopdef}
\hat W_{jk}= \sum_{1\le\mu<\nu\le M} \left(
\hat \lambda_{j\nu} \hat \lambda_{k\mu}^\dagger
- \hat \lambda_{j\mu}\hat \lambda_{k\nu}^\dagger \right)\gamma_\mu\gamma_\nu.
\end{equation}
Equation \eqref{Heff} describes cotunneling paths through the box, where the energy of the intermediate virtual
state has been approximated by $E_C$, cf.~Eq.~\eqref{condit}, and photon emission and absorption processes are encoded by the $\hat \lambda$ factors in Eq.~\eqref{hatlambda}.
For even QD occupation number $N_{\rm d}$, Eq.~\eqref{Heff} reduces to
\begin{equation}\label{heffeven}
H_{\rm cot}^{(N_{\rm d}=0,2)}=g_0 \ {\rm sgn}(N_{\rm d}-1) \ \sum_j \hat W_{jj}.
\end{equation}
For $N_{\rm d}=1$, using the notation
\begin{eqnarray}
\nonumber
\hat W_+&\equiv& \hat W_{12}, \quad \hat W_-=\hat W_+^\dagger,\quad \hat W_x=\hat W_++\hat W_-,\\ \label{wdef1}
\hat W_y&=&i(\hat W_+ -\hat W_-),\quad
\hat W_z=\hat W_{11}-\hat W_{22},
\end{eqnarray}
we find that Eq.~\eqref{Heff} can instead be expressed in the form
\begin{equation}\label{cot1}
H_{\rm cot}^{(N_{\rm d}=1)} = g_0 \sum_{a=x,y,z} \hat W_a \tau_a,
\end{equation}
with the QD Pauli operators $\tau_a$ in Eq.~\eqref{taudef}.
We emphasize that like the $\hat W_{jk}$ operators in Eq.~\eqref{Wopdef}, also the $\hat W_a$ still contain the phase fluctuation operators
due to the electromagnetic environment.
In order to realize the most general qubit-qubit exchange coupling between the QD spin $\{\tau_a\}$ and the $M=4$ Majorana box spin $(X,Y,Z)$
in the cotunneling regime, one has to specify nine independent (tunable) real-valued coupling constants.
For the $M=4$ case in Fig.~\ref{fig1}, taking into account gauge invariance --- which allows us to set one of the $\lambda_{j\nu}$ to a real value ---, the five different
complex-valued hopping parameters $\lambda_{j \nu}$ are sufficient.
On top of that, the direct tunnel amplitude between the QDs is assumed to be real-valued after setting $t_{12} = 0$ in Eq.~\eqref{Hdriv}.
To simplify the subsequent analysis, we assume that the dominant contribution to the environmental electromagnetic noise comes from the long wavelength part.
In effect, such contributions will cause dephasing of the QDs, e.g., due to the presence of a backgate electrode. This assumption is also consistent with the picture of a single bath. To good accuracy, the couplings $g_{j\nu,m}$ in Eq.~\eqref{phasedef} then do not depend on the Majorana index $\nu$, i.e., $g_{j\nu,m}=g_{j,m}$.
As a consequence, also the fluctuating phases \eqref{phasedef} become $\nu$-independent, $\theta_{j\nu}=\theta_j$. In that case, the diagonal entries $\hat W_{jj}$
are insensitive to electromagnetic noise and
the bath completely decouples for even $N_{\rm d}$, see Eq.~\eqref{heffeven}.
From now on, we therefore focus on the case of a single electron shared by the QDs, $N_{\rm d}=1$. Defining the phase operator
\begin{equation}
\theta\equiv \theta_1-\theta_2 = \sum_m (g_{1,m}-g_{2,m}) \left(b_m^{}+b_m^\dagger\right),
\end{equation}
Eq.~\eqref{cot1} then yields
\begin{equation}\label{cotfinal}
H_{\rm cot}= 2g_0 \left(e^{i\theta} W_+\tau_+ + {\rm h.c.}\right)+ g_0 W_z \tau_z.
\end{equation}
The operators $W_+$ and $W_z$ correspond to `undressed' ($\theta_{j\nu}\to 0$) versions of $\hat W_+$ and $\hat W_z$, respectively.
These operators act only on the Hilbert space sector describing Majorana states. Comparing to Eq.~\eqref{Wopdef}, we have
\begin{equation}\label{wdef2}
W_{jk} = \sum_{\mu<\nu}^M \left( \lambda_{j\nu} \lambda_{k\mu}^\ast
- \lambda_{j\mu} \lambda_{k\nu}^\ast \right)\gamma_\mu\gamma_\nu.
\end{equation}
For the device in Fig.~\ref{fig1}, the $W_{jk}$ operators can be expressed in terms of the Pauli operators $(X,Y,Z)$ in Eq.~\eqref{PauliOp}, see below.
\subsubsection{Bath correlation functions}
The equilibrium density matrix of the thermal environment is given by
\begin{equation}\label{envtrace}
\rho_{\rm env}=Z^{-1}_{\rm env} e^{-H_{\rm env}/T}\quad {\rm with} \quad Z_{\rm env}={\rm tr}_{\rm env} \ e^{-H_{\rm env}/T},
\end{equation}
with `tr$_{\rm env}$' denoting a trace operation over the environmental bosons.
Using $\braket{\hat O}_{\rm env}\equiv {\rm tr}_{\rm env} (\hat O\rho_{\rm env})$,
we define the correlation function \cite{Weiss2007}
\begin{eqnarray} \nonumber
&& J_{\rm env}(t)=\braket{[\theta(t)-\theta(0)] \theta(0)}_{\rm env} =
\int_0^\infty \frac{d\omega}{\pi} \frac{{\cal J}(\omega)}{\omega^2}\times \\
&&\quad \times \left\{ [\cos(\omega t)-1] \coth\left(\frac{\omega}{2T}\right) - i \sin(\omega t) \right\},\label{BathCorr}
\end{eqnarray}
with the spectral density
\begin{equation}\label{spectraldensity}
{\cal J}(\omega)=\pi\sum_m (g_{1,m}-g_{2,m})^2 E_m^2 \delta(\omega-E_m).
\end{equation}
Switching to the continuum limit in bath frequency space, we focus on the practically most important Ohmic case with ${\cal J}(\omega)\propto \omega$ in the low-frequency limit.
In concrete calculations, we use the model spectral density \cite{Weiss2007}
\begin{equation}\label{Ohmic}
{\cal J}(\omega) = \alpha \omega e^{-\omega/\omega_c},
\end{equation}
where $\alpha$ is a dimensionless system-bath coupling and frequencies above $\omega_c$ are exponentially suppressed. For a related discussion in the context of Majorana qubits, see Ref.~\cite{Munk2019}. The parameter $\alpha$ is related to the environmental impedance $Z(\omega)$ \cite{Devoret1990},
\begin{equation}\label{alphadef}
\alpha= \frac{e^2}{2h} {\rm Re} Z(\omega=0).
\end{equation}
We consider the case $\alpha<1$ below.
For the subsequent discussion, we rewrite $H_{\rm cot}$ in normal-ordered form relative to the phase fluctuations,
\begin{equation}\label{normalorder1}
H_{\rm cot} = H_{\rm cot}^{(0)} + V,
\end{equation}
where $H_{\rm cot}^{(0)}$ is the expectation value of $H_{\rm cot}$ with respect to phase fluctuations and $V$ represents the coupling of the combined QD-MBS system to
phase fluctuations.
Since $\langle\theta^2\rangle_{\rm env}$ diverges in the Ohmic case, we have $\langle e^{i\theta}\rangle_{\rm env}=0$, resulting in
\begin{equation}\label{normalorder2}
H_{\rm cot}^{(0)} \equiv \braket{H_{\rm cot}}_{\rm env}= g_0 W_z\tau_z.
\end{equation}
The interaction term in Eq.~\eqref{normalorder1} is then given by
\begin{equation}\label{normalorder3}
V= 2g_0 \left( e^{ i\theta} W_+ \tau_+ + {\rm h.c.}\right) .
\end{equation}
By construction, $\braket{V}_{\rm env}=0$.
Correlation functions of exponentiated phase fluctuations are given by ($s=\pm 1$)
\begin{equation}\label{xicor}
\braket{e^{is\theta (t)} e^{-is\theta(0)}}_{\rm env} = e^{J_{\rm env}(t)}
\end{equation}
with $J_{\rm env}(t)$ in Eq.~\eqref{BathCorr}.
\subsubsection{Interaction picture and Rotating Wave Approximation}
From now on, we shall switch to the interaction picture with respect to $H_{\rm d}+H_{\rm env}$.
The Hamiltonian then takes the form,
see~Eqs.~\eqref{heff0} and \eqref{normalorder1},
\begin{eqnarray}\label{Hfull}
H_{{\rm eff},I}(t) &=& H_{0,I}(t)+ V_I(t),\\ \nonumber H_{0,I}(t) &=& H_{{\rm drive},I}(t) + H^{(0)}_{{\rm cot},I}(t).
\end{eqnarray}
For simplicity, we drop the `$I$' index (for interaction picture) in what follows and focus on resonant drive conditions,
\begin{equation}\label{omegadef}
\omega_0=\epsilon_2-\epsilon_1.
\end{equation}
In the regime $\omega_0\gg T$ considered below, see Eq.~\eqref{basiccond},
we can then apply the rotating wave approximation (RWA) \cite{Gardiner2004}.
As a consequence, $H_{\rm drive}(t)\to \tilde H_{\rm drive}$ with
\begin{equation}\label{Hdriv2}
\tilde H_{\rm drive}=A\left( d_1^{\dagger} d^{}_2+ d_2^{\dagger} d^{}_1\right) = A \tau_x,
\end{equation}
resulting in a time-independent drive Hamiltonian in the interaction picture.
If the drive frequency is slightly detuned,
$\omega_0=\epsilon_2-\epsilon_1+ \delta\omega_0$,
a residual time dependence remains, $H_{\rm drive}(t)= e^{-i\delta\omega_0 t} A d_1^{\dagger} d^{}_2 +$~h.c., after applying the RWA.
However, we find that the final Lindblad equation for the dynamics of the Majorana sector in Sec.~\ref{sec2c}
is not affected to leading order in $\delta\omega_0$. A small mismatch in the resonance condition \eqref{omegadef} will therefore not obstruct our findings. We then put $\delta\omega_0=0$ from now on.
\subsection{Master equation}\label{sec2b}
In this subsection, we consider the time evolution of the reduced density matrix, $\rho(t)$, describing the coupled system defined by the MBSs and the pair of QD fermions.
After tracing over the environmental bosons, we arrive at a Lindblad master equation for the dynamics of $\rho(t)$.
In Sec.~\ref{sec2c}, we will subsequently trace over the QD fermions to obtain a Lindblad equation for the Majorana sector only.
With $\omega_0=\epsilon_2-\epsilon_1$ and $g_0=t_0^2/E_C$, we consider the regime
\begin{equation}\label{basiccond}
g_0\ll T\ll \omega_0, \quad A\alt g_0.
\end{equation}
In particular, $T\ll \omega_0$ is needed to justify the RWA, while $g_0\ll T$ is required for the Born-Markov approximation.
In addition, the regime $g_0\ll T$ allows us to neglect emission and absorption processes taking place only in the Majorana sector since the bath is then unable to resolve such transitions. Of course, we will account for boson-assisted inter-QD transitions resulting from cotunneling processes.
Equation~\eqref{basiccond} also states that we study a weakly driven system with drive amplitude $A\alt g_0$. The opposite strongly driven case is briefly discussed in App.~\ref{appA}.
We note that inelastic corrections to the drive Hamiltonian due to electromagnetic phase fluctuations, see Eq.~\eqref{envtrace}, can be neglected by the secular approximation, cf. Sec.~II.B of Ref.~\cite{Shavit2019}.
We show below that the parameters appearing in Eq.~\eqref{basiccond} will only affect the speed of approach towards the targeted dark state (or dark space) but not the state fidelity.
Moreover, our protocols turn out to be exceptionally robust under even $10\%$ mismatch in \emph{all} tunneling amplitudes which in turn may affect the state fidelity, see, e.g.,
Fig.~\ref{fig5} below. We therefore expect that, in practice, it is not necessary to fulfill the `$\ll$' inequalities in Eq.~\eqref{basiccond} in an overly strict sense.
\subsubsection{Lindblad master equation for $\rho(t)$}
The master equation governing the dynamics of the density matrix $\rho(t)$ for the combined system (QDs and Majorana sector) is obtained by following the standard derivation of
Born-Markov master equations \cite{Weiss2007,Breuer2006,Gardiner2004}.
We assume a factorized initial (time $t=0$) density matrix of the total system,
$\rho_{\rm tot}(0)=\rho(0) \otimes \rho_{\rm env}$, with $\rho_{\rm env}$ in Eq.~\eqref{envtrace}.
Starting from the von-Neumann equation for $\rho_{\rm tot}(t)$ subject to $H_{\rm eff}(t)$ in Eq.~\eqref{Hfull},
we trace over the environmental modes and apply the Born-Markov approximation \cite{Weiss2007,Breuer2006,Gardiner2004}.
As a result, $\rho(t)$ obeys the master equation
\begin{eqnarray} \label{UnwantedTerms}
&&\partial_t\rho(t)=-i\left[ H_{0}(t),\rho(t)\right] -
{\rm tr}_{\rm env}
\int_0^{\infty} d\tau \\ \nonumber &\times& \left[ V(t),\left[ V(t-\tau)+H_0(t-\tau),\rho(t)\otimes\rho_{\rm env}\right]\right],
\end{eqnarray}
where we have used that, by construction, ${\rm tr}_{\rm env}\left[ V(t),\rho(0)\otimes\rho_{\rm env}\right]=0$.
Similarly, the mixed term involving $V(t)$ and $H_{0}(t-\tau)$ vanishes identically. We are
left with the coherent evolution term due to $H_{0}$ and the double commutator containing two $V$ terms.
Unfolding the double commutator,
we arrive at a master equation of Lindblad \cite{Lindblad1976,Lindblad1983} type,
\begin{equation}\label{GeneralLindblad}
\partial_t\rho(t) = -i\left[ H_{\rm L},\rho(t)\right]+ \sum_{\pm} \Gamma_\pm \mathcal{L}[J_\pm]\rho(t) .
\end{equation}
The subscript `L' in $H_{\rm L}$ is meant to clarify that this Hamiltonian appears in a Lindblad equation.
The dissipator ${\cal L}$ acts as superoperator on $\rho$ \cite{Breuer2006},
\begin{equation}\label{Dissipator}
\mathcal{L}[J]\rho=J\rho J^\dagger -\frac{1}{2}\lbrace J^{\dagger} J,\rho \rbrace.
\end{equation}
The two \emph{jump operators} in Eq.~\eqref{GeneralLindblad} are given by
\begin{equation}\label{jumpops}
J_\pm = 2W_\pm \tau_\pm = J_\mp^\dagger,
\end{equation}
with the corresponding dissipative transition rates,
\begin{equation}\label{dissrate}
\Gamma_{\pm}= 2g_0^2\, {\rm Re} \Lambda_\pm .
\end{equation}
Here, we define the quantities
\begin{equation}\label{lambdadef}
\Lambda_\pm= \int_0^\infty dt \, e^{\pm i\omega_0 t} e^{J_{\rm env}(t)},
\end{equation}
with the bath correlation function \eqref{BathCorr}.
Their imaginary parts give Lamb shift parameters,
\begin{equation}\label{shifts2}
h_\pm = g_0^2 \, {\rm Im}\Lambda_\pm ,
\end{equation}
which appear in the Hamiltonian governing
the coherent time evolution in Eq.~\eqref{GeneralLindblad},
\begin{equation}\label{Hq}
H_{\rm L} = A \tau_x + g_0 W_z\tau_z + \sum_{\pm} h_\pm J_{\pm}^{\dagger} J_\pm^{}.
\end{equation}
The first two terms in $H_{\rm L}$ originate from $H_0$ in Eq.~\eqref{Hdriv2}, while
the third term contains the Lamb shifts \eqref{shifts2}.
Next we observe that Eq.~\eqref{BathCorr} implies the general relation
\begin{equation}\label{detbal}
J_{\rm env}\left(-t-i/T\right)=J_{\rm env}(t)
\end{equation}
in the complex-time plane.
Using Eq.~\eqref{detbal} in Eq.~\eqref{lambdadef} then results in a detailed balance relation, $\Lambda_-=e^{-\omega_0/T} \Lambda_+$. As a consequence, for arbitrary parameters, we find
\begin{equation}
\Gamma_-=e^{-\omega_0/T}\Gamma_+, \quad h_-=e^{-\omega_0/T} h_+.
\end{equation}
In particular, for $T\ll \omega_0$, the dissipative rate $\Gamma_-$ associated with the jump operator $J_-$ will be
exponentially suppressed against the rate $\Gamma_+$. The dissipative part of the Lindblad equation
\eqref{GeneralLindblad} is therefore completely dominated by the jump operator $J_+$.
It is a distinguishing feature of our DD platform
that jump operators can be directly implemented by designing \emph{unidirectional} inelastic cotunneling paths connecting pairs of QDs via the box, with the overall energy scale $g_0$.
The QDs are also directly coupled by a driven tunnel link $w(t)$, see Eq.~\eqref{Hdriv}, with overall energy scale $A$. For $T\ll \omega_0$, as far as inter-dot transitions via the box are concerned, only photon emission processes are relevant. As a consequence, only transitions from the energetically high-lying QD 2 to QD 1 may take place, corresponding to the jump operator $J_+\propto \tau_+$, see Eqs.~\eqref{taudef} and \eqref{jumpops}.
Such transitions act on the Majorana state according to the operator $W_+$. As we show below, this operator can be engineered at will by adjusting the tunneling parameters $\lambda_{j\nu}$, which in turn is possible by changing suitable gate voltages. The driving field pumps the dot electron in the opposite direction, i.e., from QD $1\to 2$, and for a small pumping rate, $A\alt g_0$, we obtain a
steady state circulation $1\to 2\to 1$ by alternating pumping and cotunneling processes. On the other hand, for $A>g_0$, pumping processes will dominate and the cotunneling channel is effectively suppressed, see App.~\ref{appA}.
To facilitate analytical progress,
we consider the case $\omega_0\ll \omega_c$. (Otherwise Eq.~\eqref{lambdadef} can be solved numerically in a straightforward manner.) One then finds \cite{Weiss2007}
\begin{equation}
J_{\rm env}(t) \simeq -2\alpha \ln\left( \frac{\omega_c}{\pi T} \sinh(\pi T t)\right) - i\pi\alpha \,{\rm sgn}(t),
\end{equation}
and with the Gamma function $\Gamma(z)$, we arrive at
\begin{eqnarray}
\label{rate2}
\Gamma_+ &\simeq& \Gamma(1-2\alpha)\sin(2\pi\alpha) \left(\frac{\omega_0}{\omega_c}\right)^{2\alpha} \frac{2 g_0^2 }{\omega_0} ,\\
\nonumber
h_+ &\simeq& \frac12 \cot(2\pi \alpha) \Gamma_+.
\end{eqnarray}
For the device in Fig.~\ref{fig1}, using the Pauli operators \eqref{PauliOp},
the jump operators $J_\pm^{}=J_\mp^\dagger$ follow
from Eq.~\eqref{jumpops} in the general form
\begin{eqnarray}\nonumber
J_{+}&=& \tilde J_+ \tau_+ , \\
\tilde J_+ &=& 2ie^{i\beta_2}|\lambda_{23}|\left(e^{-i\beta_3}|\lambda_{11}|X-e^{-i\beta_1}|\lambda_{12}|Y\right) \nonumber
\\ &-& 2i\left[e^{-i\beta_1}|\lambda_{12}\lambda_{21}|-e^{i\beta_4}|\lambda_{11}\lambda_{22}|\right]Z, \label{jumpN}
\end{eqnarray}
where the phases $\beta_{1,2,3,4}$ are indicated in Fig.~\ref{fig1}.
They are connected to the phases $\chi_{j\nu}$ in the tunneling parameters, $\lambda_{j\nu}=|\lambda_{j\nu}|e^{-i\chi_{j\nu}}$, by
the relations
\begin{equation}\label{betadef}
\beta_1 = \chi_{12}, \quad \beta_2 = \chi_{23},\quad \beta_3= \chi_{11}, \quad \beta_4=\chi_{22},
\end{equation}
with the gauge choice $\chi_{21}=0$.
In particular, $\beta_1-\beta_3$ ($\beta_2$) is the loop phase accumulated along the shortest closed tunneling trajectory involving only QD 1 (QD 2), cf.~Eq.~\eqref{jzdef}.
These phases, as well as the absolute values $|\lambda_{j\nu}|$, can be experimentally varied, e.g., by changing the voltages on nearby gates.
We emphasize that $\tilde J_+$ is fully determined by selecting the state design parameters $\lambda_{j\nu}$.
The Hamiltonian $H_{\rm L}$ then follows as
\begin{eqnarray}
H_{\rm L}&=& A\tau_x + 2g_0 \tilde J_z \tau_z + \sum_\pm h_\pm J_\pm^\dagger J_\pm^{},\nonumber \\ \label{jzdef}
\tilde J_z &=& \frac12\bar\lambda^2+\sin\beta_2|\lambda_{21}\lambda_{23}| X+\\ \nonumber
&+& \sin\left(\beta_4-\beta_2\right)|\lambda_{22}\lambda_{23}|Y+\\ \nonumber
&+&\left[\sin\beta_4|\lambda_{21}\lambda_{22}|-\sin\left(\beta_1-\beta_3\right)|\lambda_{11}\lambda_{12}|\right]Z ,
\end{eqnarray}
where $\bar \lambda^2\equiv |\lambda_{11}|^2+|\lambda_{12}|^2+|\lambda_{21}|^2+|\lambda_{22}|^2+|\lambda_{23}|^2$.
It is worth mentioning that the operators $\tilde J_\pm$ and $\tilde J_z$ act only on the Majorana subsector.
To illustrate the above general expressions, let us consider a simple example. We take stabilization parameters subject to the conditions
\begin{eqnarray}\label{parameterchoice1}
|\lambda_{11}| &=& |\lambda_{12}|, \quad \lambda_{22}=0, \\
\nonumber \beta_1 &=&- \beta_2 =\pi/2, \quad \beta_3 = \beta_4 =0.
\end{eqnarray}
Using Eq.~\eqref{jumpN}, the dominant jump operator contributing to the Lindbladian is then given by
\begin{equation}
J_+=2|\lambda_{11}| \left(2|\lambda_{23}|\sigma_+ + |\lambda_{21}|Z\right)\tau_+,
\label{NumericJumpOperator}
\end{equation}
where $\sigma_\pm=(X\pm iY)/2$. For $|\lambda_{23}|\gg |\lambda_{21}|$, the Lindbladian will then automatically drive an arbitrary Majorana state $\rho_{\rm M}$ towards $|0\rangle\langle 0|$, with the
$Z$-eigenstate $|0\rangle$ to eigenvalue $+1$, i.e., $Z|0\rangle=|0\rangle$. Here, the reduced density matrix $\rho_{\rm M}(t)$ describes the Majorana sector only, see Sec.~\ref{sec2c}.
However, the operator $\tilde J_z$ appearing in the Hamiltonian $H_{\rm L}$ still contains a small $X$ component, see Eq.~\eqref{jzdef}, which could potentially disrupt the action of the dissipator.
Nonetheless, we find below that for small $|\lambda_{21}|$, the desired state $|0\rangle$ is approached with high fidelity, regardless of the initial system state $\rho(0)$.
An optimized parameter choice for stabilizing $|0\rangle$ will be discussed in Sec.~\ref{sec3}.
\subsubsection{Numerical results}\label{specialsec}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{f2}
\end{centering}
\caption{ Driven dissipative dynamics for the setup in Fig.~\ref{fig1}, illustrating
the time-dependent expectation values of the Pauli operators $\tau_{x,y,z}$ describing the QDs, see Eq.~\eqref{taudef}. We also show the
purity, $P_s(t)$, of the system state, see Eq.~\eqref{puritydef}. All results were obtained by numerical integration of the Lindblad equation \eqref{GeneralLindblad} for the density matrix $\rho$ describing the QDs and the Majorana sector, with $H_{\rm L}$ in Eq.~\eqref{jzdef}.
We used the parameters in Eq.~\eqref{parameterchoice1}, with $T/g_0=4$, $\omega_0/g_0=40$, $\omega_c/g_0=200$, $A/g_0=0.1$, $\alpha=1/4$, $|\lambda_{11}|=|\lambda_{12}|=|\lambda_{23}|= 1$, and $|\lambda_{21}|=0.1$. Fast transient oscillations in $\langle\tau_a(t)\rangle$ are not resolved on the shown time scale, corresponding to shaded regions.
The respective dynamics in the Majorana sector is depicted in Fig.~\ref{fig3}. }
\label{fig2}
\end{figure}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.7\columnwidth]{f3}
\end{centering}
\caption{Time evolution of the Bloch vector, $(\langle X\rangle,\langle Y\rangle,\langle Z\rangle)(t)$, describing the Majorana state $\rho_{\rm M}(t)$ for the same parameters as in Fig.~\ref{fig2}. The expectation value is computed by numerically integrating the Lindblad equation. Starting from the initial $X$-eigenstate $|+\rangle$, the DD protocol stabilizes the dark state $|0\rangle$ at long times, corresponding to the north pole of the Bloch sphere. The intermediate states (with alternating colors) were obtained at times $g_0t\in\lbrace 5\times 10^{3}, 10\times 10^3,\ldots,15\times10^4\rbrace$.
\label{fig3}}
\end{figure}
We next turn to a numerical integration of Eq.~\eqref{GeneralLindblad} using the approach of Refs.~\cite{Johansson2012,Johansson2013}.
Numerical results for the above parameters are shown in Figs.~\ref{fig2} and \ref{fig3}. While the goal of the DD protocol is to stabilize a selected state in the Majorana sector,
it is useful to also study the dynamics in the QD sector, see Fig.~\ref{fig2}.
We start from a pure initial state, $\rho(0)=|\Psi(0)\rangle\langle\Psi(0)|$, with $|\Psi(0)\rangle= |+\rangle \otimes |0\rangle_{\rm d}$,
where the $\tau_z=+1$ QD eigenstate, $|0\rangle_{\rm d}$, describes an electron located in the energetically lower QD 1, with
QD 2 left empty, see Eq.~\eqref{taudef}. The initial Majorana state has been chosen as the $X$-eigenstate $|+\rangle$ with
eigenvalue $+1$. However, we have checked that the same long-time limit of $\rho(t)$ is reached for other initial states.
We define the purity of the system state as
\begin{equation}\label{puritydef}
P_s(t)= {\rm tr} \rho^2(t).
\end{equation}
The upper left panel of Fig.~\ref{fig2} shows that the purity approaches a value close to the largest possible value ($P_s=1$) at long times. Moreover, as observed from Fig.~\ref{fig3}, the DD protocol steers the Majorana state towards the pure state $|0\rangle$, i.e., towards the north pole of the corresponding Bloch sphere. For the shown example, the QD state $\rho_{\rm d}$ has most probability weight in the energetically lower QD 1. Indeed, Fig.~\ref{fig2} shows that at long times, the electron shared by the two QDs will predominantly relax to QD 1,
corresponding to the state $|0\rangle_{\rm d}$.
Nonetheless, it is of crucial importance that the occupation probability $p$ for encountering the electron in the energetically higher QD 2 (corresponding to the state $|1\rangle_{\rm d}$) remains finite at long times.
We find $p\approx 0.001$ for the parameters in Fig.~\ref{fig2}.
We conclude that
the system state factorizes at long times, $\rho(t)\simeq \rho_{\rm M}\otimes \rho_{\rm d}$ with $\rho_{\rm M}=|0\rangle \langle 0|$.
The approach of the Majorana state towards $|0\rangle$ takes place on a time scale given by the inverse of the dissipative gap of the reduced Lindbladian describing the Majorana sector only, see Sec.~\ref{sec3} below. The relaxation time scales for the QD subsystem can be longer, cp.~Figs.~\ref{fig2} and \ref{fig3}.
Finally, we remark that for the special case $\lambda_{21}=0$, the electron shared by the two QDs will \emph{not} predominantly relax to the energetically lower QD $1$. One here has only two cotunneling paths between both QDs, namely the constituents forming the operator $4|\lambda_{11}\lambda_{23}|\sigma_+$ in Eq.~\eqref{NumericJumpOperator}. Both paths interfere destructively once the Majorana island is stabilized in the state $\ket{0}$. An arbitrarily weak drive can then overcome all dissipative effects in the long-time limit.
In contrast to what happens for $\lambda_{21}\ne 0$, the QDs will thus realize an
equal-weight mixture of $|0\rangle_{\rm d}$ and $|1\rangle_{\rm d}$.
Nonetheless, the reduced Lindblad equation \eqref{LindbladMBQ} below still applies, with $p\to 1/2$ and $p_\perp\to 0$ in Eq.~\eqref{ssform}. We note that those parameters are also appropriate in
the strongly driven case, cf.~App.~\ref{appA}.
\subsection{Lindblad equation for the Majorana sector} \label{sec2c}
The above observations allow us to derive a reduced Lindblad equation,
which directly describes the dynamics of $\rho_{\rm M}(t)$ in the Majorana sector alone.
To that end, we now trace also over the QD subspace.
At long times, our numerical simulations generically show that $\rho(t)$
factorizes into a Majorana part, $\rho_{\rm M}(t)$, and a QD contribution, $\rho_{\rm d}(t)$,
\begin{equation}\label{factorize}
\rho(t \to \infty) \simeq \rho_{\rm M}(t) \otimes \rho_{\rm d}(t).
\end{equation}
The discussion in Sec.~\ref{sec2b} highlights that the
Majorana sector and the dot sector have to couple during intermediate times in order
to drive the Majorana system towards the desired target dark state (or dark space).
Once this state is stabilized, however, the dot electron can relax to the energetically favored state (up to the effects of the drive).
This argument also shows that, in accordance with our numerical observations, the specific choice for the tunneling parameters $\{\lambda_{j\nu}\}$ is only important
for determining the targeted dark state while the
disentanglement of Majorana and dot subspaces in Eq.~\eqref{factorize} represents a generic long-time feature.
For tracing over the QD part, we can effectively use a time-independent \emph{Ansatz},
\begin{equation}\label{ssform}
\rho_{\rm d}=\left( \begin{array}{cc} 1-p & p_\perp \\ p^\ast_\perp & p \end{array}\right),
\end{equation}
written in the basis $\{ |0\rangle_{\rm d},|1\rangle_{\rm d}\}$ selected by the coupling to the QDs. Here, $p\ne 0$ refers to the occupation probability of the energetically higher QD 2. This probability can be determined by numerically solving Eq.~\eqref{GeneralLindblad}, cf.~Sec.~\ref{sec2b}, or it may be treated as phenomenological parameter.
A simple estimate predicts $p\approx {\rm max}(A,g_0)/\omega_0$.
Noting that a small but finite expectation value $\langle\tau_x\rangle\ne 0$ is observed in Fig.~\ref{fig2} at long times, we
have also included an off-diagonal term $(p_\perp)$ in Eq.~\eqref{ssform}.
Inserting Eq.~\eqref{factorize} into Eq.~\eqref{GeneralLindblad} and tracing over the QD subsystem, we arrive at a
Lindblad equation for the $2\times 2$ density matrix $\rho_{\rm M}(t)$ only,
\begin{equation}\label{LindbladMBQ}
\partial_t\rho_{\rm M}(t) = -i[\tilde H_{\rm L},\rho_{\rm M} ]+ \sum_{s=\pm} \tilde\Gamma_s \mathcal{L}[\tilde J_s]\rho_{\rm M}(t),
\end{equation}
where the jump operators $\tilde J_\pm$ have been defined in Eq.~\eqref{jumpN}.
The dissipative transition rates $\tilde \Gamma_\pm$ in Eq.~\eqref{LindbladMBQ} are given by
\begin{equation}\label{mod1}
\tilde \Gamma_+ = p \Gamma_+,\quad \tilde \Gamma_- = (1-p)\Gamma_-,
\end{equation}
cf.~Eqs.~\eqref{dissrate} and \eqref{rate2}.
The coherent time evolution in Eq.~\eqref{LindbladMBQ} is governed by the Hamiltonian
\begin{equation}\label{mod22}
\tilde H_{\rm L} = 2(1-2p)g_0 \tilde J_z + \sum_\pm \tilde h_\pm \tilde J_\pm^\dagger \tilde J_\pm^{},
\end{equation}
where $\tilde J_z$ has been specified in Eq.~\eqref{jzdef} and the Lamb shifts $\tilde h_\pm$ are given by
\begin{equation}\label{mod2}
\tilde h_+ = p h_+ ,\quad \tilde h_- = (1-p) h_-.
\end{equation}
The drive amplitude $A$ then appears only implicitly through the dependence $p=p(A)$.
We note that within the RWA, no contributions $\propto p_\perp$ appear in Eq.~\eqref{LindbladMBQ}. Indeed,
the RWA allows one to neglect terms $\propto \tau_+\rho\tau_+$
which stem from $p_\perp\ne 0$.
Importantly, apart from the initial transient behavior,
all of our numerical results for the Majorana dynamics obtained from the full Lindblad equation for the combined QD-MBS system, Eq.~\eqref{GeneralLindblad}, are
quantitatively reproduced by using the simpler Lindblad equation \eqref{LindbladMBQ}. This statement is valid
for arbitrary model parameters subject to Eqs.~\eqref{condit} and \eqref{basiccond}.
We emphasize that the integration over the QD degrees of freedom as carried out above relies on the facts that (i) the convergence towards the
target state is dictated by the Majorana sector, and that (ii) the QD and MBS degrees of freedom always decouple in the long-time limit, see Eq.~\eqref{factorize}.
The latter feature has been established by extensive numerical simulations of Eq.~\eqref{GeneralLindblad}.
The reduced Lindblad equation \eqref{LindbladMBQ} is applicable
as long as transient behaviors are not of interest. In particular,
when studying, e.g., the dynamics of $\rho_{\rm M}(t)$ in the presence of time-dependent QD level
energies $\epsilon_j(t)$, Eq.~\eqref{LindbladMBQ} should only be used for very slow (adiabatic) time dependences. For rapidly varying QD level energies,
one has go back to the full Lindblad equation for the combined QD-MBS system in Eq.~\eqref{GeneralLindblad}.
\section{Dark state stabilization }\label{sec3}
Using the Lindblad master equation \eqref{LindbladMBQ} and the Choi isomorphism \cite{Albert2014} summarized
in App.~\ref{appB}, we now turn to a detailed analysis of our stabilization protocols for the single-box device in Fig.~\ref{fig1}.
The parameter values for stabilizing a specific dark state can be determined by solving the zero-eigenvalue condition of the Lindbladian, cf.~App.~\ref{appB}.
We recall that the key state design parameters of our DD protocol
are given by the complex-valued tunneling amplitude parameters $\lambda_{j\nu}$, which also define the phases $\beta_j$ in Fig.~\ref{fig1}.
In Sec.~\ref{sec3a}, we show how to stabilize Pauli operator eigenstates. In Sec.~\ref{sec3b}, we discuss magic state stabilization protocols, followed by a study of
temperature effects in Sec.~\ref{sec3c}.
We show in Sec.~\ref{sec3d} that in certain cases, a dark state can be stabilized even in the absence of any driving field.
Finally, we conclude in Sec.~\ref{sec3e} with several remarks.
\subsection{Pauli operator eigenstates}\label{sec3a}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{f4}
\end{centering}
\caption{Dark-state stabilization protocols for Pauli operator eigenstates. Left side panels (blue curves): Stabilization of $|0\rangle$. Right side panels (red curves): Stabilization of $|+\rangle$, where $X|+\rangle=|+\rangle$.
In both cases, the Majorana island has initially been prepared in the $Y$-eigenstate with eigenvalue $+1$.
We use the parameters in Eq.~\eqref{sigmazcond} with $p=1/2$, all other parameters are as in Fig.~\ref{fig2}.
With $E_C=1$~meV and $g_0/E_C=2.5\times 10^{-3}$, the time units follow as shown. As explained in the main text, for the chosen parameter set, Rabi oscillations are absent.
}
\label{fig4}
\end{figure}
We start by discussing DD protocols targeting Pauli operator eigenstates. Typical numerical results obtained by solving Eq.~\eqref{LindbladMBQ}
are illustrated in Fig.~\ref{fig4}. Following the method in App.~\ref{appB}, the $Z=\pm 1$ eigenstates can be realized by choosing
\begin{equation}\label{sigmazcond}
|\lambda_{11}|=|\lambda_{12}|,\quad \lambda_{21}=\lambda_{22}=0, \quad \beta_1-\beta_3=\pm\pi/2,
\end{equation}
with arbitrary $\lambda_{23}$ and $\beta_{2,4}$, see Eq.~\eqref{betadef}. (We note that for $\lambda_{23}=0$, the phases $\beta_{2,4}$ are not defined.)
At this point, we use the concept of a \emph{dissipative map} $\hat E$ \cite{Breuer2006}, which is defined in terms of a jump operator mapping the system onto a specific state when acting inside
the Lindblad dissipator. For example, the dissipative maps targeting the $Z=\pm 1$ eigenstates are
\begin{equation}
\hat E_{\pm}=\sigma_\pm=(X\pm iY)/2.
\end{equation}
For the stabilization parameters in Eq.~\eqref{sigmazcond}, the jump operator $\tilde J_+\propto \hat E_\pm$, with the $\pm$ sign determined by Eq.~\eqref{sigmazcond},
completely dominates the Lindbladian part of Eq.~\eqref{LindbladMBQ} at low temperatures, $T\ll\omega_0$.
The dissipative dynamics then
maps every input state to $|0\rangle$ (for the $+$ sign) or $|1\rangle$ (for the $-$ sign). At the same time, the Hamiltonian evolution in Eq.~\eqref{LindbladMBQ} comes from $\tilde H_{\rm L}\propto Z$, see Eq.~\eqref{mod22}. Evidently, this Hamiltonian commutes with the targeted state $\rho_{\rm M}(\infty)$, and therefore does not affect the dynamics towards the steady state generated by the dissipative map $\hat E_\pm$.
The Majorana state $\rho_{\rm M}(t)$ is thus automatically
steered towards the corresponding $Z$-eigenstate by the Lindbladian, with no obstruction from the Hamiltonian dynamics.
For the above protocol, the \emph{dissipative gap} is given by, cf.~App.~\ref{appB},
\begin{equation}\label{deltaz}
\Delta_z = |4\lambda_{11}\lambda_{23}|^2\sum_{s=\pm}\tilde\Gamma_{s}.
\end{equation}
In general terms, the dissipative gap is defined as the real part of the smallest non-vanishing eigenvalue of the Lindbladian (the dark state itself has eigenvalue zero) \cite{Breuer2006}.
The time scale on which the dark state will be approached is therefore given by $\Delta_z^{-1}$.
Moreover, the approach of the Bloch vector towards the dark state $|0\rangle$ is in general accompanied by damped oscillations in the $(X,Y)$ components, where $\Delta_z$ is the damping rate and the Rabi frequency follows from Eq.~\eqref{mod22} as
\begin{equation}\label{rabiz}
\Omega_z \simeq \left|2g_0(1-2p)|\lambda_{11}|^2-8|\lambda_{11}\lambda_{23}|^2\tilde h_+\right|.
\end{equation}
For the special case $\lambda_{21}=0$ with $p=1/2$, cf. Sec.~\ref{sec2b}, and noting that $\tilde h_+=0$ for $\alpha=1/4$, cf.~Eq.~\eqref{rate2},
we obtain $\Omega_z=0$ in Eq.~\eqref{rabiz}. The left panels in Fig.~\ref{fig4} therefore exhibit only damping in the $(X,Y)$ components, without Rabi oscillations.
Next, $X=\pm 1$ eigenstates are realized by choosing
\begin{equation}
|\lambda_{21}|=|\lambda_{23}|,\quad \lambda_{11}=\lambda_{22}=0,\quad \beta_2=\mp\pi/2,
\end{equation}
with the dissipative gap $\Delta_x=|4\lambda_{12}\lambda_{21}|^2\sum_s\tilde\Gamma_s.$
As shown in the right panels of Fig.~\ref{fig4}, $X$-eigenstates, e.g., the state $|+\rangle$
for eigenvalue $+1$,
can be stabilized using the setup in Fig.~\ref{fig1}. As for the $Z$-stabilization shown in the left panels, there are no Rabi oscillations for this parameter set.
Finally, for stabilizing the $Y$-eigenstates with eigenvalue $\pm 1$, one requires
\begin{equation}
|\lambda_{22}|=|\lambda_{23}|, \quad \lambda_{12}= \lambda_{21}=0,\quad \beta_2-\beta_3-\beta_4=\pm\pi/2,
\end{equation}
with the dissipative gap $\Delta_y = |4\lambda_{11}\lambda_{22}|^2\sum_s\tilde\Gamma_s$.
In all these examples, the target axis (say, $\hat e_z$ for $Z$-eigenstates) is controlled by selecting appropriate tunneling amplitude parameters $\lambda_{j\nu}$. Two links are switched off, and two are matched in amplitude such that the desired jump operator $\tilde J_+$ is implemented.
For $T\ll \omega_0$, dissipative transitions are fully governed by this jump operator which is due to inelastic cotunneling transitions from QD $2\to 1$.
Under these conditions, we find that $\tilde H_{\rm L}$ commutes with the Pauli operator $\hat\sigma$ corresponding to the target axis (e.g., $\hat\sigma=Z$ for $Z$-states).
Finally, by adjusting the phases $\beta_j$, one can select the stabilized state, say, $|0\rangle$ or $|1\rangle$.
It is a remarkable feature of our Majorana-based DD setup that the Hamiltonian $\tilde H_{\rm L}$ can be engineered to only generate $\hat\sigma$. As a consequence, the Lindbladian dissipator already drives the system to the desired dark state.
\subsection{Magic states} \label{sec3b}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{f5}
\end{centering}
\caption{ Fidelity for a stabilization protocol targeting the magic state $|m\rangle$. Here
the Majorana state follows by numerical integration of Eq.~\eqref{LindbladMBQ} using the parameters
in Eq.~\eqref{magiccond} with $|\lambda_{23}|=1$.
Other parameters are $E_C = 1$~meV, $g_0/E_C=2.5\times 10^{-3}, T/g_0=4$,
$\omega_0/g_0=40$, $\omega_c/g_0 =200$, $\alpha=1/4$, and $p = 0.01$.
Main panel: Time dependence of the fidelity for ideal parameters [Eq.~\eqref{magiccond}] (red curve), with a mismatch of order $10\%$ in all state design parameters [$|\lambda_{11}|=-0.1+1/\sqrt2,|\lambda_{21}|=+0.1+1/\sqrt{2},|\lambda_{12}|=|\lambda_{22}|=0.9,\beta_3=-\beta_2=11 \pi/20$] (blue), and a mismatch of order $20\%$ in the same parameters (orange).
Inset: Steady-state fidelity vs deviation $\Delta\beta_2$ with otherwise ideal parameters, where $\beta_2=-\frac{\pi}{2}(1+\Delta\beta_2)$.
\label{fig5}}
\end{figure}
In order to highlight the power of our DD stabilization protocols, we next consider the magic state \cite{Nielsen}
\begin{equation}\label{magicstate}
|m\rangle = e^{-i\frac{\pi}{8}Y} |0\rangle.
\end{equation}
The practical importance of this state comes from the fact that a large number of ancilla qubits approximately prepared in the state $|m\rangle$
are needed for the magic state distillation protocol. The latter is an essential ingredient for implementing the $T$-gate required for universal surface code quantum computation
\cite{Fowler2012,Vijay2015,Landau2016,Plugge2016,Nielsen}.
Targeting $|m\rangle$, the stabilization conditions now involve all tunnel links in Fig.~\ref{fig1} and are given by
\begin{eqnarray} \label{magiccond}
|\lambda_{12}|&=&|\lambda_{23}|, \quad |\lambda_{21}|=|\lambda_{11}|=|\lambda_{23}|/\sqrt2,\\ \nonumber \lambda_{22}&=&0,\quad \beta_3=\beta_1+\beta_2, \quad \beta_2 = -\pi/2.
\end{eqnarray}
We here define the \emph{fidelity} of the state $\rho_{\rm M}(t)$ with respect to a specific pure state, $\rho_{\rm M}^{(0)}=|\psi\rangle\langle \psi|$, as
\begin{equation} \label{fidelity}
F(t)={\rm tr}\left[ |\psi\rangle\langle \psi|\rho_{\rm M}(t)\right].
\end{equation}
We show numerical results for the magic state fidelity with $|\psi\rangle=|m\rangle$ in Fig.~\ref{fig5},
using the parameters in Eq.~\eqref{magiccond}.
We find $F=1$ at long times for the ideal parameter choice in Eq.~\eqref{magiccond}. Figure \ref{fig5} also illustrates the long-time fidelity when allowing for
small deviations from Eq.~\eqref{magiccond} which are inevitable in practical implementations. Remarkably, even for sizeable deviations from the ideal parameter set, the fidelity remains close to unity. By determining the spectrum of the Lindbladian, we obtain the dissipative gap as
\begin{equation}\label{dissgapm}
\Delta_m = |4\lambda_{11}\lambda_{23}|^2\sum_s\tilde \Gamma_s.
\end{equation}
Using the parameters in Fig.~\ref{fig5}, we find $\Delta_m^{-1}\simeq 80$~ns.
Even though our magic state stabilization protocol requires more parameter fine tuning than the stabilization of $|0\rangle$,
the dark state $|m\rangle$ is reached on essentially the same time scale.
\subsection{Effect of temperature}\label{sec3c}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.95\columnwidth]{f6}
\end{centering}
\caption{ Steady-state fidelity, $F(\infty)$, and purity, $P(\infty)$, vs temperature (in Kelvin) for
the state $|0\rangle$ and for the magic state $|m\rangle$. We use ideal state design parameters, see Eqs.~\eqref{sigmazcond} and \eqref{magiccond}, with all other parameters as in
Figs.~\ref{fig4} and \ref{fig5}, respectively. The numerical results for both states cannot be distinguished for these parameter choices on
the shown scales. The frequency $\omega_0$ corresponds to a temperature of $\approx 2.5$~K, while $E_C=1$~meV \cite{Lutchyn2018} corresponds to $\approx 11$~K. }
\label{fig6}
\end{figure}
We next address the effect of raising temperature within the conditions set by Eq.~\eqref{basiccond}, in particular $T\ll \omega_0$.
Figure~\ref{fig6} shows numerical results for the $T$-dependent steady state
fidelity $F(\infty)$ with respect to the states $|0\rangle$ and $|m\rangle$,
choosing ideal parameters as in Eqs.~\eqref{sigmazcond} and \eqref{magiccond}, respectively.
At very low temperatures,
the fidelity stays very close to the ideal value ($F=1$) since here only the rate
$\tilde \Gamma_+$, see Eqs.~\eqref{dissrate} and \eqref{mod1}, is significant. In this limit, corrections
to $F=1$ are exponentially small and appear to be governed by the dissipative gap, $1-F\propto \exp(-\Delta_{z/m}/T)$. The same scaling behavior also applies to the purity.
As temperature increases, the thermal excitation
rate $\tilde\Gamma_-=e^{-\omega_0/T}\tilde \Gamma_+$ cannot be neglected anymore. Focusing on the stabilization of the state $|0\rangle$, we have $\tilde J_-\propto \sigma_-$. The Lindblad dissipator $\tilde \Gamma_- {\cal L}[\tilde J_-]$ will then target the `wrong' $Z$-eigenstate $|1\rangle$. The competition between ${\cal L}[\tilde J_+]$ and ${\cal L}[\tilde J_-]$ implies that the fidelity will deteriorate as temperature increases.
This expectation is confirmed by our numerical results.
For the parameters in Fig.~\ref{fig6}, the fidelity noticeably drops once
$T$ exceeds the crossover temperature $T_c\approx 250$~mK.
Figure \ref{fig6} also shows the temperature dependent purity of the steady state,
$P(\infty) ={\rm tr}\rho_{\rm M}^2(t\to \infty)$. For $T\ll T_c$,
we find $P(\infty) \simeq 1$. As $T$ increases, however, the maximally mixed state $\rho_{\rm M}(\infty)=\frac12 \mathbb{1}$ with $F(\infty)= P(\infty)=1/2$ is approached, and consequently the purity also becomes smaller.
Finally, let us note that at elevated temperatures, the RWA will also become less accurate. One may thus need to account for dephasing effects induced by corrections beyond RWA \cite{Shavit2019}. However, for the results shown in Fig.~\ref{fig6} with $T/\omega_0<0.2$, such effects are expected to be very small.
\subsection{Stabilization without driving field} \label{sec3d}
In certain cases, it is possible to stabilize dark states even without drive Hamiltonian, $H_{\rm drive}=0$.
In this subsection, we demonstrate the feasibility of this idea for special choices of the state design parameters.
We are not aware of other DD systems allowing for dark states in the absence of driving.
In our setup, we will see that the dissipative dynamics can also generate terms that mimic the effects of a weak driving field.
To be specific, we apply the Lindblad equation \eqref{LindbladMBQ} to
setups where $\lambda_{j\nu}\ne 0$ only for $(j \nu) \in \{ 11,12,23 \}$.
In particular, since $\lambda_{21}=0$, this case corresponds to the special parameter regime discussed in Sec.~\ref{specialsec}.
For simplicity, below we drop the exponentially small contribution to the dissipator due to $\tilde J_-$.
From Eq.~\eqref{jumpN}, the only relevant jump operator is then given by
\begin{equation} \label{jplus1}
\tilde J_{+} = 2i \lambda_{23}^\ast \left( \lambda_{11} X - \lambda_{12} Y \right).
\end{equation}
In addition, we keep Lamb shift effects implicit. In particular,
they can be taken into account by renormalizing $B_z$ in Eq.~\eqref{HM1} below.
The operator $\tilde J_z$ entering $\tilde H_{\rm L}$, see Eqs.~\eqref{jzdef} and \eqref{mod22},
has the form
\begin{equation}\label{jz1}
\tilde J_z =-\sin\beta_1\left| \lambda_{11} \lambda_{12} \right| \, Z .
\end{equation}
We now study the undriven ($A=0$) scenario for two parameter sets allowing for analytical progress.
The stabilization of pure dark states may then be possible because the Hamiltonian $\tilde H_L$ can
effectively take over the role of the drive. As a result, the arguments behind the factorized form of the long-time
density matrix in Eq.~\eqref{factorize} carry over to the present case.
The frequency $\omega_0$ now simply represents the (positive) energy difference $\epsilon_2-\epsilon_1$, see Eq.~\eqref{omegadef}, instead of a drive frequency.
Moreover, we assume $p_\perp = 0$ while the probability $p$ in Eq.~\eqref{ssform} is estimated by $p\approx g_0/\omega_0$.
We note in passing a finite static contribution to the inter-QD tunnel coupling, $t_{12}\ne 0$ in Eq.~\eqref{Hdriv}, can be taken
into account here. This coupling will modify $p$ according to $p \approx{\rm max}(t_{12}, g_0)/\omega_0$.
We also recall that for $A\ne 0$, one instead finds $p=1/2$ since we have $\lambda_{21}=0$, cf.~Sec.~\ref{specialsec}.
\subsubsection*{Case 1: $\lambda_{11}=\pm i \lambda_{12}$}
The first case is defined by $\lambda_{11}=is \lambda_{12}$, with $s=\pm 1$.
We observe that the dot fermion operator $d_1$ corresponding to QD 1 is then tunnel-coupled to a nonlocal fermion formed from the Majorana operators,
$c=(\gamma_1-is \gamma_2)/2$. With $\sigma_\pm=(X\pm iY)/2$, Eqs.~\eqref{jplus1} and \eqref{jz1} simplify to
\begin{equation}\label{case1}
\tilde J_+=4i\lambda_{23}^\ast \lambda_{11}\sigma_{-s}, \quad\tilde J_z=-s |\lambda_{11}|^2 \, Z.
\end{equation}
The Lindblad equation \eqref{LindbladMBQ} is then given by
\begin{equation}\label{nodrive:LBMcase1}
\partial_t \rho_{\rm M}(t) = - i [ \tilde H_{\rm L}, \rho_{\rm M}(t) ] +
\Gamma_1 {\cal L} \left[ \sigma_{-s} \right] \rho_{\rm M}(t),
\end{equation}
where the Hamiltonian follows from Eq.~\eqref{mod22} as
\begin{equation}\label{HM1}
\tilde H_{\rm L} = -2s (1-2p) g_0 |\lambda_{11}|^2 Z=s B_z Z.
\end{equation}
We note that the Lamb shift $\tilde h_+$ can be taken into account by redefining $B_z$.
Furthermore, the rate $\Gamma_1$ in Eq.~\eqref{nodrive:LBMcase1} is proportional to $\tilde \Gamma_+$ in Eq.~\eqref{rate2}.
The only zero eigenstate of the Lindbladian is the $Z$-eigenstate $|0\rangle$ (for $s=-1$) [or $|1\rangle$ (for $s=+1$)], e.g.,
${\cal L} \left[ \sigma_+ \right] |0 \rangle \langle 0 | = 0$.
The same $Z$-eigenstate is also the lowest energy eigenstate of $\tilde H_{\rm L}$ in Eq.~\eqref{HM1}.
Using the $Z$-eigenstate basis $\{ |0\rangle , |1\rangle \}$ for $s=-1$ [and $\{|1\rangle,|0\rangle \}$ for $s=+1$], we
can parametrize the time-dependent density matrix $\rho_{\rm M}(t)$ solving Eq.~\eqref{nodrive:LBMcase1} with real-valued $x(t)$ subject to $0\le x\le 1$ and
complex-valued $y(t)$ as
\begin{equation}
\rho_{\rm M}(t) = \left( \begin{array}{cc} 1 - x(t) & y(t) \\ y^\ast(t) &x(t)\end{array}\right).
\end{equation}
The quantities $x(t)$ and $y(t)$ represent the diagonal and off-diagonal density matrix deviations, respectively, from the steady-state density matrix corresponding to the stabilized
$Z$-eigenstate. Using Eq.~\eqref{nodrive:LBMcase1}, these deviations obey the equations of motion
\begin{equation}\label{relax1}
\partial_t x = - \Gamma_1 x,\quad \partial_t y = - 2 i B_z y - \frac{\Gamma_1}{2} y,
\end{equation}
which explicitly shows the relaxation and decoherence dynamics of $\rho_{\rm M}(t)$ towards the stabilized pure
state. The above example demonstrates that the dissipative stabilization of a dark state can be achieved even in the absence of a driving field in our Majorana box setup.
\subsubsection*{Case 2: $\beta_1=0$}
Putting the phase $\beta_1$ to zero, $d_1$ is effectively coupled to a single Majorana operator, $\gamma_{\rm eff}=\gamma_1 \cos \delta + \gamma_2\sin \delta$,
with $\delta= \tan^{-1} \left| \lambda_{12} / \lambda_{11} \right|$.
One then obtains $\tilde J_z=0$. The jump operator
$\tilde J_+$ is now given by
\begin{equation}\label{case2}
\tilde J_+= B_{\perp}\sigma_+ e^{i\delta}+{\rm h.c.},\quad
B_\perp= 2i \lambda_{23}^\ast \lambda_{11}/|\cos\delta|.
\end{equation}
Noting that the Lamb shifts in $\tilde H_{\rm L}$ only give an irrelevant constant, we arrive at the Lindblad equation
\begin{equation}\label{nodrive:LBMcase2}
\partial_t \rho_{\rm M}(t) = \frac{\Gamma_2}{4} {\cal L} \left[ \sigma_{\bf n} \right] \rho_{\rm M}(t),
\end{equation}
where we define
\begin{equation}
\sigma_{\bf n} = {\bf n \,\cdot} {\bm \sigma} = \sigma_+ e^{i \delta} + \sigma_-e^{-i\delta},
\end{equation}
with the unit vector ${\bf n} = (\cos \delta, - \sin \delta, 0)$. Again, the rate $\Gamma_2$ is proportional
to the respective rate $\tilde \Gamma_+$ in Eq.~\eqref{rate2}.
For the case in Eq.~\eqref{nodrive:LBMcase2}, the Lindbladian has two zero eigenstates,
${\cal L} \left[ \sigma_{\bf n} \right] | s \rangle \langle s | =
{\cal L} \left[ \sigma_{\bf n} \right] | a \rangle \langle a | = 0$, corresponding to the
eigenstates of ${\bf \sigma}_{\bf n}$, i.e.,
$\sigma_{\bf n}| s \rangle = | s \rangle$ and $\sigma_{\bf n}| a \rangle =- | a \rangle$.
Using the $X$-eigenstates $|\pm\rangle$, one finds
\begin{equation}
| s / a \rangle =\frac{1}{ \sqrt{2}} \left( e^{i \delta} |+\rangle \pm e^{-i \delta} |-\rangle \right).
\end{equation}
In the $\{ |s\rangle,|a\rangle\}$ basis, $\rho_{\rm M}(t)$ can be parametrized as
\begin{equation}
\rho_{\rm M}(t) = \left( \begin{array}{cc} \frac12 + x(t) & y(t) \\ y^\ast(t) & \frac12-x(t)
\end{array} \right),
\end{equation}
where the real-valued parameter $x(t)$ has to satisfy $|x|\le 1/2$. Equation~\eqref{nodrive:LBMcase2} then yields
\begin{equation}
\partial_t x = 0,\quad \partial_t y = - \frac{\Gamma_2}{2} y.
\end{equation}
Clearly, there is no relaxation in the basis selected by the environment via the QDs, i.e., $x(t)$
remains constant. Only the off-diagonal elements of the density matrix are subject to decay with the rate $\Gamma_2/2.$ One can therefore prepare an arbitrary mixed state as steady state.
\subsection{Discussion}\label{sec3e}
We conclude this section with several additional points.
\subsubsection{Mixed states}
As pointed out in Sec.~\ref{sec3d}, one can also use our protocols for stabilizing mixed states, see also Ref.~\cite{Kumar2020}.
To give another example, now for $A\ne 0$, we consider changing the above phase conditions such that a mixture of Pauli eigenstates can be prepared as dark state. For instance, by choosing the state design parameters as in Eq.~\eqref{sigmazcond} but keeping $\bar\beta=\beta_1-\beta_3$ arbitrary, one obtains the dark state
\begin{equation}
\rho_{\rm M}(\infty)=\frac{1+\sin\bar\beta}{2}|0\rangle\langle 0|+\frac{1-\sin\bar\beta}{2}|1\rangle\langle 1|.
\end{equation}
The relative weight of the two components can then be altered by adjusting the phase difference $\bar\beta$.
\subsubsection{Majorana overlaps}
So far we have assumed that the overlap between different MBSs is negligibly small.
What are the effects of a finite (but small) hybridization between different MBS pairs on the above stabilization protocols?
Such terms could arise, e.g., due to the finite nanowire length \cite{Alicea2012}.
They are described by a Hamiltonian term $ H'=\sum_{\nu<\nu'}i\epsilon_{\nu\nu'}\gamma_\nu\gamma_{\nu'}$, with
hybridization energies $\epsilon_{\nu\nu'}$.
By construction, such a term survives the RWA and the Schrieffer-Wolff projection in Sec.~\ref{sec2} and thus contributes to the Hamiltonian $\tilde H_{\rm L}$ in the Lindblad equation \eqref{LindbladMBQ} without affecting the Lindbladian dissipator.
In the Pauli operator language, such terms act like a weak magnetic Zeeman field.
If the corresponding field is parallel to the target axis of the dark state,
it does not cause any dephasing. For instance, for the stabilization of the $Z$-eigenstate $|0\rangle$,
the hybridization parameters $\epsilon_{12}$ and $\epsilon_{34}$ can be tolerated as they only couple to the Pauli operator $Z$ in Eq.~\eqref{PauliOp}.
Clearly, such couplings have no detrimental effects on our stabilization protocols.
For the stabilization of arbitrary target states, however, the role of MBS overlaps is more subtle, in particular when power-law scaling of the overlap with increasing distance becomes important \cite{Aseev2019}. A detailed discussion of such effects will be given elsewhere.
\subsubsection{Readout dynamics}
For reading out a stabilized dark state, it is possible to use the same techniques suggested previously for the native Majorana qubit \cite{Plugge2017,Karzig2017,Munk2019}.
In particular, one can perform capacitance spectroscopy using
additional single-level QDs that are tunnel-coupled to MBS pairs. These QDs are used for measurements only, where
the spectroscopic signal contains an interference term which depends on the respective Pauli matrix in Eq.~\eqref{PauliOp}.
This projective readout yields the Pauli eigenvalue $\pm 1$ with a state-dependent probability \cite{Karzig2017}.
Of course, this method can also be used to prepare the Majorana island in a Pauli eigenstate before the DD protocol is started.
In order for the readout not to interfere with the DD stabilization protocol, one has to make sure that the characteristic projective measurement time
scale (see Refs.~\cite{Plugge2017,Karzig2017} for detailed expressions) is much longer than the typical
inelastic cotunneling time $\tilde\Gamma_+^{-1}$.
Similarly, single-electron pumping protocols via a pair of QDs attached to different MBSs
allow one to apply a Pauli operator to the tetron state in a topologically protected manner \cite{Plugge2017}.
\subsubsection{Beyond the horizon of a dark state}
\label{sec3e4}
So far we have discussed DD stabilization protocols targeting a desired dark state.
The dark space dimension for those protocols is $D=1$, see App.~\ref{appC}.
Since there is a unique dark state for a given choice of the state design parameters,
one could utilize a DD single-box device as a self-correcting quantum memory.
By means of adiabatic changes of the state design parameters, one can in principle steer the Majorana state on its Bloch sphere. However, for general
state manipulation protocols, it is advantageous to have access to
a dark space manifold with $D>1$, which may be engineered in systems with more than four MBSs.
We address this case in the next section.
\section{Dark space engineering }
\label{sec4}
We continue with DD protocols targeting quantum states within a dark space manifold. A degenerate manifold of dark states may
be engineered by employing a device with at least two Majorana boxes as depicted in Fig.~\ref{fig7}.
After introducing our model and the
corresponding Lindblad equation in Sec.~\ref{sec4a}, we show in Sec.~\ref{sec4b} how a dark space can be created and classified.
In Sec.~\ref{sec4c}, we then describe how to stabilize Bell states. In Ref.~\cite{ourprl}, we describe external perturbations for moving the dark state to another state within the protected dark space manifold, and we show how to create a dark space manifold realizing a `dark Majorana qubit'. In such a system, topological and DD mechanisms reinforce each other and thereby can provide exceptionally high levels of fault tolerance.
Moreover, we remark that the stabilization of Bell states can also be implemented in a hexon device (i.e., a Majorana box with six MBSs \cite{Karzig2017}), see Ref.~\cite{GauThesis}.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{f7}
\end{centering}
\caption{Schematic two-box layout for DD dark space stabilization and manipulation protocols, cp.~Fig.~\ref{fig1} for the single-box case.
The left (right) box harbors four MBSs described by $\gamma_{\nu}^L$ ($\gamma^R_\nu$).
The tunneling bridge with amplitude $t_{LR}$ connects $\gamma_4^L$ and $\gamma_2^R$.
QD 3 has independently driven tunneling bridges to QD 1 and to QD 2 (solid lines). The three QDs are operated in
the single-electron regime, $N_{\rm d}=1$.
The electromagnetic environment affects the phases of the tunnel links betweens QDs and MBSs
(dashed lines). The phases $\beta_{j}$ for this geometry are also indicated.}
\label{fig7}
\end{figure}
\subsection{Lindblad equation for two coupled boxes}\label{sec4a}
\subsubsection{Model}
Following the discussion in Sec.~\ref{sec2a}, we describe the two islands in Fig.~\ref{fig7} by $H_{\rm box}=H_{{\rm box},L}+H_{{\rm box},R}$, with $H_{{\rm box},L/R}$ as in Eq.~\eqref{Hbox}. Here, the four MBSs on the left (right) box
correspond to Majorana operators $\gamma_\nu^L$ ($\gamma_\nu^R$). Both islands are separately operated under Coulomb valley conditions.
For notational simplicity, we assume that they have the same charging energy, $E_{C,L}=E_{C,R}=E_C$.
Focusing on the long-wavelength components of the electromagnetic environment, we again work with a single bosonic bath, $H_{\rm env}=\sum_m E_m b_m^\dagger b_m^{}$, where photons couple to the QDs and MBSs via fluctuating phases, $\theta_j$, in the tunneling Hamiltonian, see Sec.~\ref{sec2a}.
The setup in Fig.~\ref{fig7} requires up to three single-level QDs,
$H_{\rm d}=\sum_{j=1}^3 \epsilon_j d_j^\dagger d_j^{}$,
where QD 3 couples to both other QDs via independently driven tunnel links.
We consider the regime $N_{\rm d}=1$, where on time scales $\delta t>1/E_C$, the three QDs share a single electron.
Using the interaction picture with respect to the dot Hamiltonian $H_{\rm d}$, the full Hamiltonian is then given by
\begin{equation}\label{Hfull2MBQ}
H(t) = H_{\rm box} + H_{\rm env}+ H_{LR} + H_{\rm drive}(t) + H_{\rm tun}(t),
\end{equation}
where a phase-coherent tunnel link couples the boxes. Without loss of generality, we assume a real-valued tunneling amplitude $t_{LR}>0$,
\begin{equation}\label{LRCoup}
H_{LR} = i t_{LR} \gamma_4^L \gamma_2^R.
\end{equation}
The drive Hamiltonian now has the form
\begin{equation}\label{Hdrive2}
H_{\rm drive}(t)=\sum_{j=1,2} 2A_j\cos\left(\omega_jt\right)e^{i\left(\epsilon_j-\epsilon_3\right)}d_j^{\dagger}d^{}_3 + {\rm h.c.},
\end{equation}
where the two driving fields have the respective amplitude $A_{1,2}$ and frequency $\omega_{1,2}$.
In analogy to Eq.~\eqref{Htun}, the QD-MBS tunnel links are described by
\begin{equation}\label{Htun2}
H_{\rm tun}(t)=t_0 \sum_{j\nu, \kappa=L/R}\lambda_{j,\nu\kappa}^{} e^{-i\phi_\kappa} e^{i\theta_j} e^{i\epsilon_jt} d^\dagger_j
\gamma_{\nu}^\kappa +{\rm h.c.},
\end{equation}
with the phase operators $\phi_{L/R}$ for the left/right Majorana island. Using the same approximations as in Sec.~\ref{spsec1},
the electromagnetic environment enters Eq.~\eqref{Htun2} through the
fluctuating phases $\theta_j$.
With the overall energy scale $t_0$, the complex-valued
parameters $\lambda_{j,\nu\kappa}$ parametrize the transparency of the tunnel contact between
$d_j$ and $\gamma_\nu^{\kappa=L/R}$. Similar to Eq.~\eqref{betadef}, the phases $\beta_j$ in Fig.~\ref{fig7}
follow from the phases of these parameters. Since $\beta_4$ can be absorbed by a renormalization of $\beta_3$ for the purposes below, we put $\beta_4=0$.
To simplify the presentation, we next assume that QDs 1 and 2 have the same
energy level, $\epsilon_1=\epsilon_2$. Moreover, we consider the case of
equal drive frequencies, $\omega_1=\omega_2\equiv \omega_0$, and identical drive amplitudes, $A_1=A_2\equiv A$,
and again impose a resonance condition, $\omega_0=\epsilon_3-\epsilon_1$.
However, in analogy to our discussion in Sec.~\ref{sec2}, we expect that
overly precise fine tuning with respect to those conditions is not necessary.
We now proceed in analogy to Sec.~\ref{sec2a} with the construction of an effective low-energy model by means of a Schrieffer-Wolff transformation
to the lowest-energy charge state in each box. We can then define Pauli operators $(X_\kappa,Y_\kappa,Z_\kappa)$ with $\kappa=L,R$
referring to the left and right box, respectively, using the
Majorana representation in Eq.~\eqref{PauliOp}.
In the present case, it is crucial to
keep all terms up to third order in the expansion parameters \eqref{condit} when accounting for cotunneling trajectories
connecting pairs of QDs, cf.~Fig.~\ref{fig7}. (For the single-box case in Sec.~\ref{sec2a},
it is sufficient to go to second order only.)
The electromagnetic environment then enters the low-energy theory via the three phase differences $\theta_j-\theta_k$ with $j<k$. This fact implies that, in general, we have six different spectral densities ${\cal J}_{jk;j'k'}(\omega)$. We model these spectral densities by the Ohmic form in Eq.~(\ref{Ohmic}),
with system-bath couplings $\alpha_{jk;j'k'}$.
For simplicity, we employ an average value $\alpha$ for these couplings below. The bath is then described by a single spectral density ${\cal J}(\omega)$ again.
Importantly, the physics is not changed in an essential manner by this approximation.
In particular, no additional jump operators appear when allowing for different $\alpha_{jk;j'k'}$.
\subsubsection{Lindblad equation}
We consider again the weak driving regime with $T\ll \omega_0$.
Under these conditions, proceeding along similar steps as in Sec.~\ref{sec2b}, one obtains a Lindblad master equation for the density matrix, $\rho(t)$,
describing both the Majorana sector and the QD degrees of freedom.
In order to arrive at a Lindblad equation for the reduced density matrix, $\rho_{\rm M}(t)$, which refers only to the Majorana sector of both boxes,
we next trace over the QD subsector, see Sec.~\ref{sec2c}.
For the QD steady-state density matrix, $\rho_{\rm d}$, we use the \emph{Ansatz}
\begin{equation}\label{ansatz2}
\rho_{\rm d}= {\rm diag}\left(\frac{1-p}{2}, \frac{1-p}{2},p\right),
\end{equation}
expressed in the basis $\{ |100\rangle,|010\rangle,|001\rangle \}$ with QD occupation states $|n_1,n_2,n_3\rangle$ for $N_{\rm d}=1$. Note that since we assumed $\epsilon_1=\epsilon_2$, the occupation probabilities of QDs 1 and 2 are equal.
The occupation probability $0<p\ll 1$ refers to the energetically highest QD 3. Equation \eqref{ansatz2} is consistent with our numerical analysis of the Lindblad equation for
$\rho(t)$, where we again find a factorized density matrix at long times, $\rho(t)\simeq \rho_{\rm M}(t)\otimes \rho_d$.
We note that the dark space turns out to be independent of the concrete value of $p$.
Going through the corresponding steps in Sec.~\ref{sec2c}, we arrive at a Lindblad equation
for $\rho_{\rm M}(t)$,
\begin{equation} \label{LindbladTwoMBQ}
\partial_t\rho_{\rm M}(t)=-i[\tilde H_{\rm L},\rho_{\rm M}(t)]+
\sum_{a=1}^6 \tilde \Gamma_a \mathcal{L}[K_a] \rho_{\rm M}(t).
\end{equation}
The six jump operators are denoted by $K_a$, with the respective dissipative transition rates $\tilde\Gamma_a$.
With $\lambda_{LR}\equiv t_{LR}/E_C \ll 1$, we obtain
\begin{eqnarray}\nonumber
K_1^{} &=& K_4^\dagger = ie^{i(\beta_3-\beta_1)}\frac{|\lambda_{1,1R}\lambda_{3,3R}|}{\lambda_{LR}} X_R \\
&&\qquad - \,e^{i\beta_3} |\lambda_{1,3L}\lambda_{3,3R}| Z_L Y_R,\nonumber\\ \label{jumpK}
K_2^{} &=& K_5^\dagger =-ie^{i(\beta_3-\beta_2)}\frac{|\lambda_{2,4R}\lambda_{3,3R}|}{\lambda_{LR}}Z_R\\
&&\qquad + \, e^{i\beta_3}|\lambda_{2,2L}\lambda_{3,3R}| X_LY_R,\nonumber\\
K_3^{} &=& K_6^\dagger = i\frac{|\lambda_{1,3L}\lambda_{2,2L}|}{\lambda_{LR}}Y_L-
ie^{i(\beta_2-\beta_1)}
\frac{|\lambda_{1,1R}\lambda_{2,4R}|}{\lambda_{LR}} Y_R\nonumber\\
&&\qquad+ \,e^{-i\beta_1} |\lambda_{1,1R}\lambda_{2,2L}| \,X_LZ_R \nonumber \\ \nonumber &&
\qquad -\,e^{i\beta_2}|\lambda_{1,3L}\lambda_{2,4R}| \,Z_LX_R.
\nonumber
\end{eqnarray}
The coherent evolution in Eq.~\eqref{LindbladTwoMBQ} is governed by the Hamiltonian
\begin{equation}\label{h2qdef}
\tilde H_{\rm L} = 2p\tilde g_0 K_z + \sum_{a=1}^6 \tilde h_a K_a^\dagger K_a,
\end{equation}
with the operator
\begin{equation} \label{kz}
K_z^{} =\sin\beta_1|\lambda_{1,1R}\lambda_{1,3L}| Z_LZ_R+
\sin\beta_2|\lambda_{2,2L}\lambda_{2,4R}| X_L X_R.
\end{equation}
We here used the energy scale
\begin{equation}\label{tildeg0}
\tilde g_0= \lambda_{LR} g_0 = \frac{t_0^2 t_{LR}}{E_C^2},
\end{equation}
which characterizes the relevant inelastic cotunneling processes in the double-box setup.
The transition rates $\tilde\Gamma_a$ follow in the form
\begin{eqnarray}
\tilde\Gamma_1 &=& \tilde \Gamma_2 = 2p\tilde g_0^2 \,{\rm Re} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\
\tilde\Gamma_3 &=& \tilde \Gamma_6 =(1-p)\tilde g_0^2\, {\rm Re} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{trans22}\\
\tilde\Gamma_4 &=&\tilde\Gamma_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde \Gamma_1,\nonumber
\end{eqnarray}
and the Lamb shifts $\tilde h_a$ are given by
\begin{eqnarray}
\tilde h_1 &=& \tilde h_2 = p\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\
\tilde h_3 &=& \tilde h_6 = \frac12 (1-p)\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{lamb22}\\
\tilde h_4 &=&\tilde h_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde h_1.\nonumber
\end{eqnarray}
For $\omega_0\ll \omega_c$, we can then make further analytical progress. Explicit
expressions for $\tilde \Gamma_{1,2}$ and $\tilde h_{1,2}$ follow by comparison with Eq.~\eqref{rate2}.
In addition, we find
\begin{eqnarray}\nonumber
\tilde \Gamma_{3,6} &\simeq& (1-p) \frac{\cos(\pi \alpha)\Gamma(\alpha)\Gamma(1-2\alpha)}{2^{1-2\alpha} \Gamma(1-\alpha)} \left( \frac{\pi T}{\omega_c}\right)^{2\alpha-1} \frac{2g_0^2}{\omega_c} ,\\
\tilde h_{3,6}&=& -\frac{1}{2}\tan(\pi \alpha) \tilde \Gamma_{3,6}.
\end{eqnarray}
By following the derivation of the reduced master equation \eqref{LindbladTwoMBQ}, we observe that the operator $K_1$ ($K_2$) comes from unidirectional transitions transferring an electron from the energetically high-lying QD 3 to QD 1 (QD 2) via the double-box setup, collecting all possible cotunneling trajectories allowed by third-order perturbation theory. Likewise, the jump operator $K_4$ ($K_5$) describes the reversed process, with a cotunneling transition from QD 1 (QD 2) to QD 3.
For $T\ll \omega_0$, the transition rates $\tilde \Gamma_{4,5}$ and Lamb shifts $\tilde h_{4,5}$ are exponentially suppressed, $\propto e^{-\omega_0/T}$,
against the respective contributions from $K_{1,2}$.
Moreover, the jump operators $K_3$ and $K_6$ in Eq.~\eqref{jumpK} describe cotunneling transitions between QDs 1 and 2. Since these QDs are not directly connected by a driven tunnel link and have the same energy, $\epsilon_1=\epsilon_2$, the corresponding rates and Lamb shifts coincide, $\tilde\Gamma_3=\tilde\Gamma_6$ and $\tilde h_3=\tilde h_6$.
Importantly, for $1/2< \alpha < 1$, these quantities are reduced by a factor $(T/\omega_0)^{2\alpha-1}\ll 1$ against $\tilde\Gamma_{1,2}$ and $\tilde h_{1,2}$, respectively.
In the remainder of this section, we shall study this parameter regime
where the most important jump operators in Eq.~\eqref{LindbladTwoMBQ} are given by $K_1$ and $K_2$. Nonetheless, we retain the other jump operators in our numerical analysis as well.
Finally, we note that all terms without the factor $\lambda^{-1}_{LR}\gg 1$ in Eqs.~\eqref{jumpK} and \eqref{kz} stem from third-order processes. While one \emph{a priori} expects that the corresponding dissipative terms in Eq.~\eqref{LindbladTwoMBQ} are suppressed against second-order contributions, by careful tuning of the link transparencies $\lambda_{j,\nu\kappa}$,
they can become of comparable magnitude.
As a consequence, all relevant cotunneling paths will then have amplitudes corresponding to third-order processes. This means that for the present two-box setup, the energy scale $g_0=t_0^2/E_C$ appearing in Eq.~\eqref{basiccond} has to be replaced by $\tilde g_0$ in Eq.~\eqref{tildeg0}.
The Lindblad equation \eqref{LindbladTwoMBQ} describing the weak driving limit is
therefore valid under the conditions
\begin{equation}\label{basiccond2}
\tilde g_0\ll T\ll \omega_0,\quad A \alt \tilde g_0.
\end{equation}
\subsubsection{Dissipative maps}
Before entering our discussion of
stabilization protocols for the layout in Fig.~\ref{fig7}, it is convenient to
introduce the dissipative maps \cite{Barreiro2011}
\begin{equation}\label{bellmap}
\hat E_{1,\pm} = (\mathbb{1}\pm Z_L Z_R) X_R,\quad
\hat E_{2,\pm} = (\mathbb{1}\pm X_L X_R) Z_R.
\end{equation}
These maps can be used to target the four Bell states,
\begin{equation}\label{bellstates}
|\psi_{\pm} \rangle =\frac{1}{\sqrt2}(|00\rangle\pm|11\rangle),
\quad|\phi_{\pm}\rangle=\frac{1}{\sqrt2}(|01\rangle\pm|10\rangle),
\end{equation}
which are eigenstates of both $Z_LZ_R=\pm 1$ and $X_LX_R=\pm 1$.
We observe that
$\hat E_{1,-}$ maps even-parity onto the respective odd-parity states,
$\hat E_{1,-}|\psi_{\pm}\rangle=|\phi_{\pm}\rangle$, while odd-parity states do not evolve in time,
$\hat E_{1,-}|\phi_{\pm}\rangle=0$.
Under this dissipative map, the system will thus be driven into the degenerate odd-parity subsector spanned by the $|\phi_\pm\rangle$ states. Similarly, $\hat E_{2,-}$ can drive the system into the antisymmetric subsector spanned by $|\phi_-\rangle$ and $|\psi_-\rangle$.
The key idea in our DD protocols below is to identify state design parameters such that the jump operators effectively
realize the needed dissipative map(s) in Eq.~\eqref{bellmap}.
Recalling that a dissipative map breaks a number of conserved quantities (and therefore symmetries) in our system, see Refs.~\cite{Albert2014,Albert2016} and
App.~\ref{appC}, we here employ this insight to either stabilize a dark space, see Sec.~\ref{sec4b} and Ref.~\cite{ourprl}, or to target protected and
maximally entangled two-qubit dark states, see Sec.~\ref{sec4c}.
\subsection{Stabilization of a dark space}\label{sec4b}
In this subsection, we briefly outline how one can stabilize a dark space in the setup of Fig.~\ref{fig7}, see also Ref.~\cite{ourprl}.
For convenience, we decouple QD 2 from the system by using the parameter choice
\begin{equation}\label{decouple2}
\lambda_{2,2L}=\lambda_{2,4R}=0,\quad \beta_2=0.
\end{equation}
We note that this is not the only possible parameter set for constructing a dark space.
As a consequence of Eq.~\eqref{decouple2}, many of the jump operators in Eq.~\eqref{jumpK} vanish identically, $K_2=K_3=K_5=K_6=0$.
The jump operator $K^{}_1=K_4^\dagger$ then yields the dissipative map $\hat E_{1,-}$ in Eq.~\eqref{bellmap} upon choosing
\begin{equation}
\beta_1=-\pi, \quad \beta_3 = -\pi/2, \quad |\lambda_{1,1R}|=\lambda_{LR} |\lambda_{1,3L}|.
\label{spacecond}
\end{equation}
Noting that $\hat E_{1,-}= X_R-i Z_L Y_R$, see Eq.~\eqref{bellmap},
we indeed arrive at $K_1\propto \hat E_{1,-}$ from Eq.~\eqref{jumpK}.
In addition, Eq.~\eqref{h2qdef} shows that under the above conditions, $\tilde H_{\rm L}$ only generates terms $\propto Z_L Z_R$ which do not obstruct the dissipative dynamics.
For $T\ll \omega_0$, we next observe that to exponential accuracy, $K_1$ is the only jump operator contributing to the Lindbladian in Eq.~\eqref{LindbladTwoMBQ}
for the parameters in Eqs.~\eqref{decouple2} and \eqref{spacecond}.
The DD protocol therefore will stabilize the system in the
odd-parity ($Z_L Z_R=-1$) Bell state manifold spanned by $\{ |\phi_+\rangle, |\phi_-\rangle \}$.
We show in Ref.~\cite{ourprl} that this
degenerate manifold has the dark space
dimension $D=4$, see also App.~\ref{appC},
which is equivalent to a degenerate qubit space \cite{Albert2014}.
It is possible to manipulate dark states within a dark space by following different strategies \cite{ourprl}.
For instance, one can adiabatically switch on a perturbation that breaks at least one conservation law.
An alternative possibility is to employ single-electron pumping protocols, in analogy to previous proposals for native Majorana qubits \cite{Plugge2017,Karzig2017}.
\subsection{Stabilizing Bell states} \label{sec4c}
We next turn to the stabilization of Bell states in the setup of Fig.~\ref{fig7}, where the couplings between QD 2 and the Majorana islands are now assumed finite again.
In that case, the jump operator $K_2$ in Eq.~\eqref{jumpK} does not vanish anymore. In the low temperature regime, the corresponding Lindbladian term in Eq.~\eqref{LindbladTwoMBQ} contributes
with the same transition rate, $\tilde\Gamma_2=\tilde\Gamma_1$, as for $K_1$, see Eq.~\eqref{trans22}.
Importantly, $K_2$ breaks additional conservation laws and thereby allows one to engineer stabilization protocols targeting maximally entangled two-qubit states. We again study the regime $1/2< \alpha<1$, where the jump operators $K_{3,6}$
give only subleading contributions.
Let us start with the Bell singlet state $|\phi_-\rangle$ in Eq.~\eqref{bellstates}, where $Z_LZ_R=-1$ and $X_LX_R=-1$.
By choosing the state design parameters as
\begin{eqnarray}\label{bellcond}
\beta_1 &=& -\pi,\quad \beta_2 = 0,\quad \beta_3 = -\pi/2,\\
|\lambda_{1,1R}|&=&\lambda_{LR}|\lambda_{1,3L}|,\quad |\lambda_{2,4R}|=\lambda_{LR}|\lambda_{2,2L}|,
\nonumber
\end{eqnarray}
we observe from Eq.~\eqref{jumpK} that $K_1\propto \hat E_{1,-}$ and $K_2\propto \hat E_{2,-}$ are directly expressed in terms of the corresponding dissipative maps, see Eq.~\eqref{bellmap}.
The Lindbladian will therefore drive the system to the dark state $|\phi_-\rangle$. The dark space dimension is thus given by $D=1$.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{f8}
\end{centering}
\caption{Fidelity for stabilizing the Bell singlet state $|\phi_-\rangle$ in the setup of Fig.~\ref{fig7}.
We show numerical results obtained from Eq.~\eqref{LindbladTwoMBQ} with the parameters
in Eq.~\eqref{bellcond} and $|\lambda_{1,3L}|=|\lambda_{2,2L}|=|\lambda_{3,3R}|=1$, using the initial state $\rho_{\rm M}(0)=|00\rangle\langle00|$.
Other parameters are $E_C=1$~meV, $\tilde g_0/E_C=10^{-5}$, $T/\tilde g_0=2, \omega_0/\tilde g_0=2\times 10^3, \omega_c/\tilde g_0=10^4, \alpha=0.99$, and $p=0.01$.
Main panel: Time dependence of $F(t)$ for ideal parameters [Eq.~\eqref{bellcond}] (red curve), and for a mismatch of order $10\%$ in all
state design parameters [$|\lambda_{1,1R}|=1.1\lambda_{LR}|\lambda_{1,3L}|, \, |\lambda_{2,4R}|=0.9\lambda_{LR}|\lambda_{2,2L}|, \, \beta_1=-1.1\pi, \, \beta_3=-9\pi/20$] (blue).
Inset: Steady-state fidelity vs deviation $\Delta\beta_1$ from the ideal value, i.e., $\beta_1=-\pi(1+\Delta\beta_1)$,
with otherwise ideal parameters.}
\label{fig8}
\end{figure}
As is shown in Fig.~\ref{fig8}, the numerical solution of Eq.~\eqref{LindbladTwoMBQ} confirms this expectation. For the stabilization parameters in Eq.~\eqref{bellcond},
the Bell singlet state is reached with nearly perfect fidelity when taking ideal parameter values.
One can rationalize the almost perfect fidelity by noting that the coherent evolution due to $\tilde H_{\rm L}$, see Eq.~\eqref{h2qdef}, involves only the operators $Z_L Z_R$ and $X_L X_R$. As a consequence, the dynamics induced by the dissipative maps $K_{1,2}\propto \hat E_{1/2,-}$ will not be disturbed.
Note that the parameters in Fig.~\ref{fig8} were chosen such that $\tilde\Gamma_1\gg\tilde\Gamma_3$ while staying in the regime specified in Eq.~\eqref{basiccond2}. Indeed,
the observed small deviations from the ideal value $F=1$, see Fig.~\ref{fig8}, can be traced back to the jump operators $K_3$ and $K_6$, which give nominally subleading but practically important contributions to the Lindblad equation.
Figure \ref{fig8} shows that the stabilization protocol is rather robust against
deviations of state design parameters from their ideal values in Eq.~\eqref{bellcond}, see Sec.~\ref{sec3}.
Following the approach in App.~\ref{appB}, we find that the dissipative gap for stabilizing $|\phi_-\rangle$ is given by
\begin{equation}\label{bellgap}
\Delta_{\rm Bell}=|2\lambda_{3,3R}|^2\left(|\lambda_{1,3L}|^2+|\lambda_{2,2L}|^2\right)\sum_{a=1,2,4,5}\tilde\Gamma_a.
\end{equation}
Due to the importance of third-order inelastic cotunneling processes, this dissipative gap is several orders of magnitude below the
corresponding gaps in the single-box case, cf.~Sec.~\ref{sec3}.
For the parameters in Fig.~\ref{fig8}, we obtain the time scale $\Delta_{\rm Bell}^{-1}\approx 0.3$~ms.
The other Bell states in Eq.~\eqref{bellstates} can be targeted by changing the phases $\beta_j$ in Eq.~\eqref{bellcond}. The jump operators $K_1$ and $K_2$ will then directly implement the desired dissipative maps, with the dissipative gap still given by Eq.~\eqref{bellgap}. For stabilization of the Bell state $|\psi_+\rangle$ ($|\psi_-\rangle$), one has to
put $\beta_1=0, \, \beta_2=\pi \, (\beta_2=0)$, and $\beta_3=\pi/2$. Similarly, $|\phi_+\rangle$ is stabilized for $\beta_1=-\pi, \beta_2 = \pi,$ and $\beta_3= -\pi/2$. We thus always have $\beta_3-\beta_1=\pi/2$, and the remaining two phases select the targeted Bell state.
In particular, $\beta_1$ selects the parity of the target state while $\beta_2$ determines the symmetric vs antisymmetric state.
\section{Summary and prospects}\label{secConc}
In this paper, we have described DD protocols in Majorana-based layouts for stabilizing as well as manipulating dark states and dark spaces.
For devices with one or two Majorana boxes coupled to driven QDs and subject to electromagnetic noise, we have shown that
in a wide parameter regime the dynamics in the Majorana sector is accurately described by Lindblad master equations.
The underlying topological nature of the Majorana states significantly boosts the power of DD schemes in several directions.
First, the role of uncontrolled environmental noise sources should be suppressed compared to topologically trivial realizations, which is a key advantage for high-dimensional dark space constructions.
Second, the fact that Pauli operators describing native Majorana qubits correspond to products of Majorana operators (pertaining to spatially separated MBSs), see Eq.~\eqref{PauliOp}, allows for unique addressability options.
Only through this feature, which is rooted in topology, it is possible to design the special unidirectional cotunneling paths which directly
implement the jump operators appearing in the Lindblad equation. In the simplest single-box case, see Fig.~\ref{fig1}, the basic pumping-cotunneling cycle involves
(i) pumping the dot electron from QD 1 to the high-lying QD 2 by means of a weak driving field, and (ii) the back transfer of the electron from QD 2 to QD 1 by cotunneling through the
box. In general, competing transfer mechanisms may also contribute to both steps, and the parameter regime has to be carefully adjusted to
minimize their impact. Taking step (ii) as example, the drive Hamiltonian in Eq.~\eqref{Hdriv}, possibly together with photon emission processes, may provide such
a competing rate. By choosing both a sufficiently small drive amplitude, $A<g_0$, and a very small direct tunnel coupling $t_{12}$ between both QDs,
these competing rates can be systematically suppressed against the cotunneling rates through the box.
We also note that in most cases of interest, the Lindbladian dissipator alone is responsible for driving the system into the desired dark state or dark space, i.e., the Hamiltonian appearing in coherent part of the Lindblad equation does not obstruct the dissipative dynamics.
For a single-box architecture, we have shown how to stabilize arbitary pure dark states, i.e., states that are fault tolerant and stable on arbitrary time scales. For multiple-box devices, one can also stabilize dark spaces, i.e., manifolds of degenerate dark states, as well as protected two-qubit Bell states. In our accompanying short paper \cite{ourprl}, we show that a two-box device allows one to implement a dark Majorana qubit, which in turn could serve as basic ingredient for dark space quantum computation schemes.
Our stabilization and manipulation protocols can be implemented with available hardware elements once a working Majorana platform becomes available.
The above concepts and ideas raise many interesting perspectives for future research. First, we expect that one can devise robust
Majorana braiding protocols \cite{Alicea2012,Leijnse2012,Beenakker2013} that are stabilized by working within a dark space manifold.
Second, for chains of many boxes, our DD stabilization protocols may allow for interesting quantum simulation
applications, e.g., a realization of the topologically nontrivial ground state of spin ladders \cite{Ebisu2019} or of the
Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain \cite{Kraus2008,Affleck1987}.
For clarifying the feasibility of such ideas, one needs to analyze the spectrum of the Lindbladian for DD multiple-box networks. We leave this endeavor to future work.
\begin{acknowledgments}
We thank A. Altland, S. Diehl, and K. Snizhko for discussions.
This project has been funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Grant No.~ 277101999, TRR 183 (project
C01), under Germany's Excellence Strategy - Cluster of Excellence Matter
and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769, and under Grant No.~EG 96/13-1. In addition, we acknowledge funding
by the Israel Science Foundation.
\end{acknowledgments}
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{"url":"https:\/\/cs.stackexchange.com\/questions\/32557\/calculating-the-runtime-for-a-recursive-algorithm","text":"# Calculating the runtime for a recursive algorithm [duplicate]\n\nIf the runtime of a recursive algorithm could be expressed as\n\n$T(n) = \\begin{cases}O(1) & n \\leq c \\\\ k * T\\left(\\frac{n}{k}\\right) + \\left(k + n * k \\right)\\end{cases}$\n\nwhat would be the result of the asymptotic analysis (the Landau class)? $k \\in \\mathbb{N}, k \\neq 0$ is a constant value.\n\nI would say that $k$ even if it is larger than $n$ is still a constant and therefore $T(n) = O(1) * T\\left(\\frac{n}{k}\\right) + O(n) = T\\left(\\frac{n}{k}\\right) * n$ for $n > c$.\n\nSo I would say that the algorithm is $O(n * log_k(n))$. Is this right? How could I prove this (formally correct)?\n\nI have two main problems with this algorithm:\n\n1) What is the runtime of $f(n) = (k + n * k)$? Can the $k$ be ignored so the runtime is $O(n)$?\n\n2) I would like to use the Master theorem. I could see $T(n)$ in the form $a \\cdot T\\left(\\frac{n}{b}\\right) + f(n)$ with $a = b = k$. But then I need to check if $f(n)$ is $\\in O(log_b a) = O(log_1 1) = ?$ This logarithmus is undefined, so can't I use the Master theorem?\n\n## marked as duplicate by FrankW, Yuval Filmus, David Richerby, D.W.\u2666, Luke MathiesonNov 5 '14 at 1:06\n\n\u2022 @FrankW I added some infos why I asked the question and what problems occurred while trying to solve it \u2013\u00a0muffel Nov 1 '14 at 18:33\n\u2022 I calculate $\\log_kk = 1$. Also, $f(n)$ doesn't have a \"runtime\", since it's not an algorithm. It's just a function. If $k > 0$ is fixed then $f(n) = \\Theta(n)$. \u2013\u00a0Yuval Filmus Nov 1 '14 at 19:04\n\nApplying the master theorem will give you $T(n) = O_k(n\\log n)$, but the interesting part is the dependence on $k$. For simplicity, let's assume that $n$ is a power of $k$, and that instead of $k+nk = k(n+1)$ you have the cleaner $kn$ (it's not going to change the answer appreciably). Then you have $$T(n) = kn + kT(n\/k) = kn + kn + k^2T(n\/k^2) = \\cdots = kn\\log_k n + nT(1).$$ Assuming, for example, that $T(1) = 0$, then we get $$T(n) = \\frac{k}{\\log k} n\\log n.$$ For general $n$ you probably get $T(n) = \\frac{k}{\\log k} n\\log n (1 \\pm o(1))$ if you sweat. This gives you the dependence on $k$.\nSince $k\/\\log k$ is increasing for $k > e$, in this scenario the best integral choice for $k$ would be either $k = 2$ or $k = 3$ (as it happens, $k = 3$ is better), which is probably the point of the exercise.\n\u2022 thank you for your great answer. Could you please add which bases the logarithms in $T(n) = \\frac{k}{\\log k} n\\log n$ have and what you did for getting this formula? \u2013\u00a0muffel Nov 2 '14 at 7:55\n\u2022 I was just wondering about the last part from $kn + kn + T(n\/k^2) = \\cdots = kn \\; log_k n + T(1)$, what exactly did you do there? In addition, when analyzing the function $f(x) = \\frac{x}{log x}$, the (rounded) minimum seems to be 3, not 2, or am I missing something? \u2013\u00a0muffel Nov 2 '14 at 16:02\n\u2022 You're right about $x\/\\log x$, it's increasing only for $x > e$. Regarding the rest, when you open up the recurrence $\\log_k n$ times then you will have $\\log_k n$ terms $kn$ and an extra term $nT(n\/k^{\\log_k n}) = nT(1)$ (I forgot the $n$). \u2013\u00a0Yuval Filmus Nov 2 '14 at 16:41\n1) $f(n) \\in O(n)$, since $k$ is a constant.\n2) If $a=b=k$, then $\\log_b a = \\log_k k$. Your specific case of $\\log_1 1$ only appears for $k=1$. And if we insert $k=1$ into the original recurrence we get $T(n) = T(n) + n + 1$, which does not make much sense as a description of a runtime. So it is sensible to simply exclude this case.","date":"2019-06-16 18:50:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.807663083076477, \"perplexity\": 236.73629025308455}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998291.9\/warc\/CC-MAIN-20190616182800-20190616204800-00241.warc.gz\"}"} | null | null |
Q: VBA function: Compile Error: Argument not optional Sub Test_Run()
MkDir "C:\ST\temp\AM"
End Sub
I am trying to create a new folder (AM) in an already existing directory which is C:\ST\temp. But it's causing error showing message: Argument not optional
A: Your MkDir function is expecting two arguments and you are only sending one.
Instead of calling
MkDir "C:\ST\temp\AM"
Change it to:
MkDir "C:\ST\temp", "AM"
and it should work.
Function MkDir(strDir As String, strPath As String)
Dim FSO As New FileSystemObject
Dim path As String
path = strPath & strDir
If Not FSO.FolderExists(path) Then
FSO.CreateFolder path
End If
End Function
| {
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A 4434 Nikulin (ideiglenes jelöléssel 1981 RD5) a Naprendszer kisbolygóövében található aszteroida. Ljudmilla Vasziljevna Zsuravljova fedezte fel 1981. szeptember 8-án.
Kapcsolódó szócikkek
Kisbolygók listája (4001–4500)
Jegyzetek
A Naprendszer kisbolygói | {
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Our Lady's High School may refer to:
Our Lady's High School, Broxburn, West Lothian, Scotland
Our Lady's High School, Cumbernauld, North Lanarkshire, Scotland
Our Lady's High School, Motherwell, North Lanarkshire, Scotland
Our Lady's Catholic High School, Fulwood, Lancashire, England
Our Lady's Catholic High School, Stamford Hill, London, England
See also
Our Lady's Secondary School (disambiguation) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 576 |
Wendy Hoose
Birds of Paradise and Random Accomplice
Genre: Comedy, Drama, Theatre
Venue: Assembly Rooms
She needs, and he is there to provide. In a bristling 21st Century comedy we have a story of our times as Jake arrives to have sex with Laura thanks to a web based sex site. Unfortunately for the eager Jake, Laura wants a bit of chat first before he gets down to the action; unfortunately he finds a couple of things missing that spoil the mood for him. From there we get a run through all the things that we shouldn't say, ought not think but blurt out anyway as Jake tries to be PC and ends up gaining a little bit of respect from Laura … a little bit … allegedly …
From the audio descriptor to the Sign Language Interpreter on the TV, this is a fantastic, excellent and superlative filled comedy where all of your prejudices are examined, pushed and then laughed at. Jake has arrived to have sex. Laura is a 21st Century girl with needs and a drawer to meet them. Once the revelation occurs the squirming and apologetic egg shell splitting performance between the two actors – Amy Conachan and James Young – is as squirmingly wonderful as it ought to be. With the third character – the audio describer – pitching in with her commentary this is how you should do inclusiveness in the theatre.
Writer Johnny McKnight has provided us with a faultless script, co directing with Robert Softley Gale we get an amazing journey into desire for women as well as sex for the disabled single parent. It is crisp and sparkling wit, well directed with every ounce of comedy brought by our actors with mostly well timed wit and repartee.
My criticisms are minor as the opening had some projected commentary that I could not easily read. There were also a couple of exchanges between Conachan and Young that dipped. But these were but minor gripes in a cracking hours' worth of banter.
Birds of Paradise and Random Accomplice know that they have a hit on their hands and Wendy Hoose is commended as part of the curated Made in Scotland programme for the Fringe. It is one of the funniest afternoons I have spent in a theatre for some time. What was even more impressive was the way that accessibility was not only part of the performance but also central to it all. Using a mixed cast is a start, to include the Sign Language Interpreter is becoming more and more integrated but to then throw in an audio describer with abundant wit and scorn as well as surtitles takes us into new territory. What it should not be is testament to the work still to be done, but it is. That this is done with such charm and humour means we should see this type of integration far more often because it is a creative method of making our theatres far more relevant to today.
Published August 22, 2015 by Donald Stewart
Birds of Paradise Theatre company | {
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Some learners prefer to learn at home as opposed to attending a University. Practical reasons such as the availability of University resources and the increasing cost of transport forces some students to complete most of their learning at home. Working from home is beneficial but the learner needs to develop a system and a routine for managing and dealing with distractions. Distractions can be a barrier to learning, and if the distractions are consistent then there is little value in learning from home.
Create an action list of the existing and potential future distractions. For each existing distraction, note how they are to be handled to reduce their probability of interfering with the learning process. For each future distraction, note how they shall be handled if they occur to reduce the effects they might have on the learning process.
Dedicating a room purely for learning would be ideal as all the equipment and books would be in that room. The learner can inform other people living in the house not to enter the room during certain time. Alternatively, locate an area of an existing room to dedicate to learning. A popular example is an area of an upstairs bedroom.
Distraction levels can increase due to modern bedrooms consisting of books, TVs and other electronic devices. Learners therefore need to be disciplined and organised through setting time periods for entertainment and learning. Ensure the learning area has appropriate air conditioning and lightening for learning needs. If need be, invest in fans, heaters and extra lighting as necessary.
Set a regular routine for learning and breaks so that the information has time to digest and for any stress or frustration levels to be maintained at a low level. Keep a regular diary of the learning activities that need to be completed, such as reading a chapter of a book, or writing notes from a series of different resources. This shall improve motivation and self discipline because a regular structure to each learning session shall be developed and followed.
Set time for any children that need looking after. Spend time with them every day after school to help them with their own schoolwork or to just spend time with them in general. Some parents find it easy to study at night, particularly if they have preschool children.
Unplug the Internet whilst working on none Internet required applications, for example when using a word processor to type the work or using a spreadsheet to analyse data. Only use the Internet during learning sessions to do research only. Also turn off all mobile phones so that any phone calls and texts messages do not distract from learning.
Position the computer so that the monitor is not reflecting light from the sun or from any lights within the room. Reflections can cause difficulty in viewing the monitor display and will cause a distraction. If this is not possible, the shut the curtains to reduce the amount of sunlight reflecting off the computer monitor.
Appropriate planning and organising will eradicate the majority of existing distractions or at least minimise their effects on learning and concentration. Plan a regular routine of learning and breaks and stick to this routine as much as possible, as this shall increase motivation and discipline. This routine, the timing of learning sessions and breaks, will be determined by the home environment such if whether or not pets and children need to be looked after. | {
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\section{Introduction}
\label{sec:intro}
The clustering of particles into hadronic jets is relevant to all current high-energy collider experiments.
The clustering procedure is described by a jet algorithm.
The available jet algorithms can be divided into two classes: Sequential recombination algorithms and cone-type algorithms.
For a comparison between experiment and theory it is essential that a jet algorithm is infrared-safe.
Infrared-safe observables can be calculated reliably within perturbation theory.
The cone-type algorithms have suffered for a long time from not being infrared-safe.
This problem has been solved with the invention of the SISCone jet algorithm,
which is a seedless infrared-safe cone jet algorithm \cite{Kidonakis:1998bk,Blazey:2000qt,Salam:2007xv}.
For a correct comparison between experiment and theory one has to use exactly the same jet algorithm in both cases.
However, the situations in which the jet algorithm needs to be used are quite different:
Experimentalists need to cluster of the order of one hundred particles, while in higher-order calculations within fixed-order
perturbation theory one usually deals only with a few partons in the range from two to five or six.
The original proposal of the SISCone algorithm in \cite{Blazey:2000qt} described a method, which takes ${\cal O}(n \cdot 2^n)$ time
to cluster $n$ particles.
This is impractical in the experimental case.
A major breakthrough was therefore the invention of a method, which performs this task in ${\cal O}(n^2 \ln n)$ time \cite{Salam:2007xv}.
In addition, the authors of ref.~\cite{Salam:2007xv} clarified and fixed several ambiguities related to residual infrared unsafety.
The implementation by the authors of ref.~\cite{Salam:2007xv} has become the reference implementation of the SISCone algorithm.
This method together with the corresponding implementation was mainly responsible for the fact,
that experimentalists nowadays use frequently the infrared-safe SISCone jet algorithm.
Let us now turn to the theoretical side:
Next-to-leading order (NLO) or next-to-next-to-leading order (NNLO) calculations are usually done with the help of the subtraction method.
This introduces a collection of subtraction terms for the real emission part.
Each subtraction term has its own configuration of parton momenta, in general different from the other ones and different from the configuration
of the matrix element it approximates.
For jet observables this implies that the jet algorithm has to be run once for each subtraction term.
For sequential recombination algorithms like the $k_\perp$- or anti-$k_\perp$-algorithms \cite{Stirling:1991ds,Catani:1993hr,Ellis:1993tq,Cacciari:2005hq,Cacciari:2008gp} this is not a problem, since the CPU time
for the jet algorithm is negligible against the CPU time required for the matrix element associated with the subtraction term.
However, in a recent comparison of different jet algorithms at NNLO in electron-positron annihilation \cite{Weinzierl:2010cw}
it turned out that the CPU time of the reference implementation of the SISCone algorithm was at the order of the CPU time of the matrix
elements.
This slowed down the computation of the NNLO prediction for the SISCone jet rates significantly compared to the NNLO predictions
for the jet rates with other jet algorithms, in particular when results for different cone sizes were calculated in parallel.
(In ref.~\cite{Weinzierl:2010cw} the results for 100 different cone sizes where calculated in parallel.)
To fix this problem, a dedicated implementation of the SISCone algorithm optimised for low particle multiplicities has been written.
These routines can be of use for any calculation within fixed-order perturbation theory using the SISCone algorithm.
In this article I report on the implementation of the SISCone algorithm optimised for low particle multiplicities.
The implementation includes a version for hadronic collisions as well as a version for electron-positron annihilation.
The implementation is written in C++. Wherever possible I kept the same syntax as the reference implementation of \cite{Salam:2007xv}.
Efficiency issues require a few modifications of the syntax, which are all documented in this article.
It should be clearly stated the implementation of this article is targeted for the case of a few final state particles.
In this case the implementation of this article can give a significant performance boost, up to a factor $70$
for two-particle configurations.
For final states where the number of particles is at the order of one hundred, the implementation of \cite{Salam:2007xv} should be used.
The cross-over occurs around $10$ final state particles.
This paper is organised as follows:
In the next section I review the SISCone jet algorithm.
Section~\ref{sect:implementation} describes the optimisation techniques used in the low parton multiplicity case.
Section~\ref{sect:howto} gives detailed information on how to use the program.
The performance is reported in section~\ref{sect:performance}.
Finally, section~\ref{sect:conclusions} contains a summary and the conclusions.
\section{The SISCone jet algorithm}
\label{sect:definition}
In this section I review the definition of the SISCone jet algorithm \cite{Salam:2007xv}.
There are two variants of the SISCone jet algorithm, one variant is adapted to hadronic collisions, while the other
variant is adapted to electron-positron collisions.
The hadronic version uses the rapidity $y$ and the azimuthal angle $\phi$ as the basic variables and corresponds to
a cylindrical geometry.
The version for electron-positron annihilation uses instead the angle between particles as the basic variable
and corresponds to a spherical geometry.
I will first describe the common features of both versions and give the specific features for each version in the end.
The SISCone jet algorithm depends on the four parameters $R$, $f$, $n_{\mathrm{pass}}$ and $v_{\mathrm{min}}$.
The most important parameters are the cone size $R$ and the overlap parameter $f$.
The parameter $n_{\mathrm{pass}}$ specifies how often the procedure for finding stable cones is maximally iterated. This parameter can be set to infinity.
The parameter $v_{\mathrm{min}}$ defines the minimal value of the quantity $v_{\mathrm{threshold}}$ (to be defined below) required for a jet.
In addition to these four numbers we have to specify a distance measure $R(i,J)$ between a particle $i$ and a cone axis corresponding
to the protojet $J$ and three functions
$v_{\mathrm{threshold}}(J)$, $v_{\mathrm{ordering}}(J)$ and $v_{\mathrm{overlap}}(J)$ related to the split merge procedure.
The two versions for hadronic collisions and electron-positron annihilation will only differ in the definition of the distance measure
and in the definition of the functions $v_{\mathrm{threshold}}$, $v_{\mathrm{ordering}}$ and $v_{\mathrm{overlap}}$.
The SISCone jet algorithm is specified as follows:
\begin{enumerate}
\item Put the set of current particles equal to the set of all particles in the event and set $i_{pass}=0$.
\item For the current set of particles find all stable cones with cone size $R$.
\item Each stable cone is added to the list of protojets.
\item Remove all particles that are in stable cones from the list of current particles and increment $i_{pass}$.
\item If $i_{pass}<n_{pass}$ and some new stable cones have been found in this pass, go back to step 2.
\item Run the split-merge procedure with overlap parameter $f$ and threshold $v_{\mathrm{min}}$.
\end{enumerate}
A set of particles defines a cone axis, which is given as the sum of the momenta of all particles in the set.
A cone is called stable for the cone size $R$,
if all particles defining the cone axis have a distance measure smaller than $R$ to the cone
axis and if all particles not belonging to the cone have a distance measure larger than $R$ with respect to the cone axis.
The four-momentum of a protojet is the sum of the four-momenta of the particles in the protojet.
This corresponds to the E-scheme.
Two protojets are called overlapping, if they share at least one particle.
Let us now turn to the split-merge procedure:
The split-merge procedure depends on the two parameters $v_{\mathrm{min}}$ and $f$, and on the three functions
$v_{\mathrm{threshold}}$, $v_{\mathrm{ordering}}$ and $v_{\mathrm{overlap}}$.
The split-merge procedure discards all protojets with $v_{\mathrm{threshold}} < v_{\mathrm{min}}$, therefore $v_{\mathrm{min}}$ is a threshold value
a jet must have.
In addition, the split-merge procedure requires an infrared-safe ordering variable for the protojets.
This variable is denoted $v_{\mathrm{ordering}}$.
Furthermore, if two protojets overlap, we need a quantity which describes the overlap. This quantity is denoted $v_{\mathrm{overlap}}$.
The split-merge procedure is defined as follows:
\begin{enumerate}
\item Remove all protojets with $v_{\mathrm{threshold}} < v_{\mathrm{min}}$.
\item Find the protojet $I$ with the highest value of $v_{\mathrm{ordering}}$.
\item Among the remaining protojets find the one ($J$) with highest value $v_{\mathrm{ordering}}$ that overlaps with $J$.
\item If there is such an overlapping jet then compute the quantities $v_{\mathrm{overlap}}(I \cap J)$ and $v_{\mathrm{overlap}}(J)$.
\begin{enumerate}
\item If $v_{\mathrm{overlap}}(I \cap J) < f v_{\mathrm{overlap}}(J)$ assign each particle that is shared between the two protojets to the protojet whose
axis is closest. Recalculate the momenta of the protojets.
\item If $v_{\mathrm{overlap}}(I \cap J) \ge f v_{\mathrm{overlap}}(J)$ merge the two protojets into a single new protojet and remove the two original
ones.
\end{enumerate}
\item Otherwise, if no overlapping jet exists, then add $I$ to the list of jets and remove it from the list
of protojets.
\item As long as there are protojets left, go back to step 1.
\end{enumerate}
It remains to specify the distance measure $R(i,J)$
and the three functions $v_{\mathrm{threshold}}(J)$, $v_{\mathrm{ordering}}(J)$ and $v_{\mathrm{overlap}}(J)$.
These quantities differ in the two versions of the SISCone algorithm.
We have
\begin{align}
&
& & \mbox{cylindrical}
& & \mbox{spherical}
\nonumber \\
& R(i,J):
& & d_{i,J} = \sqrt{ \left(y_i-y_J\right)^2 + \left( \phi_i -\phi_J \right)^2}
& & \theta_{i,J} = \arccos \frac{\vec{p}_i \cdot \vec{p}_J}{\left|\vec{p}_i \right| \left| \vec{p}_J\right|}
\nonumber \\
& v_{\mathrm{threshold}}(J):
& & p_{\perp}(J) = \left| \sum\limits_{k \in J} \vec{p}_{\perp,k} \right|
& & E(J) = \sum\limits_{k \in J} E_k
\nonumber \\
& v_{\mathrm{ordering}}(J):
& & \tilde{p}_{\perp}(J) = \sum\limits_{k \in J} \left| \vec{p}_{\perp,k} \right|
& & \tilde{E}(J) = \sum\limits_{k \in J} E_k \left( 1 + \frac{\left|\vec{p}_J\right|^2}{E_J^2} \sin^2 \theta_{k,J} \right)
\nonumber \\
& v_{\mathrm{overlap}}(J):
& & \tilde{p}_{\perp}(J) = \sum\limits_{k \in J} \left| \vec{p}_{\perp,k} \right|
& & E(J) = \sum\limits_{k \in J} E_k
\nonumber
\end{align}
In this table the indices $i$ and $k$ refer to a particle, while the index $J$ refers to the four-vector of a protojet.
In the cylindrical version (which is appropriate for hadronic collisions) the distance measure is calculated from the rapidity $y$ and
the azimuthal angle $\phi$.
The threshold function $v_{\mathrm{threshold}}(J)$ is given as the absolute value of the sum of the transverse momenta $p_{\perp,k}$ of the particles $k$
making up the protojet $J$.
The ordering function $v_{\mathrm{ordering}}(J)$ and the overlap function $v_{\mathrm{overlap}}(J)$ are given as the sum
of the absolute values of the transverse momenta $p_{\perp,k}$ of the particles $k$
making up the protojet $J$.
Since the main application of this implementation is within fixed-order perturbation theory infrared-safeness is essential.
Therefore choices of the ordering function which are not infrared-safe are not considered.
The choice for $v_{\mathrm{ordering}}(J)$ given above is the recommended one of ref.~\cite{Salam:2007xv}.
In the spherical version (which is appropriate for electron-positron annihilation) the distance measure is given by the angle between the three-momentum
$\vec{p}_i$ of particle $i$ and the three-momentum $\vec{p}_J$ of the protojet axis of the protojet $J$.
The threshold function $v_{\mathrm{threshold}}(J)$ and the overlap function $v_{\mathrm{overlap}}(J)$ are given by the sum of the energies of the
particles $k$ making up the protojet $J$.
The ordering function $v_{\mathrm{ordering}}(J)$ is given by the quantity
\begin{eqnarray}
\tilde{E}(J) & = & \sum\limits_{k \in J} E_k \left( 1 + \frac{\left|\vec{p}_J\right|^2}{E_J^2} \sin^2 \theta_{k,J} \right).
\end{eqnarray}
Note that this is the function, which is actually implemented in the code of \cite{Salam:2007xv},
and not the function
\begin{eqnarray}
\sum\limits_{k \in J} E_k \left( 1 + \sin^2 \theta_{k,J} \right).
\end{eqnarray}
\section{The implementation for low particle multiplicities}
\label{sect:implementation}
The standard implementation \cite{Salam:2007xv} of the SISCone algorithm
is highly optimised for the experimental situation, where one faces the challenge to cluster
of the order one hundred particles.
In this situation the asymptotic run-time behaviour of the algorithm is very important.
The asymptotic run-time behaviour is dominated by the time needed to find all stable cones
and is ${\cal O}(n^2 \ln n)$ for $n$ final state particles in the implementation of \cite{Salam:2007xv}.
The CPU time of the parts of the algorithm which do not grow so fast are less important if one deals with the
order of one hundred particles.
However, the situation is different if only a few final state particles need to be clustered.
In this situation the asymptotic run-time behaviour is less important, and all parts of the algorithm have to be taken
into account for the optimisation.
It turns out that it is most efficient to use for the low particle multiplicity case a method
which reduces significantly the computational cost of the parts which would be sub-dominant in the asymptotic regime.
The implementation of this paper uses the method of \cite{Blazey:2000qt} to find all stable cones. This method
grows like ${\cal O}(n \cdot 2^n)$ for $n$ final state particles.
This is acceptable since the intention is to use this implementation only in the case of small values of $n$.
The implementation is written in C++. There are a few standard optimisation techniques which are used to make the code fast:
First of all the intention is that this implementation is used within theoretical fixed-order calculations.
The authors of fixed-order numerical programs usually have their own implementation of a four-vector class, which stores the momenta
of the final state particles.
Copying the data to a similar class which is used by the jet algorithm can take a significant amount of CPU time.
This can be avoided by making the implementation of the SISCone jet algorithm a template.
The implementation of this paper works with any four-vector class, under the condition that the four-vector class
provides a few standard methods.
A simple four-vector class which has all the required methods comes with the implementation.
The input to the jet algorithm is a list (actually a {\tt std::vector}) of $n$ four-vectors, representing the $n$ final-state particles.
After clustering has been done, the content of a jet is represented by an unsigned integer.
Within this unsigned integer, the $j$-th bit corresponds to particle $j$.
A value of $1$ of the corresponding bit indicates that the particle belongs to the jet, while a value
of $0$ means that the particle does not belong to the jet.
In this way one can represent up to $64$ particles in an unsigned integer on $64$-bit machines
(and up to $32$ particles in an unsigned integer on $32$-bit machines).
However, these numbers should be taken as a hard-coded upper limit, beyond which it is not possible to use this implementation.
In practise one would like to switch already earlier on to the implementation of \cite{Salam:2007xv} with the better
asymptotic run-time behaviour.
The advantage of using unsigned integers to represent jets is given by the fact that many operations (like merging) can be carried
out through fast bit-operations.
The implementation of the spherical version of the SISCone jet algorithm uses internally the distance measure
\begin{eqnarray}
y_{i,J} & = & 1 - \cos \theta_{i,J}
\end{eqnarray}
instead of $\theta_{i,J}$. $y_{i,J}$ is a monotonic function of $\theta_{i,J}$ in the interval $[0,\pi]$.
Using $y_{i,J}$ has the advantage that the distance can be calculated without any call to a trigonometric function.
The value of $y_{i,J}$ is calculated as
\begin{eqnarray}
y_{i,J} & = & 1 - \frac{\vec{p}_i}{\left|\vec{p}_i\right|} \cdot \frac{\vec{p}_J}{\left|\vec{p}_J\right|}.
\end{eqnarray}
This requires only a call to the \verb/sqrt/-function for the normalisation of the spatial three-vectors $\vec{p}$.
All normalised spatial three-vectors are pre-computed at the start of the jet algorithm and stored in an array.
In the cylindrical case the distance measure involves the rapidity $y$ and the azimuthal angle $\phi$.
This involves a call to the \verb/log/-function and a call to the \verb/atan2/-function.
These calls are rather expensive and lead to the effect that the cylindrical version requires more computer time.
In order to avoid some of these function calls, the following optimisation is implemented in the program:
At the start of the jet algorithm, all azimuthal angles are pre-computed and stored in an array, but the rapidities
are not yet computed.
In the search for stable cones some cones can immediately be discarded based
on the information of the azimuthal angle alone.
These are the cones $J$, which have a particle $i$ belonging to the cone $J$, whose distance in azimuthal angle is already
larger than the cone size $R$:
\begin{eqnarray}
\tilde{d}_{i,J}^2 & = & \left( \phi_i - \phi_J \right)^2 > R^2.
\end{eqnarray}
These cones are never stable, since adding the rapidity distance would only increase the distance measure.
For those cones the calculation of the rapidity can be avoided.
A second optimisation technique can be applied to events, which have a total $p_\perp$-sum of zero.
This is the case for pure QCD events, but not for mixed QCD-electroweak events like $W + \mbox{jets}$.
If the total $p_\perp$-sum is zero,
only half of the values of the azimuthal angle (and of the transverse momenta) need to be computed.
The other half can be obtained from transverse momentum conservation without a call to the \verb/atan2/-function.
In the implementation this additional optimisation can be turned on by setting a flag that the total $p_\perp$-sum is zero.
\section{How to use the program}
\label{sect:howto}
The implementation of the SISCone jet algorithm is written in C++ and can be obtained from
\\
\\
{\tt http://wwwthep.physik.uni-mainz.de/\~{}stefanw/software.html}\\
\\
After unpacking, the directory will contain the four files
\begin{verbatim}
siscone_parton.h fourvector.h example.cc example_spherical.cc
\end{verbatim}
The file \verb/siscone_parton.h/ contains the implementation of the SISCone jet algorithm,
defined within the namespace \verb/siscone_jet_algorithm/.
The SISCone jet algorithm is implemented as a template class, therefore all code is contained in the header file
\verb/siscone_parton.h/.
No compilation or installation is required, the implementation can be used by including the
header file in the source code of the user.
The version of the SISCone jet algorithm which is based on the cylindrical geometry and which is appropriate for hadronic
collisions is implemented in
\begin{verbatim}
template<class T> class siscone_parton
\end{verbatim}
The version based on the spherical geometry, which is appropriate for electron-positron annihilation is implemented in
\begin{verbatim}
template<class T> class siscone_spherical_parton
\end{verbatim}
In both cases the template argument is a class implementing four-vectors and denoted by \verb/T/ in the declarations above.
The four-vector class must have a few standard methods, which are summarised in the following fragment of an example header file:
\begin{verbatim}
class fourvector {
public :
fourvector(void);
fourvector & sum_up(const fourvector & q);
fourvector & rescale(double c);
double transverse_momentum(void) const;
double rapidity(void) const;
double azimuthal_angle(void) const;
double spatial_norm(void) const;
double spatial_scalar_product(const fourvector & q) const;
double energy_component(void) const;
};
\end{verbatim}
In detail, the four-vector class has to provide a default constructor, which initialises the four components of the four-vector with zero.
The method \verb/sum_up(q)/ adds the four-vector \verb/q/ to the current four-vector.
The method \verb/rescale(c)/ multiplies all components of the four-vector by the factor \verb/c/.
The default constructor and the two methods \verb/sum_up/ and \verb/rescale/ are required by both versions of the SISCone jet algorithm.
The class \verb/siscone_parton/ which uses the cylindrical geometry requires in addition the following three methods:
A method \verb/transverse_momentum()/ which returns the transverse momentum with respect to the beam axis.
With the beam-axis along the $z$-axis the transverse momentum is given by
\begin{eqnarray}
p_\perp & = & \sqrt{p_x^2+p_y^2},
\end{eqnarray}
where $p_x$ and $p_y$ denotes the $x$- and the $y$-component of the four-vector $p$, respectively.
Secondly, a method \verb/rapidity()/ which returns the rapidity of the four-momentum. With the beam along the $z$-axis, the rapidity is given
by
\begin{eqnarray}
y & = & \frac{1}{2} \ln \frac{p_t+p_z}{p_t-p_z},
\end{eqnarray}
where $p_t$ denotes the energy component and $p_z$ denotes the $z$-component of the four-vector $p$.
Finally, a method \verb/azimuthal_angle()/ which returns the azimuthal angle $\phi$ of the four-vector with respect to the beam axis. This angle
obeys the relation
\begin{eqnarray}
\tan \phi & = & \frac{p_y}{p_x},
\end{eqnarray}
if the beam-axis is again along the $z$-axis.
The three methods \verb/transverse_momentum/, \verb/rapidity/ and \verb/azimuthal_angle/ are not needed if just the spherical version of the
SISCone jet algorithm is used.
The class \verb/siscone_spherical_parton/ which uses the spherical geometry requires in addition the following three methods:
A method \verb/spatial_norm()/ which returns the norm of the three spatial components of the four-vector:
\begin{eqnarray}
\left| \vec{p} \right|^2 & = & p_x^2 + p_y^2 + p_z^2.
\end{eqnarray}
Secondly, a method \verb/spatial_scalar_product(q)/ which returns the scalar product of the spatial components with the ones of the four-vector \verb/q/:
\begin{eqnarray}
\vec{p} \cdot \vec{q} & = & p_x q_x + p_y q_y + p_z q_z.
\end{eqnarray}
Finally, a method \verb/energy_component()/, which returns the energy component $p_t$ of the four-vector.
The three methods \verb/spatial_norm/, \verb/spatial_scalar_product/ and \verb/energy_component/
are not needed if just the cylindrical version of the
SISCone jet algorithm is used.
The file \verb/fourvector.h/ gives an example of a four-vector class which has all the required methods.
Since the required methods are rather simple, they can be implemented as inline functions.
Therefore all code relevant to the four-vector class is contained in the header file \verb/fourvector.h/.
The SISCone jet algorithm based on the cylindrical geometry is initialised by
\begin{verbatim}
siscone_parton<fourvector> jet_finder(n_max);
\end{verbatim}
The template argument \verb/fourvector/ should be replaced by the name of the appropriate four-vector class.
The variable \verb/n_max/ gives the maximal number of final state particles.
In a fixed-order perturbative calculation \verb/n_max/ is usually set by the number of final state particles in the real emission contribution.
The constructor allocates the memory which is needed for the jet algorithm.
Since the CPU time for the allocation and deallocation of memory is not negligible, it is recommended that within a Monte Carlo integration
the constructor is called exactly once.
The spherical version of the SISCone jet algorithm is correspondingly initialised by
\begin{verbatim}
siscone_spherical_parton<fourvector> jet_finder_spherical(n_max);
\end{verbatim}
In both versions the jet algorithm is invoked by a call to the method
\begin{verbatim}
int compute_jets(const std::vector<fourvector> & particles, double R,
double f, int n_pass_max=0, double v_min=0.0);
\end{verbatim}
The input parameters of the method are: a list of four-vectors of the final state particles, given as a \verb/std::vector<fourvector>/.
The input parameters \verb/R/ and \verb/f/ denote the cone size and the overlap parameter, respectively.
The maximum number of passes is given by \verb/n_pass_max/.
A zero value for \verb/n_pass_max/ is treated as infinity.
This is the default choice for \verb/n_pass_max/.
The minimal value of the threshold parameter is given by \verb/v_min/. In the cylindrical version this is the $p_\perp$-value, in the spherical version
it is the energy $E$.
The default choice for this parameter is zero.
The name and the arguments of the method \verb/compute_jets/ are identical to the ones in the implementation of \cite{Salam:2007xv}.
This should make it easy to switch between the two implementations and to use each implementation where it performs best.
The method \verb/compute_jets/ returns the number of jets $n_{\mathrm{jet}}$.
In addition, the content of the jets can be found in the data member
\begin{verbatim}
template<class T> class siscone_parton
{
public:
std::vector<unsigned> final_jets;
};
\end{verbatim}
The class \verb/siscone_spherical_parton/ has also the data member \verb/final_jets/.
After a call to \verb/compute_jets/ this vector contains $n_{\mathrm{jet}}$ elements. Each entry is an unsigned integer.
If the $j$-th bit of entry $i$ is set, it implies that particle $j$ belongs to jet $i$.
The class \verb/siscone_parton/ has in addition a method
\begin{verbatim}
public:
void set_flag_total_pt_is_zero(bool flag);
\end{verbatim}
For events with a total $p_\perp$-sum of zero, this flag can be set to \verb/true/.
The jet algorithm will then use special optimisations, which are valid if the total $p_\perp$-sum is zero.
This can be used for pure QCD events, but should not be used for mixed QCD+electroweak events like $W+\mbox{jets}$-events.
The default choice is \verb/false/.
The file \verb/example.cc/ contains a simple example program, which shows how to use the jet algorithm.
The listing of the test program is as follows:
\begin{verbatim}
#include <iostream>
#include <vector>
#include "fourvector.h"
#include "siscone_parton.h"
int main()
{
using namespace siscone_jet_algorithm;
// intialise the SISCone algorithm for maximally 4 final state particles
int n_max = 4;
siscone_parton<fourvector> jet_finder(n_max);
// define the parameters for the SISCone algorithm
double R = 0.9;
double f = 0.5;
int n_pass = 0;
double pt_min = 5.0;
// some final-state momenta
std::vector<fourvector> particles(4);
particles[0] = fourvector(1.87116,-1.08275,1.33996,-0.730343);
particles[1] = fourvector(33.3795,1.00171,28.1084,-17.9751);
particles[2] = fourvector(32.4375,-12.8963,-13.154,26.6992);
particles[3] = fourvector(22.3118,12.9774,-16.2944,-7.99375);
// call the jet algorithm
int n_jet = jet_finder.compute_jets( particles, R, f, n_pass, pt_min );
// output
std::cout << "Number of jets = " << n_jet << std::endl;
for (int j=0; j<n_jet; j++) {
std::cout << " Particle content of jet " << j << " : ";
for (int i=0; i<n_max; i++) {
if ( jet_finder.final_jets[j] & (1<<i) ) std::cout << i << " ";
}
std::cout << std::endl;
}
return 0;
}
\end{verbatim}
The program initialises first the jet algorithm for maximally four final state particles.
It then defines the parameters for the jet algorithm and a set of four final state momenta.
The call to the method \verb/compute_jets/ invokes the jet algorithm.
After the jets have been computed, the program outputs the number of jets as well as the particle content of each jet.
The program can be compiled with the command
\begin{verbatim}
g++ -o example example.cc
\end{verbatim}
Running the program will produce the output
\begin{verbatim}
Number of jets = 3
Particle content of jet 0 : 0 1
Particle content of jet 1 : 3
Particle content of jet 2 : 2
\end{verbatim}
We see that three jets have been found. Particles $0$ and $1$ have been clustered together to form jet $0$.
Jet $1$ consists of the particle $3$, while jet $2$ consists of particle $2$.
The file \verb/example_spherical.cc/ is very similar to the file \verb/example.cc/, but uses the spherical version of the SISCone
jet algorithm instead.
The example for the spherical version uses the same set of momenta of the final state particles.
Running the program \verb/example_spherical/ will produce the output
\begin{verbatim}
Number of jets = 3
Particle content of jet 0 : 0 1
Particle content of jet 1 : 2
Particle content of jet 2 : 3
\end{verbatim}
\section{Checks and performance}
\label{sect:performance}
In this section I compare the implementation of the SISCone jet algorithm of this article with the implementation
of \cite{Salam:2007xv} as a function of the number $n$ of final state particles.
The number $n$ of final state particles ranges in the comparison between $2$ and $11$.
For each $n$ I generate ${\cal O}(10^6)$ random events and I cluster the final state particles with the two implementations of
the SISCone jet algorithm.
The two implementations find exactly the same number of jets for all events.
Let us first consider in more detail the clustering based on the cylindrical geometry appropriate to hadronic collisions.
\begin{figure}
\begin{center}
\includegraphics[bb= 125 460 490 710,width=0.8\textwidth]{performance.eps}
\end{center}
\caption{
Speed-up factor as a function of the number $n$ of final state particles
for the cylindrical version of the SISCone jet algorithm.
Shown is the ratio of the CPU time of the implementation of \cite{Salam:2007xv} by the CPU time of this implementation.
The blue line shows a ratio of $1$.
The cross-over occurs between $9$ and $10$ final state particles.
}
\label{fig_performance}
\end{figure}
Fig.~\ref{fig_performance} shows the speed-up factor
as a function of the number $n$ of final state particles.
In this figure the ratio of the CPU time of the implementation of \cite{Salam:2007xv} by the CPU time of this implementation
is shown.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\
\hline
this work & $0.3$ & $1.1$ & $2.4$ & $4.4$ & $8.0$ & $15$ & $28$ & $51$ & $100$ \\
ref.~\cite{Salam:2007xv} & $4.9$ & $10$ & $15$ & $20$ & $27$ & $36$ & $42$ & $52$ & $61$ \\
\hline
\end{tabular}
\caption{\label{table_performance}
Average CPU time in $\mu\mbox{s}$ for the clustering of $n$ particles with the cylindrical version of the SISCone algorithm on a standard PC.
}
\end{center}
\end{table}
The absolute timings of the two implementations are shown in table~\ref{table_performance}.
It can be seen that the implementation of this article is faster up to nine final state particles.
A speed-up factor of $15$ is obtained for configurations with two final state particles.
This factor is for example relevant to a forthcoming NNLO calculation of inclusive single jet production in $p p$ collisions.
In this calculation most subtraction terms have configurations with two final state particles.
\begin{figure}
\begin{center}
\includegraphics[bb= 125 460 490 710,width=0.8\textwidth]{performance_spherical.eps}
\end{center}
\caption{
Speed-up factor as a function of the number $n$ of final state particles
for the spherical version of the SISCone jet algorithm.
Shown is the ratio of the CPU time of the implementation of \cite{Salam:2007xv} by the CPU time of this implementation.
The blue line shows a ratio of $1$.
The cross-over occurs between $10$ and $11$ final state particles.
}
\label{fig_performance_spherical}
\end{figure}
Let us now consider the spherical version of the SISCone jet algorithm.
Fig.~\ref{fig_performance_spherical} shows the corresponding plot for the spherical geometry, appropriate
to electron-positron annihilation.
The absolute timings of the two implementations are shown in table~\ref{table_performance_spherical}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ \\
\hline
this work & $0.1$ & $0.4$ & $1.1$ & $2.4$ & $4.7$ & $9.3$ & $16$ & $33$ & $65$ & $130$ \\
ref.~\cite{Salam:2007xv} & $6.7$ & $13$ & $19$ & $29$ & $38$ & $48$ & $61$ & $74$ & $88$ & $105$ \\
\hline
\end{tabular}
\caption{\label{table_performance_spherical}
Average CPU time in $\mu\mbox{s}$ for the clustering of $n$ particles with the spherical version of the SISCone algorithm on a standard PC.
}
\end{center}
\end{table}
Here, the speed-up factor is almost $70$ for two-particle configurations and roughly $35$ for three-particle configurations.
The implementation of this article is faster up to $10$ particles.
The speed-up factor of $35$ for three-particle configurations was essential in the recent NNLO calculation
of three-jet rates in electron-positron annihilation \cite{Weinzierl:2010cw}.
Without this speed improvement the CPU time for the jet clustering would have out-weighted the CPU time of the corresponding matrix elements.
One observes that the speed-up factor is higher in the spherical version than in the cylindrical version.
This can be understood by the following two facts:
First, as explained in section~\ref{sect:implementation} the spherical version can be coded in such a way that
no trigonometric function needs to be evaluated in the main part of the algorithm.
On the other hand, the cylindrical version requires the evaluation of \verb/log/- and \verb/atan2/-functions.
It is therefore expected that the implementation of the cylindrical version requires more CPU time compared to the
spherical version. In the implementation of this article this is indeed the case.
The second reason is rather surprising: It turns out that in the implementation of \cite{Salam:2007xv} the
cylindrical version performs better than the spherical version.
\section{Conclusions}
\label{sect:conclusions}
In this article I reported on an implementation of the SISCone jet algorithm, which is optimised for low particle multiplicities.
This implementation can be used for theoretical calculations in fixed-order perturbation theory, where the number of final state
particles is rather low.
The implementation of this article performs better then the reference implementation of \cite{Salam:2007xv}
up to roughly nine final state particles.
For a higher number of final state particles the reference implementation of \cite{Salam:2007xv} should be used.
For a low number of final state particles the implementation of this article can lead to a speed-up factor up to $70$.
\subsection*{Acknowledgements}
I would like to thank G. Salam and G. Soyez for useful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,466 |
Posted by Joan Concilio August 20, 2017 Local memories, York City/Suburban
Musical memories from York County, PA: A fall miniseries, Part 1
Often in my columns, I try to jump around to a variety of topics – talking about restaurants one week, stores another, particular towns another, and so on.
But now we're coming into fall, which will always be most notable in my mind as marching band season, and I wanted to talk some more about local music memories. So I tried to dig out ALL the submission I have on that topic, and realized I have enough for a short miniseries on musical memories from York County.
So, for the next few weeks (with a brief intermission next week to talk about something even more seasonally relevant than music), I'm going to be sharing readers' memories and photos on a variety of local musical topics, from venues to teachers to bands and beyond.
I hope you'll enjoy it!
Reader MaryAnn Hoke shared this photo, which her husband's family wants to know more about. "The only thing we know is that one of the men in this mandolin band is my husband's grandfather, George Clinton Hoke, born Aug. 22, 1892, died April 5, 1930."
To kick things off, here's a musical mystery: Of the photo you see with today's column, shared by reader MaryAnn Hoke, the only known information is that one of the mandolin players in this mandolin band was Hoke's husband's grandfather, George Clinton Hoke, who was born Aug. 22, 1892 and died April 5, 1930. Based on that, I'm sure this photo was taken in the early 1900s, probably between 1910 and 1920 or 1925. I'd love to know more for MaryAnn and her family, so if you have any ideas, please let us know!
Another topic of musical interest in the past has been the American Legion shows held in York during the 1940s and 1950s at Post No. 127. Originally, a reader named Anita in North Carolina asked about that in 2015, and I shared some memories of those shows back in June of last year.
I had a letter on the topic from George M. Trout Sr., a longtime local radio personality who is often a wealth of musical information. George noted:
"Regarding the question of the American Legion shows from Anita in North Carolina… York Post 127 was host for the shows from 1920 to 1956 or 1957, and I doubt there are many local participants left in York. I suspect I am one of the very few. Although there are probably a number of people who saw some of the shows."
George continued, "At the suggestion of a local vocal teacher, Ralph Woolley, I auditioned for the male chorus of 'Rio Rita' in 1950 and was stunned to win the lead role of Captain Jim, of the Texas Rangers. The late Bettie Jones of East York was the romantic lead."
He added, "In 1951, bagpipers and a large chorus were featured in the wonderful 'Brigadoon,' with beautiful songs like 'The Heather on the Hill' and the title song. As leading man my 'love interest' was Joyce Forman, but the talented Isabelle Tubb was the talent find of the year."
"The 1952 show was 'Bloomer Girl,'" George noted, which "starred a member of the famous theatrical family, Ethel Barrymore, who also used her married last name, Colt. Local regular actors who had roles in many shows included Harry Seyler, Alverta Keller, Don Hoke and Bill Flinchbaugh. All four have passed, I think. Emanuel Cassimatis and I were singing sons-in-law. Emanuel in later years served as a judge in the York County courts. He too has passed away. Isabelle Tubb was, again, a local star."
George said, "My last participation was the wonderful and powerful Rodgers and Hammerstein 'South Pacific.' This show still plays on many stages even today. Songs like 'Some Enchanted Evening,' 'There is Nothing Like a Dame,' 'Happy Talk,' and 'This Nearly was Mine' are still popular. My role was that of Lt. Joseph Cable, who fell in love with the beautiful native girl Liat. My 'feature' song was 'You've Got to be Taught,' dealing musically with racial situations. The lyrics were strong and an affirmation of the strong feelings of Hammerstein."
He continued, "The leading lady was Betty Jane Watson, who created the female lead in 'Oklahoma' for four years on Broadway. Yorker Dale Uffelman was the York male lead for South Pacific, with Anita Cassimatis as Bloody Mary. Strong male and female chorus members were wonderful."
And, George concluded, "I think the Legion produced two or probably three more shows, but then I was deeply involved in York High and York Catholic play-by-play and show rehearsals were out of the question. I sometimes did four and five games a week. Basketball, that is, on WORK radio, 1350. In later years I was very much back in the theater with the 'Kenley' players, but that was summer stock, and no play-by-play. Thanks to you and Anita for bringing back some wonderful career memories."
Of note here, I just put together that the Anita who asked about these shows originally was almost certainly Anita Cassimatis, as she mentioned in her original letter that her family originally owned the Ramona restaurant in York, which, thanks to a letter from another local notable, William Hoffmeyer, I realized had been owned by the Cassimatis family, specifically Judge Cassimatis' parents.
I had another memory of the Legion shows from reader Ronald L. Walter, who wrote, "Concerning the Legion shows, my sister Carolyn Walter and my brother Gerald Walter played parts in those shows. She was a soprano and he was a tenor. Actually my sister became Miss York County in 1950. Also sang on local radio – WORK."
Last up for today, I want to switch to a different musical venue – York's Valencia Ballroom. These notes come from reader Mildred Shaffer of Shrewsbury, who wrote after seeing a previous column on the Valencia:
"My memories too go back a long time. When I graduated from the Thompson business school in 1938 my first job was in the office of the Valencia. My boss was Steve Tassia. This was the Big Band era. We had a long list of patrons on our mailing list to receive notice of any upcoming band. During the two years I worked there we published a 10th anniversary book with stories and pictures of each band leader plus many pictures of the ballroom, inside and outside; the Rainbow Grill, which was downstairs; plus all of the Tassia family and the Valencia's history."
She continued, "All attendance records were roken when Kay Kyser's orchestra appeared at the Valencia on Friday, March 3, 1939. Kyser's orchestra was one of the first to play in the Valencia, filling an engagement there on March 2, 1929. Attendance on that night numbered 134. What a comparison 10 years later!"
She added, "I still have my book and get it every now and then – brings back a lot of great memories. I sold tickets for the student hop on Friday evenings. Miss Sadie had rules and regulations and it was a well-behaved crowd."
Mildred noted when she wrote her letter (which I believe was in 2015) she was turning 95, and she commented, "What a change in dancing I've seen!" I believe that, Mildred!
Have questions or memories to share? Email me at joan@joanconcilio.com or write to Ask Joan, York Daily Record/Sunday News, 1891 Loucks Road, York PA 17408. We cannot accept any phone calls with questions or information.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,020 |
Q: Difference in items height at layouts with same DP value I am running 2 emulators:
*
*360x640 mdpi (1x)
*1080x1920 xxhdpi (3x)
Using dp values when defining attributes for views should result in covering same physical space, but as the result they look different.
What might be the reason for such outcome?
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{"url":"http:\/\/erucolindo3.com\/resources\/cndcalc.php","text":"]> Statistics: Cumulative Normal Distribution Calculator\n\n# Cumulative Normal Distribution Calculator\n\nCalculates $P\\left(X for any given number x and normal random variable X with mean \u03bc and standard deviation \u03c3.","date":"2018-01-21 06:13:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.6962040662765503, \"perplexity\": 1029.4852022227512}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084890314.60\/warc\/CC-MAIN-20180121060717-20180121080717-00044.warc.gz\"}"} | null | null |
Q: how to fix "comparison between pointer and integer" error in C++? I am trying to write a program that takes in 2 string inputs and assigns a numeric value to each letter in the string input. The program then multiples the numeric values assigned to the letters in the string inputs and sees if the product of the values (mod 47) of both inputs are equal. If they are equal, then the program must output "yes" if not, then it must output "NO".
I am using a brute force method, so please feel free to suggest more efficient methods and also how to implement them. I have reached a point where my IDE only gives several warnings, but no errors. However when I run the program, I get an illegal instruction, which is essentially because of a particular function, which I will show later.
I WOULD PREFER THAT THE SOLUTIONS TO MY QUESTION DON'T INCLUDE ANY EXTRA CLASS LIBRARIES AND FILES.
THANK YOU ;)
I've tried using a brute force method e.g in this long function:
This takes in values from a function of return type int which assigns the numeric value to the 6 letter input, and returns them to a variable of return type bool, which finds and compares the final product of the two strings before returning a value of type bool. I've used the brute force method of comparing each individual letter of the string to a letter and assigning a value. I am quite sure that the first function isn't right, although I don't know how to fix it.
int parse(const char * x)
{
if (x == "A")
{
return 1;
}
if (x == "B")
{
return 2;
}
if (x == "C")
{
return 3;
}
if (x == "D")
{
return 4;
}
if (x == "E")
{
return 5;
}
if (x == "F")
{
return 6;
}
if (x == "G")
{
return 7;
}
if (x == "H")
{
return 8;
}
if (x == "I")
{
return 9;
}
if (x == "J")
{
return 10;
}
if (x == "K")
{
return 11;
}
if (x == "L")
{
return 12;
}
if (x == "M")
{
return 13;
}
if (x == "N")
{
return 14;
}
if (x == "O")
{
return 15;
}
if (x == "P")
{
return 16;
}
if (x == "Q")
{
return 17;
}
if (x == "R")
{
return 18;
}
if (x == "S")
{
return 19;
}
if (x == "T")
{
return 20;
}
if (x == "U")
{
return 21;
}
if (x == "V")
{
return 22;
}
if (x == "W")
{
return 23;
}
if (x == "X")
{
return 24;
}
if (x == "Y")
{
return 25;
}
if (x == "Z")
{
return 26;
}
}
bool returnfunc(std::string & GROUP, std::string & k)
{
const char A = GROUP[0];
const char B = GROUP[1];
const char C = GROUP[2];
const char D = GROUP[3];
const char E = GROUP[4];
const char F = GROUP[5];
int xet = parse(&A);
int r = parse(&B);
int m = parse(&C);
int z = parse(&D);
int h = parse(&E);
int j = parse(&F);
double mu = (xet * r * m * z * h * j) % 47;
const char G = k[0];
const char H = k[1];
const char I = k[2];
const char J = k[3];
const char K = k[4];
const char L = k[5];
int w = parse(&G);
int x = parse(&H);
int y = parse(&I);
int a = parse(&J);
int b = parse(&K);
int c = parse(&L);
double fin = (w * x * y * a * b * c) % 47;
{
if (mu == fin)
{
return true;
}
else
{
return false;
}
}
}
/*I expected the program to run but it didn't and I don't understand
why.*/
A: This is wrong
if (x=="A")
because it compares two pointers for pointer equality. It does not compare what those pointers are pointing at (which is what you want to do).
Here's a simpler version of your code that keeps the same structure but does not involve any pointers (for some reason newbies love pointers, given how difficult they are I've never understood why).
int parse(char x)
{
if (x == 'A')
return 1;
if (x == 'B')
return 2;
// etc etc
}
Then
int xet = parse(A);
int r = parse(B);
// etc etc
PS, the more consise way to write your parse function would be to use a lookup table and a loop
int parse(char x)
{
const char* table = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
for (int i = 0; i < 26; ++i)
if (x == table[i])
return i + 1;
return 0; // error return
}
A: int parse(const char * x) {
if (x == "A")
x is a const char* and "A" is a const char* but they will not be equal even if they point at C strings with the same characters since the comparison compares the addresses of the C strings, not the contents.
Also, if your parse function does not find a match it will leave the function without returning a value which is Undefined Behaviour.
Try this instead:
int parse(char x) {
if(x >= 'A' && x <= 'Z') return x - 'A' + 1;
else return 0;
}
And call parse with the char itself as an argument (instead of a pointer to it):
const char A = GROUP[0];
int xet = parse(A);
A: It's hard to answer this mostly because of this part of your question:
I have reached a point where my IDE only gives several warnings, but no errors.
There may be a warning that we don't know about that might be part of your problem. However, I'm going to take a stab anyway at a couple of things that are wrong.
So, first is this line right here:
int parse ( const char * x)
Now in and of itself, this isn't bad, except it's coupled with something like this:
if(x=="A")
{
return 1;
}
That is all sort of bad. x is a pointer. That means that the value store in x is an address. That means you are comparing the c string "A" to an address. That's not usually a good thing to do.
What you want is probably this instead:
if(*x == "A")
That's much better.
Well, not quite, actually, because you probably want single quotes here instead of double quotes:
if(*x == 'A')
By the way, you could probably simplify your parse function as follows, and it'd do the same thing:
int parse ( const char * x) {
return (*x - 'A') + 1; // parenthesis for clarity. Not really needed here.
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,173 |
\section{Introduction}
An important and interesting topic in the theory of Diophantine approximation is the distribution of algebraic numbers and algebraic integers \cite{BeBeGo10, Cas57, Spr67, Sch80}. In this paper we consider problems related to the distribution of points with algebraic conjugate integer coordinates in the plane.
Let us start with some useful notation. Let $n$ be a positive integer and $Q>1$ be a sufficiently large real number. Given a polynomial $P(t)=a_nt^n+\ldots+a_1t+a_0\in\mathbb{Z}[t]$ denote by $H(P)=\max\limits_{0\leq j \leq n}{|a_j|}$ the height of the polynomial $P$, and by $\deg P$ the degree of the polynomial $P$. We define the following classes of integer polynomials with bounded height and degree:
\[
\mathcal{P}_{n}(Q):=\{P\in\mathbb{Z}[t]:\,\deg P\leq n,\, H(P)\leq Q\}.
\]
\[
\tilde{\mathcal{P}}_{n}(Q):=\{P\in\mathbb{Z}[t]:\,\deg P=n,\, H(P)\leq Q,\, a_n=1\}.
\]
Denote by $\#S$ the cardinality of a finite set $S$ and by $\mu_k S$ the Lebesgue measure of a measurable set $S\subset \mathbb{R}^k$, $k\in\mathbb{N}$. Furthermore, denote by $c_j>0$, $j\in \mathbb{N}$ positive constants independent of $Q$. We are going to use the Vinogradov symbol $A\ll B$, which means that there exists a constant $c>0$ independent of $A$ and $B$ such that $A\leq c\cdot B$. We will write $A\asymp B$ when $A\ll B$ and $B\ll A$.
Now let us introduce the concept of an algebraic integer point. A point $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ is called an {\it algebraic point} if $\alpha_1$ and $\alpha_2$ are roots of the same irreducible polynomial $P\in\mathbb{Z}[t]$. If the leading coefficient $a_n$ of polynomial $P$ is equal to 1, then a point $\boldsymbol{\alpha}$ is called an {\it algebraic integer point}. The polynomial $P$ is called the minimal polynomial of the point $\boldsymbol{\alpha}$. Denote by $\deg(\boldsymbol{\alpha})=\deg P$ the degree of the point $\boldsymbol{\alpha}$ and by $H(\boldsymbol{\alpha})=H(P)$ the height of the point $\boldsymbol{\alpha}$. We denote by $\mathbb{A}^2$ (respectively $\tilde{\mathbb{A}}^2$) the set of algebraic points (respectively integer algebraic points). Furthermore, we define the following sets:
\[
\mathbb{A}_n^2(Q):=\left\{\boldsymbol{\alpha}\in \mathbb{A}^2:\, \deg{\boldsymbol{\alpha}}\leq n,\, H(\boldsymbol{\alpha})\leq Q\right\},
\]
\[
\tilde{\mathbb{A}}_n^2(Q):=\left\{\boldsymbol{\alpha}\in \tilde{\mathbb{A}}^2:\, \deg{\boldsymbol{\alpha}}= n,\, H(\boldsymbol{\alpha})\leq Q\right\}.
\]
The problem of determining the number of integer points in regions and bodies of $\mathbb{R}^k$ can be naturally generalized to estimating the number of rational points in domains of Euclidean spaces. Let $f:J_0\rightarrow \mathbb{R}$ be a continuously differentiable function defined on a finite open interval $J_0\subset\mathbb{R}$. Let us consider the following set:
\[
N_f(Q,\gamma, J):=\left\{\left(\frac{p_1}{q},\,\frac{p_2}{q}\right)\in\mathbb{Q}^2:\,0<q\leq Q,\, \frac{p_1}{q}\in J,\,\left|f\left(\frac{p_1}{q}\right)-\frac{p_2}{q}\right|<Q^{-\gamma}\right\},
\]
where $J\subset J_0$ and $0\leq\gamma < 2$. Thus, the quantity $\# N_f(Q,\gamma, J)$ denotes the number of rational points with bounded denominators lying within a certain neighborhood of the curve parametrized by $f$. The problem is to estimate the value $\# N_f(Q,\gamma, J)$. This question was considered by Huxley \cite{H96}, Vaughan, Velani \cite{VoVe06} and Beresnevich, Dickinson, Velani \cite{BeDiVe07} and with some additional restrictions on the function $f$ it was proved that
\[
\# N_f(Q,\gamma,J)\asymp Q^{3-\gamma}.
\]
This result was obtained using methods of metric number theory introduced by Schmidt in \cite{Sch80}.
The following natural extension of this question is the problem of distribution of algebraic points $\boldsymbol{\alpha}\in\mathbb{A}_n^2(Q)$ near smooth curves. Let $\varphi:J_0\rightarrow \mathbb{R}$ be a continuously differentiable function defined on a finite open interval $J_0\subset\mathbb{R}$ satisfying the conditions:
\begin{equation}\label{eq1_1}
\sup\limits_{x\in J_0}|\varphi'(x)|:= c_{4} < \infty,\qquad\#\{x\in J_0:\varphi(x)=x\} := c_{5}<\infty.
\end{equation}
Define the following set:
\[
M^n_\varphi(Q,\gamma, J):= \left\{\boldsymbol{\alpha}\in\mathbb{A}_n^2(Q):\,\alpha_1\in J, |\varphi(\alpha_1)-\alpha_2|\ll Q^{-\gamma}\right\},
\]
where $J\subset J_0$. The goal is to estimate the number $\# M^n_\varphi(Q,\gamma, J)$. A first attempt to solve this problem for $0<\gamma\leq\frac12$ has been made in \cite{BeGoKu14}. This result was complemented by lower bound of the right order for $0<\gamma<\frac34$ \cite{BeGoGu16a} and finally by lower and upper bounds of the same order for $0<\gamma<1$ \cite{BeGoGu16b}. We are going to state the final result in the following form: for any smooth function $\varphi$ with conditions \eqref{eq1_1} we have $\# M^n_\varphi(Q,\gamma, J) \asymp Q^{n+1-\gamma}$ for $Q>Q_0(n, J, \varphi,\gamma)$ and $0< \gamma < 1$.
We will consider the same problem in the case of integer algebraic points. Let
\[
\tilde{M}^n_\varphi(Q,\gamma, J):= \left\{\boldsymbol{\alpha}\in\tilde{\mathbb{A}}_n^2(Q):\,\alpha_1\in J, |\varphi(\alpha_1)-\alpha_2|<c_1Q^{-\gamma}\right\}.
\]
\begin{theorem}\label{main}
For any smooth function $\varphi$ with conditions \eqref{eq1_1} there exist positive values $c_{2},c_{3}>0$ such that
\[
c_{2}\cdot Q^{n-\gamma}<\# \tilde{M}^n_\varphi(Q,\gamma, J) < c_{3}\cdot Q^{n-\gamma}
\]
for $Q>Q_0(n, J,\varphi,\gamma)$, $0< \gamma < 1$, $n\ge 3$ and a sufficiently large constant $c_1>0$ in definition of the set $\tilde{M}^n_\varphi(Q,\gamma, J)$.
\end{theorem}
It should be noted that a lower bound of $\# \tilde{M}^n_\varphi(Q,\gamma, J)$ for $0< \gamma \leq \frac12$ was obtained earlier \cite{GoGu16}. Hence we will assume for the lower bound that $\gamma >\frac12$.
The proof of Theorem \ref{main} is based on the following idea. We consider the strip $L^n_\varphi(Q,\gamma, J):= \left\{\mathbf{x}\in\mathbb{R}^2: x_1\in J, |\varphi(x_1)-x_2|<c_1Q^{-\gamma}\right\}$ and cover it with squares $\Pi=I_1\times I_2$ with sides of length $\mu_1 I_1=\mu_1 I_2 = c_6Q^{-\gamma}$, where $c_6=c_1 / \left(\frac12+c_4\right)$ (the full description of this scheme is given in \cite{BeGoGu16b}). Thus, in order to prove Theorem \ref{main} we need to estimate the number of integer algebraic points lying in such a square $\Pi$.
Let us consider here a more general case, namely, the case of a rectangle $\Pi=I_1\times I_2$, where $\mu_1 I_i = c_6Q^{-\gamma_i}$.
\begin{theorem}\label{th1}
Let $\Pi=I_1\times I_2$ be a rectangle with a midpoint $\mathbf{d}$ and sides $\mu_1 I_i=c_6Q^{-\gamma_{i}}$, $i=1,2$. Then for $0<\gamma_1,\gamma_2< 1$, $n\ge 2$ and $Q>Q_0(n,\gamma, \mathbf{d})$ the estimate
\[
\# \left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi\right) < c_{7}\cdot Q^{n}\mu_2\Pi
\]
holds, where
\[
c_{7}=2^{3n+8}n^2\rho_n(d_1)\rho_n(d_2)\left|d_1-d_2\right|^{-1}\quad\text{and}\quad\rho_n(x)=\left(\left(|x|+1\right)^{n+1}-1\right)\cdot |x|^{-1}.
\]
\end{theorem}
The case of lower bound is more difficult. It is easy to prove that there exist rectangles $\Pi$ of size $\mu_2 \Pi \asymp Q^{-1}$ such that $\# \left(\mathbb{A}_n^2(Q)\cap\Pi)\right)= 0$, and since $\tilde{\mathbb{A}}_n^2(Q)\subset\mathbb{A}_n^2(Q)$ we have $\# \left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi)\right)= 0$ for such rectangles. It means that we cannot obtain non-zero lower bounds for all rectangles $\Pi$. In particular, it is easy to show that certain neighborhoods of algebraic points of small height and small degree do not contain any other algebraic points $\boldsymbol{\alpha}\in\mathbb{A}_n^2(Q)$ at all. In order to avoid such domains we use the concept of a {\it $(v_1,v_2)$-special} square, which has been introduced in \cite{BeGoGu16b}.
\begin{definition}
Let $\Pi=I_1\times I_2$ be a square with midpoint $\mathbf{d}$, $d_1\neq d_2$ and sides $\mu_1 I_1 = \mu_1 I_2 = c_6Q^{-\gamma}$ such that $\frac12< \gamma< 1$. We say that the square $\Pi$ satisfies the {\it $(l,v_1,v_2)$-condition} if $v_1+v_2=1$ and there exist at most $\delta^3\cdot 2^{l+3}Q^{1+2\lambda_{l+1}}\mu_2\Pi$ polynomials $P\in\mathcal{P}_2(Q)$ of the form $P(t)=a_2t^2+a_1t+a_0$ satisfying the inequalities
\[
\begin{cases}
|P(x_{0,i})|<h\cdot Q^{-v_i},\quad i=1,2,\\
\delta Q^{\lambda_{l+1}}\leq|a_2|<\delta Q^{\lambda_l}
\end{cases}
\]
for some point $\mathbf{x}_0\in \Pi$, where $\delta = 2^{-L-17}h^{-2}\cdot (d_1-d_2)^2$, $L=\left[\textstyle\frac{3-2\gamma}{1-\gamma}\right]$ and
\[
\lambda_l=
\begin{cases}
1-\frac{(l-1)(1-\gamma)}{2},\quad 1\leq l\leq L+1,\\
\gamma-\frac12,\quad l= L+2,\\
0,\quad l\ge L+3.
\end{cases}
\]
\end{definition}
\begin{definition}
The square $\Pi=I_1\times I_2$ with sides $\mu_1 I_1 = \mu_1 I_2 = c_6Q^{-\gamma}$ such that $\frac12< \gamma< 1$ is called a {\it $(v_1,v_2)$-special} square if it satisfies the {\it $(l,v_1,v_2)$-condition} for all $1\leq l\leq L+2$.
\end{definition}
The following theorem can be proved for {\it $(v_1,v_2)$-special} squares.
\begin{theorem}\label{th2}
For all {\it $\left(\textstyle\frac12,\textstyle\frac12\right)$-special} squares $\Pi=I_1\times I_2$ with midpoints $\mathbf{d}$, $d_1\neq d_2$ and sides $\mu_1 I_1 = \mu_1 I_2 =c_6Q^{-\gamma}$, where $\frac12< \gamma< 1$ and $c_6>c_0(n,\mathbf{d})$, there exists a value $c_{8}=c_{8}(n,\mathbf{d},\gamma)>0$ such that
\[
\#\left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi\right)> c_{8}\cdot Q^{n}\mu_2\Pi
\]
for $Q>Q_0(n,\mathbf{d},\gamma)$ and $n\ge 3$.
\end{theorem}
\section{Auxiliary statements}
This section contains several lemmas which will be used to prove Theorems \ref{th1} and \ref{th2}. Some of them are related to geometry of numbers, see \cite{Cas97}. The first paper discussing approximations by algebraic integers are due to Davenport and Schmidt \cite{DavSch}. Recently their approach has been further developed by Bugeaud \cite{Bug03} and we shall use ideas of this paper.
\begin{lemma}[Minkowski 2nd theorem on successive minima]\label{lm_Minkowski}
Let $K\subset\mathbb{R}^n$ be a bounded central symmetric convex body with successive minima $\tau_1,\ldots,\tau_n$ and volume $V(K)$.
Then
\[
\frac{2^n}{n!}\leq\tau_1\tau_2\ldots\tau_nV(K)\leq 2^n.
\]
\end{lemma}
For a proof, see \cite[pp. 203]{Cas97}, \cite[pp. 59]{Gruber}.
\begin{lemma}[Bertrand postulate]\label{lm_Bertran}
For any $n\in\mathbb{N}$, $n \ge 2$ there exists a prime $p$ such that $n<p<2n$.
\end{lemma}
It was proved by Chebyshev in 1850. A proof can be found, for example, in \cite[Theorem 2.4]{Nesterenko}.
\begin{lemma}[Eisenstein's criterion]\label{lm_Eisenstein}
Let $P\in\mathbb{Z}[t]$ be a polynomial of the form $P(t)=a_nt^n+a_{n-1}t^{n-1}+\ldots +a_1t+a_0$. If there exists a prime number $p$ such that:
\begin{equation}\label{eq2_1}
\begin{cases}
a_n\not\equiv 0 \mod{p},\\
a_i\equiv 0 \mod{p},\quad i=0,\ldots, n-1\\
a_0\not\equiv 0 \mod{p^2},
\end{cases}
\end{equation}
then $P$ is irreducible over the rational numbers.
\end{lemma}
For a proof see \cite{Eis}, \cite[Theorem 2.1.3]{Prasolov}.
\begin{lemma}\label{lm_polynomial}
Consider a point $x\in\mathbb{R}$ and a polynomial $P$ with zeros $\alpha_1,\alpha_2,\ldots,\alpha_n$
where $|x-\alpha_1| = \min\limits_{i} |x-\alpha_i|$. Then
\[
|x-\alpha_1| \le n\cdot|P(x)|\cdot|P'(x)|^{-1}.
\]
\end{lemma}
\begin{proof}
Evaluate the polynomial $P$ and its derivative $P'$ at the point $x\neq\alpha_i$ for $i=1,2,\ldots,n$. Since
\[
|P(x)|\cdot|P'(x)|^{-1}\leq\sum\limits_{i=1}^n|x-\alpha_i|^{-1}\leq n\cdot|x-\alpha_1|^{-1},
\]
we obtain
\[
|x-\alpha_1| \le n\cdot|P(x)|\cdot|P'(x)|^{-1}.
\]
\end{proof}
\begin{lemma}[see \cite{Fel51}]\label{lm5}
For any subset of roots $\alpha_{i_1},\ldots,\alpha_{i_s}$, $1\leq s\leq n$, of the polynomial $P(t)=a_nt^n+\ldots+a_1t+a_0$ we have $\prod\limits_{j=1}^{s}|\alpha_{i_j}|\leq (n+1)2^n\cdot H(P)\cdot |a_n|^{-1}$.
\end{lemma}
\begin{lemma}[see {\cite{BeGoGu16b}}]\label{lm_BeGoGu}
Let $\Pi=I_1\times I_2$ be a square with midpoint $\mathbf{d}$, $d_1\neq d_2$ and sides $\mu_1 I_1 = \mu_1 I_2 =c_6Q^{-\gamma}$, where $\frac12< \gamma< 1$ and $c_6>c_0(n,\mathbf{d})$. Given positive values $v_1,v_2$ such that $v_1+v_2=n-1$, let $L=L_n(Q,\delta_n,\mathbf{v},\Pi)$ be the set of points $\mathbf{x}\in\Pi$ such that there exists a polynomial $P\in\mathcal{P}_{n}(Q)$ satisfying the following system of inequalities:
\[
\begin{cases}
|P(x_i)|< h_n\cdot Q^{-v_i},\\
\min\limits_i\{|P'(x_i)|\}<\delta_n\cdot Q,\quad i=1,2,\\
\end{cases}
\]
where $h_n = \sqrt{\frac32(|d_1|+|d_2|)\cdot \max\left(1,3|d_1|,3|d_2|\right)^{n^2}}$. If $\Pi$ is a {\it $\left(\frac{v_1}{n-1},\frac{v_2}{n-1}\right)$-special} square, then
\[
\mu_2 L<\textstyle\frac14 \cdot\mu_2\Pi
\]
for $\delta_n<\delta_0(n,\mathbf{d})$ and $Q>Q_0(n,\mathbf{v},\mathbf{d},\gamma)$.
\end{lemma}
\begin{lemma}[see {\cite{BeGoGu16b}}]\label{lm6}
Let $G=G(\mathbf{d},\mathbf{K})$, where $|d_1-d_2|>\varepsilon_1>0$, be a set of points $\mathbf{b}=(b_1,b_0)\in\mathbb{Z}^2$ such that
\[
|b_1d_i+b_0|\leq K_i,\quad i=1,2.
\]
Then
\[
\# G\leq \left(4\varepsilon_1^{-1}K_1+1\right)\cdot\left(4K_2+1\right).
\]
\end{lemma}
\section{Proof of Theorem \ref{th1}}
Assume that $\#\left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi\right) \ge c_{7}\cdot Q^{n}\mu_2\Pi$. Take an integer algebraic point $\boldsymbol{\alpha}\in\tilde{\mathbb{A}}_n^2(Q)\cap\Pi$ with minimal polynomial $P$. Let us give an estimate for the polynomial $P$ at the points $d_1$ and $d_2$. Since $\alpha_i\in I_i$, we have
\[
|P^{(k)}(\alpha_{i})|\leq\sum\limits_{j=k}^{n-1}{\textstyle\frac{j!}{(j-k)!}\cdot|a_{j}|\cdot |\alpha_i|^{j-k}}+\textstyle\frac{n!}{(n-k)!}\cdot |\alpha_i|^{n-k}< \textstyle\frac{n!}{(n-k)!}\cdot \rho_n(d_i)\cdot Q
\]
for all $1\leq k\leq n$ and $Q>Q_0$. From these estimates and Taylor expansion of $P$ in the intervals $I_i$, $i=1,2$, we obtain the following inequality:
\begin{multline}\label{eq3_1}
|P(d_i)|\leq\sum\limits_{k=1}^n{\left|\textstyle\frac{1}{k!}P^{(k)}(\alpha_i)(d_i-\alpha_i)^k\right|}<\\
< \sum\limits_{k=1}^n{2^{-k}\textstyle{k\choose n}\rho_n(d_i)\cdot Q\mu_1 I_i}\leq 2^n\rho_n(d_i)\cdot Q\mu_1 I_i.
\end{multline}
Let us fix the vector $\mathbf{A}_{1}=(1,a_{n-1},\ldots,a_{2})$, where $a_{n-1},\ldots,a_{2}$ are the coefficients of the polynomial $P\in \tilde{\mathcal{P}}_{n}(Q)$. Denote by $\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1})\subset \tilde{\mathcal{P}}_n(Q)$ the subclass of polynomials $P$ with the same vector of coefficients $\mathbf{A}_{1}$ such that $P$ satisfies \eqref{eq3_1}. The number of subclasses $\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1})$ is equal to the number of vectors $\mathbf{A}_{1}$, which for $Q>Q_0$ can be estimated as follows:
\begin{equation}\label{eq3_2}
\#\{\mathbf{A}_{1}\}=(2Q+1)^{n-2}< 2^{n-1}\cdot Q^{n-2}.
\end{equation}
It should also be noted that every point of the set $\tilde{\mathbb{A}}_n^2(Q)\cap\Pi$ corresponds to some polynomial $P\in\tilde{\mathcal{P}}_n(Q)$ that satisfies \eqref{eq3_1}. On the other hand, every polynomial $P\in\tilde{\mathcal{P}}_n(Q)$ satisfying (\ref{eq3_1}) corresponds to at most $n^2$ points of the set $\tilde{\mathbb{A}}_n^2(Q)\cap\Pi$. This allows us to write
\[
c_{7}\cdot Q^{n+1}\mu_2\Pi< \# \left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi\right)\leq n^2\sum\limits_{\mathbf{A}_{1}}\# \tilde{\mathcal{P}}_n(Q,\mathbf{A}_{1}).
\]
Thus, by the estimate \eqref{eq3_2} and Dirichlet's box principle applied to vectors $\mathbf{A}_{1}$ and polynomials $P$ satisfying \eqref{eq3_1}, there exists a vector $\mathbf{A}_{1,0}$ such that
\begin{equation}\label{eq3_2}
\#\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1,0})\ge c_{7}\cdot 2^{-n+1}n^{-2}Q^{2}\mu_2\Pi.
\end{equation}
Let us find an upper bound for the value $\#\tilde{\mathcal{P}}_n(Q,\mathbf{A}_{1,0})$. To do this, we fix some polynomial $P_0\in\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1,0})$ and consider the difference between the polynomials $P_0$ and $P_j\in\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1,0})$ at the points $d_i$, $i=1,2$. From the estimate \eqref{eq3_1} it follows that
\[
|P_0(d_i)-P_j(d_i)|=|(a_{0,1}-a_{j,1})d_i+(a_{0,0}-a_{j,0})|\leq 2^{n+1}\rho_n(d_i)\cdot Q\mu_1 I_i.
\]
Thus the number of different polynomials $P_j\in\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1,0})$ does not exceed the number of integer solutions of the following system:
\[
|b_1d_i+b_0|\leq 2^{n+1}\rho_n(d_i)\cdot Q\mu_1 I_i,\quad i=1,2.
\]
Now, let us use Lemma \ref{lm6} for $K_i=2^{n+1}\rho_n(d_i)\cdot Q\mu_1 I_i$. Since $\mu_1 I_i = c_6 Q^{-\gamma_i}$ and $\gamma_i < 1$, we have $K_i\ge 2^{n+1}\rho_n(d_i)c_6\cdot Q^{1-\gamma_i}>\max\{\varepsilon_1,1\}$ for $Q>Q_0$. This implies
\[
j \leq 2^{2n+8}|d_1-d_2|^{-1}\rho_n(d_1)\rho_n(d_2)\cdot Q^2\mu_2\Pi.
\]
It follows that $\#\tilde{\mathcal{P}}_{n}(Q,\mathbf{A}_{1,0})\leq 2^{2n+8}|d_1-d_2|^{-1}\rho_n(d_1)\rho_n(d_2)\cdot Q^2\mu_2\Pi$, which contradicts inequality \eqref{eq3_2} for $c_{7}=2^{3n+8}n^2\rho_n(d_1)\rho_n(d_2)|d_1-d_2|^{-1}$. This leads to
\[
\#\left(\tilde{\mathbb{A}}_n^2(Q)\cap\Pi\right)< c_{7}\cdot Q^{n}\mu_2\Pi.
\]
\section{Proof of Theorem \ref{th2}}
Since $d_1\neq d_2$ we can assume that for $Q>Q_0$ the following inequality
\[
|x_1-x_2|>\varepsilon=\textstyle\frac{|d_1-d_2|}{2}
\]
is satisfied for every point $\mathbf{x}\in\Pi$.
In order to prove the Theorem \ref{th2} we use Lemma \ref{lm_BeGoGu}. Given positive constants $u_1$ and $u_2$ satisfying the condition $u_1+u_2=n-2$ let $L=L_{n-1}(Q,\delta,\mathbf{u},\Pi)$ be the set of points $\mathbf{x}\in\Pi$ such that the following system of inequalities
\begin{equation}\label{eq4_1}
\begin{cases}
|P(x_i)|< h_{n-1}\cdot Q^{-u_i},\\
\min\limits_i\{|P'(x_i)|\}<\delta\cdot Q,\quad i=1,2,\\
\end{cases}
\end{equation}
has a solution in polynomials $P\in\mathcal{P}_{n-1}(Q)$. Lemma \ref{lm_BeGoGu} implies that the measure of the set $L$ can be estimated as
\[
\mu_2 L\leq\textstyle\frac14\cdot \mu_2\Pi
\]
for $\delta<\delta_0(n-1,\mathbf{d})<1$ and $Q>Q_0(n-1,\mathbf{u},\mathbf{d},\gamma)$.
Let us consider the set $B=\Pi\setminus L$. Using Minkowski's linear form theorem \cite[Ch. 2, \S 3]{Sch80} for every point $\mathbf{x}\in \Pi$ there exists a polynomial $P\in \mathcal{P}_{n-1}(Q)$ such that
\[
|P(x_i)|\leq h_{n-1}\cdot Q^{-u_i},\quad i=1,2.
\]
Thus, we can assert that for every point $\mathbf{x} \in B$ there exists an irreducible polynomial $P\in\mathcal{P}_{n-1}(Q)$ such that
\[
\begin{cases}
|P(x_{i})|<h_{n-1}\cdot Q^{-u_i},\\
|P'(x_{i})|>\delta\cdot Q,\quad i=1,2,
\end{cases}
\]
and $\mu_2\, B \ge\textstyle\frac34\cdot \mu_2\Pi$.
Consider an arbitrary point $\mathbf{x} \in B$ and let us examine the successive minima
$\tau_1,\ldots,\tau_n$ of the compact convex set defined by
\begin{equation}\label{eq4_2}
\begin{cases}
|a_{n-1}x_{i}^{n-1}+\ldots+a_1x_{i}+a_0|\leq h_{n-1}Q^{-u_i},\\
|(n-1)a_{n-1}x_{i}^{n-2}+\ldots+2a_2x_{i}+a_1| \leq Q,\quad i=1,2,\\
|a_{n-1}|,\ldots,|a_2|\leq Q.
\end{cases}
\end{equation}
Assume that $\tau_1\leq\delta$. Then for sufficiently small $\delta$ there exists a polynomial $P\in\mathcal{P}_{n-1}(Q)$ such that the inequalities
\[
\begin{cases}
|P(x_{i})| < \delta h_{n-1} Q^{-u_i}<h_{n-1}Q^{-u_i},\\
|P'(x_{i})|<\delta Q,\quad i=1,2,\\
H(P) < Q
\end{cases}
\]
hold. This leads to a contradiction, since $\mathbf{x}\not\in L$. Thus $\tau_1>\delta$. Since the volume of the compact convex set defined by the inequalities \eqref{eq4_2} is at least $2^n$, it follows from Lemma \ref{lm_Minkowski} that $\tau_1\ldots\tau_{n}\leq 1$ and $\tau_{n}\leq \delta^{-n+1}$. Thus, by definition of successive minima, we can choose $n$ linearly independent polynomials $P_j(t)=a_{j,n-1}t^{n-1}+\ldots+a_{j,1}t+a_{j,0}\in\mathcal{P}_{n-1}(Q)$, $1\leq j\leq n$, satisfying
\begin{equation}\label{eq4_3}
\begin{cases}
|P_j(x_{i})|\leq \delta^{-n+1}h_{n-1}Q^{-u_i},\\
|P_j'(x_{i})| \leq \delta^{-n+1}Q,\quad i=1,2,\\
|a_{j,k}|\leq \delta^{-n+1}Q, \quad 4\leq k\leq n-1.
\end{cases}
\end{equation}
Using well-known estimates from the geometry of numbers, see \cite[pp. 219]{Cas97}, we obtain for the polynomials $P_j$, $1\leq j\leq n$ the inequality:
\[
\Delta = \det|(a_{j,k-1})^n_{j,k=1}|\leq n!.
\]
For a prime $p$ not dividing $\Delta$, Lemma \ref{lm_Bertran} yields
\begin{equation}\label{eq4_4}
n! <p< 2n!.
\end{equation}
Consider the system of linear equations for the $n$ variables $\theta_1,\ldots,\theta_n$
\begin{equation}\label{eq4_5}
\begin{cases}
x_{i}^n+p\sum\limits_{j=1}^{n}{\theta_j P_j(x_{i})}=p(n+1)\delta^{-n+1}h_{n-1}Q^{-u_i},\\
nx_{i}^{n-1}+p\sum\limits_{j=1}^{n}{\theta_j P_j'(x_i)}=pQ+ p\sum\limits_{j=1}^{n}{|P_j'(x_{i})|},\quad i=1,2,\\
\sum\limits_{j=1}^{n}{\theta_j a_{j,k-1}}=0,\quad 5\leq k\leq n.
\end{cases}
\end{equation}
It should be mentioned that in case $n=3$ the values $|P_j'(\alpha_{j,1})|$ and $|P_j'(\alpha_{j,2})|$ are equal, where $\alpha_{j,1}$ and $\alpha_{j,2}$ are the roots of the polynomial $P_j$. It means that one of the equations numbered 2 and 3 can be removed.
In order to fined the determinant of this system, we transform it as follows.
Multiply the equation numbered as $k=5, 6, \ldots, n$ by $p\cdot x_1^{k-1}$ (respectively by $p \cdot x_2^{k-1}$) and subtract it from the first (respectively the second) equation of the system \eqref{eq4_5}. Similarly multiply the equation numbered as $k=5, 6, \ldots, n$ by $p\cdot (k-1)x_1^{k-2}$ (respectively by $p \cdot (k-1)x_2^{k-2}$) and subtract it from the third (respectively the fourth) equation. After these transformations the determinant of the system \eqref{eq4_5} can be written as
\[
\hat{\Delta}(\mathbf{x})=p^4\cdot
\begin{vmatrix}
\sum\limits_{k=0}^3a_{1,k}x_1^k & \dots & \sum\limits_{k=0}^3a_{n,k}x_1^k \\
\sum\limits_{k=0}^3a_{1,k}x_2^k & \dots & \sum\limits_{k=0}^3a_{n,k}x_2^k \\
\sum\limits_{k=1}^3k\cdot a_{1,k}x_1^{k-1} & \dots & \sum\limits_{k=1}^3k\cdot a_{n,k}x_1^{k-1} \\
\sum\limits_{k=1}^3k\cdot a_{1,k}x_2^{k-1} & \dots & \sum\limits_{k=1}^3k\cdot a_{n,k}x_2^{k-1} \\
a_{1,4} & \dots & a_{n,4}\\
\vdots & \ddots & \vdots \\
a_{1,n-1} & \dots & a_{n,n-1}
\end{vmatrix}.
\]
Let us transform the first four rows of this matrix as follows. Multiply the third (respectively the fourth) row by $\frac13 x_1$ (respectively by $\frac13 x_2$) and subtract it from the first (respectively the second) row. Then we subtract the first (respectively the third) row from the second (respectively the fourth) row and obtain the following determinant:
\[
\hat{\Delta}(\mathbf{x})=\textstyle\frac{p^4(x_2-x_1)^2}{9}\cdot\begin{vmatrix}
a_{1,2}x_1^2+2a_{1,1}x_1+3a_{1,0} & \dots & a_{n,2}x_1^2+2a_{n,1}x_1+3a_{n,0} \\
a_{1,2}(x_2+x_1)+2a_{1,1} & \dots & a_{n,2}(x_2+x_1)+2a_{n,1} \\
3a_{1,3}x_1^2+2a_{1,2}x_1+a_{1,1} & \dots & 3a_{n,3}x_1^2+2a_{n,2}x_1+a_{n,1} \\
3a_{1,3}(x_2+x_1)+2a_{1,2}& \dots & 3a_{n,3}(x_2+x_1)+2a_{n,2} \\
a_{1,4} & \dots & a_{n,4}\\
\vdots & \ddots & \vdots \\
a_{1,n-1} & \dots & a_{n,n-1}
\end{vmatrix}.
\]
Now let us subtract the second row multiplied by $x_1$ from the first row. Similarly, subtract the fourth row multiplied by $\frac12$ from the third row. Then subtract the third row multiplied by $\frac{x_2+x_1}{x_1^2}$ from the fourth row, and finally subtract the fourth row multiplied by $x_1x_2$, $x_2+x_1$ and $\frac32x_1-\frac12x_2$ from the first, the second and the third row respectively. We obtain the equation
\begin{equation}\label{eq4_6}
\hat{\Delta}(\mathbf{x})=p^4(x_2-x_1)^4\cdot
\begin{vmatrix}
a_{1,0} & \dots & a_{n,0}\\
\vdots & \ddots & \vdots \\
a_{1,n-1} & \dots & a_{n,n-1}
\end{vmatrix}
=p^4(x_2-x_1)^4\Delta >0,
\end{equation}
since the polynomials $P_j$, $1\leq j\leq n$ are linearly independent and $|x_1-x_2|>\varepsilon>0$. By \eqref{eq4_6} the system \eqref{eq4_5} has a unique solution
$(\theta_1,\ldots,\theta_n)$.
Consider $n$ integers $s_1,\ldots,s_n$ satisfying
\begin{equation}\label{eq4_7}
|\theta_j - s_j|\leq 1,\quad 1\leq j \leq n.
\end{equation}
and construct the following polynomial with integer coefficients
\[
P(t)=t^n+p\cdot\sum\limits_{j=1}^n{s_jP_j(t)}=t^n+p\cdot(a_{n-1}t^{n-1}+\ldots+a_1t+a_0),
\]
where $a_k=\sum\limits_{j=1}^{n}{s_ja_{j,k}}$, $0\leq k\leq n-1$.
The polynomial $P$ is irreducible if it satisfies the conditions of Lemma \ref{lm_Eisenstein}. Let us show that there exists a suitable combinations of the coefficients $s_j$. Clearly, the first and the second condition of \eqref{eq2_1} hold for any $s_j$. It remains to show that
$a_0=s_1a_{1,0}+\ldots+s_na_{n,0}$ is not divisible by $p$. Since $p$ doesn't divide $\Delta$, there exists a number $1\leq j\leq n$ such that $a_{j,0}$ is not divisible by $p$. From the condition \eqref{eq4_7}, we have two possible values for $s_j$, which can be denoted as $s_j^1$, $s_j^2=s_j^1+1$. Since $a_{j,0}$ is not divisible by $p$, either $a_0^1=s_1a_{1,0}+\ldots+a_{j,0}s_j^1+\ldots+a_{n,0}s_n$
or $a_0^2=s_1a_{1,0}+\ldots+a_{j,0}s_j^2+\ldots+a_{n,0}s_n$ is also not divisible by $p$. Therefore, choosing $s_j$ in this manner yields an irreducible polynomial $P$.
We finally derive bounds for $|P(x_i)|$, $|P'(x_i)|$ and $H(P)$. By the inequalities \eqref{eq4_3}, \eqref{eq4_5} and \eqref{eq4_7} we obtain the following estimates:
\begin{equation}\label{eq4_8}
p\delta^{-n+1}h_{n-1}Q^{-u_i}\leq|P(x_i)|\leq p(2n+1)\delta^{-n+1}h_{n-1}Q^{-u_i},\quad i=1,2,
\end{equation}
\begin{equation}\label{eq4_9}
pQ\leq |P'(x_i)|\leq (p+2pn\delta^{-n+1})Q,\quad i=1,2.
\end{equation}
We now estimate the height $H(P)$.
By equation $5$ to $n$ of the system \eqref{eq4_5}, inequalities \eqref{eq4_3} and \eqref{eq4_7} we have:
\begin{equation}\label{eq4_10}
|a_k|\leq n\delta^{-n+1}Q,\quad 4\leq k\leq n-1.
\end{equation}
It remains to estimate $|a_0|$, $|a_1|$, $|a_2|$ and $|a_3|$. By \eqref{eq4_8}-- \eqref{eq4_10} and inequalities $|x_i|\leq |d_i|+\frac12$ we get:
\begin{multline}\label{eq4_11}
|a_3x_i^3+a_2x_i^2+a_1x_i+a_0|\leq |P(x_i)|+\sum\limits_{k=4}^{n}{\left(|d_i|+1\right)^k\cdot |a_k|}<c_{9,i}Q,\\
|3a_3x_i^2+2a_2x_i+a_1|\leq |P'(x_i)|+\sum\limits_{k=4}^{n}{k\left(|d_i|+1\right)^k\cdot |a_k|}<\\
<c_{10,i}Q,\quad i=1,2,
\end{multline}
where
\[
c_{9,i}=\begin{cases}
h_{n-1},\quad n = 3,\\
2n\delta^{-n+1}h_{n-1}\left(|d_i|+1\right)^{n},\quad n > 3;
\end{cases}
c_{10,i}=\begin{cases}
p+2pn\delta^{-n+1}h_{n-1},\quad n = 3,\\
4pn^2\delta^{-n+1}h_{n-1}\left(|d_i|+1\right)^{n},\quad n > 3.
\end{cases}
\]
Consider the system of linear equations for $a_0$, $a_1$, $a_2$ and $a_3$
\begin{equation}\label{eq4_12}
\begin{cases}
a_3x_i^3+a_2x_i^2+a_1x_i+a_0=l_{1,i},\\
3a_3x_i^2+2a_2x_i+a_1=l_{2,i},\quad i=1,2.\\
\end{cases}
\end{equation}
Since the determinant of the system \eqref{eq4_12} does not vanish, there exists a unique solution. We solve the system \eqref{eq4_12} subject to the estimates \eqref{eq4_11} and inequalities $|x_i|\leq |d_i|+\frac12$. We obtain
\[
|a_k|<c_{11}Q,\quad 0\leq k\leq 3.
\]
Hence, by \eqref{eq4_4} and \eqref{eq4_10}, we find that
\begin{equation}\label{eq4_13}
H(P)<\max\{c_{11}, n\delta^{-n+1}\}Q=Q_1.
\end{equation}
Consider the roots $\alpha_{1},\ldots,\alpha_{n}$ of the polynomial $P$, where $|x_i-\alpha_{i}|=\min\limits_{j}{|x_i-\alpha_{j}|}$, $i=1,2$.
By Lemma \ref{lm_polynomial}, the following estimates hold
\[
|x_i-\alpha_{i}|\leq n|P(x_i)|\cdot|P'(x_i)|^{-1},\quad i=1,2.
\]
By \eqref{eq4_8} and \eqref{eq4_9}, we have
\begin{equation}\label{eq4_14}
|x_i-\alpha_{i}|< n(2n+1)\delta^{-n+1}h_{n-1}Q^{-u_i-1}<c_{12}Q^{-u_i-1},\quad i=1,2.
\end{equation}
where $c_{12}=n(2n+1)\delta^{-n+1}h_{n-1}$. Let us prove that $\alpha_1,\alpha_2 \in \mathbb{R}$ for $u_1=u_2=\frac{n-2}{2}$. Assume the converse: let $\alpha_i\in\mathbb{C}$, then the number $\overline{\alpha_i}$ being complex conjugate to $\alpha_i$ is also a root of the polynomial $P$. Hence, by \eqref{eq4_13}, \eqref{eq4_14} and Lemma \ref{lm5} we conclude that
\[
|P(x_i)|=\prod\limits_{j=1}^n|x_i-\alpha_j|\leq c_{12}^2Q^{-n}\cdot c_{13}\cdot Q=c_{13}c_{12}^2\cdot Q^{-n+1}.
\]
This inequality contradicts \eqref{eq4_8} for $Q>Q_0$.
Let us choose a maximal system of algebraic integer points $\Gamma = \{\boldsymbol{\gamma}_1,\ldots,\boldsymbol{\gamma}_t\}\subset\tilde{\mathbb{A}}_n^2(Q_1)$ satisfying the condition that rectangles $\sigma(\boldsymbol{\gamma}_k)=\{|x_i-\gamma_{k,i}|<c_{12}Q^{-\frac{n}{2}},i=1,2\}$, $1\leq k\leq t$ do not intersect.
Furthermore, let us introduce the expanded rectangles
\[
\sigma'(\boldsymbol{\gamma}_k)=\left\{|x_i-\gamma_{k,i}|<2c_{12}Q^{-\frac{n}{2}},i=1,2\right\},\quad k = 1,\ldots, t,
\]
and show that
\begin{equation}\label{eq4_16}
B\subset\bigcup_{k=1}^t \sigma'(\boldsymbol{\gamma}_k).
\end{equation}
To prove this fact, we are going to show that for any point $\mathbf{x}_1 \in B_1$ there exists a point $\boldsymbol{\gamma}_k\in\Gamma$ such that $\mathbf{x}_1\in\sigma'(\boldsymbol{\gamma}_k)$.
Since $\mathbf{x}_1 \in B_1$, there is an algebraic integer point $\boldsymbol{\alpha}\in\tilde{\mathbb{A}}_n^2(Q_1)$ satisfying the inequalities \eqref{eq4_14}. Thus, either $\boldsymbol{\alpha}\in\Gamma$ and $\mathbf{x}_1\in\sigma'(\boldsymbol{\alpha})$, or there exists a point $\boldsymbol{\gamma}_k\in\Gamma$ satisfying
\[
|\alpha_i-\gamma_{k,i}|\leq c_{12}Q^{-\frac{n}{2}},\quad i=1,2,
\]
which implies that $\mathbf{x}_1\in\sigma'(\boldsymbol{\gamma}_k)$. Hence, from \eqref{eq4_16} and the estimate $\mu_2\,B\ge\frac34\cdot\mu_2\,\Pi$ we have
\[
\textstyle\frac34\cdot \mu_2\Pi \leq\mu_2 B\leq \sum\limits_{k=1}^t{\mu_2\sigma_1(\boldsymbol{\gamma}_k)}\leq t\cdot 2^4c_{12}^2Q^{-n},
\]
which yields the estimate
\[
\#\left(\tilde{\mathbb{A}}_n^2(Q_1)\cap\Pi\right)\ge t \ge c_{8}\cdot Q^{n}\mu_2\Pi.
\]
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,247 |
\section{Introduction}
Collaboration is a fundamental and increasingly common feature in scientific research. Collaborative research has been associated with higher productivity, with higher impact, and, ultimately, with higher quality: from an economic perspective, collaboration allows the division of labor leading to reduced costs and time saving, consent the access to scientific funding, to expensive (possibly large-scale) equipment, and to unique scientific data. From a bibliometric perspective, collaborative works are generally more visible and more cited by other scholars; moreover, they are rated higher by peer reviewers with respect to papers written in isolation, although notable exceptions exist \citep{CF10}.
\begin{comment}
Academic collaboration is fundamental in many research fields. It arises at different levels, ranging from individual scholars collaborating in writing papers to larger collaborations involving research structures or even entire world regions. Collaborating researchers can derive scientific advantages by sharing knowledge, expertise and techniques as well as jointly controlling the accuracy and the significance of results. Furthermore, collaboration enhances the visibility of results: a multi-authored contribution is brought to the attention of a larger number of researchers through personal contacts of each contributor. Collaboration, however, has not only advantages. A collaborative work needs deep integration among co-authors, since the final result should be a coherent and uniform piece of work. If integration among authors fails, the quality of the outcome definitely declines.
\end{comment}
In this paper, we study collaboration in computer science using a network science approach. The field of \textit{network science} -- the holistic analysis of complex systems through the study of the structure of networks that wire their components -- exploded in the last decade, boosted by the availability of large databases on the topology of various real networks, mainly the Web and biological networks \citep{N10}. The network science approach has been successfully applied to analyse disparate types of networks, including technological, information, social, and biological networks. Here, we use co-authorship in publications as a proxy for scientific collaboration and build two differently grained network representations of collaboration in computer science: an author-paper \textit{affiliation network}, which is a bipartite graph with two types of nodes for authors and papers and links running from authors to papers that they wrote. We use affiliation networks to investigate the distribution of scientific productivity and that of collaboration level.
A coarser and highly informative alternative representation is the \textit{collaboration network}, in which the nodes represent authors and the links are collaborations between authors in publications. A collaboration network is a type of social network, since co-authorship in publication can be interpreted as a social relationship between authors: in most cases, two authors that have written a paper together do know each other quite well, at least from a scientific perspective. This is particularly true in disciplines, like computer science, where the typical paper has few co-authors and the share of single-authored papers is not large.\footnote{This is true to a less degree for disciplines like medicine, biology and experimental physics, where the average number of authors per paper is significantly lager than in computer science. On the other hand, in arts, humanities and some social sciences, a significant share of contributions are written by a single author and hence are not collaborative works.} We study the large-scale structure of the collaboration network for computer science, investigating properties like reachability and average separation distance among scholars, distribution of the number of scholar collaborators, network resilience and dependence on star collaborators, network clustering, and network assortativity by number of collaborators.
In the computer science publication culture, conferences are important publication sources, and journals often publish deeper versions of papers already presented at conferences. This is a peculiarity of computer science that makes it an original research discipline: in all other sciences, indeed, journals are the primary publication source, while monographs are the standard publication type in most social sciences, arts and humanities. This singularity motivated us to analyse separately the large-scale structure of two sub-networks of the whole collaboration graph, namely the \textit{conference} and the \textit{journal} collaboration networks. It is worth observing that the role of conferences in computer science is currently heartily discussed in the computer science literature (see \citet{F10-CACM} and references therein).
This is the first part of our investigation of collaboration in computer science using a network science approach. In the second part of our contribution, we make a longitudinal (time-resolved) study of the network properties analysed in this paper, to get a \textit{dynamic picture} of how bibliometric and collaboration patterns evolved over time in the last half-century of computer science \citep{F11b}.
\section{Related literature}
Academic collaboration has been extensively studied in \textit{bibliometrics}, the branch of information and library science that quantitatively investigates the process of publication of research achievements \citep{S83,LPS92,KM97,BG00,SO07,CF10}. Bibliometricians observed that collaboration intensity neatly varies across disciplines. The intensity of research collaboration is negligible in arts and humanities, while social scientists often work in team, but collaborations are smaller in scale and formality compared to science disciplines. By contrast, collaborative work is heavily exploited in science, in particular in physics, medicine, and biology. Collaboration is, however, moderate in mathematics, computer science, and engineering. Moreover, collaboration generally pays in terms of impact, measured with the popular bibliometric practice that tallies the number of citations that a work receives from other papers. Furthermore, collaborative works are generally valued higher by peer experts. Both impact and quality of papers are further enhanced when the affiliations of authors are heterogeneous. Interestingly, in computer science a little collaboration, but not more than that, seems fruitful to obtain more valuable papers.
Collaboration has been also investigated under the network analysis umbrella. Sociologists have the longest tradition of quantitative study of social networks \citep{M34,DGG41,WF94,S00}. However, the notion of academic collaboration network, a particular type of social network, first appeared in 1969 in a brief note by mathematician \citet{G69}. Goffman defined the \textit{Erd\H{o}s number} for a given mathematician as the length of the shortest path on the mathematics collaboration network connecting the mathematician with Paul Erd\H{o}s.\footnote{Paul Erd\H{o}s was a notably eccentric Hungarian mathematician that is currently the most prolific and the most collaborative among mathematicians. He wrote more than 1400 papers cooperating with more than 500 co-authors \citep{G97}. Erd\H{o}s was an itinerant mathematician, living most of his life out of a suitcase visiting those colleagues willing to give him hospitality in exchange for collaboration in the writing of papers (\textit{``Another roof, another proof''}, he was used to say).} The idea of Erd\H{o}s number and hence of collaboration network, however, was informally already present in the mathematics community before 1969, since in Goffman's note we can read:
\begin{quote}
\textit{I was told several years ago that my Erd\H{o}s number was 7. It has recently been lowered to 3. Last year I saw Erd\H{o}s in London and was surprised to learn that he did not know that the function v(Erd\H{o}s; .) was being considered. When I told him the good news that my Erd\H{o}s number had just been lowered, he expressed regret that he had to leave London the same day. Otherwise, an ultimate lowering might have been accomplished.
}
\end{quote}
The note of Goffman is followed by a brief series of (occasionally sarcastic) papers of colleagues of him, including one written by Paul Erd\H{o}s himself, speculating on some theoretical properties of the collaboration graph in mathematics \citep{H71,E72,O79}.
Newman was the first to experimentally study large-scale collaboration networks with the aid of modern network analysis toolkit. He analysed the structural properties of collaboration networks for biomedicine, physics \citep{N01a,N01b}, and mathematics \citep{N04}, as well as the temporal evolution of collaboration networks in physics and biomedicine \citep{N01c}. \citet{BJNRSV02} studied the evolution in time of collaboration networks in neuroscience and mathematics. The temporal dynamics of mathematics collaboration networks is also investigated by \citet{G02}. \citet{M04} studied the structure and the temporal evolution of a social science collaboration network.
As for studies concerning the computer science collaboration network, \citet{HZLG08} considered publications from 1980 to 2005 extracted from the CiteSeer digital library. The dataset consists of 451,305 papers authored by 283,174 distinct researches. The authors studied properties at both the network level and the community level and how they evolve in time. \citet{BBNDFS09} focused on the structure and dynamics of collaboration in research communities within computer science. They isolated 14 computing areas, selected the top tier conferences for each area, and extracted publication data for the chosen conferences from DBLP 2008. The dataset contains 83,587 papers, 76,598 authors, and 194,243 collaboration pairs. They used network analysis metrics to find differences in the research styles of the areas and how these areas interrelate in terms of author overlap and migration. \citet{MZLA09} made a geographical analysis of collaboration patterns using network analysis. They considered publications from 1954 to 2007 for members of 30 research institutions (8 from Brasil, 16 from North America, and 6 from Europe) and focused on the differences in collaboration habits among these geographical areas. The dataset, extracted from DBLP, contains 352,766 papers and 176,537 authors. \citet{N01a} studied the collaboration graph for computer science as well, using the NCSTRL library, a database of preprints published in computer science during 1991-2001 and submitted by 160 participating institutions. Unfortunately, as acknowledged by Newman himself, the coverage of the used dataset (13,169 papers and 11,994 authors) is rather limited and hence the sample is not representative of the set of computing publications. Finally, the temporal evolution of the collaboration graph for the database community is studied by \citet{EL05}. The dataset is extracted from DBLP and contains 38,773 publications written by 32,689 authors from 1968 to 2003 covering 19 journals and 81 conferences closely related to the database community. Table~\ref{networks} contains a summary of network statistics for different disciplines including the results found in this paper.
Our investigation differs from the mentioned previous studies on computer science for the following reasons:
\begin{itemize}
\item we build the largest computer science affiliation and collaboration networks ever investigated;
\item with the support of the affiliation network representation of collaboration, we study bibliometric properties for computer science, like author productivity and collaboration level in papers;
\item with the aid of the collaboration network we study meaningful large-scale network properties; in particular, the size of biconnected components, the concentration of collaboration using the Lorenz curve and the Gini coefficient, the collaboration network resilience and dependence on star collaborators have never been examined before for computer science;
\item we investigate separately the networks emerging from scholar collaborations in conference and journal papers.
\end{itemize}
\begin{table}
\begin{center}
\begin{footnotesize}
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}llrrrrrrrrr}
\textbf{disc} & \textbf{source} & \textbf{nodes} & \textbf{edges} & \textbf{deg} & \textbf{com} & \textbf{dis} & \textbf{dia} & \textbf{tra} & \textbf{clu} & \textbf{mix} \\ \hline
MAT & Mat.\ Rev.\ & 253,339 & 496,489 & 3.92 & 0.82 & 7.57 & 27 & 0.15 & 0.34 & 0.12 \\ \hline
PHY & arXiv & 52,090 & 245,300 & 9.27 & 0.84 & 6.19 & 20 & 0.45 & 0.56 & 0.36 \\ \hline
BIO & Medline & 1,520,251 & 11,803,064 & 15.53 & 0.92 & 4.92 & 24 & 0.09 & 0.60 & 0.13\\ \hline
NEU & -- & 209,293 & -- & 11.54 & 0.91 & 6.00 & -- & -- & 0.76 & -- \\ \hline
SOC & Soc.\ Abs.\ & 128,151 & -- & -- & 0.53 & 9.81 & -- & 0.19 & -- & -- \\ \hline
CS & CiteSeer & 283,174 & -- & 5.56 & 0.66 & 7.10 & 26 & -- & 0.63 & 0.28 \\ \hline
CS & DBLP & 688,642 & 2,283,764 & 6.63 & 0.85 & 6.41 & 23 & 0.24 & 0.75 & 0.17\\ \hline
CS-C & DBLP & 503,595 & 1,584,108 & 6.29 & 0.85 & 6.54 & 23 & 0.24 & 0.75 & 0.16\\ \hline
CS-J & DBLP & 356,822 & 987,059 & 5.53 & 0.77 & 7.26 & 25 & 0.37 & 0.77 & 0.30\\ \hline
\end{tabular*}
\end{footnotesize}
\end{center}
\caption{Structural properties of discipline collaboration networks. Column names are abbreviated as follows: disc (discipline: MAT (mathematics \citep{G02,N04,N10}), PHY (physics \citep{N01a,N10}), BIO (biomedicine \citep{N01a,N10}), NEU (neuroscience \citep{BJNRSV02}), SOC (social science \citep{M04}), CS (computer science \citep{HZLG08} and this paper), CS-C (computer science conferences; this paper), CS-J (computer science journals; this paper)), source (bibliographic source), nodes (number of nodes), edges (number of edges), deg (average node degree), com (percentage of the largest connected component), dis (average geodesic distance), dia (largest geodesic distance), tra (transitivity coefficient), clu (clustering coefficient), mix (assortative mixing). A dash sign indicates that the data is not available.}
\label{networks}
\end{table}
\section{Methodology}
Data were collected from \textit{The DBLP Computer Science Bibliography} (DBLP, for short) \citep{DBLP}. The DBLP literature reference database was developed within the last 15 years by Dr.\ Michael Ley at Trier University, Germany. DBLP is internationally respected by informatics researchers for the accuracy of its data. As of today, DBLP contains more than 1.6 million entries covering computer and information science.
Each publication record in DBLP has a key that uniquely identifies the publication and a property that represents the publication type, such as journal article, conference article, book, book chapter, and thesis. Moreover, it contains a semi-structured list of bibliographic attributes describing the publication, like authors, title, and year of publication. This list veries according to the publication type \citep{Ley09}.
DBLP is particularly careful with respect to the quality of its data, and is especially sensible to the \textit{name problem}, which includes the cases of a scholar with several names (synonyms) and that of several scholars with the same name (homonyms) \citep{RWLWK06}. DBLP uses full names and avoids initials as much as possible. This reduces, but does not eliminate, the name problem. Furthermore, it uses effective heuristics on the collaboration graph to identify possible cases of synonyms or homonyms. For instance, if two lexicographically similar names are assigned to authors that have a distance of two in the collaboration graph, that is, these authors never directly collaborated in a paper but they have a common collaborator, then these names are identified as possible synonyms and they are further manually investigated. Furthermore, if the list of co-authors of an author splits in two or more clusters of highly interconnected authors, but with no collaborations among authors of different clusters, then we might have a case of homonym, and an additional manual check is performed.
DBLP can be used free of charge. Data can be accessed using a Web interface or through automatic HTTP requests, and the entire dataset can be downloaded in XML format to run experiments on top of it.
We downloaded the XML version of DBLP bibliographic dataset in early 2010 (637.9 MB) and filtered all publications from 1936 to 2008 inclusive.\footnote{We excluded year 2009 since for it the bibliography has not reached the same level of completeness as for previous years.} On top of this database we built the following networks:
\begin{comment}
The obtained XML database contains 1,238,458 publications, of which 740,465 (60\%) are conference papers and 476,061 (38\%) are journal papers. The remaining 2\% consists of proceedings (11992), book chapters (8348), books (1480), theses (98), and Web publications\footnote{We excluded author Web pages, which are catalogued as special WWW publications.} (14).
\end{comment}
\begin{itemize}
\item \textit{Author-paper affiliation network}. This is a bipartite graph with two types of nodes: authors and papers. There is an edge from an author to a paper if the author has written the paper. See an example in Figure~\ref{affiliation}. Affiliation networks are the most complete representations for the study of collaboration \citep{N10}; in particular, on top of such bipartite representations, one can investigate both author-oriented and paper-oriented properties. The resulting affiliation network contains 731,333 author nodes, 1,216,526 paper nodes, and 3,112,192 crossing edges.
\item \textit{Collaboration network}. A collaboration network is an undirected graph obtained from the projection of the author-paper affiliation network on the author set of nodes. Nodes of the collaboration network represent authors and there is an edge between two authors if they have collaborated in at least one paper. An example is given in Figure~\ref{affiliation}. Clearly, the collaboration network is a coarser representation with respect to the affiliation network; for instance, if three authors are mutually linked in the collaboration network, then it is not clear, from the analysis of the collaboration network alone, whether they have collaborated in a single paper or in three different ones. Nevertheless, the collaboration network is highly informative since many collaboration patterns can be captured by analysing this form of representation. Furthermore, the collaboration network is the main (mostly unique) representation of collaboration that has been studied in the network science literature.
The resulting collaboration network contains 688,642 nodes (authors) and 2,283,764 edges (collaborations).\footnote{We excluded from collaboration networks isolated nodes, that are authors that have never collaborated. The share of these authors is about 6\% of the total number of authors.} This is, to our knowledge, the largest computer science collaboration network and the second largest discipline collaboration network ever studied, second only to the Medline collaboration network for biomedicine investigated in \citep{N01a};
\item \textit{Conference collaboration network}. In the conference collaboration network, two scholars are linked if they have collaborated in at least one conference paper. The resulting network has 503,595 nodes and 1,584,108 edges;
\item \textit{Journal collaboration network}. In the journal collaboration network, two scholars are connected if they have co-authored at least one journal paper. The resulting network contains 356,822 nodes and 987,059 edges.
\end{itemize}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.25, angle=0]{affiliation.eps}
\caption{A toy example of an author-paper affiliation network (left graph). Authors (circle nodes) match the papers (square nodes) that they wrote. We also show the corresponding collaboration network (right graph). In this case, two authors are connected if they wrote at least one paper together.}
\label{affiliation}
\end{center}
\end{figure}
We saved the collaboration networks in GraphML format (an XML syntax for graphs). We loaded them in the R environment for statistical computing \citep{R} and analysed the structure of the networks using the R package \textit{igraph} developed by G\'{a}bor Cs\'{a}rdi and Tam\'{a}s Nepusz. On the other hand, we never materialized the (much larger) affiliation network. Instead, we used XQuery, the standard XML query language, and BaseX \citep{basex}, a light-speed native XML database, to extract the relevant properties from the XML version of the DBLP database.
\section{Analysis}
In this section we show how a structural analysis of the affiliation and collaboration networks for computer science uncovers interesting properties of the publication process of the discipline.
\subsection{Scientific productivity and collaboration level}
In this section we investigate two typical bibliometric distributions for the set of papers of a discipline: the distribution of the number of papers per author (scientific productivity) and the distribution of the number of authors per paper (collaboration level). It is worth noticing that these distributions can not be extracted from the collaboration network, since this network does not represent the links between authors and papers, but it contains only the links among collaborating authors. The correct network to investigate in this case is the author-paper affiliation network. We recall that this network is a bipartite graph with two node types representing authors and papers; edges match authors with papers they wrote.
The distribution of the number of papers per author corresponds to the distribution of the node degree for nodes of type author in the author-paper affiliation network. Indeed, the degree (number of adjacent nodes) of a node of type author on the affiliation network is precisely the number of papers published by the author. In substantial agreement with one of the oldest bibliometric laws -- Lotka's law of scientific productivity \citep{L26} -- the distribution of the number of papers per author is highly skewed, with most of the authors that produced a small number of contributions and few prolific ones that published a large volume of papers. In Table~\ref{Lotka} we show the relative frequency of authors that wrote a given number of papers. The table shows only the first 10 numbers of papers, but the distribution has a long tail ending at 528, the number of papers of the most prolific author. This asymmetry in scientific productivity is not characteristic of computer science but it has been noticed in many fields; for instance, \citet{M04} found a similar pattern in the productivity of social scientists, with 65.8\% of them with 1 paper, 15.1\% with 2 papers, 6.5\% with 3 papers, 3.7\% with 4 papers, 2.2\% with 5 papers, and the remaining 6.7\% with 6 or more contributions.
\begin{table}
\begin{center}
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}l|llllllllll}
\hline
\textbf{\# of papers} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
\textbf{\% of authors} & 53.2\% & 15.8\% & 7.7\% & 4.7\% & 3.2\% & 2.3\% & 1.8\% & 1.4\% & 1.1\% & 0.9\% \\ \hline
\end{tabular*}
\end{center}
\caption{The scientific productivity of computer scientists. The table shows the relative frequency of authors (second row) that wrote a given number of papers (first row, from 1 to 10 papers).}
\label{Lotka}
\end{table}
As for the distribution of the number of authors per paper, it corresponds to the distribution of the node degree of nodes of type paper in the author-paper affiliation network. We found that the average computer science paper has 2.56 authors. This figure is significantly lower than the average collaboration level in other scientific fields like physics, chemistry, biology and medicine, but it is higher than the average collaboration level in social sciences and humanities (see \citet{CF10} for the collaboration level of different disciplines). Hence, computer science stays in a peculiar intermediate position where a little collaboration (2 or at most 3 scholars), but not more than that, seems to be optimal.
Table~\ref{tab.collaboration} shows the relative frequency of papers having a given number of authors, distinguishing between conference and journal papers. We observe that conference papers are more collaborative (2.69 authors on average) than journal papers (2.35 authors on average). In particular, notice that 19\% of the conference papers are single-author works, while this share is significantly higher, 30\%, for the papers in journals.
\begin{table}
\begin{center}
\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill}}l|llllllllll}
\hline
\textbf{\# of authors} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline
\textbf{\% of papers} & 23.3\% & 32.8\% & 23.5\% & 11.6\% & 4.6\% & 1.9\% & 0.8\% & 0.4\% & 0.2\% & 0.1\%
\\ \hline
\textbf{\% of conf.} & 19.2\% & 32.4\% & 25.4\% & 13.3\% & 5.4\% & 2.2\% & 0.9\% & 0.4\% & 0.2\% & 0.1\%
\\ \hline
\textbf{\% of jour.} & 29.7\% & 33.4\% & 20.5\% & 9.0\% & 3.4\% & 1.4\% & 0.7\% & 0.4\% & 0.2\% & 0.1\%
\\ \hline
\end{tabular*}
\end{center}
\caption{The collaboration level of computer science papers. The table shows the relative frequency of papers (second row for all papers, third row for conference papers, and fourth row for journal papers) having a given number of authors (first row, from 1 to 10 authors).}
\label{tab.collaboration}
\end{table}
\subsection{Connected components} \label{components}
A \textit{connected component} of an undirected graph is a maximal subset of nodes such that any node in the set is reachable from any other node in the set by traversing a path of intermediate nodes. A connected component of a collaboration graph is hence a maximal set of authors that are mutually reachable through chains of collaborators.
It is reasonable to assume that scientific information flows through paths of the collaboration networks; we expect, indeed, that two authors that collaborated in some paper are willing to exchange scientific information with a higher probability than two scholars that never collaborated. Having a large connected component in the collaboration graph, of the order of the number of scholars, is a desirable property for a discipline that signals its maturity: theories and experimental results can reach, via collaboration chains, the great majority of the scholars working in the field, and thus scholars are scientifically well-informed and can incrementally build new theories and discover new results on top of established knowledge. Furthermore, the connectedness of a discipline is welcome in the view of proofs of theorems (and validation of experimental results) as a social human process and a community project \citep{DLP79}. Of course, collaboration represents only one way to spread scientific information; the processes of journal publishing and conference attendance make also notable contributions in this direction.
On the other hand, a high level of discipline connectedness might also have negative effects, since it favors the globalization and the standardization of results, and hence the publication of mainstream contributions at the expense of more innovative papers that explore research directions outside the established core subjects. Moreover, the independent discovery of the same theories and results by different groups of scientists, which is more likely when the discipline community is disconnected, increases the confidence of the whole community in the validity of these theories and results.
The computer science collaboration network is widely connected. The largest component counts 583,264 scholars, that is 85\% of the entire network. It is a \textit{giant component}, since it collects the great majority of nodes. There are two second largest components, the size of which, only 40 nodes, is negligible compared to that of the giant component. The third largest component has 30 nodes, and there are components for each size smaller than 30. In total we have 34,691 connected components, most of which have small sizes: 18,244 of them have size 2, while 8354 have size 3, and 3854 have size 4; hence 88\% of the components have size at most 4. The distribution of the size of the connected components that are different from the giant component has a long tail in which most components have small size and a few of them have large size.
Interestingly, the relative size of the giant component of the collaboration network for computer science matches quite well that for physics, and it is a bit higher than that for mathematics. With respect to computer science, the networks for biomedicine and neuroscience are better connected, while the cohesion of social science is lower (see Table~\ref{networks}). This means that research collaboration is more effective in medical disciplines than in social as well as hard sciences.
\begin{comment}
The figure found by \citet{HZLG08} for computer science using the CiteSeer dataset is significantly smaller that our finding and rather odd when compared with other similar disciplines. This might indicate that the coverage of the CiteSeer dataset is less representative of the publication dataset of the computing community.
\end{comment}
A \textit{biconnected component} of an undirected graph is a maximal subset of nodes such that for each pair of nodes there are \textit{two} independent (disjoint) paths connecting them. It follows that the removal of a single node from a biconnected component does not destroy the connectivity of the component. A biconnected component is hence more tightly connected than a connected component. Information flowing on a biconnected component has more chance to reach a target node of the component since there exist two independent paths from any component node to the target node.
The largest biconnected component of the collaboration graph for computer science counts 418,001 nodes, or 61\% of the entire network, and it covers a share of 72\% of the largest connected component. The second-largest biconnected component has only 32 nodes. As a comparison, the physics collaboration has a biconnected component of 59\% of the entire network \citep{NG08}, and for social science the biconnected component occupies a share of 23\% of the network space.
\begin{comment}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{cc.eps}
\caption{The distribution of the size of the non-giant connected components of the collaboration graph. The size of the component is given on the $x$ axis (from 2 to 40), and the number of components of that size is plotted on the $y$ axis.}
\label{cc}
\end{center}
\end{figure}
\end{comment}
The conference collaboration network is also well connected, with 429,193, that is 85\% of the authors belonging to the giant component. The distribution of the size of secondary components has a long tail, with the second largest component counting 44 nodes. The journal collaboration network is somewhat less connected: 273,861, that is 77\% of the authors lie in the giant component. Again, smaller components distribute with a long tail in terms of size, with a second largest component of 37 elements. Hence, the conference collaboration network is more connected than the journal counterpart, indicating that information has a broader reach when flowing via conference collaboration links.
\subsection{Geodesic distances} \label{geodesic}
A high level of connectedness in the collaboration network means that scientific information -- theorems and experimental results -- can reach almost the whole community via collaboration paths. Connectedness, however, does not tell us the whole story, since it says nothing about how fast the information flows. Information flows faster along shorter paths. In this respect, there exists a substantial difference if the average path connecting two scholars has length, say, six edges, or one hundred links.
We may assume that information preferentially flows along \textit{geodesics}, which are shortest paths in terms of number of edges on a graph.\footnote{The terms geodesic comes from geodesy, the science of measuring the size and shape of Earth; in geodesy a geodesic is the shortest route between two points on the Earth's surface.} A \textit{geodesic distance} between two nodes is defined as the length (number of edges) of any geodesic (shortest path) connecting the nodes -- notice that a geodesic is not necessarily unique. The average geodesic distance is the mean geodesic distance among all pairs of nodes of a graph. If the graph is not connected, then there are node pairs that are not reachable. In this case the mean is typically computed on the subset of connected pairs only. The largest geodesic distance in the graph is called the \textit{diameter} of the graph. It tells us how far are two connected nodes in the worst case.
We computed the geodesic distances for all pairs of nodes in the computer science collaboration network and took the average over the subset of connected pairs (the pairs with a defined distance). Since the graph has 688,642 nodes, the number of node pairs is 237,113,557,761, of which 72\% are connected by a path.\footnote{The computation of all-pairs shortest paths is computational intensive. In the unweighted case, it takes $O(n m)$, where $n$ and $m$ are the number of nodes and the number of edges of the graph, respectively. Notice that our collaboration graph is sparse, being $m \simeq \, 3.32 n$, hence the computational complexity is of the order of $n^2$. Using the igraph R package, the computation took more than 65 hours, that is 1 microsecond per pair of nodes on average.} This figure matches the relative size of the giant component, which was found to be about 0.85 (see Section~\ref{components}). Indeed, if we randomly pick two nodes in the graph, the probability that they fall in the giant component is $0.85^2 \simeq 0.72$. Since the sizes of the other components are negligible compared to that of the giant component, this probability is a close approximation of the probability that two nodes are connected by a path.
Figure~\ref{distances} shows the share of geodesics having a given length. Notice that geodesics have typically very short lengths compared to the number of nodes: 19\% of geodesics have length 5, 33\% have length 6, and 26\% have length 7. The average geodesic distance is 6.41, and, interestingly, distances normally distribute around this peak. The largest distance, the diameter of the computer science collaboration graph, is also remarkably small: 23 (there are 8 different geodesics with this length). Hence, computer scientists are separated on average by 6 collaboration links, a figure that matches well the legendary 6 degrees of separation found by the experimental psychologist Stanley Milgram in the 1960s with his popular small-world experiment \citep{Mi67}. These are additional good news for the computing community: not only the collaboration network is mostly connected, but the average distance is short, and the longest one is not that longer. This means that scientific information can spread \textit{quickly} on the great majority of the computing community through its collaboration network.
The fact that distances normally distribute is interesting because it means that the average distance of 6 links represents a typical value of all distances in the network. Furthermore, since the distribution of distances drops off rapidly around the mean, the time-consuming computation of the exact average distance can be approximated by computing the average distance on a relatively small random sample of node pairs. To demonstrate this, we estimated the average distance on a sample of 10,000 node pairs belonging to the giant component of the network. The outcome is extremely close to the real distance of 6.415: the approximated distance is 6.427, with a 95\% confidence interval of [6.401, 6.453].
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.50, angle=-90]{distances.eps}
\caption{The distribution of geodesic distances. The distances are shown on the $x$ axis (from 1 to the network diameter 23) and the height of the corresponding bar is the percentage of geodesics with that distance.}
\label{distances}
\end{center}
\end{figure}
Is the computer science collaboration network a \textit{small world}? \citet{WS98} define a social network a small world if typical distances grow roughly logarithmically in the number of nodes of the network. More precisely, a network of $n$ nodes and $m$ edges is a small world if the average geodesic distance is roughly $d = \log n / \log k$, where $k = 2m / n$ is the average node degree. Plugging into the formula the corresponding values for our collaboration network we have $k = 6.63$ and $d = 7.10$. Recalling that we measured an average distance of $6.41 < 7.10$, we conclude that the collaboration network of computer science is indeed a small world.
Comparing the found mean geodesic distance for computer science collaborations with that of other disciplines (see Table~\ref{networks}), we notice that the separation distance for computer science is comparable with that for physics and neuroscience. Moreover, biomedicine has a lower collaboration distance, indicating that collaborations in this field are more densely intertwined. On the other hand, mathematics and, in particular, social science collaboration distances are higher, meaning that collaborations in these disciplines are less frequent and less effective.
The conference collaboration network matches well the whole network in terms of geodesic distances. A share of 73\% of node pairs are connected with an average geodesic distance of 6.54 and a largest geodesic distance of 23. In particular, 31\% of all shortest paths have length exactly 6. On the other hand, separation distances on the journal collaboration network are larger: the typical distance is now 7.26 and the largest is 25. The largest share of paths, 27\%, have length 7. Hence, scholars on the journal collaboration network have on average 7 degrees of separation instead of 6. Furthermore, only 59\% of the node pairs are connected. Summing up, the conference collaboration network is not only more widely but also more densely connected than the journal counterpart.
\citet{KSS97} have proposed to use paths on social networks among scientists as \textit{referral chains} to establish contacts with domain experts. In the simplest case, suppose I am searching for a piece of information and I am aware that you are a domain expert that most likely can answer my query. If I do not know you personally, it might be useful to know that we have a common collaborator that can arrange an introduction. In general, as we have seen, there exists a referral chain of intermediate collaborators connecting almost any scholars in computer science. Furthermore, this chain is short in the average case. We might use the people in this chain in order to smoothly get in contact with the target scientist. To be sure, the chance of success depends on the path length but also on the strength of the intermediate path links. If, for instance, I can reach you through either collaborator A or collaborator B, and I have published with A a number of 100 papers and with B just one article, common sense suggests to use A as a broker. In other terms, one might reasonably argue that the intensity of the scientific relationship between two scholars is proportional to the number of papers they have written together. We can implement such an intuition by labelling each edge $(x,y)$ of the collaboration graph with a positive weight $1/k$, where $k$ is the number of papers that $x$ and $y$ have written together.\footnote{\citet{N01b} proposes to consider also the cardinality of the author set of the co-authored papers in order to define the collaboration weight. The plausible intuition is that the intensity of the scientific relationship is higher if two scholars collaborated on a paper in which they are the sole authors than if they wrote the paper with many other collaborators. We do not consider this factor, however, since the typical computer science paper has a small number of authors (typically 2), much smaller than in other experimental sciences in which tens or even hundreds of names can sign a paper.} The edge label can be interpreted as a scientific distance among scholars: the more papers two authors have written together, the closer they are scientifically.
The weighted collaboration graph naturally induced the notion of \textit{weighted geodesic}: a shortest path in terms of path weight, defined as the sum of the weights of the path edges. The \textit{weighted geodesic distance} is hence the weight of a weighted geodesic. Notice that a weighted geodesic is not necessarily unique. Moreover, we expect that it differs from its unweighted counterpart. This opens an interesting question connected to the above mentioned referral chain issue: is preferable a short and weighty path, or a longer and lighter one in order to reach a given domain expert? Notice that a short path has the obvious advantage of having few intermediate scholars to bother, but a light path is desirable since the intermediate links are stronger and more reliable.
In order to investigate whether weighted and unweighted shortest paths are significantly different on the computer science collaboration network, we conducted the following experiment. We extracted a random sample of 10,000 node pairs belonging to the giant component of the computer science collaboration network and computed, for each pair of nodes, the weighted geodesic distance as well as the length of the weighted geodesic. The average weighted geodesic distance is 3.15, and the average length of the weighted geodesics is 11.27. Hence, light paths are much longer, almost twice longer, than the typical geodesic path, which is about 6 edges long. We furthermore generated 10,000 random node pairs belonging to the giant component and computed, for each of them, the (unweighted) geodesic distance as well as the weight of the geodesic. The average geodesic distance is 6.43, and the average weight of the geodesics is 5.07. Thus, short paths are significantly heavier than the typical weighted geodesic path, which weights about 3. As conjectured, for the computer science collaboration network, weighted shortest paths and unweighted shortest paths are different referral chains; the information seeker is hence faced with the dilemma of following short and weighty (unreliable) chains or long and light (reliable) paths in order to get in contact with the coveted expert.
\subsection{Node degree distribution} \label{degree}
A property of the full-scale structure of a network that is typically investigated is the distribution of the network node degrees. We recall that the \textit{degree} of a node is the number of neighbours of the node. In a collaboration network, the degree is the number of unique collaborators of a scholar. For any natural number $k$, the quantity $p_k$ is the fraction of nodes having degree $k$. This is also the probability that a randomly chosen node in the network has degree $k$. The quantities $p_k$ represent the \textit{degree distribution} of the network.
In most real networks, the degree distribution is highly \textit{right skewed}: most of the nodes (\textit{the trivial many}) have low degrees while a small but significant fraction of nodes (\textit{the vital few}) have an extraordinarily high degree. A highly connected node, a node with remarkably high degree, is called \textit{hub}. Since the probability of hubs, although low, is significant, the degree distribution, when plotted, shows a \textit{long tail}, which is much fatter than the tail of a Gaussian or exponential model.
This asymmetric shape of the degree distribution has important consequences for the processes taking place on networks. The highly connected nodes, the hubs of the networks, are generally responsible for keeping the network connected. In other words, the network falls apart if the hubs are removed from the network. On the other hand, since hubs are rare, a randomly chosen node is most likely not a hub, and hence the removal of random nodes from the network has a negligible effect on the network cohesion. Substantially, networks with long tail degree distributions are resilient to random removal of nodes (failure) but vulnerable to removal of the the hub nodes (attack). In Section~\ref{resilience} we will investigate the resilience of the collaboration network under removal of nodes.
\begin{comment}
For instance, consider the \textit{configuration model}, a generalization of the random graph model in which one specifies a particular, arbitrary degree distribution and then forms a graph that has that distribution but is otherwise random. In a graph generated according to this model, a giant component exists if and only if
\begin{equation}
\frac{\langle k^2 \rangle - \langle k \rangle}{\langle k \rangle} > 1
\end{equation}
where where $k_i$ is the degree of node $i$, $\langle k \rangle = \sum_i k_{i}$ is the mean degree, and $\langle k^2 \rangle = \sum_i k_{i}^{2}$ is the mean-square degree (see \citep{N10}, page 456). If the degree distribution has a long tail, the mean-square degree $\langle k^2 \rangle$ turns out to be large compared to the mean degree $\langle k \rangle$, so that the above disequation is easily satisfied and most of the nodes of the network are connected.
\end{comment}
Hubs are also important for the spread of information or of any other quantity flowing on the network. In fact, hubs play a dual role in information diffusion over the network: on the one hand, since they are highly connected, they quickly harvest information, on the other hand, and for the same reason, they effectively spread it. In a network with hub nodes, the probability that each node spreads the information to its neighbours need not be large for the information to reach the whole community.
\begin{comment}
Consider again the configuration model. Imagine a scenario in which nodes receive some kind of information and might decide to spread it over the network through their adjacent nodes. Suppose each node that received the information communicates it with independent probability $r$ to its immediate neighbours. Is this information going to become widely popular over the network? This happens if and only if (see \citep{N10}, page 646)
\begin{equation}
r > \frac{\langle k \rangle}{\langle k^2 \rangle - \langle k \rangle}
\end{equation}
Again, for networks with long tail degree distribution the value $\langle k^2 \rangle$ is big compared to $\langle k \rangle$, and hence the probability $r$ that each individual spreads the information need not be large for it to reach the whole community.
\end{comment}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{degree.eps}
\caption{The degree distribution for computer science collaboration network. Insets: the degree distributions for conference (top) and journal (bottom) collaboration networks.}
\label{fig:degree}
\end{center}
\end{figure}
The degree distribution for the collaboration network in computer science is depicted in Figure~\ref{fig:degree}. The distribution has in fact a long tail: roughly half of the scholars have one, two, or three unique collaborators. The other half of the scholars distribute over a slow decreasing long tail. There are, for instance, 350 scholars with 50 collaborators, and 40 scholars with 100 collaborators. The tail is in fact longer than shown in the figure, with 28 authors with more than 300 collaborators and the most collaborative computer scientist with 595 unique co-authors. The degree distributions for conference and journal articles show a similar pattern (see insets of Figure~\ref{fig:degree}), but the tail for the journal degree distribution is shorter (maximum degree is 260) than the conference counterpart (maximum degree is 481).
To quantitatively study the asymmetry of the degree distribution, we investigate the skewness and the concentration of the distribution. \textit{Skewness} measures the symmetry of a distribution. A distribution is symmetric if the values are equally distributed around its mean, it is right skewed if it contains many low values and a relatively few high values, and it is left skewed if it comprises many high values and a relatively few low values. As a rule of thumb, when the mean is larger than the median the distribution is right skewed and when the median dominates the mean the distribution is left skewed. The mean degree for the whole collaboration network is 6.63, it is 6.29 for the conference collaboration network, and it is 5.53 in the journal case. The median degree is always 3, and the 3rd quartile is 7 for the whole and conference networks, and 6 for the journal network. A numerical indicator of skewness is the third standardized central moment of the distribution: positive values for the skewness indicator correspond to right skewness, negative values correspond to left skewness, and values close to 0 mean symmetry. The skewness indicator is 8.04 for the whole collaboration network, 7.64 for the conference network, and 6.02 for the journal network. It follows that the analysed degree distributions are right skewed, and the conference distribution is more asymmetric than the journal counterpart.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{lorenz.eps}
\caption{The Lorenz collaboration concentration curves for the whole (solid curve), conference (dashed curve), and journal (dotted curve) collaboration networks. The share of most collaborative scholars collecting a given percentage of collaboration is plotted.}
\label{fig:lorenz}
\end{center}
\end{figure}
\textit{Concentration} measures how the character (in our context, the collaborations) is equally distributed among the statistical units (the scholars). The two extreme situations are equidistribution, in which each statistical unit receives the same amount of character (each scholar has the same number of collaborators) and maximum concentration, in which the total amount of the character is attributed to a single statistical unit (there exists a super-star collaborator that co-authored with all other scholars, and each other scholar collaborated only with this super-star). We analyse the concentration of collaborations among computer scientists, that is, the concentration of edges attached to nodes in the collaboration graph. Figure~\ref{fig:lorenz} depicts the Lorenz concentration curves representing the concentration of collaboration in the whole, conference, and journal networks. Each concentration curve is obtained by sorting scholars in decreasing order with respect to the number of collaborators. Then, the share of most collaborative scholars (or network nodes) collecting a given percentage of collaboration (or network edges) is plotted. It is clear that the concentration of collaboration is far from the equidistribution situation, which is illustrated by the straight line with slope 1. For instance, the most collaborative 1\% of the scholars harvest 13\% of the collaborations, the 5\% of them collect one-third (33\%) of the collaborations, and the 10\% of them attract almost half (46\%) of the collaborations.
A numerical indicator of concentration is the \textit{Gini coefficient}, which is the ratio between the area contained between the concentration curve and the equidistribution line and the area representing maximum concentration. The index ranges between 0 and 1 with 0 representing equidistribution and 1 representing maximum concentration. The Gini coefficient is 0.56 for the whole network, 0.54 for the conference network, and 0.53 for the journal network. Notice that journal collaborations are slightly less concentrated than conference collaborations, but the three concentration curves are very close.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{cdf.eps}
\caption{The complementary cumulative distribution function for the degree distribution of the whole collaboration network. Insets: the same for the conference collaboration network (top) and for the journal collaboration network (bottom).}
\label{fig:cdf}
\end{center}
\end{figure}
To be sure, the most popular long tail probability distribution is the \textit{power law}. For a degree distribution, it states that the probability $p_k$ of having a node with $k$ neighbours is $C k^{-\alpha}$, where $C$ is a normalization constant and $\alpha$ is an exponent parameter. A convenient method to visualize and detect a power law behaviour is to plot the complementary cumulative distribution function (CCDF) on log-log scales (both axes are on logarithmic scales). If a distribution follows a power law, then so does the CCDF of the distribution, but with exponent one less than the original exponent (see \citep{N10}, page 252). When plotted on log-log scales, a power law appears as a straight line. Figure~\ref{fig:cdf} plots the CCDFs for the networks at hand; all show a clear upward curvature, a sign that they do not match the power law model on the entire domain.
In practice, few empirical phenomena obey power laws on the entire domain. More often the power law applies only for values greater than or equal to some minimum location. In such case, we say that the \textit{tail} of the distribution follows a power law. \citet{CSN09} developed a principled statistical framework for discerning and quantifying power law behaviour and analysed 24 real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power law distribution in previous studies. Only 17 of them passed the test with a p-value of at least 0.1, and all of them show the best adherence to the model when a (limited) suffix of the distribution is considered. We applied the techniques developed by \citet{CSN09} to detect a power law behaviour in the degree distribution of the computer science collaboration network. As expected, the degree distributions do not follow a power law on the entire regime. Nevertheless, the degree distribution for the whole collaboration network has a power law tail starting from degree 111 ($\alpha = 4.4$, p-value = 0.11). The tail contains, however, only 1098 highly collaborative scholars, which correspond to 0.16\% of all authors. The degree distribution for the conference network does not follow a power law in any significant portion of its tail. Finally, the degree distribution for the journal network matches a power law from degree 105 ($\alpha = 5.79$, p-value = 0.67); the tail is 178 scholars long, or 0.05\% of the entire distribution. There is, however, a longer (551 scholars, 0.15\%) but statistically less significant power law distributed tail starting from degree 77 ($\alpha = 4.91$, p-value = 0.09). All in all, two of the analysed networks (the whole and the journal one) have a power law distributed tail, but the relative size of the tail is in both cases rather limited. We conclude that the process of \textit{preferential attachment} -- the attitude of scholars to collaborate preferentially with highly collaborative peers, which is one of the possible causes for the power law behaviour \citep{P76,BA99}, is not a valid explanation for the generation of the computer science collaboration network.
\subsection{Network resilience} \label{resilience}
\textit{Percolation}\footnote{The name comes from percolation studies in physics.} is one of the simplest \textit{processes} taking places on networks. The process progressively removes nodes, as long as the edges connected to these nodes, from the network, and studies how the connectivity of the network changes. In particular, one wants to find the fraction of nodes to remove from the network in order to disintegrate its giant component into small disconnected pieces. If such a fraction is relatively large, then the network is said to be resilient (or robust) to the process of percolation. Our study of percolation on collaboration networks is shaped by the following questions:
\begin{itemize}
\item What is the best removal strategy to destroy the overall connectivity of the collaboration network?
\item What is the tipping point in the percolation process after which the network consists of only small disconnected clusters?
\item Are the most collaborative scholars responsible for keeping the network connected?
\end{itemize}
To address the above questions, we performed the following computer experiment. We implemented a computer procedure that progressively removes nodes from the collaboration network. At each step the procedure removes an increasing number of nodes from the original network and, after each removal, it computes the relative size of the giant (largest) component of the resulting sub-network. More precisely, the procedure initially generates, according to a given strategy that will be discussed below, a number of $5000 \cdot 20 = 100,000$ nodes (about 15\% of the total number of nodes) to remove from the original graph. At each step $i$ from $1$ to $20$, the procedure removes the first $n_i = 5000 \cdot i$ nodes of the generated ones from the original network as well as the incident edges and computes the share of the giant component of the resulting network.
The procedure can choose the removal nodes according to three strategies: (i) \textit{random-driven percolation}, in which randomly chosen nodes are removed, (ii) \textit{degree-driven percolation}, in which the nodes are removed in decreasing order of node degrees, and (iii) \textit{eigenvector-driven percolation}, in which the nodes are removed in decreasing order of node eigenvector centrality scores.\footnote{Eigenvector centrality scores correspond to the values of the dominant eigenvector of the graph adjacency matrix. These scores may be interpreted as the solution of a linear system in which the centrality of a node is proportional to the centralities of the nodes connected to it.} To avoid possible biases in random-driven percolation due to the non-deterministic output of the random generator, we repeated the experiment a large number of times and took the average of the results.\footnote{A small bias might be introduced also in the other two cases, whenever one has to partially remove nodes with the same degree or eigenvector centrality. We did not consider this issue in our experiment.}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{robust.eps}
\caption{Effect of percolation on the connectivity of the computer science collaboration network. The relative size of the giant component is shown as an increasing fraction of nodes are removed according to three different percolation strategies.}
\label{fig:robust}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40, angle=-90]{robust2.eps}
\caption{Effect of random-driven and eigenfactor-driven percolation on the connectivity of the computer science collaboration network.}
\label{fig:robust2}
\end{center}
\end{figure}
Figure~\ref{fig:robust} clearly shows that the most effective strategy (among the surveyed ones) to destroy the connectivity of the collaboration network is to percolate the highly collaborative scholars: after the removal of about 12\% of the most collaborative scholars, the giant component of the collaboration network, which initially contains 85\% of the nodes, falls below 10\%, and it soon vanishes when the removal fraction is a little higher (15\%). The effect of removing scholars with high eigenvector centrality is much less substantial: the largest component is still considerable, about 62\% of the network, when a fraction of 15\% of the most central nodes are removed. Finally, the random removal of nodes has a negligible effect on the giant component of the network: after the percolation of a share of 15\% of randomly chosen nodes the network is still highly connected, with 80\% of the nodes belonging to the giant component. For a comparison, \citet{N10} found that to destroy the connectivity in the physics collaboration network it is necessary to remove a share between 20\% and 30\% among the highly collaborative physicists.
The shape of the degree-driven percolation curve shows clear curvatures. In particular, we can distinguish three main phases in the percolation process. An initial phase, up to a removal fraction of 7\%, in which the decrease of connectivity is limited (3-4 percentage points at each step). In this phase, although we severely attack the network by removing its most important hubs, the effect is somewhat reduced since the collaboration frame is still densely intertwined. That is, each node pair in the giant component is connected by more independent paths, and the removal of some of them do not prevent reachability.\footnote{As we have seen in Section~\ref{components}, the relative size of the largest biconnected component of the network is quite large.} In a following phase, which extends up to a removal share of 12\%, the reduction of connectivity is more notable (5-9 percentage points). In this phase, the size of the largest component is below 50\% and its collaboration frame is weaker and more vulnerable. Hence, the effect of the attack is more devastating. In the last segment, up to a removal percentage of 15\%, the relative size of the giant component is below 10\% and it goes rapidly to 0; in this phase the network consists of small disconnected clusters, none of which strongly dominates the others. Hence, for degree-driven percolation the tipping point after which only small disconnected clusters exist is around 15\%.
The overall effect of random-driven and eigenvector-driven percolations of network connectivity is shown in Figure~\ref{fig:robust2}. The shape of the random-driven percolation curve shows a clear upward curvature, meaning that the effect of random removal has a higher impact on connectivity when a significant fraction of the nodes have been already removed, as in the degree-driven case. Most of the nodes must be removed before connectivity is lost, and the tipping point where the network falls apart into small pieces is around 90\%, far beyond the percolation threshold for the degree-driven case. The eigenvector-driven percolation curve is instead linear up to its tipping point around 50\%, which lies between the percolation thresholds of the degree-driven and random-driven processes.
A similar analysis for the second-largest component shows that its size during the percolation process is never significant (always below 1\%). This means that, when a giant component exists, relatively small pieces belonging to the periphery of the giant component separate during the percolation process, and it never happens that the giant component splits into two fragments of similar size. Only when the network is divided into small disconnected clusters, the size of the giant and sub-giant components are comparable.
We are left to the last question posed at the beginning of the present section: are the star collaborators, the hubs of the collaboration network, responsible for the connectivity of the overall network? Our answer, maybe surprisingly, in negative. The hubs are nodes with an \textit{extraordinary} number of neighbours. Let us define hubs as those nodes with a degree higher than or equal to the 99th percentile in the degree distribution, that is the 1\% of nodes with highest degree. These are the 7036 scholars of the network with at least 54 collaborators. Recall that the average scholar has between 6 and 7 collaborators, hence hub scholars have a number of collaborators 8 times higher than the average scholar. The removal of these super-star collaborators has, in fact, a negligible effect on the connectivity of the network: the size of the giant component decreases form 85\% to 81\%. On the other hand, as we have seen, to dismantle the network we need to remove at least a share of 15\% of top collaborative scholars, that is all authors with a degree higher than 11. In our assessment, scholars with 11 unique collaborators are not collaboration hubs (I have 15 unique collaborators, and I really do not feel I am a collaboration star in computer science). This conclusion matches the findings of \citet{M04} for the social science collaboration network. Hence, the computer science collaboration network is not glued together by star collaborators and, while such actors are likely very influential within their local communities, they do not control information diffusion on the \textit{whole} computer science collaboration network.
As for conference and journal collaboration networks, the results are similar but the tipping points are lower. In particular, the tipping points for the journal network are below those for the conference network. For instance, the journal network falls to pieces after 7\% of the most collaborative scholars are inhibited, while the conference network crumbles when 9\% of the most collaborative scholars are removed. This means that the journal network is more fragile and its connectivity is more dependent on star collaborators.
\subsection{Clustering and mixing}
In common parlance, clustering, known also as transitivity, measures the average probability that ``the neighbour of my neighbour is also my neighbour''. The definition of clustering can be formalized as follows. Let a \textit{connected triple} be a triple of nodes $x$, $y$ and $z$ such that $x$ is linked to both $y$ and $z$. That is, $y$ and $z$ have a common neighbour, $x$. A \textit{triangle} is a triple of nodes such that all pairs of nodes are connected by an edge. Notice that a triangle codes for three connected triples, each one centered at one vertex of the triangle. The \textit{global clustering coefficient} of a network can be defined as
\begin{equation} \label{clustering1}
T = 3 \frac{N_\bigtriangleup}{N_\wedge}
\end{equation}
where $N_\bigtriangleup$ is the number of triangles and $N_\wedge$ is the number of connected triples in the network. The factor 3 constrains the coefficient to lie in the range between 0 and 1. Thus, $T=1$ implies perfect transitivity, that is, a network whose components are \textit{complete} graphs,\footnote{An undirected graph is said \textit{complete} if every pair of different nodes is connected by an edge.} while $T=0$ implies no triangles, which happens, for instance, in a network whose components are \textit{tree} graphs.\footnote{A \textit{tree} is an undirected graph that is both connected and acyclic.}
Local clustering refers to a single node. For a vertex $i$, its \textit{local clustering coefficient} $C_i$ is the fraction of neighbours of $i$ that are connected, that is the number of pairs of neighbours of $i$ that are connected divided by the number of pairs of neighbours of $i$. By taking the average local clustering over all nodes of a network we have an alternative (but different) definition of global clustering coefficient \citep{WS98}:
\begin{equation} \label{clustering2}
C = \frac{1}{n} \sum_i C_i
\end{equation}
It is worth noticing that these two definitions of clustering -- $T$ and $C$ expressed respectively by Equations~\ref{clustering1} and~\ref{clustering2} -- are not equivalent and can give substantially different results for a given network. We prefer definition $T$ because it has an intuitive interpretation -- the average probability that two neighbours of a node are themselves neighbours. To distinguish between them, in the following we will refer to $T$ as transitivity coefficient and to $C$ as clustering coefficient.
The transitivity coefficient of the computer science collaboration network is 0.24. This means that, on average, the chance that two scholars that share a common collaborator wrote a paper together is almost one-fourth. This is a rather high probability, indeed. As a comparison, the transitivity coefficient for a random network of the same size is $9.6 \cdot 10^{-6}$, and that for a network with the same degree distribution of our collaboration network but otherwise random is $1.4 \cdot 10^{-4}$; both values are orders of magnitude lower that what we computed on the real collaboration network.\footnote{The transitivity coefficient of a random graph with $n$ nodes and $m$ edges is $m / m_{*}$, where $m_* = n (n-1) / 2$ is the maximum number of edges of the graph. The transitivity coefficient of a random graph with degree sequence $k$ is $1/n \, (\langle k^2 \rangle - \langle k \rangle)^2 / \langle k \rangle^3$, where $\langle k \rangle$ is the mean degree and $\langle k^2 \rangle$ is the mean-square degree.} This large discrepancy is a clear sign of real social effect at work in the context of academic collaboration: authors with a common collaborator have several good reasons to write a paper together, for instance they are probably working on very close topics or they scientifically know each other through the common collaborator.
The transitivity coefficient for the conference and journal collaboration networks are 0.24 and 0.37, respectively. These values might indicate that journal collaboration establishes a stronger relationship among authors, so that authors having a common journal collaborator are professionally and maybe socially closer than authors sharing a common conference collaborator, and hence more inclined to collaborate themselves. In computer science publishing a paper in a journal is, generally, harder than writing a paper for a conference \citep{F10-CACM}, and this might explain the difference in strength between conference and journal co-authorship. Comparing with other disciplines, the magnitude of transitivity in computer science is comparable with that in sociology, larger than that in mathematics and biomedicine, and lower than that in physics (Table~\ref{networks}).
The clustering coefficient for the computer science collaboration network is 0.75 (0.75 for conference collaboration, and 0.77 for journal collaboration), much larger than the transitivity coefficient. This confirms, once again, that the two clustering measures are far different. Digging deeper, we computed the local clustering coefficient for each node and noticed that almost half of the nodes (48\%) have local clustering coefficient equal to 1, and this explains the large value of the clustering coefficient. Recall that a node $i$ has local clustering $C_i = 1$ if its immediate neighbours form a complete graph (a clique). Notice that the neighbourhood clique can be extended by adding node $i$ itself. We noticed that most of these special cliques have small size. The most frequent pattern (37\%) is a clique of size 3, that is a triple $i$, $j$, $k$ of scholars such that $j$ and $k$ are the only collaborators of $i$ and they are themselves collaborators (not necessarily on a paper with $i$). Moreover, 26\% of these special cliques have size 4, 15\% of them have size 5, 8\% of them have size 6, and the distribution decreases slowly with a long tail ending with size 114. This large maximum value, however, is easily explained by the existence of a paper with 114 co-authors.\footnote{A paper with such large number of authors is quite unnatural in computer science. In fact, this hyper-authored paper is an article in bioinformatics, an area at the intersection of biology and computer science.}
\begin{comment}
Notice that this is not necessarily the largest clique in the network, but only a lower bound.\footnote{Finding the largest clique in a graph is an NP-complete problem, that is, most likely, there exists no efficient (polynomial) algorithm to solve the problem.} Indeed, there might be a larger clique such that the clique is not the neighbourhood of any of the nodes in the clique.
\end{comment}
Social networks differ from most other types of networks, including technological and biological networks, at least in two aspects \citep{NP03}. First, the transitivity coefficient is higher for social networks. Second, they show positive correlation between the degrees of adjacent nodes, while other networks have negative correlation. \textit{Assortative mixing} is the tendency of nodes to connect to other nodes that are like them in some way. In particular, assortative mixing by degree is the tendency of nodes to connect to other nodes with a similar degree. In our context, we have assortative mixing by degree if scholars collaborate preferentially with other scholars with similar number of collaborators. We have disassortative mixing by degree if collaborative scholars co-author with hermits and vice versa. We have no mixing at all if none of these patterns is clearly visible.
A quantitative measure of the magnitude of mixing by degree can be computed using the \textit{Pearson correlation coefficient} applied to the degree sequences of nodes connected by an edge. The coefficient ranges between -1 and 1, where negative values indicate disassortative mixing, positive values indicate assortative mixing, and values close to 0 indicate no mixing. The coefficient is 0.17 for the whole network, 0.16 for the conference one, and 0.30 for the journal one. The values are statistically significant. Hence, collaboration networks in computer science confirm assortative mixing by degree, as other social networks: collaborative computer scientists tend to collaborate with other collaborative computer scientists, and solitary authors match preferentially with other solitary authors. Notice the higher value for collaboration in journal papers, indicating that this collaboration pattern is stronger in this case. These findings are useful to picture the structure of the collaboration network. A network that is assortative by degree has a \textit{core-periphery structure}: a dense core of high-degree nodes is surrounded by peripheral low-degree ones.
In fact, the correlation is even stronger. Pearson correlation coefficient is an appropriate measure of correlation when the data samples roughly follow a normal distribution. In this case, the mean of the samples, which is used in the computation of the coefficient, represents a characteristic scale for the network. However, we have seen in Section~\ref{degree} that our collaboration networks are \textit{scale-free}: their degree distribution is highly right-skewed and the distribution mean does not represent a typical value of the number of collaborators of a scholar. To correct for the bias introduced in the Pearson correlation coefficient by the use of asymmetric degree distributions, we can either make a logarithmic transformation of the degree sequences before using the Pearson coefficient formula, or use a non-parametric correlation method, like the Spearman one. Both methods give the same results: the correlations increase to 0.25, 0.21, and 0.36 in the whole, conference, and journal collaboration network, respectively.
\section{Conclusions}
We have analysed collaboration in computer science using a network science approach. Substantially, we have found that the scientific productivity of computer scientists is highly asymmetric, in agreement with Lotka's law of scientific productivity. The collaboration level in computer science papers is rather moderate with respect to other scientific fields, indicating that a little collaboration of two or at most three authors is optimal in computer science. However, conference papers are more collaborative than journal ones. This suggests that collaboration is more important when there are stringent deadlines for the production of a paper, like those imposed by computer science conferences.
The computer science collaboration network is a widely connected small world, hence scientific information flows along collaboration links very quickly and it potentially reaches almost all scholars in the discipline. This signals the reached scientific maturity of the relatively young field of computer science. The distribution of collaboration among computer science scholarsis highly skewed and concentrated, with few star collaborators responsible for a relatively high share of collaborations. The collaboration network is, however, resilient to the removal of these star collaborators, meaning that the connectivity of the network does not crucially depend on them. These is good news for the computer science community, since it means that this restricted circle of influential scholars with many contacts do not control the diffusion of information on the whole discipline, although they are probably very influential within their local communities.
Finally, while the conference collaboration network is more widely and densely connected than the journal counterpart, journal collaboration establishes a stronger social relationship among authors, also because, as observed above, the typical journal paper has fewer authors than the average conference contribution. The journal network is more dependent on star collaborators, and these highly collaborative authors prefer to collaborate with other star collaborators, leading to a core-periphery structure in the journal graph. These patterns might indicate that conferences are better to widely and quickly communicate scientific results, while journals are optimal to establish stronger and longer scientific relationships with other scholars.
\bibliographystyle{elsarticle-harv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,475 |
\section{INTRODUCTION}
\label{sec:I}
\subsection{Motivation}
The Coulomb interaction between electrons in solids leads to a
virtually unlimited variety of phenomena, such as magnetic correlations
and magnetic order, high-temperature superconductivity, metal-insulator
transitions, phase separation and stripes, and the formation of exotic
quantum liquid phases.
The latter include Luttinger liquids, quantum critical points, and
fractional quantum Hall states.
Interacting electron systems usually exhibit very distinct behavior on
different energy scales. Composite objects and collective phenomena emerge
at scales far below the bare energy scales of the microscopic Hamiltonian.
For example, in cuprate high-temperature superconductors one bridges three
orders of magnitude from the highest scale, the bare Coulomb interaction,
via the intermediate scale of short-range magnetic correlations, down to
the lowest scale of $d$-wave superconductivity and other ordering phenomena
(see Fig.~\ref{htscscales}).
\begin{figure}[ht]
\centerline{\includegraphics[width = 6cm]{fig1.eps}}
\caption{(Color online)
Important energy scales in high-temperature superconductors
of the cuprate family. Magnetic interactions and superconductivity are
generated from the kinetic energy (hopping) and the Coulomb
repulsion.}
\label{htscscales}
\end{figure}
This diversity of scales is a major obstacle to a straightforward
numerical solution of microscopic models, since the most
interesting phenomena emerge only at low temperatures and in systems with
a large size. It is also hard to deal with by conventional many-body
methods, if one tries to treat all scales at once and within the same
approximation, for example by summing a subclass of Feynman diagrams.
Perturbative approaches which do not separate different scales are plagued
by infrared divergences, and are therefore often inapplicable even at
weak coupling, especially in low dimensions.
It is thus natural to treat degrees of freedom with different energy
scales successively, descending step by step from higher to lower scales.
This is the main idea behind the {\em renormalization group} (RG).
\subsection{RG for interacting Fermi systems}
Renormalization group methods have a long tradition in the theory
of interacting Fermi systems.
Already in the 1970s, various versions of the RG have been used to
deal with infrared singularities arising in one-dimensional Fermi
systems \cite{Solyom79}.
Naturally, the RG was also applied to (mostly bosonic) effective
field theories describing critical phenomena at continuous
classical or quantum phase transitions in interacting Fermi systems
\cite{Fradkin91,Sachdev99}.
Renormalization group approaches dealing with interacting fermions
in arbitrary dimensions $d$ have been developed much later.
Due to the extended (not point-like) geometry of the Fermi surface
singularity in dimensions $d>1$, the renormalization group flow
cannot be reduced to a finite number of running couplings.
However, the main reason for the delayed development of a
comprehensive RG approach for interacting Fermi systems in higher
dimensions was probably not this difficulty, but rather a lack of
motivation.
The few infrared singularities appearing in three-dimensional
Fermi systems could usually be handled by simple resummations of
perturbation theory \cite{Abrikosov63,Nozieres64}.
Triggered by the issue of non-Fermi liquid behavior in two-dimensional
systems, and the related discussion on the validity of perturbation
theory, systematic RG approaches to interacting Fermi systems in
arbitrary dimensions have been developed by various groups in the
early 1990s.
Aiming at a mathematical control of interacting Fermi systems,
\textcite{FT1,FT2}, and independently \textcite{Benfatto90a,
Benfatto90b}, have formulated a rigorous fermionic version of
Wilson's momentum-shell RG \cite{Wilson74}.
Important rigorous results have indeed been obtained in
one-dimensional \cite{Benfatto94} and two-dimensional
\cite{FMRT92,FST1,FKT03,FKT04,Disertori00,BGM06} systems.
The essential message from these results is that no hitherto
unknown instabilities or non-perturbative effects occur in Fermi
systems with sufficiently weak short-range interactions, at least
in the absence of special features such as Van Hove singularities
at the Fermi level.
The Wilsonian RG for interacting Fermi systems was popularized among
(non-mathematical) physicists by \textcite{Shankar91,Shankar94}
and \textcite{Polchinski93}, who presented some of the main ideas
in a pedagogical style.
In particular, they provided an intuitive RG perspective of Fermi
liquid theory.
Subtleties associated with the singularities of the interaction
vertex for forward scattering were clarified a bit later
\cite{Chitov95,Metzner98}.
A Hamiltonian-based RG interpretation of Fermi liquid theory was
presented by \textcite{Hewson94}, who discussed not only translation
invariant systems but also models for magnetic impurities in metals.
As an alternative to the Wilsonian RG one may also use flow
equations for Hamiltonians based on infinitesimal unitary
transformations, which make the Hamiltonian successively more
diagonal \cite{Wegner94}.
This approach has been used successfully for quantum impurity
models and other systems \cite{Kehrein06}.
A weak coupling truncation of the flow equations has been
applied to identify instabilities of the two-dimensional Hubbard
model \cite{Grote02}.
There is much current interest in RG methods for correlated fermions
in non-equilibrium. The perturbative RG \cite{Rosch01,Mitra06},
Wilson's numerical RG \cite{Anders05}, as well as Wegner's flow
equation approach \cite{Kehrein06} were extended to non-equilibrium,
and real-time RG methods were developed
\cite{Schoeller00a, Schoeller00b,Schoeller09a}.
\subsection{Functional renormalization group}
The Wilsonian RG is not only useful for a deeper and partially
even rigorous understanding of interacting fermion systems.
A specific version of Wilson's RG known as {\em exact} or
{\em functional} RG turned out to provide a valuable framework
for computational purposes.
Approximations derived from exact functional flow equations
have played an increasingly important role in the last decade.
These developments are the central topic of this review.
Exact flow equations describe the evolution of a generating
functional for all many-particle Green or vertex functions
as a function of a flow parameter $\Lam$, usually an infrared
cutoff.
They can be derived relatively easily from a functional integral
representation of the generating functional.
Exact flow equations have been known since the early years of
the RG, starting with the work of \textcite{Wegner73}.
\textcite{Polchinski84} employed an exact flow equation to
formulate a relatively simple proof of renormalizability of
the $\Phi^4$-theory in four dimensions.
Renormalizability proofs can be further simplified by using a
Wick-ordered variant of Polchinski's equation \cite{Wieczerkowski88}.
For computational purposes the exact flow equation for the
effective action, first derived in the context of bosonic
field theories by \textcite{Wetterich93} turned out to be most
convenient.
The effective action $\Gam^{\Lam}[\phi]$ is the generating
functional for one-particle irreducible vertex functions.
The latter are obtained by taking derivatives with respect to
the source field $\phi$.
The flow parameter $\Lam$ describes a regularization of the
underlying bare action, which regularizes infrared divergencies
in perturbation theory.
The regularization is removed at the end of the flow, say for
$\Lam \to 0$.
The initial regulator (for $\Lam = \Lam_0$) can be chosen such that
$\Gam^{\Lam_0}[\phi]$ is given by the bare action.
The flow of $\Gam^{\Lam}[\phi]$ then provides a smooth interpolation
between the bare action of the system and the final effective action
$\Gam[\phi]$, from which any desired information can be extracted.
This flow is determined by an exact functional differential equation
\cite{Wetterich93,Morris94,Ellwanger94}.
Expanding in the fields one obtains a hierarchy of flow equations
for the one-particle irreducible vertex functions.
The advantage of that hierarchy compared to others, obtained, for
example, from Polchinski's equation, is that self-energy feedback
is included automatically and no one-particle reducible terms
appear.
The expression {\em functional} RG stems from the feature that
the exact flow equations describe the flow of a functional or
(equivalently) of a hierarchy of functions.
An important difference compared to Wilson's original formulation
is that a complete set of source fields is kept in the flowing
generating functionals, not only those corresponding to scales
below $\Lam$.
Hence, the full information on the properties of the system
remains accessible, not only the low energy or long wavelength
behavior.
Exact flow equations can be solved exactly only in special cases,
where the underlying model can also be solved exactly (and more
easily) by other means.\footnote{An instructive example is provided
by the exact solution of the Tomonaga-Luttinger model via functional
RG flow equations \cite{Schuetz05}.}
However, the functional RG is a valuable source for devising powerful
new approximation schemes, which can be obtained by truncating the
hierarchy and/or by a simplified parametrization of the Green or
vertex functions.
These approximations have several distinctive advantages: i) they have
a renormalization group structure built in, that is, scales are handled
successively and infrared singularities are thus treated properly;
ii) they can be applied directly to microscopic models, not only to
effective field theories which capture only some asymptotic behavior;
iii) they are physically transparent, for example one can see
directly how and why new correlations form upon lowering the scale;
iv) one can use different approximations at different scales.
Small steps from a scale $\Lam$ to a slightly smaller scale $\Lam'$
are much easier to control than an integration over all degrees of
freedom in one shot, and one can take advantage of the flexibility
provided by the choice of a suitable flow parameter.
Approximations derived from exact flow equations have been applied
in many areas of quantum field theory and statistical physics
\cite{Berges02}.
In the context of interacting Fermi systems, functional RG methods
were first used for an unbiased stability analysis of the
two-dimensional Hubbard model
\cite{Zanchi98,Zanchi00,Halboth00a,Honerkamp01d}.
Since then, approximations derived within the functional RG
framework have been applied to numerous interacting fermion
systems.
\subsection{Scope of the review}
This review provides a thorough introduction to the functional
RG in the context of interacting Fermi systems.
It should serve as a manual and reference for many-body theorists
who wish to apply approximations based on the functional RG to
their own problem of interest.
We will first describe the functional RG framework and derive in
particular the exact flow equations, which are the starting point
for approximations.
We will discuss general aspects related to the flow equations
such as the choice of cutoffs, power counting and truncations.
Links to the use of flow equations in the mathematical literature
will be pointed out along the way.
We will then review some of the most interesting applications of
truncated functional RG equations (see table of content).
Our aim is not to deliver an exhaustive overview of all applications,
but rather to show via selected applications how the functional RG
method can be fruitfully used.
The functional RG was recently extended to Fermi systems out of
equilibrium \cite{Jakobs03,Gezzi07,Jakobs07a,Jakobs10a,Jakobs10c,
Karrasch10a,Karrasch10c}.
In the derivation of the flow equations in Sec.~II we restrict
ourselves to the equilibrium formalism.
Functional RG flow equations for non-equilibrium Keldysh Green and
vertex functions can be derived in close analogy
\cite{Jakobs03,Gezzi07,Jakobs10b,Karrasch10b}.
The necessary extensions are briefly mentioned when discussing the
application of this method to finite bias steady state transport
through correlated quantum wires and quantum dots in Sec.~VI.
A number of reviews with a focus on the functional RG are already
available.
Mathematically rigorous developments until the end of the last
millenium were summarized in a book by \textcite{Salmhofer99},
a large portion of which is dedicated to interacting Fermi
systems.
Examples of approximations derived from the exact flow equation
for the effective action with many applications in quantum field
theory and statistical physics were presented in the review
article by \textcite{Berges02}.
A detailed introduction to the functional RG in a textbook style
supplemented by selected applications (including Fermi systems)
can be found in the recent book by \textcite{Kopietz10}.
\section{FUNCTIONAL FLOW EQUATIONS}
\label{sec:II}
In this section we present the general functional RG framework.
The reader should be familiar with the functional integral formalism
for quantum many-body systems, as described in classic textbooks
such as \textcite{Negele87}.
After introducing the generating functionals for Green and vertex
functions in Sec.~II.A, we derive the exact functional flow
equations in Sec.~II.B.
The flow equation (\ref{floweqGam}) for the effective action
$\Gam^{\Lam}$ is the central equation of this review.
Expanding in the fields we derive the hierarchy of flow equations
for vertex functions in Sec.~II.C, which is the starting point
for approximations.
Possible choices of flow parameters are reviewed in Sec.~II.D.
The general structure of the RG hierarchy and power counting
are discussed in Sec.~II.E, with various references
to the closely related mathematical literature.
Sec.~II.F is dedicated to flow equations for observables and
correlation functions.
Coupled flow equations for fermions and bosons, which are useful
for studies of spontaneous symmetry breaking and quantum
criticality, are derived in Sec.~II.G.
\subsection{Generating functionals}
\label{sec:functionals}
We consider a system of interacting fermions which can be
described by Grassmann fields $\psi$, $\bar\psi$, and an action
of the form
\begin{equation} \label{bareaction}
\cS[\psi,\psib] =
- (\psib, G_0^{-1} \psi) + V[\psi,\psib] \; ,
\end{equation}
where $V[\psi,\psib]$ is an arbitrary many-body interaction,
and $G_0$ is the propagator of the non-interacting system.
The bracket $(.,.)$ is a shorthand notation for the sum
$\sum_x \psib(x) \, (G_0^{-1} \psi)(x)$, where
$(G_0^{-1} \psi)(x) = \sum_{x'} G_0^{-1}(x,x') \, \psi(x')$.
The Grassmann field index $x$ collects the quantum numbers
of a suitable single-particle basis set and imaginary time
or frequency.
In case of continuous variables, the sum over $x$ includes
the appropriate integrals.
Prefactors such as temperature or volume factors depend on
the representation (e.g.\ real or momentum space) and are
therefore not written in this general part.
A two-particle interaction has the general form
\begin{equation} \label{bareinteraction}
V[\psi,\psib] =
\frac{1}{4} \sum_{x_1,x_2 \atop x'_1,x'_2}
V(x'_1,x'_2;x_1,x_2) \,
\psib(x'_1) \psib(x'_2) \psi(x_2) \psi(x_1) \; .
\end{equation}
In particular,
for spin-$\frac{1}{2}$ fermions with a single-particle basis
labeled by momentum $\bk$ and spin orientation $\sg$, one has
$x = (k_0,\bk,\sg)$, where $k_0$ is the fermionic Matsubara
frequency.
If the bare part of the action is translation and spin-rotation
invariant, the bare propagator has the diagonal and
spin-independent form $G_0(x,x') =
\delta_{k_0k'_0} \delta_{\bk\bk'} \delta_{\sg\sg'} G_0(k_0,\bk)$
with
\begin{equation} \label{g0k}
G_0(k_0,\bk) = \frac{1}{ik_0 - \xi_{\bk}} \; ,
\end{equation}
where $\xi_{\bk} = \eps_{\bk} - \mu$ is the single-particle
energy relative to the chemical potential.
Connected Green functions can be obtained from the generating
functional \cite{Negele87}
\begin{equation} \label{calG}
\cG[\eta,\etab] =
- \ln \int \cD\psi \cD\psib \,
e^{-\cS[\psi,\psib]} \, e^{(\etab,\psi) + (\psib,\eta)} \; ,
\end{equation}
where
$\int \cD\psi \cD\psib \dots =
\int \prod_x d\psi(x) d\psib(x) \dots \;$.
Completing squares yields the identity
\begin{equation} \label{gauss}
\int \cD\psi \cD\psib \, e^{(\psib,G_0^{-1} \psi)}
e^{(\etab,\psi) + (\psib,\eta)} =
\cZ_0 \; e^{(-\etab,G_0 \eta)} \; ,
\end{equation}
where $\cZ_0 =
\int \cD\psi \cD\psib \, e^{(\psib, \, G_0^{-1} \psi)}$
is the partition function of the non-interacting system.
Hence $\cG[\eta,\etab] = -\ln\cZ_0 + (\etab, G_0 \eta)$ in
the non-interacting case $V[\psi,\psib] = 0$.
For vanishing source fields, $\cG[0,0] = - \ln\cZ$, where
\begin{equation} \label{Z}
\cZ =
\int \cD\psi \cD\psib \, e^{-\cS[\psi,\psib]}
\end{equation}
is the partition function of the interacting system.
The connected $m$-particle Green functions are given by
\begin{eqnarray} \label{G2m}
\hskip -5mm
&& G^{(2m)}(x_1,\dots,x_m;x'_1,\dots,x'_m) =
\nonumber \\
&& - \bra \psi(x_1) \dots \psi(x_m)
\psib(x'_m) \dots \psib(x'_1) \ket_c =
\nonumber \\
&& (-1)^m \! \left.
\frac{\partial^{2m} \cG [\eta,\etab]}{\partial\etab(x_1) \dots
\partial\etab(x_m)
\partial\eta(x'_m) \dots
\partial\eta(x'_1)}
\right|_{\eta,\etab = 0} , \hskip 5mm
\end{eqnarray}
where $\bra \dots \ket_c$ is the connected average of the
product of Grassmann variables between the brackets.
The one-particle Green function $G^{(2)}$ is the propagator of
the interacting system, which we will usually denote without the
superscript by $G$.
Expanding $\cG[\eta,\etab]$ in the fields yields a formal power
series with the connected Green functions as coefficients,
\begin{widetext}
\begin{equation} \label{Gexpansion}
\cG[\eta,\etab] = - \ln\cZ + (\etab, G \eta) +
\frac{1}{(2!)^2} \sum_{x_1,x_2,x'_1,x'_2}
G^{(4)}(x_1,x_2;x'_1,x'_2) \,
\etab(x_1) \etab(x_2) \eta(x'_2) \eta(x'_1) + \dots
\; .
\end{equation}
\end{widetext}
Renormalization group equations are most conveniently formulated
for the Legendre transform of $\cG[\eta,\etab]$, the socalled
{\em effective action}
\begin{equation} \label{Gampsi}
\Gam[\psi,\psib] =
(\etab,\psi) + (\psib,\eta) + \cG[\eta,\etab] \; ,
\end{equation}
with $\psi = - \partial\cG/\partial\etab$ and
$\psib = \partial\cG/\partial\eta \,$, which generates
one-particle irreducible vertex functions \cite{Negele87}
\begin{eqnarray} \label{Gam2m}
&& \Gam^{(2m)}(x'_1,\dots,x'_m;x_1,\dots,x_m) =
\nonumber \\[2mm]
&& \left.
\frac{\partial^{2m} \Gam[\psi,\psib]}
{\partial\psib(x'_1) \dots \partial\psib(x'_m)
\partial\psi(x_m) \dots \partial\psi(x_1)}
\right|_{\psi,\psib = 0} . \hskip 5mm
\end{eqnarray}
In the non-interacting case one obtains
$\Gam[\psi,\psib] = - \ln\cZ_0 - (\psib, G_0^{-1} \psi)$.
The Legendre correspondence between the functionals $\cG$ and $\Gam$
yields relations between the connected Green functions $G^{(2m)}$
and the vertex functions $\Gam^{(2m)}$.
In particular,
\begin{equation} \label{Dyson}
\Gam^{(2)} = G^{-1} = G_0^{-1} - \Sg \; ,
\end{equation}
where $\Sg$ is the self-energy.
The connected two-particle Green function is related to the
two-particle vertex by
\begin{eqnarray} \label{G4Ga4}
&& G^{(4)}(x_1,x_2;x'_1,x'_2) =
\sum_{y_1,y_2,y'_1,y'_2} \! G(x_1,y'_1) \, G(x_2,y'_2) \,
\nonumber \\
&& \times \, \Gam^{(4)}(y'_1,y'_2;y_1,y_2) \, G(y_1,x'_1) \, G(y_2,x'_2)
\; ,
\end{eqnarray}
while the three-particle Green function
$G^{(6)} = G^3 \Gam^{(6)} G^3 + G^3 \Gam^{(4)} G \Gam^{(4)} G^3$
involves $\Gam^{(4)}$ and $\Gam^{(6)}$.
More generally, the connected $m$-particle Green functions are
obtained by adding all possible trees that can be formed with
vertex functions of equal or lower order and $G$-lines
\cite{Negele87}.
The effective action obeys the reciprocity relations
\begin{equation} \label{reciproc1}
\frac{\partial\Gam}{\partial\psi} = - \etab \; , \quad
\frac{\partial\Gam}{\partial\psib} = \eta \; .
\end{equation}
The second functional derivatives of $\cG$ and $\Gam$ with respect
to the fields are also reciprocal \cite{Negele87}.
We define the matrices of second derivatives at finite fields
\begin{widetext}
\begin{equation} \label{bG2}
\bG^{(2)}[\eta,\etab] = - \left( \begin{array}{cc}
\frac{\partial^2\cG}{\partial\etab(x)\partial\eta(x')} &
- \frac{\partial^2\cG}{\partial\etab(x)\partial\etab(x')} \\[3mm]
- \frac{\partial^2\cG}{\partial\eta(x)\partial\eta(x')} &
\frac{\partial^2\cG}{\partial\eta(x)\partial\etab(x')}
\end{array} \right)
= - \left( \begin{array}{cc}
\bra \psi(x) \psib(x') \ket & \bra \psi(x) \psi(x') \ket \\
\bra \psib(x) \psib(x') \ket & \bra \psib(x) \psi(x') \ket
\end{array} \right) \, ,
\end{equation}
and
\begin{equation} \label{bGam2}
\bGam^{(2)}[\psi,\psib] = \left( \begin{array}{cc}
\frac{\partial^2\Gam}{\partial\psib(x')\partial\psi(x)} &
\frac{\partial^2\Gam}{\partial\psib(x')\partial\psib(x)} \\[3mm]
\frac{\partial^2\Gam}{\partial\psi(x')\partial\psi(x)} &
\frac{\partial^2\Gam}{\partial\psi(x')\partial\psib(x)}
\end{array} \right) =
\left( \begin{array}{cc}
\bar\partial\partial\Gam[\psi,\psib](x',x) &
\bar\partial\bar\partial\Gam[\psi,\psib](x',x) \\
\partial\partial\Gam[\psi,\psib](x',x) &
\partial\bar\partial\Gam[\psi,\psib](x',x)
\end{array} \right) \; ,
\end{equation}
\end{widetext}
where the matrix elements in the second matrix of the last equation
are just a more conventient notation for those in the first matrix.
The reciprocity relation for the second derivatives reads
\begin{equation} \label{reciproc2}
\bGam^{(2)}[\psi,\psib] =
\left( \bG^{(2)}[\eta,\etab] \right)^{-1} \; .
\end{equation}
Note that anomalous components are involved as long as the
source fields are finite.
Only at $\eta=\etab=0$ and $\psi=\psib=0$, and in the absence
of $U(1)$ charge symmetry breaking one has the simple relation
$\Gam^{(2)} = \big( G^{(2)} \big)^{-1}$.
Another useful generating functional is the
{\em effective interaction} \cite{Salmhofer99}
\begin{equation} \label{effpot}
\cV[\chi,\chib] =
- \ln \left\{ \frac{1}{\cZ_0}
\int \cD\psi \cD\psib \, e^{(\psib,G_0^{-1}\psi)}
e^{-V[\psi+\chi,\psib+\chib]} \right\} \; .
\end{equation}
A simple substitution of variables yields the relation
\begin{equation} \label{VG}
\cV[\chi,\chib] =
\cG[\eta,\etab] + \ln\cZ_0 - (\etab,G_0 \eta) \; ,
\end{equation}
where $\chi = G_0 \eta$ and $\chib = G_0^t \etab$.
Here $G_0^t$ is the transposed bare propagator, that is,
$G_0^t(x,x') = G_0(x',x)$.
Hence, functional derivatives of $\cV[\chi,\chib]$ with
respect to $\chi$ and $\chib$ generate connected Green
functions with bare propagators amputated from external
legs in the corresponding Feynman diagrams.
The term $\ln\cZ_0 - (\etab,G_0 \eta)$ cancels the
non-interacting part of $\cG[\etab,\eta]$ such that
$\cV[\chi,\chib] = 0$ for $V[\psi,\psib] = 0$.
The effective interaction $\cV$ can also be expressed via
functional derivatives, instead of a functional integral:
\begin{eqnarray} \label{expV}
e^{-\cV[\chi,\chib]} &=&
\frac{1}{\cZ_0}
\int \cD\psi \cD\psib \, e^{(\psib,G_0^{-1}\psi)} \,
e^{-V[\psi+\chi,\psib+\chib]} \nonumber \\
&=& \frac{1}{\cZ_0} \,
e^{-V[\partial_{\etab},\partial_{\eta}]}
\int \cD\psi \cD\psib \, e^{(\psib,G_0^{-1}\psi)}
\nonumber \\
&& \times \left.
e^{(\etab,\psi+\chi) + (\eta,\psib+\chib)} \right|_{\eta,\etab=0}
\nonumber \\
&=& e^{-V[\partial_{\etab},\partial_{\eta}]}
\left. e^{(\etab,G_0 \eta)} e^{(\etab,\chi) + (\eta,\chib)}
\right|_{\eta,\etab=0}
\nonumber \\
&=& e^{-V[\partial_{\etab},\partial_{\eta}]}
\left. e^{(\partial_{\chi},G_0 \partial_{\chib})}
e^{(\etab,\chi) + (\eta,\chib)}
\right|_{\eta,\etab=0}
\nonumber \\
&=& e^{\Delta_{G_0}} \, e^{-V[\chi,\chib]} \; ,
\end{eqnarray}
with the functional Laplacian
\begin{equation} \label{DeltaG0}
\Delta_{G_0} =
\big( \partial_{\chi}, G_0 \partial_{\chib} \big) =
\sum_{x,x'} \frac{\partial}{\partial\chi(x)} \, G_0(x,x') \,
\frac{\partial}{\partial\chib(x')} \; .
\end{equation}
It is sometimes convenient (see Sec.~II.G) to combine the
fields $\psi$ and $\psib$ in a Nambu-type field
\begin{equation} \label{nambuPsi}
\Psi(x) = \left( \begin{array}{c} \psi(x) \\
\psib(x) \end{array} \right) \; ,
\end{equation}
and similarly for the source fields $\eta$ and $\etab$,
\begin{equation} \label{nambuH}
H(x) = \left( \begin{array}{c} {\phantom -} \eta(x) \\
- \etab(x) \end{array} \right) \; .
\end{equation}
The minus sign in the definition of $H$ makes sure that the
source term $(\etab,\psi) + (\psib,\eta)$ appearing in the
definition of $\cG$, and also in the Legendre transform relating
$\cG$ and $\Gam$, can be written concisely as $(\bar H, \Psi)$.
In Nambu notation, the matrices of second derivatives of
$\cG$ and $\Gam$ have the compact form
\begin{equation} \label{bG2H}
\bG^{(2)}[H] =
- \frac{\partial^2 \cG}{\partial\bar H(x) \partial H(x')}
\end{equation}
and
\begin{equation} \label{bGam2Psi}
\bGam^{(2)}[\Psi] =
\frac{\partial^2 \Gam}{\partial\Psib(x') \partial\Psi(x)}
\; ,
\end{equation}
respectively.
\subsection{Exact fermionic flow equations}
\label{sec:floweqs}
In this section we derive exact flow equations describing
the evolution of the generating functionals defined above, as
a function of a flow parameter $\Lam$ which parametrizes a
modification of the bare propagator $G_0$.
Usually $\Lam$ is an infrared cutoff or another scale
dependence.
For example, in a translation invariant system one may impose
a momentum cutoff, modifying $G_0$ to
\begin{equation} \label{G0Lamk}
G_0^{\Lam}(k_0,\bk) =
\frac{\theta^{\Lam}(\bk)}{ik_0 - \xi_{\bk}} \; ,
\end{equation}
where $\theta^{\Lam}(\bk)$ is a function that vanishes for
$|\xi_{\bk}| \ll \Lam$ and tends to one for $|\xi_{\bk}| \gg
\Lam$. In this way the infrared singularity of the propagator
at $k_0 = 0$ and $\xi_{\bk} = 0$ (corresponding to the
non-interacting Fermi surface in $\bk$-space) is cut off at
the scale $\Lam$.
A simple choice for $\theta^{\Lam}(\bk)$, which was often
used in numerical solutions of truncated flow equations, is
\begin{equation} \label{thetaLamk}
\theta^{\Lam}(\bk) = \Theta(|\xi_{\bk}| - \Lam) \; ,
\end{equation}
where $\Theta$ is the step function. With this choice
momenta close to the Fermi surface are strictly excluded,
as illustrated in Fig.~\ref{fig:cutoff} for a two-dimensional
lattice fermion system.
\begin{figure}[ht]
\centerline{\includegraphics[width = 4.5cm]{fig2.eps}}
\caption{(Color online)
Momentum space region around the Fermi surface excluded
by a sharp momentum cutoff for fermions with a tight-binding
dispersion on a two-dimensional square lattice (lattice
constant = one).}
\label{fig:cutoff}
\end{figure}
Alternatively, one may also use a smooth cutoff function.
In the absence of translation invariance it is more convenient
to use a frequency cutoff instead of a momentum cutoff.
The cutoff excludes ''soft modes'' below the scale $\Lam$ from
the functional integral.
Instead of a cutoff one may also choose other flow parameters
such as temperature. The various possibilities will be discussed
more extensively in Sec.~\ref{sec:flowparameters}.
For the derivation of the flow equations it does not matter how
$G_0^{\Lam}$ depends on $\Lam$.
The bare action constructed with $G_0^{\Lam}$ (instead of $G_0$)
will be denoted by $\cS^{\Lam}[\psi,\psib]$, and the generating
functionals introduced in Sec.~\ref{sec:functionals} by
$\cG^{\Lam}[\eta,\etab]$, $\cV^{\Lam}[\chi,\chib]$, and
$\Gam^{\Lam}[\psi,\psib]$, respectively.
The original functionals $\cG$, $\cV$ and $\Gam$ are recovered
in the limit $\Lam \to 0$.
In the presence of a cutoff, Eq.~(\ref{expV}) becomes
\begin{equation} \label{expVLam}
e^{-\cV^{\Lam}} = e^{\Delta_{G_0^{\Lam}}} \, e^{-V} \; .
\end{equation}
At the highest energy scale $\Lam_0$ one has $G_0^{\Lam_0} = 0$,
and thus $\cV^{\Lam_0} = V$.
Hence, $\cV^{\Lam}$ interpolates smoothly between the bare
interaction $V$ and the generating functional $\cV$.
Introducing the soft mode propagator
\begin{equation} \label{barG0Lam}
\bar G_0^{\Lam} = G_0 - G_0^{\Lam} \; ,
\end{equation}
which has support on scales {\em below} $\Lam$, we can write
\begin{equation} \label{duality}
e^{-\cV} = e^{\Delta_{G_0}} e^{-V} =
e^{\Delta_{\bar G_0^{\Lam}} + \Delta_{G_0^{\Lam}}} \,
e^{-V} =
e^{\Delta_{\bar G_0^{\Lam}}} \, e^{-\cV^{\Lam}} \; .
\end{equation}
$\cV^{\Lam}$ obviously plays a dual role: It is the generating
functional for (amputated) Green functions of a system with a
cutoff $\Lam$, and at the same time the interaction for the
remaining low energy degrees of freedom
\cite{Morris94,Salmhofer99}.
The effective interaction satisfies the following exact
renormalization group equation \cite{Salmhofer99,Brydges88}
\begin{equation}\label{floweqV}
\frac{d}{d\Lam} \cV^{\Lam}[\chi,\chib] =
- \left( \frac{\partial\cV^{\Lam}}{\partial\chi} \, ,
\dot{G}_0^{\Lam} \,
\frac{\partial\cV^{\Lam}}{\partial\chib} \right)
- {\rm tr} \left( \dot{G}_0^{\Lam} \,
\frac{\partial^2 \cV^{\Lam}}
{\partial\chib \partial\chi} \right) \; ,
\end{equation}
where $\dot{G}_0^{\Lam} = \frac{d}{d\Lam} G_0^{\Lam}$ and
tr denotes the trace ${\rm tr} A = \sum_x A(x,x)$.
Its derivation is simple:
\begin{eqnarray} \label{floweqVderiv}
\frac{d}{d\Lam} {\cV^{\Lam}} &=&
- e^{\cV^{\Lam}} \frac{d}{d\Lam} e^{-\cV^{\Lam}} \nonumber \\
&=& - e^{\cV^{\Lam}} \frac{d}{d\Lam}
\big( e^{\Delta_{G_0^{\Lam}}} e^{-V} \big) =
- e^{\cV^{\Lam}} \Delta_{\dot{G}_0^{\Lam}} \, e^{-\cV^{\Lam}}
\nonumber \\
&=& \mbox{right-hand side of Eq.~(\ref{floweqV})} \; .
\nonumber
\end{eqnarray}
In the second step we have used Eq.~(\ref{expVLam}).
With the initial condition
\begin{equation} \label{VLam0}
\cV^{\Lam_0}[\chi,\chib] = V[\chi,\chib] \; ,
\end{equation}
the RG equation determines the flow of $\cV^{\Lam}$ uniquely
for all $\Lam < \Lam_0$.
The initial value $\Lam_0$ must be chosen such that
$G_0^{\Lam_0}$ vanishes.
For a sharp momentum cutoff, $\Lam_0$ can be chosen as the
maximal value of $|\xi_{\bk}|$; for a frequency cutoff
$\Lam_0 = \infty$.
An expansion of the functional $\cV^{\Lam}[\chi,\chib]$ in the
renormalization group equation (\ref{floweqV}) in powers of $\chi$
and $\chib$ leads to the fermionic analog of Polchinski's
\cite{Polchinski84} flow equations for amputated connected
$m$-particle Green functions $V^{(2m) \Lam}$.
From the flow equation for $\cV^{\Lam}$, Eq.~(\ref{floweqV}), and
the relation (\ref{VG}) applied to $\cV^{\Lam}$ and $\cG^{\Lam}$,
one obtains an exact flow equation for $\cG^{\Lam}$:
\begin{equation} \label{floweqG}
\frac{d}{d\Lam} \cG^{\Lam}[\eta,\etab] =
\left( \frac{\partial \cG^{\Lam}}{\partial\eta} ,
\dot{Q}_0^{\Lam}
\frac{\partial \cG^{\Lam}}{\partial\etab} \right) +
{\rm tr} \left(\dot{Q}_0^{\Lam} \,
\frac{\partial^2 \cG^{\Lam}}
{\partial\etab \partial\eta} \right) \; ,
\end{equation}
where $Q_0^{\Lam} = (G_0^{\Lam})^{-1}$ and the dot denotes a
$\Lam$-derivative.
This flow equation can also be derived more directly, by
applying a $\Lam$-derivative to the functional integral
representation of $\cG^{\Lam}$.
The flow equations for $G^{(2m)\Lam}$ and $V^{(2m)\Lam}$ generate,
among others, also one-particle reducible terms, which require
some special care.
In this respect the flow equations for one-particle irreducible
vertex functions $\Gam^{(2m)\Lam}$, obtained from the
scale-dependent effective action,
\begin{equation} \label{effactionLam}
\Gam^{\Lam}[\psi,\psib] =
(\etab^{\Lam},\psi) + (\psib,\eta^{\Lam}) +
\cG^{\Lam}[\eta^{\Lam},\etab^{\Lam}] \; ,
\end{equation}
are easier to handle.
Note that $\eta^{\Lam}$ and $\etab^{\Lam}$ are $\Lam$-dependent
functions of $\psi$ and $\psib$, as they are determined by the
$\Lam$-dependent equations
$\psi = - \partial\cG^{\Lam}/\partial\etab$ and
$\psib = \partial\cG^{\Lam}/\partial\eta$.
Since the $\Lam$-dependence does not change the structure of the
action as a function of the fields, all standard relations
between the connected Green functions $G^{(2m)}$ and the vertex
functions $\Gam^{(2m)}$ carry over to the ones for $G^{(2m)\Lam}$
and $\Gam^{(2m)\Lam}$.
The $\Lam$-derivative of $\Gam^{\Lam}$ can be written as
$\frac{d}{d\Lam} \Gam^{\Lam}[\psi,\psib] =
(\frac{d}{d\Lam} \etab^{\Lam},\psi)
+ (\psib,\frac{d}{d\Lam} \eta^{\Lam})
+ \frac{d}{d\Lam} \, \cG^{\Lam}[\eta^{\Lam},\etab^{\Lam}]$,
where the derivative in front of $\cG^{\Lam}$ acts also
on the $\Lam$-dependence of $\eta^{\Lam}$ and $\etab^{\Lam}$.
Due to the relations
$\partial\cG^{\Lam}/\partial\eta = \psib$ and
$\partial\cG^{\Lam}/\partial\etab = - \psi$, most terms
cancel and one obtains
\begin{equation} \label{dLamdG}
\frac{d}{d\Lam} \Gam^{\Lam}[\psi,\psib] =
\frac{d}{d\Lam} \left.
\cG^{\Lam}[\eta^{\Lam},\etab^{\Lam}]
\right|_{\eta^{\Lam},\etab^{\Lam} \; \mbox{fixed}} \; .
\end{equation}
Inserting the flow equation (\ref{floweqG}) for $\cG^{\Lam}$ and
using the reciprocity relations (\ref{reciproc1}) and
(\ref{reciproc2}), one obtains the exact functional flow
equation for the effective action
\begin{equation} \label{floweqGam}
\frac{d}{d\Lam} \Gam^{\Lam}[\psi,\psib] =
- \big(\psib, \dot Q_0^{\Lam} \psi \big)
- \frac{1}{2} {\rm tr} \left[ \dot \bQ_0^{\Lam} \left(
\bGam^{(2)\Lam}[\psi,\psib] \right)^{-1} \right] \; .
\end{equation}
Here $\bGam^{(2)\Lam}[\psi,\psib]$ is the matrix of second
functional derivatives defined in Eq.~(\ref{bGam2}), and
\begin{equation} \label{bQ0Lam}
\bQ_0^{\Lam} = \left( \begin{array}{cc}
Q_0^{\Lam} & 0 \\ 0 & - Q_0^{\Lam t}
\end{array} \right) =
\mbox{diag} (Q_0^{\Lam},- Q_0^{\Lam t}) \; ,
\end{equation}
where $Q_0^{\Lam t}(x,x') = Q_0^{\Lam}(x',x)$.
Alternative definitions of the effective action $\Gam^{\Lam}$,
differing by interaction-independent terms, have also been used.
One variant is to normalize the functional integral defining
$\cG^{\Lam}$ at $V=0$, dividing by $\cZ_0^{\Lam}$.
This yields an additional contribution $\ln\cZ_0^{\Lam}$ to
$\cG^{\Lam}$ and to its Legendre transform $\Gam^{\Lam}$.
In the flow equation for $\Gam^{\Lam}$ this leads to an
additional term ${\rm tr}(\dot Q_0^{\Lam} G_0^{\Lam})$,
which is field independent and therefore does not couple
to the other contributions \cite{Salmhofer01}.
Another variant is \cite{Ellwanger94,Berges02}
\begin{equation} \label{GamRLam}
\Gam_R^{\Lam}[\psi,\psib] =
\Gam^{\Lam}[\psi,\psib] + (\psib,R^{\Lam}\psi) \; ,
\end{equation}
where $R^{\Lam} = Q_0^{\Lam} - Q_0$. The additional
quadratic term cancels the first (trivial) term in the
flow equation (\ref{floweqGam}) for $\Gam^{\Lam}$, and one
obtains the equivalent flow equation
\begin{equation} \label{floweqGamR}
\frac{d}{d\Lam} \Gam_R^{\Lam}[\psi,\psib] =
- \frac{1}{2} {\rm tr} \left[ \dot \bR^{\Lam} \left(
\bGam_R^{(2)\Lam}[\psi,\psib]
+ \bR^{\Lam} \right)^{-1} \right] \; ,
\end{equation}
where
$\bR^{\Lam} =
\mbox{diag} \left( R^{\Lam},- R^{\Lam t} \right)$.
The functional $\Gam_R^{\Lam}$ and its analog for bosonic
fields is known as {\em effective average action} in the
literature \cite{Berges02}.
Both $\Gam_R^{\Lam}$ and $\Gam^{\Lam}$ tend to the same
effective action $\Gam$ in the limit $\Lam \to 0$, where
$R^{\Lam}$ vanishes.
At the initial scale $\Lam_0$, one has
$\Gam_R^{\Lam_0}[\psi,\psib] = \cS[\psi,\psib]$,
while
\begin{eqnarray} \label{GamLam0}
\Gam^{\Lam_0}[\psi,\psib]
&=&
- (\psib, Q_0^{\Lam_0} \psi) + V[\psi,\psib]
=
\cS^{\Lam_0}[\psi,\psib]
\nonumber\\
&=&
\cS[\psi,\psib] - (\psib,R^{\Lam_0}\psi) \; .
\end{eqnarray}
Hence, $\Gam_R^{\Lam}$ has the attractive feature that it
interpolates smoothly between the (unregularized) bare action
$\cS$ and the final effective action $\Gam$, while $\Gam^{\Lam}$
interpolates between the {\em regularized} bare action $\cS^{\Lam_0}$
and $\Gam$.
On the other hand, the functional $\Gam^{\Lam}$ has the
advantage that its second functional derivative directly yields
the inverse propagator $(G^{\Lam})^{-1}$ without the need to
add $R^{\Lam}$.
In Appendix \ref{sec:wick} we present yet another version of
exact flow equations, based on a {\em Wick ordered} effective
interaction. That version also contains one-particle
reducible contributions, but it has the distinct advantage that
the vertices are connected by propagators with an energy scale
at or below $\Lam$. This facilitates a systematic power counting
\cite{Salmhofer99}, and also a numerical evaluation of flow
equations, since the integration regions shrink upon lowering
$\Lam$.
It is instructive to compare the functional RG flow equations
with the traditional Wilsonian momentum shell RG \cite{Wilson74},
which was applied to Fermi systems by \textcite{Shankar91,Shankar94}
and \textcite{Polchinski93}.
In the commonly used version of Wilson's RG, the flow of the
effective action is computed only for soft fields, that is, for
fields with energy or momentum variables {\em below} the scale $\Lam$,
while in the functional RG the effective action with unrestricted
source fields is computed.
This allows for a direct calculation of correlations functions
with arbitrary external variables such as momenta or Matsubara
frequencies.
Furthermore, in the traditional implementations of Wilson's RG
the integration of degrees of freedom is combined with a rescaling
of momenta and fields, which is chosen such that the momentum
space and certain terms in the quadratic part of the action remain
invariant during the flow.
This facilitates the classification of interactions as relevant,
marginal or irrelevant, and helps to identify fixed points of
the flow.
The functional RG flow equations derived above do not involve
any rescaling.
Rescaling momentum space in a shell around the Fermi surface
requires a non-linear transformation in dimensions $d > 1$,
which spoils the simple linear form of momentum conservation
\cite{Shankar94,Metzner98,Kopietz01}, and is therefore of
questionable value.
Power-counting can be done also without rescaling, as shown in
Sec.~II.E.
Rescaling of the fields can be implemented easily by a simple
substitution of variables \cite{Shankar94,Kopietz01}.
However, in many applications of the functional RG, quantitative
results including power-laws with anomalous scaling dimensions
are obtained simply by direct calculation of the (unscaled)
physical quantities.
\subsection{Expansion in the fields}
\label{sec:expansion}
\subsubsection{Hierarchy of flow equations}
\label{sec:hierarchy}
The functional flow equation for the effective action can be
expanded in powers of the fields.
To this end we expand the effective action as
\begin{equation} \label{GamA}
\Gam^{\Lam}[\psi,\psib] =
\sum_{m=0}^{\infty} \cA^{(2m)\Lam}[\psi,\psib] \; ,
\end{equation}
where $\cA^{(2m)\Lam}[\psi,\psib]$ is homogeneous of degree $2m$
in the fields,
\begin{widetext}
\begin{equation} \label{A2m}
\cA^{(2m)\Lam}[\psi,\psib] = \frac{(-1)^m}{(m!)^2}
\sum_{x_1,\ldots,x_m \atop x'_1,\ldots,x'_m}
\Gam^{(2m)\Lam}(x'_1,\dots,x'_m;x_1,\dots,x_m) \,
\psib(x'_1) \dots \psib(x'_m) \psi(x_m) \dots \psi(x_1) \; ,
\end{equation}
\end{widetext}
for $m \geq 1$.
The field-independent constant $\cA^{(0)\Lam}$ yields the grand
canonical potential:
\begin{equation} \label{grandpot}
\cA^{(0)\Lam} = T^{-1} \Omega^{\Lam} \; .
\end{equation}
Here we have restored the explicit temperature factor, since it
is independent of the representation of the fields.
To expand the inverse of $\bGam^{(2)\Lam}$ on the right hand side
of the flow equation, we isolate the field-independent part of
$\bGam^{(2)\Lam}$ as
\begin{equation} \label{bGam2dec}
\bGam^{(2)\Lam}[\psi,\psib] =
(\bG^{\Lam})^{-1} - \tilde\bSg^{\Lam}[\psi,\psib] \; ,
\end{equation}
where
\begin{equation} \label{bGLam}
\bG^{\Lam} =
\left(
\left. \bGam^{(2)\Lam}[\psi,\psib] \right|_{\psi,\psib = 0}
\right)^{-1} =
\mbox{diag} (G^{\Lam},- G^{\Lam t}) \;
\end{equation}
is the full propagator, and (cf. Eq.~(\ref{bGam2}))
\begin{equation} \label{bGam2tilde}
\tilde\bSg^{\Lam}[\psi,\psib] =
- \left( \begin{array}{cc}
\bar\partial\partial\Gam^{\Lam}[\psi,\psib] &
\bar\partial\bar\partial\Gam^{\Lam}[\psi,\psib] \\
\partial\partial\Gam^{\Lam}[\psi,\psib] &
\partial\bar\partial\Gam^{\Lam}[\psi,\psib]
\end{array} \right) +
\big( \bG^{\Lam} \big)^{-1} \; .
\end{equation}
Note that $\tilde\bSg^{\Lam}[\psi,\psib]$ contains all
contributions to $\bGam^{(2)\Lam}[\psi,\psib]$ which are
at least quadratic in the fields. We can now expand
$\left( \bGam^{(2)\Lam} \right)^{-1} =
\big(1 - \bG^{\Lam} \tilde\bSg^{\Lam} \big)^{-1}
\, \bG^{\Lam}$
as a geometric series. Inserted in (\ref{floweqGam}),
this yields
\begin{eqnarray} \label{floweqGamexp}
\frac{d}{d\Lam} \Gam^{\Lam}[\psi,\psib] =
- {\rm tr} \left( \dot Q_0^{\Lam} G^{\Lam} \right) -
\big(\psib, \dot Q_0^{\Lam} \psi \big) + \hskip 1.5cm
\nonumber \\
\frac{1}{2} {\rm tr} \left[ \bS^{\Lam}
\left( \tilde\bSg^{\Lam}[\psi,\psib] +
\tilde\bSg^{\Lam}[\psi,\psib] \, \bG^{\Lam}
\tilde\bSg^{\Lam}[\psi,\psib] + \dots \right) \right] \; ,
\nonumber \\
\end{eqnarray}
where
\begin{equation} \label{bSLam}
\bS^{\Lam} =
\mbox{diag} (S^{\Lam},-S^{\Lam t}) =
- \bG^{\Lam} \dot{\bQ}_0^{\Lam} \bG^{\Lam} \; .
\end{equation}
Using the Dyson equation
$(G^{\Lam})^{-1} = Q_0^{\Lam} - \Sg^{\Lam}$, the socalled
{\em single-scale propagator} $S^{\Lam}$ can also be written as
$\Lam$-derivative of the propagator at fixed self-energy,
\begin{equation} \label{SLam}
S^{\Lam} =
\frac{d}{d\Lam} \left. G^{\Lam}
\right|_{\Sg^{\Lam} \; \mbox{fixed}} \; .
\end{equation}
The expansion of the flow equation in powers of $\psi$, $\psib$
is now straightforward and leads to a hierarchy of flow equations
for $\Sg^{\Lam}$, the two-particle vertex $\Gam^{(4)\Lam}$,
and the higher-order vertices $\Gam^{(6)\Lam}$, $\Gam^{(8)\Lam}$,
etc.
The first three equations in this hierarchy are shown
diagrammatically in Fig.~\ref{fig:floweq1pi+}.
Note that only one-particle irreducible one-loop diagrams
contribute, and internal lines are dressed by self-energy
corrections.
The hierarchy does not close at any finite order, since the
flow of each vertex $\Gam^{(2m)\Lam}$ receives a contribution
from a tadpole diagram involving $\Gam^{(2m+2)\Lam}$, and
$m$-particle vertices with arbitrary $m$ are generated by the
flow, irrespective of their presence in the bare action.
\begin{figure} [ht]
\centerline{\includegraphics[width = 7cm]{fig3.eps}}
\caption{(Color online)
Diagrammatic representation of the flow equations for
the self-energy $\Sg^{\Lam}$, the two-particle vertex
$\Gam^{(4)\Lam}$, and the three-particle vertex $\Gam^{(6)\Lam}$
in the one-particle irreducible version of the functional RG.
Lines with a dash correspond to the single scale propagator
$S^{\Lam}$, the other lines to the full propagator $G^{\Lam}$.}
\label{fig:floweq1pi+}
\end{figure}
Let us derive explicitly the first two flow equations from the
hierarchy.
Comparing coefficients of quadratic contributions (proportional
to $\psib\psi$) to the exact flow equation yields
\begin{equation} \label{floweqA2}
\frac{d}{d\Lam} \cA^{(2)\Lam} =
- (\psib,\dot Q_0^{\Lam} \psi)
- {\rm tr} \left(
S^{\Lam} \bar\partial \partial \cA^{(4)\Lam} \right) \; .
\end{equation}
Inserting Eq.~(\ref{A2m}), and using
$\Gam^{(2)\Lam} = Q_0^{\Lam} - \Sg^{\Lam}$, one obtains the
flow equation for the self-energy,
\begin{equation} \label{floweqSigma}
\frac{d}{d\Lam} \Sg^{\Lam}(x',x) =
\sum_{y,y'} S^{\Lam}(y,y') \, \Gam^{(4)\Lam}(x',y';x,y)
\; .
\end{equation}
Comparing coefficients of quartic contributions (proportional
to $(\psib\psi)^2$) yields
\begin{widetext}
\begin{eqnarray} \label{floweqA4}
\frac{d}{d\Lam} \cA^{(4)\Lam} &=&
\frac{1}{2} {\rm tr} \left(
S^{\Lam} \bar\partial\partial\cA^{(4)\Lam} G^{\Lam}
\bar\partial\partial\cA^{(4)\Lam} +
S^{\Lam t} \partial\bar\partial\cA^{(4)\Lam} G^{\Lam t}
\partial\bar\partial\cA^{(4)\Lam} \right)
\nonumber \\
&-& \frac{1}{2} {\rm tr} \left(
S^{\Lam} \bar\partial\bar\partial\cA^{(4)\Lam} G^{\Lam t}
\partial\partial\cA^{(4)\Lam} +
S^{\Lam t} \partial\partial\cA^{(4)\Lam} G^{\Lam}
\bar\partial\bar\partial\cA^{(4)\Lam} \right)
- {\rm tr} \left(S^{\Lam}
\bar\partial\partial\cA^{(6)\Lam} \right) \, .
\hskip 8mm
\end{eqnarray}
Inserting Eq.~(\ref{A2m}), one obtains the flow equation
for the two-particle vertex,
\begin{eqnarray}\label{floweqGamma4}
\frac{d}{d\Lam} \Gam^{(4)\Lam}(x'_1,x'_2;x_1,x_2) &=&
\sum_{y_1,y'_1} \sum_{y_2,y'_2}
G^{\Lam}(y_1,y'_1) \, S^{\Lam}(y_2,y'_2) \nonumber \\
&\times& \Big\{
\Gam^{(4)\Lam}(x'_1,x'_2;y_1,y_2)
\Gam^{(4)\Lam}(y'_1,y'_2;x_1,x_2) \nonumber \\
&-& \Big[ \Gam^{(4)\Lam}(x'_1,y'_2;x_1,y_1)
\Gam^{(4)\Lam}(y'_1,x'_2;y_2,x_2)
+ (y_1 \leftrightarrow y_2, y'_1 \leftrightarrow y'_2) \Big]
\nonumber \\
&+& \Big[ \Gam^{(4)\Lam}(x'_2,y'_2;x_1,y_1)
\Gam^{(4)\Lam}(y'_1,x'_1;y_2,x_2)
+ (y_1 \leftrightarrow y_2, y'_1 \leftrightarrow y'_2) \Big]
\Big\} \nonumber \\
&-& \sum_{y,y'} S^{\Lam}(y,y') \,
\Gam^{(6)\Lam}(x'_1,x'_2,y';x_1,x_2,y) \; .
\end{eqnarray}
\end{widetext}
Note that there are several distinct contributions involving
two two-particle vertices, corresponding to the familiar
particle-particle, direct particle-hole, and crossed
particle-hole channel, respectively, as shown diagrammatically
in Fig.~\ref{fig:floweqgamma4.eps}.
\begin{figure}[ht]
\centerline{\includegraphics[width = 4cm]{fig4.eps}}
\caption{Contributions to the flow of the two-particle vertex
with particle-particle and particle-hole channels written
explicitly, without the contribution from $\Gam^{(6)\Lam}$.}
\label{fig:floweqgamma4.eps}
\end{figure}
Similarly, one can obtain the flow equation for $\Gamma^{(6)}$
and all higher vertices.
Since $\Gam[\psi,\psib]$ at $\psi = \psib = 0$ is essentially
(up to a factor $T$) the grand canonical potential $\Omega$,
the flow equation (\ref{floweqGam}), evaluated at vanishing
fields, yields also a flow equation for the grand canonical
potential:
\begin{equation} \label{floweqOmega}
\frac{d}{d\Lam} \Omega^{\Lam} =
- T \, {\rm tr} \left( \dot Q_0^{\Lam} G^{\Lam} \right) \; .
\end{equation}
The flow equation (\ref{floweqGam}) and the ensuing equations
for the vertex functions can be easily generalized to cases
with $U(1)$-symmetry breaking by allowing for off-diagonal
elements in the matrices $\bQ_0^{\Lam}$, $\bG^{\Lam}$ and
$\bS^{\Lam}$.
\subsubsection{Truncations}
\label{sec:truncations}
The exact hierarchy of flow equations for the vertex functions
can be solved only for systems which can also be solved more
directly, that is, without using flow equations.
Usually truncations are unavoidable.
A natural truncation is to neglect the flow of all vertices
$\Gam^{(2m)\Lam}$ beyond a certain order $m_0$.
We call this the {\em level-$m_0$ truncation}.
The structure of the resulting equations and general properties
of their solution will be discussed in Sec.~II.E.
Note that the level-$m_0$ truncation contains all perturbative
contributions to order $m_0$ in the bare two-particle
interaction.
In practice, in applications to physically interesting systems,
vertices $\Gam^{(2m)\Lam}$ with $m > 3$ have so far
been neglected, and the contributions from $\Gam^{(6)\Lam}$
to the flow of $\Gam^{(4)\Lam}$ are usually restricted to
self-energy corrections (see below) or discarded completely.
In particular, the analysis of competing instabilities
(see Sec.~III) is based entirely on a level-2 truncation given
by the flow equation (\ref{floweqGamma4}) for the two-particle
vertex, with $\Gam^{(6)\Lam}$ replaced by zero, where in
addition the self-energy feedback is neglected.
This seemingly simple approximation captures the complex
interplay of fluctuations in the particle-particle and
particle-hole channel, which leads to interesting effects
such as the generation of $d$-wave superconductivity from
antiferromagnetic fluctuations.
In the quantum transport phenomena reviewed in Sec.~VI, the
self-energy as given by the flow equation (\ref{floweqSigma})
plays a crucial role. Some of the phenomena described there
are already obtained by a level-1 approximation where the
flowing two-particle vertex in Eq.~(\ref{floweqSigma}) is
approximated by the bare one.
That truncation might look like a Hartree-Fock approximation,
but it is in fact very different, and it works well in cases
where Hartree-Fock fails completely.
The truncated flow equations are still rather complicated.
They involve the flow of functions, not just a limited number
of running couplings. For example, the effective two-particle
interaction in a translation invariant system is a function
of three independent momentum and energy variables.
Hence, a simplified parametrization of effective interactions
is necessary even for a numerical solution.
A useful strategy is to neglect dependences which become
irrelevant in the low-energy limit, that is, whose contributions
to the flow scale to zero.
Contributions to the effective action are called {\em ``relevant''},
{\em ``marginal''}, and {\em ``irrelevant''}, if their importance
increases, stays fixed, or decreases, respectively, upon lowering
the scale $\Lam$.
This classification can be obtained from power counting.
To this end, one traditionally considers a renormalization group
transformation where one rescales momenta and fields after the
integration over fields in a momentum shell of width $d\Lam$ such
that a certain quadratic part of the action (the Gaussian fixed
point) remains invariant \cite{Wilson74}.
From the behavior of the other terms of the action under this
transformation one can assess directly whether they increase,
remain invariant, or decrease compared to the quadratic part.
For Fermi systems in dimensions $d > 1$ the conventional RG
transformation is not applicable, since the reduction of
momentum space by the mode elimination cannot be compensated
by a linear rescaling of momenta \cite{Shankar91,Shankar94}.
However, one can perform the power counting more directly by
estimating the scale dependences of Feynman diagrams on the
right hand side of the flow equations.
As described in Sec.~II.E.3, this can be done rigorously and
to all orders.
At the crudest level the power counting is independent of
dimensionality and corresponds to what one would get from
the above-mentioned RG transformation applied to one-dimensional
systems \cite{Shankar94}, that is:
(i) the self-energy has a relevant piece describing a Fermi
surface shift, while linear dependences on frequency and
momentum perpendicular to the Fermi surface are marginal;
(ii) a regular two-particle interaction is marginal; its
dependences on frequencies and momenta perpendicular to the
Fermi surface are irrelevant, such that one can parametrize
it by its static value on the Fermi surface;
(iii) regular $m$-particle interactions with $m \geq 3$ are
irrelevant.
This basic classification does not depend on dimensionality
because the bare propagator $G_0$, Eq.~(\ref{g0k}), is singular
on a $(d-1)$-dimensional surface, such that the codimension
of the singularity in the $(d+1)$ dimensional space spanned
by momentum and frequency is always two.
One should, however, not jump to the conclusion that the
$m \geq 3$ terms can simply be discarded from the RG hierarchy
in general.
This is because effective interactions with $m \geq 3$ may diverge
for small $\Lam$ even in case of finite two-particle interactions.
For example, the first contribution to the flow of $\Gam^{(6)\Lam}$
in Fig.~3 generates a three-particle interaction of order
$\Lam^{-1}$ if the external momenta add up to zero at each vertex.
When inserted into the equation for $\Gam^{(4)\Lam}$ in Fig.~3, this
may give rise to a marginal term of third order in $\Gam^{(4)\Lam}$.
For $d=1$, this term is indeed marginal.
However, if $d \geq 2$ and the Fermi surface is curved, this and
other contributions are suppressed below the basic power counting
estimate due to geometrically reduced integration volumes
\cite{FT1,Shankar94}.
This improved power counting is described in App.~B.3.
It can also be used to give a precise, scale-dependent meaning to
nesting of the Fermi surface.
A less obvious effect is that this improvement becomes uniform,
that is, independent of the external momenta, in graphs with
overlapping loops \cite{FST1, Salmhofer98a}, so that their contribution
gets further suppressed (see also App.~B.3). It is this effect which
implies that the derivative of the self-energy is not marginal, but irrelevant
for curved Fermi surfaces in dimension $d \ge 2$.
Moreover, it justifies truncated flows beyond the weak-coupling regime,
as follows. Consider again the first contribution to the flow of
$\Gam^{(6)\Lam}$, shown also in Fig.~\ref{fig:3rd} (a).
When this term is inserted in the equation for $\Gam^{(4)\Lam}$,
the two lines can be joined in two ways, shown in Fig.\ \ref{fig:3rd} (b) and (c).
The graph in (b) gets no extra small factor, but
the graph in (c) has overlapping loops and for positively curved Fermi surfaces
in $d=2$, its contribution gets suppressed by an additional small factor
$\sim \frac{\Lambda}{\Lambda_I} \log \frac{\Lambda_I}{\Lambda}$
at scales below a scale $\Lambda_I$, which depends only on the geometry of
the constant energy surfaces of the initial dispersion $\eps_{\bk}$ \cite{FST2}.
This suppression holds uniformly for all values of the external momenta.
For $d=3$, a similar bound holds, without the logarithm.
Similar (in general, weaker) estimates are shown in \cite{FST1} for general non-nested
regular Fermi surfaces in $d \ge 2$ and for Fermi surfaces with Van Hove singularities
in \cite{FS1,FS2}.
The contribution from graph (c) remains small compared to the second-order term if
$|\Gam^{(4)\Lam}|\, \frac{\Lambda}{\Lambda_I} \log\frac{\Lambda_I}{\Lambda}$
is small.
Note that this condition does {\em not} require $|\Gam^{(4)\Lam}| $
itself to be small: curvature effects justify dropping these terms
beyond the weak-coupling regime, provided that the above condition is satisfied.
This will be used in Sec.~\ref{sec:III}.
The detailed argument and a discussion of the consequences for the
functional RG flow, are given in Sections 1 and 5 of \cite{Salmhofer01}.
\begin{figure}[ht]
\centerline{\includegraphics[width = 8cm]{fig5.eps}}
\caption{(a) Third order graph contributing to $\Gamma^{(6)\Lam}$.
(b) Tadpole contraction.
(c) Contraction to form a graph with overlapping loops.}
\label{fig:3rd}
\end{figure}
In the theory of interacting Fermi systems, one is not only interested
in low-energy fixed points and scaling, but also in the behavior at
intermediate scales, and formally irrelevant terms may play an
important role.
There are cases where one would like to know the full temperature,
momentum, or frequency dependence of physical quantities, because a
low-energy expansion contains insufficient information.
One of the advantages of the functional RG framework is that such
dependences can be computed directly.
In many situations, a comparison to standard resummations of the
perturbation expansion is desirable, and it is also an interesting
question to what extent such resummations can be reproduced by
truncations of the functional RG flow equations.
A very important observation regarding this was made by
\textcite{Katanin04a}, who showed that a partial inclusion of the
six-point vertex in the flow allows to recover approximations
of the type Hartree-plus-ladder summations (in cases where these
approximations are a good starting point).
This also allows to continue fermionic flows into symmetry-broken
phases (see Sec.~\ref{sec:IV}).
If we drop the eight-point vertex from the equation for
$\Gamma^{(6)\Lam}$, it is determined by a Feynman graph containing
three four-point vertices, depicted in Fig.~\ref{fig:3rd}a.
When backsubstituted into the equation for the four-point vertex,
two external legs get contracted in all possible ways.
We have just discussed that the contribution from the graph in (c)
is suppressed by improved power counting.
When two legs of a single-four-point vertex are contracted
to form a tadpole (see Fig.~\ref{fig:3rd}b), the value of the
thus obtained subgraph is $\dot \Sigma^\Lambda$, by the flow equation for
the self-energy. Thus a factor $G^\Lambda \dot \Sigma^\Lambda G^\Lambda$
appears in the integral for the value of the graph.
By Dyson's equation,
\begin{equation}
\dot G^\Lambda =
G^\Lambda \dot \Sigma^\Lambda G^\Lambda +
S^\Lambda ,
\end{equation}
so this can be combined with the second-order contribution to
replace $S^\Lambda$ by $\dot G^\Lambda$.
If all other effects of the six-point function, corresponding to
graphs of the type shown in Fig.~\ref{fig:3rd}c are dropped, the
equation for the four-point function gets changed to one where
the product
$G^\Lambda(k) S^\Lambda(k') + S^\Lambda(k) G^\Lambda(k')$
is replaced with
$G^\Lambda(k) \dot G^\Lambda(k') + \dot G^\Lambda(k) G^\Lambda(k')
= \frac{d}{d\Lambda} \left(G^\Lambda(k) G^\Lambda (k') \right)$.
If one now restricts further to a single channel in the four-point
equation, it becomes explicitly solvable by a ladder summation in
that channel.
Backsubstitution in the equation for the self-energy gives the
corresponding Hartree-type term.
This is explained in \cite{Katanin04a}, and, also in its extension
to flows with symmetry breaking, in \cite{Salmhofer04}.
\subsection{Flow parameters}
\label{sec:flowparameters}
In the derivation of the exact functional flow equation, the scale
dependence of the bare propagator $G_0^{\Lam}$ was not specified.
The derivation holds for any choice of $G_0^{\Lam}$, provided all
functions involved are indeed differentiable with respect to $\Lam$,
and provided that the resulting flow equation is well defined.
These conditions are not trivial; in fact, badly chosen flow
parameters may lead to divergences on the right hand side of the
flow equation.
On the other hand, one can exploit the flexibility provided by the
choice of the $\Lam$-dependence to ones own advantage.
Besides regularity issues, the scale dependence of $G_0^{\Lam}$
is only constrained by the initial condition
\begin{equation} \label{G0initial}
G_0^{\Lam_0} = 0 \; ,
\end{equation}
and the final condition
\begin{equation} \label{G0final}
G_0^{\Lam \to 0} = G_0 \; .
\end{equation}
The functional $\Gam = \Gam^{\Lam \to 0}$ reached at the end of
the exact flow is independent of the choice of $G_0^{\Lam}$.
However, in most practical calculations, where approximations are
unavoidable, a judicious choice of $G_0^{\Lam}$ is mandatory.
Important aspects related to the choice of $G_0^{\Lam}$ are:
regularization of infrared singularities, minimization of truncation
errors, respecting symmetries, technical convenience.
In the following we will review the most frequently used cutoff
schemes along with their merits and drawbacks.
\subsubsection{Momentum and frequency cutoffs}
\label{sec:cutoffs}
For the sake of a concise discussion, let us focus on translation
and spin-rotation invariant one-band systems, such that the bare
propagator $G_0$ can be written as a simple function of frequency
and momentum as in Eq.~(\ref{g0k}).
The scale dependence can then be introduced by multiplying
$G_0$ with a suitable cutoff function $\theta^{\Lam}$,
\begin{equation} \label{multcutoff}
G_0^{\Lam}(k_0,\bk) =
\theta^{\Lam}(k_0,\bk) \, G_0(k_0,\bk) \; ,
\end{equation}
with $\theta^{\Lam_0} = 0$ and $\theta^{\Lam \to 0} = 1$.
To regularize the infrared divergence of $G_0$ at zero frequency
and for momenta on the Fermi surface ($\xi_{\bk} = 0$), the cutoff
function $\theta^{\Lam}(k_0,\bk)$ has to vanish sufficiently
quickly for $k_0 \to 0$, $\xi_{\bk} \to 0$ at fixed $\Lam > 0$.
The most frequently used cutoff functions are either pure momentum
cutoffs of the form $\theta^{\Lam}(\bk) = \vartheta(|\xi_{\bk}|/\Lam)$
or frequency cutoffs $\theta^{\Lam}(k_0) = \vartheta(|k_0|/\Lam)$,
where $\vartheta(x)$ is a function that vanishes for $x \ll 1$ and
tends to one for $x \gg 1$.
Mixed momentum and frequency cutoffs of the form
$\theta^{\Lam}(k_0,\bk) = \vartheta[(k_0^2 + \xi_{\bk}^2)/\Lam^2]$
are preferred in the mathematical literature, as they facilitate
power counting and rigorous estimates.
A technical advantage of momentum cutoffs compared to frequency
cutoffs is that Matsubara sums on the right hand side of the
flow equations can often be performed analytically.
Furthermore, a momentum cutoff does not spoil the analytic structure
of propagators and vertex functions in the complex frequency plane.
However, there are also serious drawbacks, which are specific to Fermi
systems.
Once self-energy effects are taken into account, the Fermi surface is
usually deformed in the course of the flow, such that the momentum
cutoff has to be continuously adapted to the new Fermi surface, which
complicates the flow equations considerably.
Second, particle-hole excitations with a small momentum transfer $\bq$
are suppressed by the momentum cutoff for
$|\xi_{\bk+\bq} - \xi_{\bk}| < 2\Lam$.
As a consequence, the limit of vanishing momentum transfer $\bq \to 0$
in interaction vertices and response functions does not commute with
the limit $\Lam \to 0$ \cite{Metzner98}.
In other words, forward scattering interactions and the response to
homogeneous fields can be obtained only by taking the limit $\bq \to 0$
at the end of the flow, at $\Lam = 0$ \cite{Honerkamp01c}.
This is a serious drawback in stability analyses (see Sec.~\ref{sec:III}),
where one compares the increase of the effective interaction in
different momentum channels (including forward scattering), or different
susceptibilities, upon lowering $\Lam$ until a divergence occurs in at
least one channel at a finite scale $\Lam_c > 0$.
A frequency cutoff has the advantage that it does not interfere with
Fermi surface shifts, and that particle-hole processes with small
momentum transfers are captured smoothly by the flow \cite{Husemann09a}.
It can also be used in systems without translation invariance
\cite{Andergassen04}, where a momentum cutoff is less useful since
the propagator is not diagonal in momentum space.
However, a frequency cutoff affects the analytic properties of
propagators and vertex functions in the complex frequency plane.
Depending on the sort of truncation used, this may pose a serious
problem if one likes to continue results to real frequency.
For a frequency cutoff the initial cutoff is $\Lam_0 = \infty$.
Since the contributions to the self-energy flow are of order $\Lam^{-1}$
at large $\Lam$, one has to retain the convergence factor $e^{ik_0 0^+}$
on the right hand side of the flow equation \cite{Andergassen04},
analogously to the convergence factor in the perturbation expansion
of the self-energy \cite{Negele87};
for a rigorous justification, see \textcite{Pedra08}.
For a bare propagator $G_0(k_0,\bk) = (ik_0 - \xi_{\bk})^{-1}$ and a
multiplicative cutoff as in Eq.~(\ref{multcutoff}), the full
propagator has the form
\begin{equation} \label{GLam}
G^{\Lam}(k_0,\bk) =
\frac{\theta^{\Lam}(k_0,\bk)}{ik_0 - \xi_{\bk} -
\theta^{\Lam}(k_0,\bk) \Sg^{\Lam}(k_0,\bk)} \; ,
\end{equation}
and the single-scale propagator
$S^{\Lam} = - G^{\Lam} \dot Q_0^{\Lam} G^{\Lam}$,
see Eq.~(\ref{bSLam}), reads
\begin{equation} \label{SLam2}
S^{\Lam}(k_0,\bk) =
\frac{(ik_0 - \xi_{\bk}) \partial_{\Lam} \theta^{\Lam}(k_0,\bk)}
{\left[ ik_0 - \xi_{\bk} -
\theta^{\Lam}(k_0,\bk) \Sg^{\Lam}(k_0,\bk) \right]^2} \; .
\end{equation}
For a sharp cutoff function such as
$\theta^{\Lam}(k_0) = \Theta(|k_0| - \Lam)$, the single-scale
propagator seems ill-defined, since
$\partial_{\Lam} \Theta(|k_0| - \Lam) = - \delta(|k_0| - \Lam)$,
so that a delta peak in the numerator of Eq.~(\ref{SLam2})
coincides with a discontinuity (due to the step function) in the
denominator.
However, this ambiguity can be easily removed by viewing the
step function $\Theta(x)$ as a limit of increasingly sharp
regularized step functions $\Theta_{\eps}(x)$, where the
discontinuity is smeared over a width $\eps$ \cite{Morris94}.
With $\delta_{\eps}(x) = \partial_x \Theta_{\eps}(x)$, a simple
substitution of variables yields
\begin{equation} \label{morrislemma}
\delta_{\eps}(x) \, f(x,\Theta_{\eps}(x))
\stackrel{\eps \to 0}{\longrightarrow}
\delta(x) \int_0^1 du f(0,u) \; ,
\end{equation}
for any continuous function $f$. Note that the right hand side
is unique, that is, it does not depend on the shape of the
smeared step function $\Theta_{\eps}(x)$ for $\eps > 0$.
For a sharp frequency cutoff
$\theta^{\Lam}(k_0) = \Theta(|k_0| - \Lam)$, for example,
the single-scale propagator thus simplifies to
\begin{equation} \label{SLamsharp}
S^{\Lam}(k_0,\bk) =
- \frac{\delta(|k_0| - \Lam)}
{ik_0 - \xi_{\bk} - \Sigma^{\Lam}(k_0,\bk)} \; ,
\end{equation}
as long as it does not appear in products where other factors
are also discontinuous at $|k_0| = \Lam$.
Otherwise, for example in products of the form
$S^{\Lam}(k_0,\bk) [G^{\Lam}(k_0,\bk)]^m$, one has to apply
Eq.~(\ref{morrislemma}) to the entire product.
A sharp cutoff has the obvious technical advantage that the
integration over the cutoff variable ($k_0$ or $\xi_{\bk}$)
can be carried out analytically, thanks to the delta-function
in the numerator of $S^{\Lam}$.
On the other hand, a sharp cutoff generates discontinuities in
the momentum or frequency dependences of the vertex functions,
corresponding to a pronounced non-locality of the effective
action \cite{Morris94}, which is often not amenable to a
simple parametrization.
At finite temperature the flow equations are ill-defined for
a sharp frequency cutoff, since the Matsubara frequencies are
discrete: $k_0 = (2n+1)\pi T$ with integer $n$.
Continuous cutoff functions at $T>0$ are conveniently chosen
such that the $\Lam$-derivative is non-zero only in a frequency
range of width $2\pi T$, since then only two frequencies
contribute to the Matsubara sum on the right hand side of the
flow equation \cite{Enss05b}.
There are useful cutoff schemes which are formulated more
naturally by {\em adding}\/ a regulator function $R^{\Lam}$ to
the inverse propagator (instead of multiplying):
\begin{equation}
Q_0^{\Lam}(k_0,\bk) =
\big[ G_0^{\Lam}(k_0,\bk) \big]^{-1} =
Q_0(k_0,\bk) + R^{\Lam}(k_0,\bk) \; ,
\end{equation}
with $R^{\Lam_0} = \infty$ and $R^{\Lam \to 0} = 0$.
In particular, regulator functions of the form \cite{Litim01}
\begin{equation} \label{litimcutoff}
R^{\Lam}(\bk) =
- Z^{\Lam} [\sgn(\xi_{\bk})\Lam - \xi_{\bk}]
\Theta(\Lam - |\xi_{\bk}|) \; ,
\end{equation}
or its frequency dependent analogue,
$R^{\Lam}(k_0) =
i Z^{\Lam} [\sgn(k_0) \Lam - k_0] \Theta(\Lam - |k_0|)$,
have some distinct advantages.
The prefactor $Z^{\Lam}$ is initially one and is then
determined by a momentum (or frequency) derivative of the
flowing self-energy $\Sg^{\Lam}(k_0,\bk)$.
The Litim cutoff satisfies a criterion of ''optimal''
regularization of the infrared singularity of the propagator
\cite{Litim01}.
For simple truncations it also leads to a very convenient
form of the integrands, facilitating the integrations.
It is easy to choose the cutoff function in a way that does
not affect the global symmetries of the system, such as global
charge conservation or global spin rotatation invariance.
However, {\em local}\/ conservation laws are typically spoiled.
The corresponding Ward identities are modified by cutoff
dependent additional terms, which vanish only in the limit
$\Lam \to 0$ \cite{Enss05a}.
It is very hard to devise truncations which satisfy the
modified Ward identities at each scale $\Lam$, and hence
truncated flows often violate Ward identities also in the
limit $\Lam \to 0$ \cite{Katanin04a}.
In these cases it is better to compute only independent
quantities from the flow, and determine the remaining
quantities, which are fixed by local conservation laws,
via the Ward identity.
\subsubsection{Temperature and interaction flows}
\label{sssec:tflow}
For fermion systems the infrared singularity of the bare
propagator can also be regularized by temperature, instead
of a cutoff, since the fermionic Matsubara frequencies stay
away from zero at a distance $\pi T$.
A flow equation with temperature as a flow parameter can be
obtained from the general flow equation derived in
Sec.~\ref{sec:floweqs}, if one manages to shift all temperature
dependences of the bare action to the quadratic part.
This is indeed possible by a simple rescaling of the fields
\cite{Honerkamp01c}.
Let us consider a translation invariant system of
spin-$\frac{1}{2}$ fermions for definiteness, where the
fields depend on a momentum $\bk$, a spin index $\sg$, and
a Matsubara frequency $\om_n = (2n+1)\pi T$.
Rescaling the fields as
$\psi'_{\sg}(n,\bk) = T^{3/4} \psi_{\sg}(\om_n,\bk)$ and
$\psib'_{\sg}(n,\bk) = T^{3/4} \psib_{\sg}(\om_n,\bk)$
removes all explicit $T$-factors from the (quartic) interaction
in the bare action.
The temperature dependence is thereby shifted entirely to the
quadratic part of the action, given by the inverse bare
propagator for the rescaled fields,
\begin{equation} \label{QprimeT}
Q_0^T(n,\bk) = \frac{T^{1/2}}{i\om_n - \xi_{\bk}} \; .
\end{equation}
The effective action $\Gam^T[\psi',\psib']$ for the rescaled
fields obeys the exact flow equation Eq.~(\ref{floweqGam}),
with temperature as the flow parameter.
The unscaled vertex functions $\Gam^{(2m)}$ are recovered
from the vertex functions $\Gam^{(2m)T}$ by multiplying with
$T^{3m/2}$.
The temperature flow has several advantageous features.
First, it generates directly a temperature scan of the
computed quantities. In cutoff schemes one has to run a full
flow for each temperature separately.
Second, the temperature flow includes particle-hole
excitations with small momentum transfers uniformly at each
scale.
Third, local symmetries and the corresponding Ward identities
are respected at each step at least for the exact flow,
which makes the still difficult issue of Ward identities in
truncated flows at least more transparent.
A particularly simple choice of a flow parameter is provided
by a uniform factor $\lam$ scaling the bare propagator
\cite{Honerkamp04},
\begin{equation} \label{G0lam}
G_0^{\lam} = \lam G_0 \; ,
\end{equation}
with $\lam_0 = 0$, and $\lam \to 1$ at the end of the flow.
By a simple rescaling of the fields one can see that this is
equivalent to multiplying the bare quartic interaction with
a factor $\lam^2$, which means that the interaction is scaled
up continuously from $0$ to its full strength in the course
of the flow. Hence the name ''interaction flow'' for this
scheme.
In the absence of self-energy feedback the interaction flow
has the technical advantage that the propagator has the same
form at each scale, such that certain loop integrals need
to be done only once.
However, the global scaling of the propagator does not
regularize the infrared singularities, such that one easily
runs into infrared divergences.
Nevertheless, for suitable problems and simple truncations
the interaction flow has been shown to yield results similar
to flows with a cutoff, and with less computational effort
\cite{Honerkamp04}.
\subsection{General properties of the RG equations}
\label{sec:powercount}
\newcommand{{\cal H}}{{\cal H}}
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\newcommand{a}{a}
\newcommand{b}{b}
\newcommand{c}{c}
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\newcommand{{\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud}}{{\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud}}
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\newcommand{v_{\rm F,min}}{v_{\rm F,min}}
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\newcommand{e_1}{e_1}
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In this section we discuss the general structure of the RG hierarchy
of equations and provide power counting bounds for its solution.
These bounds are simple, but mathematically exact, and they
provide a strict sense to the notion of relevant and irrelevant terms.
We shall also briefly discuss improved power counting bounds,
which provide sharper estimates for bulk Fermi systems in $d \ge 2$
and exhibit the role of Fermi surface geometry.
The generating functionals were introduced to obtain the
Green functions and vertex functions of the model by differentiation,
cf.\ (\ref{G2m}) and (\ref{Gam2m}).
In the framework of the RG as an iterated convolution,
they acquire an independent importance.
Indeed, in many situations in bosonic field theory, an expansion
in the fields is avoided in favor of a gradient expansion
\cite{Berges02} or other types of parametrization (see also Sec.~V),
and the flow may lead to a non-analytic function of the fields.
Functions of Grassmann variables are, however, defined only
by power series expansions in these variables, so in this case
the meaning of the RG is strictly that of the infinite hierarchy.
This is only a seeming disadvantage because by the anticommutation
properties of Grassmann variables, the fully regularized
functionals (as they appear in the flow equations) have convergent
expansions in the fields
\cite{FMRS,GKGN,Lesniewski87,Abdesselam98,FKT98,Salmhofer00,FKT02}.
In contrast, the expansion in the fields of bosonic functionals
is almost always divergent, even in the regularized theory.
Convergent expansions then take the form of cluster expansions
that distinguish between regions of small and large fields
(see, for example, \textcite{BKFT10} and references therein).
\subsubsection{Inductive structure of the RG hierarchy}
The functional $\tilde\bSg^{\Lam}[\psi,\psib]$
appearing in (\ref{floweqGamexp}) has an expansion
similar to (\ref{GamA}), namely
$\tilde\bSg^{\Lam}[\psi,\psib] (x',x) =
\sum_{m\ge 1} \tilde\bSg^{(2m)\Lam}[\psi,\psib] (x',x)$,
where $\tilde\bSg^{(2m)\Lam}$ is homogeneous of degree $2m$
in the fields, hence has a representation with coefficient
functions $\tilde\bSg^{(2m)\Lam}$ similar to (\ref{A2m}).
By definition (\ref{bGam2tilde}) of $\tilde\bSg$, the
$\tilde\bSg^{(2m)\Lam}$ are determined by
$\Gamma^{(2m+2)\Lam}$, for example
\begin{widetext}
\begin{eqnarray}
\left( \tilde\bSg^{(2m)\Lam} (x',x) \right)_{11} \,
(x'_1, \dots, x'_m;x_1, \ldots, x_m) =
- \Gam^{(2m+2)\Lam} (x',x'_1,\dots, x'_m; x_1,,\dots,x_m,x)
\, .
\end{eqnarray}
Here the indices refer to the matrix structure of
(\ref{bGam2tilde}). The other matrix elements are
given by similar expressions.
We use this to expand (\ref{floweqGamexp}) in homogeneous parts
in $\psi$ and $\psib$ and compare coefficients. This gives
\begin{eqnarray}
\sfrac{d}{d \Lam} \cA^{(2m)\Lam} [\psi,\psib]
&=&
\sfrac12 \; {\rm tr}
\left(
\bS^\Lambda \tilde\bSg^{(2m)\Lambda} [\psi,\psib]
\right)
+
\sfrac12 {\rm tr}
\left(
\bS^\Lambda \tilde\bSg^{(2)\Lambda} [\psi,\psib]
\bG^\Lambda \tilde\bSg^{(2m-2)\Lambda} [\psi,\psib]
\right)
\nonumber\\
&+&
\sfrac12
\sum_{p \ge 2}
\sum_{m_0, \ldots, m_p \ge 1 \atop m_0+ \ldots + m_p =m}
{\rm tr}
\left(
\bS^\Lambda \tilde\bSg^{(2m_0)\Lambda} [\psi,\psib]
\prod\limits_{q=1}^p
\bG^\Lambda \tilde\bSg^{(2m_q)\Lambda} [\psi,\psib]
\right) \, .
\label{mqexp}
\end{eqnarray}
\end{widetext}
In the sum over $p$, each of $m_0, \ldots , m_p$ is at least one
because $\tilde\bSg$ only contains field-dependent terms, and
$m_0 + \ldots + m_p = m$ must hold since $\cA^{(2m)\Lambda}$
is homogeneous of degree $2m$ in the fields $[\psi,\psib]$.
These two conditions imply that $p \le m$ and that
$m_q \le m-p$ for all $0 \le q \le p$,
so that for every given $m$, the sum only runs over finitely many terms.
Since the coefficient in $\cA^{(2m)\Lambda}$ is $\Gamma^{(2m)\Lambda}$
and $\tilde\bSg^{(2m)\Lambda} \sim \Gamma^{(2m+2)\Lambda}$,
comparing coefficients of powers of $\psi$ and $\psib$ in (\ref{mqexp})
gives a hierarchy of differential equations for the $\Gamma^{(2m)\Lambda}$,
labelled by $m$.
We rewrite (\ref{mqexp}) as
\begin{eqnarray}\label{HSL}
\sfrac{d}{d \Lambda}
\Gamma^{(2m)\Lambda}
&=&
{\cal H}^\Lambda \Gamma^{(2m+2)\Lambda}
+
{\cal K}^\Lambda \left(\Gamma^{(4)\Lambda}\right)
\;\Gamma^{(2m)\Lambda}
\nonumber\\
&+&
\sum_{p=2}^m
{\cal L}^\Lambda_p \left(\Gamma^{(<2m)\Lambda}\right) \, .
\end{eqnarray}
The three summands on the right hand side are obtained from
the three terms in (\ref{mqexp}),
and it is understood that both sides are functions of $2m$
variables $x_1, \ldots, x_{2m}$. The action of the operator ${\cal H}^\Lambda$
on $\Gamma^{(2m+2)\Lambda}$ is linear,
as is that of ${\cal K}^\Lambda(\Gamma^{(4)\Lambda})$
on $\Gamma^{(2m)\Lambda}$, while ${\cal L}^\Lambda_p$ is nonlinear in the
lower-$m$ vertex functions
$\Gamma^{(<2m)\Lambda}= \Gamma^{(4)\Lambda}, \ldots, \Gamma^{(2m-2)\Lambda}$.
Specifically, the action of ${\cal H}^\Lambda$
is given by a tadpole-type contraction and summation, the action of
${\cal K}^\Lambda(\Gamma^{(4)\Lambda})$
is given by the evaluation of a one-loop diagram
formed from $\Gamma^{(2m)\Lambda}$ and the four-point function
$\Gamma^{(4)\Lambda}$, and ${\cal L}^\Lambda_p$ is given by a sum
over one-loop diagrams involving $p+1$ vertex functions,
each of which has $m_q < m$.
Thus ${\cal H}^\Lambda$, ${\cal K}^\Lambda(\Gamma^{(4)\Lambda})$
and ${\cal L}^\Lambda_p$ also
depend on $\Lambda$ and on the self-energy $\Sigma^\Lambda$
via the propagators $S^\Lambda$ and $G^\Lambda$.
The ${\cal H}^\Lambda$-term couples the higher vertex function $\Gamma^{(2m+2)\Lambda}$
into the equation for $\Gamma^{(2m)\Lambda}$.
Thus the hierarchy does not close among finitely many $m$,
and therefore truncations need to be employed to obtain solutions.
\subsubsection{Truncated hierarchies and their iterative solution}
If for some $m_0 \ge 1$, the initial vertex functions
$\Gamma^{(2m)\Lambda_0}$ vanish for all $m > m_0+1$,
one may employ the approximation of setting
$\Gamma^{(2m)\Lambda} = \Gamma^{(2m)\Lambda_0}$
for all $m \ge m_0 +1$. That is, all vertices with $m > m_0 +1$
remain zero, and the $(m_0+1)$-particle vertex is kept fixed
at its initial value.
This {\em level-$m_0$ truncation} reduces the infinite hierarchy
to a system of finitely many differential equations for
$(\Gamma^{(2m)\Lambda})_{m \le m_0}$.
The vertex $\Gamma^{(2m_0+2)\Lambda_0}$ enters in the equation for $\Gamma^{(2m_0)\Lambda}$.
Specifically, in the level-$1$ truncation, the two-particle vertex $\Gamma^{(4)\Lambda}$ is fixed to its bare value
$\Gamma^{(4)\Lambda_0}$, and the self-energy is
the solution of (\ref{floweqSigma}).
The level-$2$ truncation is given by (\ref{floweqGamma4}),
with $\Gamma^{(6)\Lambda}$ fixed to its initial value
$\Gamma^{(6)\Lambda_0}$ (which may vanish),
together with (\ref{floweqSigma}).
The term ${\cal K}^\Lambda (\Gamma^{(4)\Lambda}) \Gamma^{(4)\Lambda}$
is quadratic in $\Gamma^{(4)\Lambda}$.
In the level-$m_0$ truncation of the hierarchy, with $m_0 > 2$,
and {\em at given $\Sigma^\Lambda$ and
$(\Gamma^{(2m')\Lambda})_{m'<m_0}$},
Eq.\ (\ref{HSL}) for $\Gamma^{(2m_0)\Lambda}$
becomes a linear inhomogeneous differential equation
for $\Gamma^{(2m_0)\Lambda}$, which can be solved by
an operator version of the standard method of variation of the constant:
when all sums and integrals corresponding to the traces in
(\ref{mqexp}) are written out, it takes the form of a linear
integro-differential equation which is, viewed more abstractly,
a linear ordinary differential equation in a suitable space of
functions, to which standard techniques apply.
Together with the initial condition $\Gamma^{(2m)\Lambda_0}$,
this determines $\Gamma^{(2m)\Lambda}$ uniquely in terms
of $(\Gamma^{(2m')\Lambda})_{m'<m_0}$.
Backsubstitution of this solution into the ${\cal H}^\Lambda$-term for the
equation for $\Gamma^{(2m_0-2)\Lambda}$ then yields an equation
for $\Gamma^{(2m_0-2)\Lambda}$, which can be solved,
in terms of the not yet determined lower vertex functions
$(\Gamma^{(2m')\Lambda})_{m'<m_0-1}$.
Proceeding downwards in $m$ in this way, one can formally solve
the truncated hierarchy, with the final equation determining
$\Sigma^\Lambda$. We write ``formally'' here because after
at most two steps of this iteration, the differential equations become
nonlinear, so that existence of the solution is typically known only for short flow
times, and because the question of blowup of solutions is rather nontrivial.
Indeed, we shall see below that blowup generically occurs in RG
equations if relevant terms have not been taken into account.
This phenomenon is related to the infrared divergences of
unrenormalized perturbation theory.
The major advantage of the RG method
is that the growing terms can be identified and studied long before they get singular,
and then removed by taking into account appropriately chosen relevant parts
in the flowing action.
Increasing $m_0$ to improve the accuracy is then a natural
strategy for approximation of the true solution; however,
explicit and numerical calculations can be done only for
small $m_0$, because the number of variables increases rapidly with $m$.
Nevertheless, one can get useful information in the form of bounds
for the maximal possible value of the vertex
functions (or other norms that measure their size).
This is done in the following section.
\subsubsection{Running coupling expansion and power counting}
\label{ssec:runcoup}
We turn to the standard situation of a model with two-body interactions,
where the initial interaction of the fermion
system is quartic, i.e.\ $\Gamma^{(2m)\Lambda_0} = 0 $ for all $m \ge 3$.
We also assume that this interaction is short-range so that its Fourier
transform is bounded (e.g.\ an unscreened Coulomb interaction is long-range).
In a perturbative expansion in powers of the initial four-point interaction
$\Four{\Lambda_0} = \Gamma^{(4)\Lambda_0}$, the vertices are given by
sums over irreducible graphs.
An irreducible Feynman graph formed with $r$ four-legged vertices
can have at most $2r$ external legs, so that in order $r$ in that expansion,
all vertex functions with $m > r$ vanish.
As we shall now explain, one can solve the RG hierarchy in terms of a similar
expansion in the {\em scale-dependent} four-point function
$\Four{\Lambda}=\Gamma^{(4)\Lambda}$, again by integrating
the RG hierarchy downwards in scale, but keeping the $2m$-point functions
for $m > 2$ only to a fixed order in $\Four{\Lambda}$. The equation for
$\Four{\Lambda}$ itself then becomes an integro-differential equation with
a power $r$ nonlinearity on the right hand side (the equation for
$\Sigma^\Lambda$ remains unchanged).
This leads in a natural way to power counting estimates
for the higher $2m$-point functions in terms of the maximal value of
the four-point vertex that occurs in the flow.
We denote the $O\left( (\Four{\Lambda})^r \right)$ contribution to
$\Gamma^{(2m)\Lambda}$ by $\Gamma_r^{(2m)\Lambda}$.
Its scale derivative equals
\begin{eqnarray} \label{mprimeflow}
\sfrac{d}{d\Lambda} \Gamma_r^{(2m)\Lambda}
&=& {\cal H}^{\Lambda} \Gamma_r^{(2m+2)\Lambda} +
{\cal K}^{\Lambda} (\Four{\Lambda})\, \Gamma_{r-1}^{(2m)\Lambda}
\nonumber \\
&+&
\sum_{p \ge 2}
\sum {}'\;
{\cal L}^\Lambda_p
\left(
\Gamma_{r_0}^{(2m_0)\Lambda},
\ldots,
\Gamma_{r_p}^{(2m_p)\Lambda}
\right) . \hskip 5mm
\end{eqnarray}
The primed sum runs over all sequences $(m_0, \ldots m_p)$
and all sequences $(r_0, \ldots, r_p)$ with
$m_q \ge 1$ and $r_q \ge 1$ for all $1 \le q \le p$,
$\; m_0 + \ldots + m_p = m+ p$, and $r_0 + \ldots + r_p =r$.
The solution of the RG hierarchy for an initial quartic interaction has
the property that $\Gamma_r^{(2m)\Lambda}=0$ for all $m > r$.
Therefore, for $m=r$, the ${\cal H}^\Lambda$ term drops out of (\ref{HSL}),
and all remaining terms contain only $\Four{\Lambda}$ or terms of order
at most $r-1$ in $\Four{\Lambda}$. Thus, given these lower-order
$\Gamma$'s, $\Gamma_r^{(2r)\Lambda}$ can be obtained by integration.
Then the right hand side of the equation for $m=r-1$ is determined, so
$\Gamma_r^{(2r-2)\Lambda}$ can be determined, and so on.
Successive backsubstitution then leads to a system of equations where
$\frac{d}{d\Lambda} \Gamma_r^{(2m)\Lambda}$ gets contributions
from a sum of graphs with $r$ vertices of type $\Four{\Lambda'}$, where
$\Lambda_0 \ge \Lambda' \ge \Lambda$, and propagators $G^{\Lambda''}$
and $S^{\Lambda'''}$, and all the intermediate scales $\Lambda'$ etc.\ are
integrated.
Thus the equation becomes nonlocal in the flow parameter $\Lambda$
but the right hand side is known once $\Four{\Lambda}$ and $\Sigma^\Lambda$
are known. $\Four{\Lambda}$ is given by a degree $r$ nonlinear equation,
with a similar graphical background as discussed above, and $\Sigma^\Lambda$
by the standard self-energy equation (\ref{floweqSigma}).
While more restricted than the level-$m$ truncation,
the running coupling scheme also captures effects
that cannot be seen in any fixed order of bare perturbation theory,
such as screening or asymptotic freedom of certain coupling functions.
We use this inductive structure to derive basic power counting bounds
for the vertex functions in terms of the flowing four-point function,
for a $d$-dimensional bulk fermion system ($d \ge 1$).
For simplicity, we focus on spin-$\frac12$ fermions with translation-invariant
action, so that we can use $x_i = (k_i,\sg_i) = (k_{0,i},\bk_i,\sg_i)$,
as discussed at the beginning of Section \ref{sec:functionals}.
We also assume that the symmetries of the action remain unbroken.
These specific assumptions are for presentation only; power counting can
be done without them.
By translation invariance,
$\Gamma_r^{(2m)\Lambda} \left((k_1,\sg_1), \ldots (k_{2m},\sg_{2m})\right)
= \delta \left( \sum_i k_i \right)\,
\hat \Gamma_r^{(2m)\Lambda}({\underline{k}},{\underline{\sigma}})$,
where the delta function forces conservation of the spatial momentum
$\bk$ (up to reciprocal lattice vectors) and conservation of the frequency
variable $k_0$, and we have introduced the abbreviations
${\underline{\sigma}} = (\sg_1,\ldots \sg_{2m})$ and ${\underline{k}} = (k_1, \ldots, k_{2m-1} )$.
For $\Lambda=\Lambda_0$,
the function $\hat \Gamma_r^{(2m)\Lambda}$ is smooth and bounded
because the initial interaction is short-range,
and this stays so during the flow above critical scales.
Consider the maximal size of the vertex functions,
$\Vert \Gamma_r^{(2m)\Lambda} \Vert =
\sup\limits_{{\underline{k}}, {\underline{\sigma}}}
| \hat \Gamma_r^{(2m)\Lambda}({\underline{k}},{\underline{\sigma}}) |$.
Then, for $m \ge 3$,
\begin{equation} \label{poco}
\Vert \Gamma_r^{(2m)\Lambda} \Vert \le \gamma_r^{(2m)}\;
{\snorm{\Lambda}}^{r-m+1}
{f_\Lambda}^r \;
\Lambda^{2-m} \; ,
\end{equation}
where $\gamma_r^{(2m)}$ is independent of $\Lambda$ and $\beta$,
\begin{equation}
f_\Lambda = \sup\limits_{\Lambda \le \ell \le \Lambda_0}
\Vert\Four{\ell}\Vert
\end{equation}
is the maximal value of the four-point coupling on all scales
between $\Lambda$ and $\Lambda_0$, and
\begin{equation}
\snorm{\Lambda} = \max\limits_\alpha \sum\limits_{\alpha'} \;
\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} k \; |\hat S^\Lambda_{\alpha,\alpha'} (k) | \; ,
\end{equation}
with
$\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} k \dots = T \sum_{k_0} \int \frac{d^dk}{(2\pi)^d} \dots$
(for the general power counting, we do not need to assume that the
propagator is diagonal in the spin indices $\alpha,\alpha'$, so $\hat S^\Lambda$
also carries these indices).
The dependence of $\snorm{\Lambda}$ on $\Lambda$ is determined
by the shape of the Fermi surface.
As shown in Appendix \ref{appssec:propbounds}, $\snorm{\Lambda}$
is of order one for regular Fermi surfaces. If the Fermi surface
contains Van Hove points, $\snorm{\Lambda}$ grows logarithmically
in $\Lambda$ for $\Lambda \to 0$.
At first sight, one may worry about the factor $\Lambda^{2-m}$,
which diverges for $m \ge 3$ as $\Lambda \to 0$. For the maximum value
of the vertex functions, it is indeed true -- and easily verified in
examples -- that there are always particular values of the external
momenta where these vertex functions become very large in $\beta = 1/T$
(and diverge at zero temperature).
However, this happens only on a ``small'' set of momenta.
For a general $m$-point function, it is very involved to determine this
set, but this is not necessary for power counting.
One can use the $L^1$ norm instead, i.e.\ consider
$\Vert \Gamma_r^{(2m)\Lambda} \Vert_1 =
\sum_{{\underline{\sigma}}} \int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} k_1 \ldots {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} k_m \;
|\Gamma_r^{(2m)\Lambda} ({\underline{k}},{\underline{\sigma}})|$.
Using generalizations of (\ref{poco}),
one can then show that if $f_\Lambda$ remains finite
\begin{equation}
\Vert
\Gamma_r^{(2m)\Lambda}
\Vert_1
\le
c_r^{(2m)} \; f_\Lambda^r
\end{equation}
with constants $c_r^{(2m)}$ that are independent of $\Lambda$, $\beta$
and the system size $L$ (see \textcite{Salmhofer99}, Section 4.4.3).
This implies that, even in the limit $\beta \to \infty$,
the $2m$-point vertices can become singular only on a set of
zero Lebesgue measure in momentum space. In general, this set can be
rather complicated, but, loosely speaking, it will have codimension at
least one.
It is one of the appealing features of the flow-equation RG that exact
statements like (\ref{poco}) can be proven in a few lines,
see Appendix \ref{app:powercounting}.
The argument given there also implies an at-most logarithmic growth of
the coefficients in the equation for $f_\Lambda$ itself.
The self-energy then comes out of order $f_\Lambda$,
provided that renormalization is done correctly, see
Section \ref{sssec:selfenergy}.
At small $f_\Lambda$, the size of the vertices $\Gamma^{(2m)\Lambda}$
with $m \ge 3$ is thus determined by $f_\Lambda$.
The terms with $m \ge 3$ are the RG-irrelevant ones, $m=2$ is marginal,
and $m=1$ is relevant.
This classification is explained in detail in
Appendix \ref{app:powercounting}.
In a Taylor expansion of the vertex functions in the Matsubara
frequency around zero, and in momentum around the Fermi surface,
additional small factors arise, which cancel the small denominators
of the propagators; at the same time, the vertex function is replaced
by a differentiated one.
Hence, the flow obtained by projecting the frequencies to zero and the
momenta to the Fermi surface gives the dominant contribution for small
$\Lambda$.
This is expected from a simple counting of bare scaling dimensions,
and can be established more rigorously by power-counting arguments
similar to those used above and in Appendix \ref{app:powercounting}.
In the case of a curved Fermi surface in $d \ge 2$, $f_\Lambda$ indeed
stays small in a weakly interacting system above a BCS-like temperature
(see, e.g., \textcite{Salmhofer98b}), indicating the absence of
symmetry-breaking.
At zero temperature, $f_\Lambda$ grows as $\Lambda$ decreases, and
the four-point function has singularities at points corresponding to
nesting vectors of the Fermi surface;
for details, see Appendix \ref{ssec:imppoco}.
This growth of $f_\Lambda$ with decreasing scale $\Lambda$ is
usually called the ``flow to strong coupling'' in RG studies,
and is described in more detail in Section \ref{sec:III}.
A singularity of the two-particle vertex in momentum space
means that the interaction becomes long-range in position space.
This is associated with the formation of critical fluctuations,
and in case of spontaneous breaking of a continuous symmetry,
with the appearance of Goldstone bosons (see Section \ref{sec:IV}).
\subsubsection{Self-energy and Fermi surface shift}
\label{sssec:selfenergy}
The self-energy is important for all effective one-particle properties
of the system, and it can cause drastic effects, as compared to the
non-interacting fermions.
Accordingly, in the RG flow, the self-energy is a relevant term.
In absence of symmetry breaking, it modifies the inverse
propagator to $ik_0 - \xi_{\bk} - \Sigma (k_0,\bk)$.
The long-distance behaviour of the fermionic propagator is determined
by the behaviour of this function around its zero set. A Taylor expansion
around $k_0=0$ gives
\begin{eqnarray}
ik_0 - \xi_{\bk} - \Sigma (k_0,\bk) =
\frac{ik_0- e_{\bk}}{Z_{\bk}} + \rho(k_0,\bk)
\end{eqnarray}
with
$Z_{\bk}^{-1} = 1 + i (\partial_0 \Sigma) (0,\bk)$,
$Z_{\bk}^{-1} e_{\bk} = \xi_{\bk} + \Sigma (0,\bk)$
and a Taylor remainder $\rho$. If $\rho$ vanishes
faster than linearly in $k_0$ as $k_0 \to 0$, we thus obtain
an effective description in terms of quasiparticles with
dispersion relation $e_\bk$, hence ``interacting'' Fermi surface
$\left\{\bk: \xi_{\bk} + \Sigma(0,\bk) = 0 \right\}$, Fermi velocity
$\nabla e_{\bk}$, and quasiparticle weight $Z_{\bk}$.
If $Z_{\bk}$ is bounded and nonvanishing for all $\bk$,
the long-distance decay of the fermion propagator in position space
is the same as for the free theory.
The question whether $\Sigma$ is smooth enough for the above to hold
is nontrivial. In the one-dimensional Luttinger model,
$\partial_0 \Sigma(0,k_F)$ diverges, and the small-$k_0$ behaviour of
the self-energy, $\Sigma (k_0,k_F) \sim |k_0|^\alpha$ with $\alpha < 1$
depending on the interaction strength, implies that the self-energy effects
dominate at small $k_0$ and the decay in position space becomes more rapid,
so that the occupation number density $n(\bk)$ becomes a continuous function of
$\bk$ even at zero temperature \cite{Giamarchi04}.
An even more drastic change is spontaneous symmetry breaking, where
the propagator cannot be written any more in the simple form given above
(see Section \ref{sec:IV}).
In the RG flow, $\Sigma$ is replaced by the $\Lambda$-dependent
self-energy $\Sigma^\Lambda = (G_0^\Lambda)^{-1} - (G^\Lambda)^{-1}$.
An important phenomenon in this
context is the shift in the Fermi surface entailed by $\Sigma^\Lambda$.
In terms of power counting, this shift is the most relevant term.
Cutting off the propagator around the free Fermi surface then fails to
regularize the propagator,
which leads to spurious singularities in the RG flow.
A convenient method to avoid this is to introduce a counterterm.
In the context of the bare perturbation expansion, the counterterm
method was already described in \cite{Nozieres64}.
The main idea is to anticipate the form that the propagator takes
at the end of the flow and to rearrange the flow such that this form,
not the bare one is used as the starting point for the RG analysis,
hence the Fermi surface is fixed to that of the interacting system in
the flow.
The difference between the two dispersion functions appears as a (finite)
counterterm. To obtain a one-to-one relation between the model given by
the Hamiltonian and the one with fixed interacting Fermi surface,
one has to solve a self-consistency equation.
\textcite{FT1} used the counterterm method for the radius shift of
a circular Fermi surface in a RG flow.
\textcite{FST1,FST2,FST3,FST4} generalized this to the case of
non-circular curved Fermi surfaces, solved the self-consistency equation,
and showed that $Z_{\bk}$ remains finite for $d \ge 2$ to all orders.
The corresponding fixed-point problem for the Fermi surface was also
considered by \textcite{Ledowski03}.
The role of Van Hove singularities was analyzed by \textcite{FS1,FS2}.
\textcite{FT2} also used the counterterm method to
derive the equation for the superconducting gap from an RG flow;
for further work in that direction, see also Section \ref{sec:IV}.
In alternative to counterterms, one can try to avoid a momentum space
cutoff altogether \cite{Honerkamp01b,Honerkamp01c,Husemann09a}, or to
use an adaptive scheme (see the appendix in \textcite{Honerkamp01d} and
\textcite{BGM06,Salmhofer07}).
\subsection{Flow equations for observables and correlation functions}
\label{sec:responsefcts}
All observables of the fermionic system are given by polynomials
in the fields, so they can be calculated from the connected Green
functions $G^{(2m)\Lambda}$, hence by the above-mentioned tree
relations also from the irreducible vertex functions
$\Gamma^{(2m)\Lambda}$.
It is nevertheless convenient, and due to the limitations of
approximations often mandatory, to calculate the flow of
observables and their correlation (or response) functions
by separate flow equations, which we derive and discuss now.
For simplicity we restrict the presentation to the particularly
important class of observables that are correlations of fermionic
bilinears.
Charge-invariant bilinears are of the form
\begin{equation}
\cB(x)
=
\sum_{y,y'} \psib (y) \, B(x; y,y') \psi (y') .
\end{equation}
Charge-non-invariant bilinears are of the form
\begin{equation}
\cB(x)
=
\sum_{y,y'}
\left(
\psi (y) \, B(x; y,y') \psi (y')
+
\psib (y) \, \tilde B(x; y,y') \psib (y')
\right) .
\end{equation}
The functions $B$ and $\tilde B$ determine the spatial and spin
structure of these bilinears.
For translation-invariant systems, we can choose a momentum
representation where $x = (k_0,\bk)$ and $y = (p_0,\bp,\sigma)$,
as explained above Eq.~(\ref{g0k}).
With the notations $p = (p_0,\bp)$ and
$\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} p = T \sum_{p_0} \int d\bp$,
a charge invariant bilinear is of the form
\begin{equation}
\cB(k) =
\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} p\;
\psib_\sigma (p)\,
b_{\sigma,\sigma'} (p,k) \, \psi_{\sigma'} (p+k) \, .
\end{equation}
The frequency $k_0$ is an integer multiple of $2\pi T$.
The case $k = 0$ and $b = \sigma_i$, where $\sigma_i$ denotes
the $i^{\rm th}$ Pauli matrix, corresponds to a uniform spin
density. The case $k_0 =0$, $\bk = (\pi,\pi, \ldots, \pi)$
and $b = \sigma_i$ corresponds to a staggered spin density.
Similarly, the same choices of $k$ but with $b_{\sigma,\sigma'}
= \delta_{\sigma,\sigma'}$ correspond to charge densities.
Other choices of $\bk$ can be used to test tendencies towards
noncommensurate magnetic or charge ordering.
Cooper pair fields correspond to the non-charge invariant
combinations
\begin{eqnarray}
\cB(k)
&=&
\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} p\;
\big[\,
\psib_\sigma (p)\,
\Delta_{\sigma,\sigma'} (p,k) \, \psib_{\sigma'} (-p+k)
\nonumber\\
&+&
\psi_\sigma (p)\,
\overline{\Delta}_{\sigma',\sigma} (-p+k,k) \, \psi_{\sigma'} (-p+k)\,
\big]\; .
\end{eqnarray}
Again, the simplest choice is uniform singlet pairing,
where $k=0$ and
$\Delta_{\sigma,\sigma'} (p,k) = \Delta (\bp) \varepsilon_{\sigma,\sigma'}$. In this case
$\Delta (\bp)$ is the gap function.
Triplet pairing, extended Cooper pairs, and spatially nonuniform gaps, are described
by suitable generalizations.
A convenient way of generating correlation functions of the
bilinears $\cB$ is to couple the $\cB (x)$ to external source fields
$J (x)$, i.e.\ to add a term $\left(J,\cB\right) = \sum_x J(x) \cB(x)$
to the action. The external field $J$ is not an integration variable,
so it can be regarded as a (functional) parameter on which
$\cG$ depends. Writing $\cG=\cG(J,\eta,\etab)$, we then have
\begin{equation}
\langle \cB(x) \, \cB(y) \rangle
-
\langle \cB(x) \rangle \, \langle \cB(y) \rangle
=
- \frac{\partial^2 \cG (J,\eta,\etab)}{\partial J(x) \partial J(y)} \,
\Big\vert_{J=0\atop\eta,\etab=0} \; .
\end{equation}
In presence of $J$, the effective action
$\Gamma = \Gamma (J, \psi,\psib)$,
as well as all other quantities appearing in the fermionic
Legendre transform (\ref{Gampsi}), depend on $J$ as well.
Since relations such as
$\frac{\partial \Gamma}{\partial \psib} (J, \psi,\psib)=\eta (J, \psi,\psib)$
remain valid for any $J$, straightforward differentiation yields
\begin{equation}\label{gaGaJ}
\frac{\partial^2 \cG (J, \eta,\etab) }{\partial J(x) \partial J(y)} \,
\Big\vert_{J=0 \atop \eta,\etab=0}
=
\frac{\partial^2 \Gamma (J, \psi,\psib) }{\partial J(x) \partial J(y)} \,
\Big\vert_{J=0 \atop \psi,\psib=0} \; .
\end{equation}
Graphically, this relation is intuitive in that the bilinears
always couple to the (effective) vertices by two lines,
and the fermionic vertices are all even, so that the graphs that
contribute are automatically irreducible.
Again, because $J$ plays the role of a parameter,
the flow equation (\ref{floweqGam}) is unchanged.
Flow equations for the response functions are then
obtained simply by expanding $\Gamma$ in the
fields $J$ and comparing coefficients. This again leads
to a hierarchy of equations for the vertex functions
$\Gamma^{(2m,n)}$ that have $2m$ fermionic and
$n$ external bosonic lines. The $J$-independent term
corresponds to the standard fermionic hierarchy for the
$\Gamma^{(2m,0)} = \Gamma^{(2m)}$, which therefore remains
unchanged.
When counting powers in the fermionic fields,
each $J$ corresponds to a bilinear, so that the truncation
$\Gamma^{(2m)} = 0$ for $m \ge m_0$ for the fermionic
vertices corresponds to a truncation $\Gamma^{(2m,n)} = 0$
for $m+n \ge m_0$.
The flow equations remaining after a truncation for $m_0 = 3$
are shown diagrammatically in Figure \ref{fig:response2}.
\begin{figure}[ht]
\centerline{\includegraphics[width = 5cm]{fig6.eps}}
\caption{The truncation of the hierarchy for the response function
that corresponds to keeping only the irreducible two-particle
vertex in the fermionic hierarchy.}
\label{fig:response2}
\end{figure}
Clearly, the two-point correlation
$\langle \cB(x)\, \cB(y)\rangle
- \langle \cB(x) \rangle\, \langle \cB(y)\rangle$
of any fermionic bilinear $\cB$ only involves the fermionic
two- and four-point functions, hence could simply be calculated
from the knowledge of $\Gamma^{(2)}$ and $\Gamma^{(4)}$
by the reciprocity relation and (\ref{G4Ga4}).
Eq.\ (\ref{gaGaJ}) shows that the route via external fields
in the one-particle irreducible equations is strictly equivalent
to this if the hierarchy is treated exactly.
When making truncations to the hierarchy and other approximations,
the two are no longer the same.
Anomalous scaling dimensions of fermionic bilinears (or other
composite objects) are captured easily by separate flow equations
for these quantities, while they are hard to obtain from
$\Gam^{(2)}$ and $\Gam^{(4)}$, if the latter are computed from
a truncated flow equation.
An instructive example is given by the calculation of the
density profile near a static impurity in a Luttinger liquid
in \textcite{Andergassen04}.
\subsection{Flow equations for coupled boson-fermion systems}
\label{sec:bosefermi}
The focus of this review is on fermion systems.
However, even if the bare action involves only fermionic fields,
bosonic degrees of freedom are frequently generated as fermion
composites and order parameter fields. For example, Cooper pairs
and the order parameter in a superconductor are bosons.
Technically, bosonic fields are introduced in an originally purely
fermionic theory by a Hubbard-Stratonovich decoupling of an
interaction between fermions \cite{Popov87}.
Often the fermionic fields are subsequently intregrated out, such
that an effective action involving only bosons remains.
Otherwise one has to deal with a coupled theory of fermions and
bosons.
In this section we generalize the flow equations derived in
Sec.~\ref{sec:floweqs} to interacting boson-fermion systems.
Flow equations for coupled boson-fermions systems have been
derived by various groups, with slight differences in the
notation \cite{Berges02,Kopietz10}.
We first introduce some notation for bosons, and write down the
bosonic analogues of some of the most important equations from
Sec.~\ref{sec:floweqs}.
Bosonic particles are described by complex fields $\phi$.
It is convenient to combine $\phi$ and its complex conjugate
$\phi^*$ in a bosonic Nambu field
\begin{equation} \label{nambuPhi}
\Phi(x) = \left( \begin{array}{l} \phi(x) \\
\phi^*(x) \end{array} \right) \; .
\end{equation}
The generating functional for connected Green functions can be
written as \cite{Negele87}
\begin{equation} \label{calGb}
\cG[H] =
- \ln \int \cD\Phi \, e^{-\cS[\Phi]} \, e^{(H^*,\Phi)}
\; ,
\end{equation}
where $\cS[\Phi]$ is the bare action, and
\begin{equation} \label{nambuHb}
H(x) = \left( \begin{array}{l} h(x) \\
h^*(x) \end{array} \right) \; .
\end{equation}
the source field.
Connected Green functions are obtained as functional derivatives
\begin{eqnarray} \label{G2mb}
&& G^{(2m)}(x_1,\dots,x_m;x'_1,\dots,x'_m) =
\nonumber \\[1mm]
&& \bra \phi(x_1) \dots \phi(x_m)
\phi^*(x'_m) \dots \phi^*(x'_1) \ket_c =
\nonumber \\[1mm]
&& - \left.
\frac{\partial^{2m} \cG[H]}
{\partial h^*(x_1) \dots \partial h^*(x_m)
\partial h(x'_m) \dots \partial h(x'_1)}
\right|_{H = 0} . \hskip 5mm
\end{eqnarray}
The effective action is defined as Legendre transform
\begin{equation} \label{GamPhi}
\Gam[\Phi] = (H^*,\Phi) + \cG[H] \; ,
\end{equation}
with $\Phi = - \partial\cG/\partial H^*$.
Functional derivatives of $\Gam[\Phi]$ yield the bosonic
$m$-particle vertex functions
\begin{eqnarray} \label{Gam2mb}
&& \Gam^{(2m)}(x_1,\dots,x_m;x'_1,\dots,x'_m) =
\nonumber \\[1mm]
&& \left.
\frac{\partial^{2m} \Gam[\Phi]}
{\partial\phi^*(x_1) \dots \partial\phi^*(x_m)
\partial\phi(x'_m) \dots \partial\phi(x'_1)}
\right|_{\Phi = 0} . \hskip 5mm
\end{eqnarray}
The matrices of second derivatives at finite fields
\begin{eqnarray} \label{bG2b}
\bG^{(2)}[H] &=&
- \frac{\partial^2 \cG}{\partial H^*(x) \partial H(x')}
\nonumber \\[1mm]
&=& \left( \begin{array}{cc}
\bra \phi(x) \phi^*(x') \ket & \bra \phi(x) \phi(x') \ket \\
\bra \phi^*(x) \phi^*(x') \ket & \bra \phi^*(x) \phi(x') \ket
\end{array} \right)
\end{eqnarray}
and
\begin{equation} \label{bGam2b}
\bGam^{(2)}[\Phi] =
\frac{\partial^2 \Gam}{\partial\Phi^*(x) \partial\Phi(x')}
\end{equation}
obey the reciprocity relation
$\bGam^{(2)}[\Phi] = (\bG^{(2)}[H])^{-1}$.
Endowing the bare propagator $G_0$ with a cutoff or another
scale dependence, one can derive exact flow equations for the
generating functionals in complete analogy to the fermionic
case.
In particular, the flow equation for the effective action
$\Gam^{\Lam}[\Phi]$ has the form
\begin{equation} \label{floweqGamb}
\frac{d}{d\Lam} \Gam^{\Lam}[\Phi] =
\frac{1}{2} (\Phi^*,\dot\bQ_0^{\Lam} \Phi) +
\frac{1}{2} {\rm tr} \left[ \dot\bQ_0^{\Lam}
\left( \bGam^{(2)\Lam}[\Phi] \right)^{-1} \right] \; ,
\end{equation}
where
$\bQ_0^{\Lam} = \mbox{diag} (Q_0^{\Lam}, Q_0^{\Lam t})$
with $Q_0^{\Lam} = (G_0^{\Lam})^{-1}$.
Note that the first term on the right hand side can also be
written as $(\phi^*,\dot Q_0^{\Lam} \phi)$.
The above flow equation is equivalent to the frequently
used flow equation for the effective average action
\cite{Berges02}
\begin{equation} \label{GamRLamb}
\Gam_R^{\Lam}[\Phi] = \Gam^{\Lam}[\Phi] -
\frac{1}{2} (\Phi^*,\bR^{\Lam} \Phi) \; ,
\end{equation}
with $R^{\Lam} = Q_0^{\Lam} - Q_0$, which reads
\cite{Wetterich93}
\begin{equation} \label{floweqGamRb}
\frac{d}{d\Lam} \Gam_R^{\Lam}[\Phi] =
\frac{1}{2} {\rm tr} \left[ \dot \bR^{\Lam} \left(
\bGam_R^{(2)\Lam}[\Phi]
+ \bR^{\Lam} \right)^{-1} \right] \; .
\end{equation}
Order parameters are often associated with real (not complex)
bosonic fields. In that (simpler) case the above equations
are still valid if one replaces the complex Nambu fields
$\Phi$ and $H$ by the real fields $\phi$ and $h$.
A generalization to coupled fermion-boson systems is now
straightforward.
Bosonic and fermionic fields are conventiently collected in
a ''super-field''
\begin{equation} \label{Xi}
\Xi = \left( \begin{array}{c}
\Phi \\ \Psi \end{array} \right) \; ,
\end{equation}
where $\Phi$ and $\Psi$ are the bosonic and fermionic Nambu
fields defined above (see Sec.~II.A).
The conjugate super-field is given by
\begin{equation} \label{Xib}
\Xib = \left( \begin{array}{l}
\Phi^* \\ \Psib \end{array} \right) \; .
\end{equation}
The generating functional for connected Green functions
involving both bosons and fermions reads
\begin{equation} \label{calGbf}
\cG[H_b,H_f] =
- \ln \int \cD\Phi \cD\Psi \,
e^{-\cS[\Phi,\Psi]} \, e^{(H_b^*,\Phi) + (\bar H_f,\Psi)}
\; ,
\end{equation}
where $\cS[\Phi,\Psi]$ is the bare action, and $H_b$ and $H_f$
are the Nambu source fields for bosons and fermions, respectively.
Functional derivatives with respect to the source fields generate
connected Green functions with an arbitrary number of bosonic and
fermionic fields, the only general constraint being that the number
of fermion fields is always even.
The effective action $\Gam[\Phi,\Psi]$ is given by the
Legendre transform
\begin{equation} \label{GamPhiPsi}
\Gam[\Phi,\Psi] =
(H_b^*,\Phi) + (\bar H_f,\Psi) + \cG(H_b,H_f) \; ,
\end{equation}
where $\Phi = - \partial\cG/\partial H_b^*$ and
$\Psi = - \partial\cG/\partial \bar H_f$.
The source fields may also be collected in a super-field
\begin{equation} \label{calH}
\cH = \left( \begin{array}{c}
H_b \\ H_f \end{array} \right) \; .
\end{equation}
The Legendre transform can then be written more
concisely as $\Gam[\Xi] = (\bar\cH,\Xi) + \cG(\cH)$.
The matrix of second functional derivatives of $\cG$ at finite
fields
\begin{eqnarray} \label{bG2bf}
\bG^{(2)}[\cH] &=&
- \frac{\partial^2 \cG}{\partial\bar\cH(x) \partial\cH(x')}
\nonumber \\[1mm]
&=& \left( \begin{array}{cc}
\bra \Phi(x) \Phi^*(x') \ket & - \bra \Phi(x) \Psib(x') \ket \\
\bra \Psi(x) \Phi^*(x') \ket & - \bra \Psi(x) \Psib(x') \ket
\end{array} \right)
\end{eqnarray}
involves also mixed boson-fermion propagators, which vanish
only for $\cH = 0$.
The matrix of second derivatives of the effective action
\begin{eqnarray} \label{bGam2bf}
\bGam^{(2)}[\Xi] &=&
\frac{\partial^2 \Gam}{\partial\Xib(x)\Xi(x')}
\nonumber \\[1mm]
&=& \left( \begin{array}{cc}
\dps \frac{\partial^2\Gam}{\partial\Phi^*(x)\partial\Phi(x')} &
\dps \frac{\partial^2\Gam}{\partial\Phi^*(x)\partial\Psi(x')} \\[3mm]
\dps \frac{\partial^2\Gam}{\partial\Psib(x)\partial\Phi(x')} &
\dps \frac{\partial^2\Gam}{\partial\Psib(x)\partial\Psi(x')}
\end{array} \right) \; . \hskip 5mm
\end{eqnarray}
is related to $\bG^{(2)}[\cH]$ by the reciprocity relation
$\bGam^{(2)}[\Xi] = (\bG^{(2)}[\cH])^{-1}$.
A flow of the generating functionals is generated by modifing the
bare propagators for bosons and fermions, $G_{b0}$ and $G_{f0}$,
such that they depend on some scale parameter $\Lam$.
We denote the scale dependent bare propagators by $G_{b0}^{\Lam}$
and $G_{f0}^{\Lam}$, and their inverse by $Q_{b0}^{\Lam}$
and $Q_{f0}^{\Lam}$.
The generalization of the exact flow equations for the effective
action in purely bosonic or fermionic systems to coupled boson-fermion
systems reads
\begin{eqnarray} \label{floweqGambf}
\frac{d}{d\Lam} \Gam^{\Lam}[\Phi,\Psi] &=&
\frac{1}{2} (\Phi^*,\dot\bQ_{b0}^{\Lam} \Phi) -
\frac{1}{2} (\Psib,\dot\bQ_{f0}^{\Lam} \Psi)
\nonumber \\
&+& \frac{1}{2} {\rm Str} \left[ \dot\bQ_0^{\Lam}
\left( \bGam^{(2)\Lam}[\Phi,\Psi] \right)^{-1} \right] \; ,
\end{eqnarray}
where
\begin{equation} \label{bQ0Lambf}
\bQ_0^{\Lam} = \left( \begin{array}{cc}
\bQ_{b0}^{\Lam} & 0 \\ 0 & \bQ_{f0}^{\Lam}
\end{array} \right) \; .
\end{equation}
The supertrace Str incorporates a minus sign in the fermionic sector.
The flow equation (\ref{floweqGambf}) is equivalent to the flow
equation \cite{Berges02}
\begin{equation} \label{floweqGamRbf}
\frac{d}{d\Lam} \Gam_R^{\Lam}[\Phi,\Psi] =
\frac{1}{2} {\rm Str} \left[ \dot \bR^{\Lam} \left(
\bGam_R^{(2)\Lam}[\Phi,\Psi]
+ \bR^{\Lam} \right)^{-1} \right] \, ,
\end{equation}
with $\bR^{\Lam} = \bQ_0^{\Lam} - \bQ_0$,
for the effective average action
\begin{equation}
\Gam_R^{\Lam}[\Phi,\Psi] = \Gam^{\Lam}[\Phi,\Psi] -
\frac{1}{2}(\Phi^*,\bR_b^{\Lam} \Phi) +
\frac{1}{2}(\Psib,\bR_f^{\Lam} \Psi) \, .
\end{equation}
The expansion of the exact functional flow equation (\ref{floweqGambf})
proceeds in complete analogy to the purely fermionic case.
Inserting
\begin{equation} \label{bGam2decbf}
\bGam^{(2)\Lam}[\Phi,\Psi] =
(\bG^{\Lam})^{-1} - \tilde\bSg^{\Lam}[\Phi,\Psi] \; ,
\end{equation}
with
\begin{equation} \label{bGLambf}
\bG^{\Lam} =
\left(
\left. \bGam^{(2)\Lam}[\Phi,\Psi] \right|_{\Phi = \Psi = 0}
\right)^{-1} =
\mbox{diag} (\bG_b^{\Lam},\bG_f^{\Lam}) \; ,
\end{equation}
into the functional flow equation (\ref{floweqGambf}), one obtains
\begin{widetext}
\begin{eqnarray} \label{floweqGambfexp}
\frac{d}{d\Lam} \Gam^{\Lam}[\Phi,\Psi] &=&
\frac{1}{2} {\rm Str} \left( \dot \bQ_0^{\Lam} \bG^{\Lam}
\right) +
\frac{1}{2} (\Phi^*,\dot\bQ_{b0}^{\Lam} \Phi) -
\frac{1}{2} (\Psib,\dot\bQ_{f0}^{\Lam} \Psi)
\nonumber \\ &-&
\frac{1}{2} {\rm Str} \left[ \bS^{\Lam}
\left( \tilde\bSg^{\Lam}[\Phi,\Psi] +
\tilde\bSg^{\Lam}[\Phi,\Psi] \, \bG^{\Lam}
\tilde\bSg^{\Lam}[\Phi,\Psi] + \dots \right) \right] \; ,
\end{eqnarray}
\end{widetext}
with the single-scale propagator
\begin{equation} \label{SLambf}
\bS^{\Lam} = - \bG^{\Lam} \dot\bQ_0^{\Lam} \bG^{\Lam} =
\mbox{diag}(\bS_b^{\Lam},\bS_f^{\Lam}) \; .
\end{equation}
The expansion in powers of the fields is now straightforward
and leads to a hierarchy of flow equations for all vertex
functions. The first few terms are shown diagrammatically in
Fig.~\ref{fig:floweq1pi_bf}.
\begin{figure}[ht]
\centerline{\includegraphics[width = 8cm]{fig7.eps}}
\caption{Diagrammatic representation of the flow equations for
the (fermionic and bosonic) self-energies and some of the
interaction vertices in a coupled boson-fermion theory.
Solid lines denote fermionic, dashed lines bosonic propagators.
Propagators with a dash are single-scale propagators.}
\label{fig:floweq1pi_bf}
\end{figure}
The flow equations derived above are also valid in case of
$U(1)$-symmetry breaking, if one allows for off-diagonal elements
in the matrices $\bQ_{b0}^{\Lam}$, $\bQ_{f0}^{\Lam}$,
$\bG_b^{\Lam}$, $\bG_f^{\Lam}$ etc.
\cite{Berges02,Schuetz06}.
Coupled flow equations for fermions and bosonic Hubbard-Stratonovich
fields are particularly convenient to treat fluctuations associated
with spontaneous symmetry breaking (see Sec.~IV) and quantum
criticality (see Sec.~V), but they may also be used to study
Luttinger liquids and other symmetric states in interacting Fermi
systems \cite{Schuetz05,Bartosch09a,Ledowski07a,Ledowski07b}.
\section{COMPETING INSTABILITIES}
\label{sec:III}
\newcommand{\Lambda_*}{\Lambda_*}
\newcommand{T_*}{T_*}
\renewcommand{\bQ}{{\bf Q}}
\newcommand{\kappa}{\kappa}
In this section we describe how one can apply the level-$2$ truncation
of the fermionic RG, mainly without self-energy corrections,
to two-dimensional fermion systems,
and study the interplay of ordering tendencies.
In resummations of perturbation theory,
their manifestation are singularities in the four-point function
and in certain susceptibilities.
In the RG, the precursor to a singularity
is the growth of some parts of the vertex function
(often termed ``flow to strong coupling'').
Since singularities in the vertex function change the power counting drastically,
this truncated flow then has to be stopped before a singularity happens,
at a scale $\Lambda_* > 0$,
where one can read off the dominant interactions and infer a tentative phase diagram
(in this, susceptibilities are used to compare the strength of different
ordering tendencies and to determine $\Lambda_*$).
As discussed in Sec.~\ref{sec:truncations}, curvature effects of the Fermi
surface imply that the truncations discussed here can be used also when the
interaction is no longer small, provided that the power counting
improvement factor times the interaction strength remains small.
To obtain a true phase diagram, however, one needs to integrate over all
degrees of freedom, also those with scales below $\Lambda_*$.
This has been achieved in some cases (see Section \ref{sec:IV}), but much remains
to be done.
This first step of monitoring the flow to strong coupling above $\Lambda_*$,
as described in this section, is important for the following reasons.
(1) It allows to determine the effective interaction just above transition scales
from the given microscopic model
without any additional a priori assumptions about the nature of symmetry-breaking,
and thereby provides an initial condition for the integration at scales below $\Lambda_*$.
(2) It exhibits how the interplay of the scale-dependent scattering processes
on different parts of the Fermi surface gradually builds up the effective
interaction.
(3) It has by now become a versatile tool for analyzing models with
an elaborate microscopic structure, such as multiple bands.
\subsection{Hubbard model and $N$-patch RG schemes}
\label{sec:2dHubbard}
The Hubbard model and its extensions have become standard in correlated fermion systems:
on the square lattice as a candidate model for high-temperature superconducting cuprates
\cite{Anderson97,Fulde91}, in a multiband-generalization, for the newly discovered iron
superconductors \cite{Miyake10}, on triangular lattices for organic crystals
\cite{kino,mckenzie}, on the honeycomb lattice for graphene \cite{herbut,lopez-sancho}.
The Hamiltonian for the simplest one-band Hubbard model reads
\begin{equation}\label{hubbard2dsq}
H = - \sum_{i,j,s} t_{i-j} c_{i,s}^\dagger c_{j,s} +
U \sum_i n_{i,\uparrow} n_{i, \downarrow }
\end{equation}
where $t_{i-j}=t_{j-i}$ is the hopping amplitude between sites $i$ and $j$ and
$U$ is the Hubbard on-site repulsion.
We consider here mainly the case with only nearest-neighbor hopping
$t$ and next-to-nearest neighbor hopping $t'$ on a square lattice.
Additional hopping terms can be added if a more detailed description of the band
structure is required, and other interaction terms may be added.
The chemical potential $\mu$ and $t$ and $t'$ determine the band structure
$\xi_{\bk} = -2t ( \cos k_x + \cos k_y ) - 4t' \cos k_x \cos k_y - \mu$,
and hence the shape of the Fermi surface.
Resummations of perturbation theory in $U$ suggest singularities in different
channels, arising from Fermi surface nesting and Van Hove singularities
\cite{Schulz87}, hence competing effects, which are best treated by RG methods.
After two-patch studies, which provided a very crude approximation to
the momentum dependence of the four-point vertex
\cite{Guinea,Furukawa98,Lederer,Dzyaloshinskii87,Schulz87},
more careful analyses with momentum-dependent vertices were done
using the Polchinski \cite{Zanchi97,Zanchi98,Zanchi00}, the Wick ordered
\cite{Halboth00a,Halboth00b}, and the one-particle irreducible flow equations
\cite{Honerkamp01d}, all with a momentum space regulator.
To include ferromagnetism, the temperature flow was introduced by
\textcite{Honerkamp01b,Honerkamp01c} and \textcite{Honerkamp01a}, and further
developed by \textcite{Katanin03}.
The results of these studies at Van Hove filling were confirmed using a refined
parametrization of the wavevector dependence \cite{Husemann09a}.
The decoupling of the various ordering tendencies in the limit of small $U$ very
close to the instability and the influence of non-local interactions were discussed
by \textcite{Binz02,Binz03}.
In the general RG setup of Section \ref{sec:II}, the fermion fields now carry
a spin index $s$ and a multiindex $K$ consisting of Matsubara frequencies $\omega$,
wavevectors $\bk$, and possibly a band index $b$.
To avoid bias, the action is required to retain all symmetries of the initial action.
This implies (see \textcite{Honerkamp01d,Salmhofer01}) that
\begin{widetext}
\begin{eqnarray}
\Gamma^{(4)\Lambda}_{s_1s_2s_3s_4} (K_1,K_2;K_3,K_4) &=&
V^\Lambda(K_1,K_2;K_3,K_4) \delta_{s_1s_3}\delta_{s_2s_4} -
V^\Lambda(K_2,K_1;K_3,K_4) \delta_{s_1s_4}\delta_{s_2s_3}
\label{fierz}
\end{eqnarray}
for a spin-rotation invariant system.
By lattice- and time-translation invariance, $K_4$ is fixed by $K_1,K_2$ and $K_3$
in the one-band model (in multiband models, the fourth band index $b_4$ still
remains free). We therefore abbreviate notation to $V^\Lambda(K_1,K_2,K_3)$.
In the truncation $\Gamma^{(6)\Lambda} =0$, the flow equations for the self-energy
and for the coupling function become
\begin{equation}
\sfrac{d}{d\Lambda} \Sigma^\Lambda (K)
= - \int dK' \,\left[ 2 V^\Lambda (K,K',K) - V^\Lambda(K,K',K') \right] \,
S^\Lambda (K') \, , \quad
\sfrac{d}{d\Lambda} V^\Lambda
= {\cal T}^\Lambda_{PP} + {\cal T}^\Lambda_{PH,d} +
{\cal T}^\Lambda_{PH,cr} \label{sigmavdot}
\end{equation}
with the particle-particle term $ {\cal T}^\Lambda_{PP}$ and the direct and crossed
particle-hole terms ${\cal T}^\Lambda_{PH,d} $ and ${\cal T}^\Lambda_{PH,cr}$:
\begin{eqnarray}
{\cal T}^\Lam_{PP} (K_1,K_2;K_3,K_4)
&=&
\int dK \, V^{\Lam}(K_1,K_2,K) \; L^{\Lam}(K,-K+K_1+K_2) \,
V^{\Lam}(K,-K+K_1+K_2 ,K_3) \label{PPdia} \; ,
\\
{\cal T}^{\Lam}_{PH,d} (K_1,K_2;K_3,K_4)
&=&
\int dK\, \biggl[- 2 V^{\Lam}( K_1,K,K_3 ) \, L^{\Lam}(K,K+K_1-K_3) \,
V^{\Lam}(K+K_1-K_3,K_2,K) \nonumber
\\ && \qquad +
V^{\Lam}(K_1,K,K+K_1-K_3) \, L^{\Lam}(K,K+K_1-K_3) \, V^{\Lam}(K+K_1-K_3,K_2,K)
\nonumber \\ && \qquad +
V^{\Lam}(K_1,K,K_3) \, L^{\Lam}(K,K+K_1-K_3) \, V^{\Lam}(K_2,K+K_1-K_3,K)
\biggr]\; ,
\label{PHddia}
\\
{\cal T}^{\Lam}_{PH,cr}(K_1,K_2;K_3,K_4)
&=&
\int dK \, V^{\Lam}(K_1,K+K_2-K_3,K) \, L^{\Lam}(K,K+K_2-K_3) \,
V^{\Lam}(K,K_2,K_3 ) \; .
\label{PHcrdia}
\end{eqnarray}
Here
$L^{\Lam}(K,K') = S^\Lam(K) G^{\Lam}(K') + G^{\Lam}(K) S^{\Lam}(K')$
is the product of single-scale propagators $S^\Lambda$ and full propagators
$G^\Lambda$ with momentum assignments corresponding to the diagrams in
Fig.~\ref{1loopRGDE}.
\end{widetext}
For the Hubbard Hamiltonian (\ref{hubbard2dsq}), the initial condition is
$V^{\Lam_0}(K_1,K_2,K_3) = U$.
Other interactions can be dealt with by modifying this initial condition.
The truncation $\Gam^{(6)\Lam} = 0$ is justified only for a sufficiently
small bare coupling, since a contribution to $\Gam^{(6)\Lam}$ is generated
at third order in the two-particle interaction, which leads to third order
contributions to the flow of $V^{\Lam}$ (see Sec.~II).
In most studies the self-energy feedback into the flow of $V^{\Lam}$ was
also neglected, since it also affects the flow only at third order in
$V^{\Lam}$.
The coupling function $V^\Lambda (K_1,K_2,K_3)$ depends on three wavevectors and
three Matsubara frequencies, so that the RG equation for a two-dimensional system
is a differential equation in a 9-dimensional space.
As discussed in Section~\ref{sec:powercount}, its most singular part sits at
zero Matsubara frequency. Hence one may neglect the frequency dependence.
Then $V^\Lambda$ defines an effective Hamiltonian.
Similarly, the $\bk$-dependence is most important in the angular direction
along the Fermi surface.
This dependence can then be taken into account by a discretization,
i.e. by devising patches in the Brillouin zone in which the coupling function is
kept constant.
\textcite{FMRT92} showed that using $N$ patches leads to a natural $N$-vector model
in two dimensions. \textcite{Zanchi98,Zanchi00} were the first to use it in studies
of the Hubbard model.
Usually one forms elongated patches that extend roughly perpendicular to the
Fermi surface but are rather narrow parallel to the Fermi surface
(see Fig.~\ref{setup}).
The coupling function is then computed for wavevectors $\bk_1$ to $\bk_3$ at
the Fermi surface in the center of the patches.
We label the patches by $\kappa_i=1, \dots N$. The function $V^\Lambda$ is thus approximated
by $O(N^3)$ interpatch couplings $V^\Lambda (\kappa_1,\kappa_2,\kappa_3)$.
Even if $\bk_1,\bk_2$ and $\bk_3$ are on the Fermi surface, $\bk_4$ can be anywhere.
In the calculation of the loop integrals it is however necessary to assign a patch
number $\kappa_4$ to $\bk_4$, which amounts to an approximation of projecting $\bk_4$
on the Fermi surface.
Note that this projected $N$-patch discretized coupling function
$V^\Lambda (\kappa_1,\kappa_2,\kappa_3)$ then has fewer symmetries;
for instance $V^\Lambda (\kappa_1,\kappa_2,\kappa_3) \not= V^\Lambda (\kappa_2,\kappa_1,\kappa_4)$
in general, as in the latter object $\bk_3$ is not necessarily on the
Fermi surface.
For sufficiently large $N$, this discretization captures the angular variation
of the coupling function along the Fermi surface with good precision.
The results obtained within this approximation, described in the following,
have been found to be robust when the dependence on frequencies $\omega_i$
\cite{Honerkamp07,Klironomos06} and the component of $\bk_i$ transversal
to the Fermi surface \cite{Halboth00a,Honerkamp01a,Honerkamp04} are included.
\textcite{Katanin09} performed a flow to third order in the scale-dependent
four-point-vertex (see Section \ref{ssec:runcoup}), with the frequency
dependence in the same approximation as \textcite{Honerkamp03}.
\begin{figure}
\centerline{\includegraphics[width = 0.4\columnwidth]{fig8a.eps}
\includegraphics[width=0.55\columnwidth]{fig8b.eps}}
\centerline{\includegraphics[width = 0.8\columnwidth]{fig8c.eps}}
\caption{Top row: The coupling function $V^\Lambda(K_1,K_2,K_3)$ with the
spin convention, and the diagrams entering in the flow equation for
the self-energy (middle and right diagram). Middle and bottom row:
The diagrams for the flow of the coupling function.
The internal lines are either full propagators $G^\Lambda$ or single-scale
propagators $S^\Lambda$.}
\label{1loopRGDE}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{0.65\columnwidth}{!}{\includegraphics{fig9.eps} }
\end{center}
\caption{(Color online)
$N$-patch discretization of the Brillouin zone for the one-band
Hubbard model on the 2D square lattice. The colored region is a
patch in which the coupling function is approximated as a constant.}
\label{setup}
\end{figure}
\subsection{Results for the two-dimensional Hubbard model}
Starting from the initial condition given by the Hubbard model,
the flow is run from $\Lambda_0$ down to a characteristic scale $\Lambda_*$,
where the largest coupling reaches some multiple $\alpha$ of the bandwidth.
The choice of $\alpha$ varies widely in the literature; the discussion here is based on
the comparably cautious choice $\alpha =2$ or $3$, as well as on the consistency check
that the results do not change drastically as $\alpha$ is changed.
The characteristic scale $\Lambda_*$ corresponds to a temperature $T_*$.
If $T$ is clearly above $T_*$, the flow can be integrated to scale zero
without any instabilities. $T_*$ is only an upper bound for the temperature
where ordering can set in because of order parameter fluctuations
at scales below $\Lambda_*$.
In two dimensions they are so strong that long-range order that breaks
continuous symmetries does not occur at any $T>0$,
thus ``ordering'' is to mean either short-range order with a very large
correlation length, or ordering in a related system with a small coupling in the third
direction, as is present in most materials.
\subsubsection{Antiferromagnetism and Superconductivity}
The results discussed here are obtained with a slightly smeared-out step-function
as cutoff on $\bk$ (no cutoff on the frequencies) and by dropping the self-energy.
\noindent
{\em Antiferromagnetism.}
For $t'=0$ and $\mu=0$,
the band is half-filled and the Fermi surface a perfect square.
Every vector connecting parallel sides of the Fermi surface is a nesting
vector, and $\nabla \xi_{\bk} = 0$ at $(\pi,0)$ and $(0,\pi)$.
This strongly enhances particle-hole terms at wavevector $\bQ = (\pi, \pi)$.
A random-phase approximation summation of these bubbles results in a divergent static
spin susceptibility at $\bQ$ for any $U>0$ at sufficiently low $T$, indicating the
formation of an antiferromagnetic (AF) spin-density wave (SDW), in accordance with
mean-field studies \cite{Fulde91}.
The basic RG results at low $T$ are shown for $U=2t$ in Fig.~\ref{comp0_120}.
The labelling of the $N=32$ patches along the Fermi surface can be read off
Fig.~\ref{comp0_120} a).
Fig.~\ref{comp0_120} b) shows $V^\Lambda$ as a function of the patch indices
$\kappa_1$ and $\kappa_2$, at $\Lambda_* \sim 0.16t$ and with $\kappa_3 = 1$
(i.e.\ $\bk_3$ near $(-\pi,0)$).
Strongly enhanced repulsive interactions appear as a vertical line at $\kappa_2=24$
(i.e.\ for $\bk_2 - \bk_3 = \bQ$), almost $\kappa_1$-independent,
and as a horizontal line at $\kappa_1=24$ (corresponding to $\bk_1 - \bk_3= \bQ$) with
only a weak dependence on $\kappa_2$, roughly half as large as the vertical feature.
In an extrapolation where the regular profiles are narrowed down to delta functions
with an appropriate prefactor $J$,
$V^\Lambda (\kappa_1,\kappa_2,\kappa_3) =
\frac{J}{4} ( 2 \delta_{\bk_2- \bk_3,\bQ} + \delta_{\bk_1- \bk_3,\bQ})$,
corresponding to a mean-field AF spin interaction Hamiltonian
$J \sum_{\langle i,j \rangle } e^{i \bQ\cdot({\bf R}_i-{\bf R}_j)} {\bf S}_i \cdot {\bf S}_j$,
with ${\bf S}_i = \frac12 c^+_i \mbox{\boldmath$\sigma$} c^{\hphantom{+}}_i$.
The effective Hamiltonian consisting of the low-scale hopping term and this interaction
exhibits AF long-range order at sufficiently low $T$.
An analysis of the flow of susceptibilities\cite{Halboth00a,Honerkamp01d} as described
in Sec.~\ref{sec:responsefcts} confirms this picture.
The extrapolation to a mean-field Hamiltonian is a drastic oversimplification,
in which the spin fluctuations are lost, but they are retained in the $V^\Lambda$ obtained
by the RG flow.
As the leading instability is clearly exposed by this analysis,
one can also resort to a bosonized description that treats the collective infrared
physics \cite{Baier04}.
\noindent
{\em $d$-wave Cooper pairing. }
For $t'=-0.3t$ and $\mu = -1.2t$, the Fermi surface still contains the saddle
points $(\pi,0)$ and $(0,\pi)$
but is curved away from these points (Fig.~\ref{comp0_120}c)).
Now Cooper pair scattering dominates, well visible in Fig.~\ref{comp0_120}d)
on the diagonal lines $\bk_1+ \bk_2=0$ ( $|\kappa_1-\kappa_2|=N/2$ in terms of patch indices).
It is attractive when the incoming pair $\bk_1, -\bk_1$ is near the same saddle point
$(\pm \pi, 0)$ as the outgoing pair $\bk_3, - \bk_3$, and repulsive when incoming and
outgoing pairs are at different saddle points.
This is the symmetry of the formfactor $d(\bk) = d_0 (\cos k_x - \cos k_y)$ for
$d_{x^2-y^2}$-Cooper pairing. In an extrapolation as above,
$V^\Lambda (\bk_1, \bk_2, \bk_3)$ gives rise
to the mean-field Hamiltonian
\[ H^\Lambda_{d\mathrm{SC}} = V^{\phantom{\Lambda}}_{d\mathrm{SC}} \sum_{\bk,\bk' } \,
\, d(\bk) d(\bk') \, c^\dagger_{\bk',\uparrow} c^\dagger_{-\bk',\downarrow}
c_{-\bk,\downarrow} c_{\bk,\uparrow} \, . \]
which has a $d$-wave singlet-paired ground state.
This $d$-wave pairing instability was found in a number of studies using different
functional RG schemes \cite{Zanchi98,Zanchi00,Halboth00a,Halboth00b,Honerkamp01d,
Tsai01,Honerkamp01b,Honerkamp01c,Honerkamp01a}, in a rather large parameter region.
This constitutes convincing evidence that the weakly coupled Hubbard model possesses
a $d$-wave superconducting ground state.
\begin{figure}
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{fig10.eps} }
\caption{(Color online)
$N$-patch functional RG data obtained with the momentum-shell functional RG
for the repulsive Hubbard model on the 2D square lattice.
Upper plots: $\mu=0$, $t'=0$ and initial $U=2t$,
lower plots: $\mu=1.2t$, $t'=-0.3t$, $U=3t$.
To the left: Fermi surfaces for the two cases and the $N=32$ discretization
points for the two incoming $\bk_1$, $\bk_2$ and the 1st outgoing wavevector $\bk_3$.
To the right: the coupling function $V^{\Lambda_*} (\kappa_1,\kappa_2,\kappa_3)$ with $\kappa_3=1$ and
$\kappa_1$ and $\kappa_2$ moving around the Fermi surface.
The colorbars on the right indicate the values of the interactions.}
\label{comp0_120}
\end{figure}
\noindent
{\em Interplay of AF and SC. }
In Fig.~\ref{comp0_120}d),
the sign structure of the $d$-wave term goes together, and fits perfectly with,
enhanced repulsive interactions near $\kappa_1 = 8$ and $\kappa_2 = 24$,
which are the remnants of the SDW feature in Fig.~\ref{comp0_120}b).
Their larger width is due to the Fermi surface curvature. As $\Lambda$ is decreased,
these SDW features appear first, due to approximate nesting at high scales, and then
create an attractive component in the $d_{x^2-y ^2}$-pairing channel, which then grows
as $\Lambda$ is lowered further, while the SDW is cut off by Fermi surface curvature,
as discussed also in Appendix \ref{ssec:imppoco}.
When the SDW-enhancing terms are removed by hand from the right hand side of the
RG equation, the $d$-wave terms are suppressed as well.
Thus the $d$-wave pairing interaction is induced by AF-spin fluctuations that
appear on higher scales.
At fixed $U$, $t$ and $t'$, there is a sizable interval of $\mu$ for which
the Fermi surface remains close to the saddle points. Since both AF-SDW and
$d$-wave SC are driven by repulsive scattering between $(\pi,0)$ and $(0,\pi)$,
both grow and reinforce one another. In the {\em saddle point regime},
it becomes impossible to single out one over the other in the truncation used here.
By analogy with the quasi-one-dimensional ladder systems, it has been argued
that in this regime, the Fermi surface gets truncated
\cite{Furukawa98,Honerkamp01d,Laeuchli04}.
\subsubsection{Ferromagnetism vs.\ Superconductivity}
At the Van Hove filling, ferromagnetic (FM) tendencies are enhanced by the logarithmic
divergence of the density of states, and the Stoner criterion for the bare
interaction suggests an FM ordered state at arbitrarily small
$U$. However, the Van Hove singularities also make the $O(U^2)$ Cooper pair scattering
$\log^2$-divergent, hence put the two terms into direct competition.
As discussed in Section \ref{sec:cutoffs},
the momentum-shell cutoff artificially suppresses FM. For this reason,
the $T$-flow (see Section \ref{sssec:tflow}) was invented
\cite{Honerkamp01b,Honerkamp01c}, and we discuss results
obtained by $T$-flow here.
The main difference to the AF/SC scenario discussed above is
that at zero transfer momentum, scattering processes driving
FM must have the opposite sign from those driving singlet SC,
hence mutually suppress one another. This simple picture is confirmed
by the RG with momentum-dependent vertices, in a study where
$t'$ and $\mu$ are varied at fixed $U$ and $t$, such that the Fermi surface
always contains the saddle points: near to $t' = - t/3$,
$T_*$ gets strongly suppressed, hinting at a quantum critical point between
the dSC and FM phases (lower left plot in Fig.~\ref{pds}).
These results were later confirmed by a two-particle self-consistent approach
\cite{Hankyevich2003} and in the so-called $\Omega$-scheme, which employs a
soft infrared regulator on the Matsubara frequencies \cite{Husemann09a};
see the lower right plot in Fig.~\ref{pds}.
In the latter study, the $N$-patch scheme was replaced by a parametrization
of the vertex functions in terms of exchange bosons.
The much higher value of $\Lambda_*$ in the transitional regime near $t' = - t/3$
is believed to be due to a form factor that was not fully resolved there.
\subsubsection{Charge instabilities}
The effective interaction develops a pronounced momentum dependence
also in the charge sector.
In the forward scattering channel, this amounts to the formation
of non-uniform contributions to the Landau interaction.
If strong enough, the latter can lead to a {\em Pomeranchuk instability}
\cite{Pomeranchuk58}, that is, a symmetry-breaking deformation of the
Fermi surface.
In particular, the antiferromagnetic peak drives the combination of
couplings $V^\Lambda_c (\kappa_1,\kappa_2,\kappa_3) =
2V^\Lambda (\kappa_1,\kappa_2,\kappa_3) - V^\Lambda( \kappa_2,\kappa_1,\kappa_3)$
at certain $\bQ = \bk_3-\bk_1$.
Near to $\bQ\approx {\bf 0}$ and $\bQ \approx (\pi, \pi)$,
\begin{equation}
V^\Lambda_c (\kappa_1,\kappa_2,\kappa_3) \approx
- f_d(\bk_1) f_d(\bk_2) V_d (\bk_3-\bk_1) \, ,
\end{equation}
where $f_d(\bk)$ has the same symmetries as $d (\bk) = \cos k_x - \cos k_y$,
but is more strongly peaked near the saddle points.
For $\bQ=(\pi,\pi)$ the corresponding mean-field state is the
{\em $d$-density wave state}, which breaks time-reversal invariance
\cite{Nayak01} and gaps the single-particle states, except at nodal points
on the Brillouin zone diagonal.
For forward scattering, $\bQ={\bf 0}$, the mean-field state only breaks
the lattice rotational symmetry of the electronic dispersion and hence of
the Fermi surface.
This tendency to form a {\em nematic} state \cite{Fradkin10}
via a $d$-wave Pomeranchuk instability driven by forward scattering
interactions was discovered using functional RG \cite{Halboth00b}.
Although the Pomeranchuk instability is not leading in the flow
for the Hubbard model \cite{Honerkamp02}, a nematic state can coexist
with the superconducting state \cite{Neumayr2003,Yamase07},
and it may get less suppressed by fluctuations since it breaks no
continuous symmetry.
The $d$-wave Pomeranchuk instability has been investigated as a
possible source of nematicity of the electronic state in relation with
experiments on various correlated electron systems
\cite{Honerkamp05a,Yamase06,Yamase09,Metlitski10,Okamoto10}.
\begin{figure}
\centering
\resizebox{0.48\columnwidth}{!}{\includegraphics{fig11a.eps} }
\hfill
\resizebox{0.47\columnwidth}{!}{\includegraphics{fig11b.eps} }
\resizebox{0.48\columnwidth}{!}{\includegraphics{fig11c.eps} } \hfill
\resizebox{0.48\columnwidth}{!}{\includegraphics{fig11d.eps} }
\caption{(Color online)
Leading instabilities as found by $N$-patch functional RG in the $t$-$t'$-Hubbard model.
Left upper plot:
$T_*$ vs.\ $\mu$ for band filling larger than unity, at $t'=-0.3t$ and $U=3t$.
There is a high-energy-scale AF SDW instability with a weaker $d_{x^2 -y^2}$-wave
pairing instability when the AF-SDW is cut off. Data from \textcite{Honerkamp01a}.
Right upper plot: Data for the same $t'$ and $U$ on the 'hole-doped' side with band
fillings smaller than one, from \textcite{Honerkamp01d}.
Now there is a broad crossover 'saddle point regime' between the nesting-driven
AF-SDW instability and the $d_{x^2 -y^2}$-wave pairing regime.
Lower left plot: $T_*$ vs. $t'$ at the Van Hove filling where the Fermi surface
contains the
points $(\pi,0)$ and $(0,\pi)$. For large $t'$ one finds a ferromagnetic instability.
Data from \textcite{Honerkamp01b,Honerkamp01c} obtained with the $T$-flow.
Right lower plot: $\Omega_*$ vs.\ $t'$ at Van Hove filling, now obtained with the
simplified vertex parametrization of \textcite{Husemann09a} and with a soft
frequency regulator $\Omega$.}
\label{pds}
\end{figure}
\subsubsection{Flows with self-energy effects}
We briefly summarize functional RG studies where the self-energy has been included.
If a frequency-independent vertex function $V^\Lambda$ is directly inserted in
the right hand side of Eq.~(\ref{floweqSigma}),
then $\Sigma^\Lambda$ is real and independent of the frequency,
hence only changes the dispersion.
This was taken into account in the appendix of \textcite{Honerkamp01d}, where the
adaptive scale decomposition method later detailed in \textcite{Salmhofer07} was used.
To keep the density fixed, $\mu$ is adjusted as a function of $\Lambda$.
Since the interaction grows in the flow, it is a nontrivial check
of the validity of the truncation that the feedback from the interaction
does not lift the low-kinetic-energy modes
to high energies, which would drastically shift the Fermi surface and lead to
spurious divergences.
The first study by \textcite{Honerkamp01d} showed that the Fermi surface tends
to become flatter as $\Lambda$ decreases, but that it indeed shifts very little
before the flow is stopped at $\Lambda_*$.
Thus including the real part of the self-energy does not lead to any essential changes
in the AF/SC scenario described above. However, correlations that only feed on the immediate
vicinity of the saddle points, like FM, are affected more strongly, and a full analysis
of the coupled flow of self-energy and vertex directly at the saddle points remains an
open problem, in spite of partial results \cite{FS2}.
The imaginary part and the frequency-dependence of the self-energy can be
approximated by inserting the integrated flow of the interaction vertex in the
self-energy equation \cite{Honerkamp01a}.
This effectively includes two-loop frequency-dependence effects,
and captures the $T^2$-dependence of the quasiparticle scattering rate
in a Fermi-liquid situation and
the exponent of the vanishing quasiparticle weight in the Luttinger liquid
up to second order in the bare couplings \cite{Honerkamp03}.
For the 2D Hubbard model, the quasiparticle lifetime and renormalization
factor was calculated in \textcite{Honerkamp01a,Honerkamp03},
exhibiting a strongly $\bk$-dependent
quasiparticle degradation as $\Lambda_*$ is approached. This trend was also found by
\textcite{Zanchi01} in a slightly different approximation for the self-energy, and is
also robust in a more elaborate treatment \cite{Katanin09}, where the six-point vertex
was included partially.
The anisotropy of the quasiparticle
lifetime was found to have a non-Fermi-liquid temperature dependence and to correlate
with the strength of the generated $d$-wave pairing interaction \cite{Ossadnik08},
similar to what is observed experimentally in overdoped cuprates. More refined studies of
the frequency-dependence revealed, however, that a simple parametrization in terms of a
quasiparticle weight is insufficient \cite{Katanin04b,Rohe05}.
It was shown that near $\Lambda_*$,
the small-$|\omega|$-behavior of $\Sigma^\Lambda (\omega,\bk)$ leads to a
split-up of the quasiparticle peak.
All these findings are
consistent with an anisotropic break-up of the Fermi surface that one would like to
connect with the phenomenology of the high-$T_c$ cuprates \cite{Honerkamp01d,Lee06},
but a quantitative comparison is difficult due to the strongly coupled nature of the
cuprates.
\subsection{Pnictide superconductors}
The functional RG has been very useful in the study of the newly discovered iron
pnictide superconductors \cite{Norman08,Ishida09,Hirschfeld10}.
Here the functional RG may work even
better, as the pnictides are less strongly correlated than the high-$T_c$ cuprates. This
can already be inferred from the experimental phase diagram, where one only finds
metallic antiferromagnetic phases (if at all), but never Mott insulating
antiferromagnetism. Theoretical works that try to assess the
iron $d$-orbital onsite-interaction strengths find values that put the materials into the
range of weak to moderate correlations \cite{Anisimov09,Miyake10}. Regarding the electronic
structure, the pnictides are more complex than the cuprates.
At least three of the five
iron $d$-orbitals have non-negligible weight near the Fermi level \cite{Mazin08,Daghofer10}.
Therefore, even if one is only interested in the vicinity of the Fermi
surface, the multi-band character has to be kept.
The Fermi surface (see Fig.~\ref{feas} b)) is divided into two hole pockets,
centered around the origin of the Brillouin zone at $\bk=0$, and two
electron pockets around $\bk=(\pi,0)$ and $\bk=(0,\pi)$ in the unfolded zone
corresponding to the small unit cell with one iron atom (or $\bk=(\pi,\pi)$ in the folded
zone corresponding to the large unit cell with two iron atoms). As pointed out
early \cite{Mazin08,Kuroki08}, there is approximate nesting of electron- and hole pockets
which enhances particle-hole susceptibilities with the wavevector connecting these
pockets. In addition, depending on the parameters and approximations \cite{Ikeda10}, there
can be a third hole pocket at $(\pi,\pi)$ in the unfolded zone.
The first $N$-patch studies of the pnictides were performed by \textcite{Wang09a,Wang09b,Wang10}
for a five-band model. These authors obtained a sign-changing $s$-wave pairing
instability driven by AF fluctuations as the dominant pairing instability. Further they
found strongly anisotropic gaps around the electron pockets, with possibility of node
formation.
The basic structure of the phase diagram with the sign-changing pairing gap between
electron- and hole-pockets can be understood already from simplified few-patch RG
approaches \cite{Chubukov08}. This would however predict isotropic gaps around these
pockets \cite{Platt09}. To understand the gap anisotropy one has to take into account the
multi-orbital nature of the electronic spectrum in the iron pnictides, as was done already
in the initial studies \cite{Wang09a,Wang09b,Wang10}. In order to understand this point,
let us start with a single-particle Hamiltonian in wavevector-Fe-$d$-orbital space
\begin{equation} H = \sum_{\bk,s,o} h(\bk)_{oo'} c^\dagger_{\bk,o,s} c_{\bk,o',s}
\end{equation} where the matrices $h(\bk)_{oo'} $ take into account intra- and
inter-orbital terms for orbital index $o=o'$ or $o\not=o'$ respectively. $s$ is the spin
quantum number. The energy bands are obtained by a unitary transformation from orbital to
band operators (index $b$), $c_{\bk,b,s} = \sum_{o} u_{bo}(\bk) c_{\bk,o,s}$.
The standard choice for the interaction between the electrons is to introduce
orbital-dependent intra- and inter-orbital onsite repulsions, plus Hund's rule and pair
hopping terms. While these local terms lead to $\bk$-independent interactions in the
orbital basis, parametrized by a tensor $V_{o1,o2,o3,o4}$, after the transformation to
bands one arrives at a $\bk$-dependent interaction function
\begin{widetext}
\begin{eqnarray} V_{b1,b2,b3,b4} (\bk_1,\bk_2,\bk_3,\bk_4) &= & \sum_{o1,o2,o3,o4}
V_{o1,o2,o3,o4}
\; u_{b1,o1} (\bk_1) u_{b2,o2} (\bk_2) u^*_{b3,o3} (\bk_3) u^*_{b4,o4} (\bk_4) \, .
\label{omu} \end{eqnarray} \end{widetext}
The combination of $u_{bo}$s behind the interaction tensor is sometimes called the
'orbital make-up' \cite{Maier09,Graser09}.
These prefactors cause a marked $\bk$-structure already in the initial interaction
which is then renormalized during the functional RG flow.
It turns out that this orbital make-up has an essential
influence on the competition between different channels in the flow and is responsible
for the gap anisotropies found in the multi-band functional RG studies by
\textcite{Wang09a,Wang09b,Wang10} and in subsequent functional RG studies
\cite{Thomale10,Platt10}. A typical result for the predicted pairing
gaps is shown in Fig.~\ref{feas} a). Note that according to the functional RG
analysis, the pairing state should be strongly doping-dependent
\cite{Thomale09,Thomale10,Thomale11}.
Summarizing this brief section, the iron superconductors pose an interesting problem
where the functional RG has been instrumental in
obtaining the main ordering tendencies in good agreement with
current experiments. For future
research, one goal should be to make the functional RG a useful bridge between ab-initio
descriptions providing the effective model at intermediate energy scales and the many-body
effects seen in the experiments at low scales. In particular it will be interesting to relate
experimentally observed materials trends in, e.g., the gap structure or the energy scales
of the different systems, to changes in the microscopic Hamiltonian taken from ab-initio
descriptions. Furthermore, the functional RG studies may have to be extended to include the
dispersion orthogonal to the iron-pnictide planes, as this would yield additional
possibilities for nodes in the gap function \cite{Norman08,Ishida09,Hirschfeld10,Platt11}.
\begin{figure}
\centering
\resizebox{0.99\columnwidth}{!}{\includegraphics{fig12.eps} }
\caption{(Color online)
Functional renormalization group results for the iron pnictide compound LaFeAsO
at moderate hole doping. a) Superconducting form factor as the outcome of functional RG,
plotted versus the position on the hole pockets at $\Gamma$ and $M$ and electron pockets
at $X$, numbered as depicted in b).
The competing fluctuations manifest themselves in diverging ordering susceptibilities
at low RG scales as shown in c), including in particular spin density wave
(SDW), superconductivity (SC), Pomeranchuk (PI) and charge density wave (CDW)
instabilities \cite{Thomale10}.}
\label{feas}
\end{figure}
\label{sec:pnictides}
\subsection{Other systems}
\label{sec:othersys}
Besides the above-described two larger fields of application,
the functional RG truncations described in this Section have also been
employed in a number of other models in strongly correlated electron physics. Here we
briefly list some of these activities.
In relation to possible unconventional superconductivity in organic crystals and
layered cobaltates, Hubbard-type models on the triangular lattice have been studied
\cite{Tsai01,Honerkamp03tria}. At large $U$, the spin exchange interaction between the
sites of the triangular lattice is geometrically frustrated, leading to a much weaker
appearance of antiferromagnetism and a possible non-magnetic insulating
phase \cite{Morita02,Sahebsara08,Yoshioka09}.
At weak coupling and for nearest-neighbor hopping,
Fermi surface nesting is absent, so that near to or at half band filling,
only low-scale Kohn-Luttinger-like superconducting instabilities occur out of an
innocuous Fermi liquid.
However, there appears to be a strong dependence on details of the microscopic modelling.
To study interaction effects in graphene, the $N$-patch functional RG has
been applied to the extended Hubbard model on the honeycomb lattice.
In nominally undoped graphene, the Fermi surface becomes a set of {\em Dirac points}
where the density of states vanishes, and
no instabilities are found for sufficiently small interactions.
If the interaction strength exceeds a certain value,
various instabilities driven by particle-hole fluctuations between the two Dirac
points \cite{Honerkamp08} are found.
Interestingly, for larger second-nearest neighbor
interactions, there is the possibility of an instability towards a quantum spin Hall
phase \cite{Raghu08}. However,
a spin-liquid phase for
intermediate strength of the Hubbard onsite repulsion that was recently found in
quantum Monte Carlo calculations \cite{Meng2010} is not reflected in the functional RG
results on this level of approximation.
When the Fermi level is moved away from the Dirac points, the functional RG again
detects pairing instabilities.
In the case of dominant nearest neighbor repulsion, the leading pairing
tendency is in the $f$-wave triplet channel \cite{Honerkamp08}.
The unbiasedness of the functional RG, and the access it gives to $\bk$- and
$\omega$-dependence of vertex functions, is also of great use in (quasi-)one-dimensional models.
The half-filled extended Hubbard model in one dimension has been studied in the search for
bond-order-wave phases, which could indeed be found with a refined patching of the
$\bk$-dependence of the interaction away from the Fermi points \cite{Tam06}.
For quasi-1D models with a small transverse hopping in a second direction the change
from a gapless Luttinger liquid in a strictly one-dimensional
situation to Fermi liquid instabilities
toward ordering can be monitored as a function of the transverse
hopping \cite{Honerkamp03}.
The Fermi surface in coupled metallic chains was studied by
\textcite{Ledowski05,Ledowski07a,Ledowski07b}.
The possibility of triplet pairing driven by density wave
fluctuations has been explored in such situations \cite{Nickel05,Nickel06}.
In these quasi-one-dimensional systems, including the frequency dependence of the interaction
vertex is numerically more feasible than in two dimensions. This has been used to study
the interplay of phonon-mediated and direct electron-electron interactions for chains
\cite{Tam07C}, ladders \cite{Tam07L} and systems with small transverse
hopping \cite{Bakrim10}.
Many-fermion lattice Hamiltonians can also be realized with ultracold atoms in optical
lattices, opening up new directions.
For example,
mixtures of more than two hyperfine states \cite{HonerkampHofstetter04}
and boson-mediated pairing on two-dimensional lattices \cite{Mathey06,Mathey07,Klironomos07}
have been investigated using fermionic $N$-patch methods.
Another promising development is the application of the functional RG to quantum spin
systems \cite{reuther1}. Here, an
auxiliary-fermion representation is used for the spins in generalized Heisenberg models,
and the functional RG can be formulated in terms of these fermions.
As important difference to systems of itinerant electrons, in the quantum spin system
the kinetic energy for the pseudo-fermions is zero and the interactions only depend on
one spatial or wavevector variable. This allows one to keep the full frequency dependence
of the self-energy and interaction vertex on the imaginary axis, in the usual truncation
where the six-point vertex is neglected.
The Katanin modification (\textcite{Katanin04a}, see also Section II.C.2)
of the flow hierarchy turns out to be crucial here.
If it is employed, the auxiliary-fermion functional RG describes the transitions
from N\'eel order to collinear order
through an intermediate paramagnetic phase in the $J_1$-$J_2$ spin-$1/2$ model on the
square lattice as function of $J_1/J_2$ in good agreement with numerical
approaches. Furthermore, similar systems on the triangular lattice \cite{reuther2} and
with longer-ranged couplings \cite{reuther3} were studied. The success of a relatively simple
truncation in such an intrinsically strongly coupled system is explained by these authors
in that the diagrams summed in this flow contain the leading
contributions in both $1/N$- and $1/S$-expansions plus particle-particle diagrams,
hence those contributions that are believed to be most important.
\section{SPONTANEOUS SYMMETRY BREAKING}
\label{sec:IV}
In many interacting Fermi systems a symmetry of the bare action
is spontaneously broken at sufficiently low temperatures and, in
particular, in the ground state.
In the fermionic flow equations, the common types of spontaneous
symmetry breaking such as magnetic order or superconductivity are
associated with a divergence of the effective two-particle
interaction at a finite scale $\Lam_c > 0$, in a specific
momentum channel.
In Sec.~III we discussed several examples for such divergences.
The truncation for the effective two-particle vertex leading to
the $N$-patch scheme described and used in Sec.~III is insufficient
to describe the symmetry-broken phase.
To continue the flow below the scale $\Lam_c$, an appropriate
order parameter has to be introduced.
There are two distinct ways of implementing spontaneous symmetry
breaking in the functional RG.
In one approach the fermionic flow is computed in presence of
a small (ideally infinitesimal) symmetry breaking term added to
the bare action, which is promoted to a finite order parameter
below the scale $\Lam_c$ \cite{Salmhofer04}.
A relatively simple truncation of the exact flow equation
captures spontaneous symmetry breaking in mean-field models
such as the reduced BCS model exactly, although the effective
two-particle interactions diverge.
Another possibility is to decouple the interaction by a bosonic
order parameter field, via a Hubbard-Stratonovich transformation,
and to study the coupled flow of the fermionic and order
parameter fields \cite{Baier04}.
In case of competing instabilities a reliable calculation based
on either of the above-mentioned routes to symmetry breaking is
quite involved.
For a rough estimate of order parameters and phase diagrams, one
may also neglect low energy fluctuations and combine flow
equations at high scales with a mean-field treatment at
low scales.
In this functional RG + mean-field approach, one stops the flow
of the effective two-particle interaction at a scale
$\Lam_{\rm MF} > \Lam_c$, that is, before it diverges.
The remaining low energy degrees of freedom are treated in mean-field
approximation, with a reduced effective interaction extracted from
the effective two-particle vertex $\Gam^{(4)\Lam_{\rm MF}}$.
In a first application of this ``poor man's'' approach to
symmetry breaking the interplay and possible coexistence of
antiferromagnetism and $d$-wave superconductivity in the (repulsive)
two-dimensional Hubbard model were studied \cite{Reiss07}.
As in any hybrid method, the results depend quantitatively on
the choice of the intermediate scale $\Lam_{\rm MF}$ (except for
mean-field models), and there is no unique criterion for this choice.
We now review the purely fermionic and the Hubbard-Stratonovich
approaches to spontaneous symmetry breaking in the functional
RG.
The methods will be explained for the case of a superconductor
as a prototype for continuous symmetry breaking, and the reader
is referred to the literature on applications involving other
order parameters.
\subsection{Fermionic flows}
\label{sec:fermi-flows}
The effective action $\Gam^{\Lam}$ obtained from the exact flow
equation or from symmetry-conserving truncations thereof
exhibits the same symmetries as the bare action $\cS$.
To analyze spontaneous symmetry breaking, one therefore has
to add a symmetry-breaking term $\delta\cS$ to the bare action
and compute the flow of $\Gam^{\Lam}$ in presence of this
term.
In case of spontaneous symmetry breaking an arbitrarily small
symmetry breaking term is promoted to a finite order parameter
at a scale $\Lam_c$, which survives until the end of the flow.
A crucial issue is then to find a managable truncation of the
exact flow equation which captures the essential features of
the flow into the symmetry broken phase.
This is non-trivial since the effective two-particle interactions
driving the symmetry breaking become large.
Indeed, truncations based on neglecting vertices $\Gam^{(2m)\Lam}$
with $m > 2$ in the hierarchy of flow equations fail miserably.
A benchmark for truncations is the requirement that they should
at least provide a decent solution for mean-field models.
This requirement is met by an approximation introduced by
\textcite{Katanin04a} to implement Ward identities in truncated
flow equations.
Katanin's truncation, which was described already in Sec.~II.C,
consists of two coupled flow equations for the self-energy
$\Sg^{\Lam}$ and the two-particle vertex $\Gam^{(4)\Lam}$,
see Fig.~\ref{fig:katanin}.
\begin{figure}
\centerline{\includegraphics[width = 8cm]{fig13.eps}}
\caption{Coupled flow equations for the self-energy and the
two-particle vertex determining the fermionic flow with
symmetry breaking.}
\label{fig:katanin}
\end{figure}
They are almost identical to the first two equations in the
hierarchy described in Sec.~II.C, with $\Gam^{(6)\Lam} = 0$,
but in the flow equation for $\Gam^{(4)\Lam}$ the single-scale
propagator $S^{\Lam}$ is replaced by
$\partial_{\Lam} G^{\Lam} = S^{\Lam} +
G^{\Lam} \partial_{\Lam} \Sg^{\Lam} G^{\Lam}$.
This modification takes tadpole contributions obtained from
contractions of the three-particle vertex $\Gam^{(6)\Lam}$ into
account.
It is easy to see that the Katanin truncation solves mean-field
models for symmetry breaking such as the Stoner model for
ferromagnetism or the reduced BCS model exactly
\cite{Salmhofer04}.
The exact self-energy in such models is given by the
Hartree-Fock term $\Sg = V G$ (schematically), where $V$ is the
bare interaction, and the two-particle vertex by a ladder sum
of the form
$\Gam^{(4)} = V (1 - GG V)^{-1}$.
These equations hold also in presence of a cutoff $\Lam$.
Applying $\Lam$-derivatives one finds immediately that
$\Sg^{\Lam}$ and $\Gam^{(4)\Lam}$ obey flow equations
of the (schematic) form
$\partial_{\Lam} \Sg^{\Lam} = \Gam^{(4)\Lam} S^{\Lam}$
and
$\partial_{\Lam} \Gam^{(4)\Lam} = \Gam^{(4)\Lam}
\partial_{\Lam}(G^{\Lam} G^{\Lam}) \Gam^{(4)\Lam}$,
which corresponds exactly to Katanin's truncation.
To be more specific, we now consider the case of singlet
superconductivity, where the continuous $U(1)$ symmetry
associated with charge conservation is spontaneously
broken, while spin-rotation invariance remains conserved.
Superconductivity can be induced by adding a term of the
form
\begin{equation}
\delta\cS = \sum_k \left[
\Delta_0(k) \psib_{\up}(k) \psib_{\down}(-k) +
\Delta_0^*(k) \psi_{\down}(-k) \psi_{\up}(k) \right] \; ,
\end{equation}
with a (generally complex) external pairing field
$\Delta_0(k)$, to the bare action.
It is convenient to use Nambu spinors $\Psi_{\alf}(k)$ and
$\Psib_{\alf}(k)$ with
$\Psib_{+}(k) = \psib_{\up}(k)$,
$\Psi_{+}(k) = \psi_{\up}(k)$,
$\Psib_{-}(k) = \psi_{\down}(-k)$,
$\Psi_{-}(k) = \psib_{\down}(-k)$.
The effective action as a functional of the Nambu fields,
truncated beyond two-particle terms, has the form
\begin{widetext}
\begin{eqnarray}
\Gam^{\Lam}[\Psi,\Psib] &=&
\Gam^{(0)\Lam} -
\sum_k \sum_{\alf_1,\alf_2} \Gam_{\alf_1 \alf_2}^{(2)\Lam}(k) \,
\Psib_{\alf_1}(k) \Psi_{\alf_2}(k) \nonumber \\
&+& \frac{1}{4} \sum_{k_1,\dots,k_4} \sum_{\alf_1,\dots,\alf_4}
\Gam_{\alf_1 \alf_2 \alf_3 \alf_4}^{(4)\Lam}(k_1,k_2,k_3,k_4) \,
\Psib_{\alf_1}(k_1) \Psib_{\alf_2}(k_2)
\Psi_{\alf_3}(k_3) \Psi_{\alf_4}(k_4) \; .
\label{Gam_nambu}
\end{eqnarray}
\end{widetext}
Due to spin-rotation invariance only terms with an equal
number of $\Psi$ and $\Psib$ fields contribute.
The Nambu propagator
$\bG^{\Lam} = (\bGam^{(2)\Lam})^{-1}$
can be written as a $2 \times 2$ matrix of the form
\begin{equation}
\bG^{\Lam}(k) =
\left( \begin{array}{cc}
G_{++}^{\Lam}(k) & G_{+-}^{\Lam}(k) \\
G_{-+}^{\Lam}(k) & G_{--}^{\Lam}(k)
\end{array} \right) =
\left( \begin{array}{cc}
G^{\Lam}(k) & F^{\Lam}(k) \\
F^{*\Lam}(k) & -G^{\Lam}(-k)
\end{array} \right) .
\label{G_nambu}
\end{equation}
It is instructive to discuss the flow of the superconducting
gap and the two-particle vertex for the reduced BCS model
\cite{Salmhofer04}, which is defined by an action of the form
\begin{eqnarray}
\cS[\psi,\psib] &=& \!
\sum_{k,\sg} (-ik_0 + \xi_{\bk}) \,
\psib_{\sg}(k) \psi_{\sg}(k)
\nonumber \\
&+& \! \sum_{k,k'} V(k,k') \,
\psib_{\up}(k) \psib_{\down}(-k)
\psi_{\down}(-k') \psi_{\up}(k') . \hskip 8mm
\end{eqnarray}
Note that the interaction is restricted to particles with
strictly opposite momenta and spins. It is well-known that
mean-field theory solves this model exactly in the
thermodynamical limit \cite{Haag62,Muehlschlegel62}.
The restricted momentum dependence of the bare interaction
carries over to similar restrictions for the effective
two-particle vertex $\Gam^{(4)\Lam}$ in Eq.~(\ref{Gam_nambu}).
Only two independent components appear, namely
\begin{eqnarray}
V^{\Lam}(k,k') &=& \Gam_{+-+-}^{(4)\Lam}(k,k',k',k) \; , \\
W^{\Lam}(k,k') &=& \Gam_{++--}^{(4)\Lam}(k,k',k',k) \; .
\label{VW}
\end{eqnarray}
The first component is a normal interaction between two
particles, and its initial value $V^{\Lam_0}(k,k')$ is the
bare interaction.
The second component is an anomalous term describing the
creation of four particles. It is initially zero, but is
generated by charge symmetry breaking terms in the course
of the flow.
Another anomalous term describing the destruction of four
particles is given by the complex conjugate of $W^{\Lam}(k,k')$.
The diagonal element of the Nambu self-energy vanishes for
the reduced BCS model, while the off-diagonal element is
given by the gap function $\Delta^{\Lam}(k)$.
For the special case of a momentum-independent s-wave
interaction $V$, the flow equations obtained from the
procedure described above are particularly simple.
Choosing a momentum-independent and real bare gap
$\Delta_0 > 0$, the flowing quantities $\Delta^{\Lam}$,
$V^{\Lam}$ and $W^{\Lam}$ are real and momentum-independent,
too. Their (exact) flow is given by
\begin{equation}
\frac{d}{d\Lam} \Delta^{\Lam} =
- (V^{\Lam} + W^{\Lam}) \sum_k \frac{d}{d\Lam} \left.
F^{\Lam} \right|_{\Delta^{\Lam} \; {\rm fixed}} \; ,
\end{equation}
where
$\frac{d}{d\Lam} F^{\Lam}|_{\Delta^{\Lam} \; {\rm fixed}}$
is the anomalous Nambu single-scale propagator, and
\begin{eqnarray}
\frac{d}{d\Lam} (V^{\Lam} \pm W^{\Lam}) &=&
- (V^{\Lam} \pm W^{\Lam})^2 \nonumber \\
&\times& \sum_k \frac{d}{d\Lam}
\left[ |G^{\Lam}(k)|^2 \mp |F^{\Lam}(k)|^2 \right]
\, . \hskip 8mm
\end{eqnarray}
A typical flow for an attractive bare interaction $V<0$
is shown in Fig.~\ref{fig:gapflow},
for two different choices of the bare gap $\Delta_0$.
\begin{figure}[ht]
\centerline{\includegraphics[width = 9cm]{fig14.eps}}
\caption{(Color online)
Flow for a reduced BCS model with a constant density
of states at zero temperature;
the band width is one and the bare interaction $V=-0.3$.
Left: Flow of the gap $\Delta^{\Lam}$; the thick line is
for a bare gap $\Delta_0 = 2.4 \cdot 10^{-4}$ and the thin
line for $\Delta_0 = 6 \cdot 10^{-8}$, in units of the band
width.
Right: Flow of the linear combinations $V^{\Lam} + W^{\Lam}$
(solid lines) and $V^{\Lam} - W^{\Lam}$ (dashed lines) of
normal and anomalous vertices. Thick lines are again for
$\Delta_0 = 2.4 \cdot 10^{-4}$ and thin lines for
$\Delta_0 = 6 \cdot 10^{-8}$.}
\label{fig:gapflow}
\end{figure}
The gap increases monotonically from the initial value
$\Delta_0$ upon lowering $\Lam$, and reaches a finite value
$\Delta \gg \Delta_0$ for $\Lam \to 0$.
A finite $\Delta_0$ regularizes the square-root singularity
in the gap flow at $\Lam = \Lam_c$.
The normal vertex $V^{\Lam}$ reaches a large negative value
at the critical scale $\Lam_c$, while the anomalous vertex
$W^{\Lam}$ becomes large and positive.
The linear combination $V^{\Lam} + W^{\Lam}$, which drives
the gap flow, is also strongly negative at $\Lam_c$, but it
saturates at a moderately negative value for $\Lam$ below
$\Lam_c$.
By contrast, $V^{\Lam} - W^{\Lam}$ decreases monotonically
and reaches a final value of order $1/\Delta_0$ for
$\Lam \to 0$, which diverges for $\Delta_0 \to 0$.
This divergence is the mean-field remnant of the Goldstone
mode associated with the broken continuous symmetry.
In the case of a discrete broken symmetry, the effective
interaction becomes large only at the critical scale, while
no large components remain for $\Lam \to 0$.
This has been exemplified in a study of the RG flow of a
mean-field model for a commensurate charge density wave
\cite{Gersch05}.
The performance of the Katanin truncation for models with full
(not reduced) interactions has not yet been fully explored,
since an accurate parametrization of the flowing vertex is
quite demanding.
However, the results obtained so far are encouraging.
Staying with superconductivity as a an example, the (Nambu)
vertex contains 16 components, most of them corresponding to
anomalous interactions. In addition to the anomalous terms
appearing already in the reduced BCS model, there are
anomalous interactions corresponding to the creation of three
particles and destruction of one particle, and vice versa
\cite{Salmhofer04,Gersch08}.
Making full use of spin-rotation invariance, all the Nambu
components can be actually expressed by only three independent
functions of momenta and frequencies \cite{Eberlein10}.
The main challenge is an adequate parametrization of the
(three-fold) momentum and frequency dependence, since
singularities associated with symmetry-breaking and the
Goldstone mode appear in the course of the flow.
Surprisingly, in a test case study for the weakly attractive
Hubbard model, a rather crude parametrization using the
$N$-patch discretization described in Sec.~III turned out to
yield a reasonable flow into the superconducting phase, with
results for the gap in good agreement with results obtained
earlier by other means \cite{Gersch08}.
This is encouraging, but the low energy fluctuations are
clearly not well described in such a parametrization.
To deal with the singular momentum and frequency dependence
in the Cooper channel (and possibly also in the forward
scattering channel), the channel decomposition devised by
\textcite{Husemann09a} seems very useful, since it allows one
to isolate singular dependences in functions of only one
momentum and frequency variable, similar to a description of
singular interactions by exchange bosons.
The channel decomposition has been formally extended already
to the superconducting state \cite{Eberlein10}, but a concrete
calculation beyond mean-field models has not yet been performed.
In systems with a first order phase transition one may miss
the symmetry broken phase if one tests only for local
stability of the symmetric phase by offering a small symmetry
breaking field, since the latter may be metastable.
However, one can escape from the metastable state by adding
a scale dependent symmetry breaking counterterm $R^{\Lam}$
to the effective action, which has to be choosen sufficiently
large at the beginning of the flow and fades out for
$\Lam \to 0$, such that the system is ultimately not modified.
Formally this is just another choice of regularization within
the general framework described in Sec.~II.B.
The counterterm method has been implemented for the exactly
soluble test case of a charge-density wave mean-field model
by \textcite{Gersch06}.
Popular approximations also beyond mean-field theory can be
retrieved from the functional RG by keeping a suitable
subset of contributions.
In particular, the Eliashberg theory for frequency-dependent
(usually phonon-induced) pairing interactions can be obtained
as an approximation to the exact flow equations both in the
symmetric \cite{Tsai05} and in the symmetry-broken state
\cite{Honerkamp05}.
This is achieved by keeping the Cooper channel for zero total
momentum and frequency and the crossed particle-hole channel
for zero transfer in the flow of the interaction, and the Fock
term for the self-energy.
\subsection{Flows with Hubbard-Stratonovich fields}
\label{sec:hs-flows}
Collective order parameter fluctuations associated with spontaneous
symmetry breaking in interacting many-body systems are often treated
by introducing an auxiliary order parameter field via a
Hubbard-Stratonovich transformation \cite{Popov87}.
A combination of the functional RG with the Hubbard-Stratonovich route
to spontaneous symmetry breaking in an interacting Fermi system was
first used by \textcite{Baier04}. They studied the formation of an
antiferromagnetic state in the repulsive two-dimensional Hubbard
model at half-filling and managed to recover the low-energy collective
behavior (described by a non-linear sigma model) from a truncated
set of coupled flow equations for the fermions and the order parameter
field.
In the following we describe the method for the case of a superfluid
phase, summarizing the work of several groups.
We consider an interacting continuum or lattice Fermi system with a
local attraction $V < 0$.
For continuum systems a suitable ultraviolet regularization is
necessary.
A local attraction can act only between particles with opposite
spin and leads to singlet pairing.
It is thus natural to decouple this interaction by a
Hubbard-Stratonovich transformation with a complex bosonic field
$\phi(q)$ corresponding to the bilinear composite of fermionic
fields $V \sum_k \psi_{\down}(-k) \psi_{\up}(k+q)$.
This leads to an action of the form
\begin{eqnarray} \label{S_super}
\cS[\phi,\psi,\psib] &=&
- \sum_{k,\sg} \psib_{\sg}(k)
\left(ik_0 - \xi_{\bk} \right) \psi_{\sg}(k)
\nonumber \\
&+& \frac{m_b}{2} \sum_q \phi^*(q) \phi(q)
\nonumber \\
&+& \sum_{k,q} \left[
\psib_{\up}(k+q) \psib_{\down}(-k) \phi(q) +
{\rm h.c.} \right] \, , \hskip 5mm
\end{eqnarray}
where $\phi^*$ is the complex conjugate of $\phi$ and $m_b = -1/V > 0$.
Spontaneous symmetry breaking can now be studied by using the flow
equation for the effective action $\Gam^{\Lam}[\phi,\psi,\psib]$
for coupled bosonic and fermionic fields derived in Sec.~II.G.
Relatively simple truncations capture several non-trivial fluctuation
effects.
Effective interactions beyond quartic order in the fields are
generally neglected. Also boson-fermion vertices beyond the order
appearing already in the bare action are discarded.
The truncations are usually formulated as an ansatz for the
effective average action
\begin{eqnarray}
\Gam_R^{\Lam}[\phi,\psi,\psib] &=&
\Gam^{\Lam}[\phi,\psi,\psib] - \mbox{regulator term}
\nonumber \\
&=& \Gam_b^{\Lam}[\phi] + \Gam_f^{\Lam}[\psi,\psib] +
\Gam_{bf}^{\Lam}[\phi,\psi,\psib] \, , \hskip 5mm
\end{eqnarray}
which obeys the initial condition $\Gam_R^{\Lam_0} = \cS$, see
Sec.~II.G.
The ansatz used for the bosonic part is guided by the usual
strategy of a double expansion in $\phi$ and gradients
(see, for example, \textcite{Tetradis94}):
\begin{equation} \label{4b:Gam_b}
\Gam_b^{\Lam}[\phi] =
\sum_x \, U_{\rm loc}^{\Lam}(\phi(x)) \, + \,
\mbox{gradient terms} \, ,
\end{equation}
where $x = (x_0,x_1,\dots,x_d)$ collects imaginary time and
real space coordinates.
Note that we use the same letter $\phi$ for the real space and
momentum space representations of the bosonic field.
The shape of the local potential $U_{\rm loc}^{\Lam}(\phi)$
depends on the scale. For $\Lam$ above a critical scale $\Lam_c$
it has the convex form
\begin{equation}
U_{\rm loc}^{\Lam}(\phi) =
m_b^{\Lam} |\phi|^2 + u^{\Lam} |\phi|^4 \, ,
\end{equation}
with a minimum at $\phi = 0$.
For $\Lam < \Lam_c$ the potential assumes a mexican hat shape
\begin{equation}
U_{\rm loc}^{\Lam}(\phi) =
u^{\Lam} \left[ |\phi|^2 - |\alf^{\Lam}|^2 \right]^2 \, ,
\end{equation}
with a circle of minima at $|\phi| = |\alf^{\Lam}|$, where
$\alf^{\Lam}$ is the (flowing) bosonic order parameter.
The regime $\Lam > \Lam_c$ is called the {\em symmetric}
regime. At $\Lam = \Lam_c$ the bosonic mass $m_b$ vanishes.
In the {\em symmetry-broken} regime, for $\Lam < \Lam_c$, the order
parameter $\alf^{\Lam}$ rises continuously from zero to a finite
value. Its flow can be computed by tracing the minimum of the
flowing potential $U_{\rm loc}^{\Lam}$ or, equivalently, by the
condition that the bosonic one-point vertex $\Gam_b^{(1)\Lam}$
vanishes.
For the gradient terms in $\Gam_b^{\Lam}[\phi]$ various choices
have been made.
The simplest one \cite{Birse05,Diehl07a,Krippa07} compatible with
the $U(1)$ symmetry has the form of an inverse bare propagator for
free bosons,
\begin{eqnarray} \label{gradterms}
&& \sum_x \left[
Z_b^{\Lam} \phi^*(x) \partial_{x_0} \phi(x) -
A_b^{\Lam} \phi^*(x) \nabla^2 \phi(x) \right] =
\nonumber \\
&& \sum_q \phi^*(q) \left[-i Z_b^{\Lam} q_0 + A_b^{\Lam} \bq^2
\right] \phi(q) \, ,
\end{eqnarray}
where $\nabla = (\partial_{x_1},\dots,\partial_{x_d})$.
For lattice fermions one may replace $\bq^2$ by a periodic
dispersion $\om_{\bq}$ which is proportional to $\bq^2$ only
at small $\bq$ \cite{Strack08}.
The term linear in $q_0$ is absent in particle-hole symmetric
systems \cite{Strack08}, such that contributions of order
$q_0^2$ become important.
Additional gradient terms have to be taken into account to
fully capture the effects of the Goldstone mode, as discussed
below.
The normal fermionic part of the effective action is usually
kept in its bare form, sometimes adjusted by renormalization
factors for the frequency and momentum dependences.
In the symmetry broken regime, an anomalous term is generated,
such that $\Gam_f^{\Lam}$ becomes
\begin{eqnarray} \label{ansatzGam_f}
\Gam_f^{\Lam}[\psi,\psib] &=&
- \sum_{k,\sg} \psib_{\sg}(k)
(iZ_f^{\Lam} k_0 - A_f^{\Lam} \xi_{\bk}) \psi_{\sg}(k)
\nonumber \\
&+& \sum_k \left[
\Delta^{\Lam}(k) \psib_{\up}(k) \psib_{\down}(-k) +
{\rm h.c.} \right] \, .
\end{eqnarray}
For a local interaction the $k$-dependence of the gap function
$\Delta^{\Lam}(k)$ is very weak and in simple truncations fully
absent.
Quartic terms corresponding to effective two-fermion interactions
are absent in the bare action by virtue of the Hubbard-Stratonovich
decoupling, but are generated again in the course of the flow.
These generated terms are neglected in lowest order truncations,
and sometimes they are treated by a dynamical decoupling procedure
called dynamical bosonization, see below.
For the effective boson-fermion interaction one also maintains
the bare form of a local 3-point function,
\begin{eqnarray}
\Gam_{bf}^{\Lam}[\phi,\psi,\psib] &=&
\sum_{k,q} g^{\Lam} \left[
\psib_{\up}(k+q) \psib_{\down}(-k) \phi(q) +
{\rm h.c.} \right]
\nonumber \\
&+& \mbox{anomalous terms} \; ,
\end{eqnarray}
with anomalous terms of the form $\psi\psi\phi$ and
$\psib\psib\phi^*$ contributing only in the symmetry-broken
regime. The coupling $g^{\Lam}$ is frequently referred to as
``Yukawa coupling''.
The anomalous terms in the boson-fermion interaction are usually
neglected. If taken into account, they remain indeed rather small
\cite{Strack08}.
Instead of using a $U(1)$-symmetric ansatz for the effective action,
one may also start from the hierarchy of flow equations for the
vertex functions and implement the $U(1)$-symmetry by Ward identities
\cite{Bartosch09}.
Even with the simple ansatz (\ref{gradterms}) for the bosonic
gradient terms, the effective action described above yields sensible
results not only at weak coupling, but actually in the entire regime
from BCS superfluidity to Bose Einstein condensation of tightly
bound pairs \cite{Diehl07a}.
In particular, the transition temperature $T_c$ increases
exponentially with the interaction in the weak coupling regime,
reaches a maximum, and finally saturates in the strong coupling
limit, as it should.
The bosonic interaction $u^{\Lam}$ and also the bosonic
renormalization factors $A_b^{\Lam}$ and $Z_b^{\Lam}$ vanish in
the limit $\Lam \to 0$ \cite{Birse05,Krippa07}.
This fluctuation effect reflects the drastic renormalization of
longitudinal order parameter correlations, which are well-known
from the interacting Bose gas in dimensions $d \leq 3$
(see, for example, \textcite{Pistolesi04}).
Note that for a nodeless gap function the low-energy behavior
of a fermionic superfluid is equivalent to that of an interacting
Bose gas, since fermionic excitations are fully gapped.
However, with the simple ansatz (\ref{gradterms}) the transverse
order parameter fluctuations corresponding to the Goldstone mode
are also strongly renormalized, which is not correct.
To distinguish between longitudinal and transverse fluctuations,
one may fix the phase of the order parameter $\alf^{\Lam}$ such
that $\alf^{\Lam}$ is real, decompose the complex order parameter
field in real and imaginary parts $\phi(q) = \sg(q) + i\pi(q)$
with $\sg(-q) = \sg^*(q)$ and $\pi(-q) = \pi^*(q)$, and introduce
different renormalization factors for $\sg$ and $\pi$ fields
\cite{Pistolesi04}.
Using this decomposition, the correct infrared behavior was
obtained by \textcite{Strack08} where, however, the cancellation
of singular contributions to the renormalization factors for
the transverse $\pi$ fields was implemented by hand.
To capture this cancellation intrinsically, one has to include
an additional $U(1)$ symmetric gradient term of the form
$[\sg(\partial_{x_0},\nabla)\sg + \pi(\partial_{x_0},\nabla)\pi]^2$
\cite{Tetradis94,Strack09}.
The fermionic flow based on the Katanin truncation described in
Sec.~IV.A reproduces the exact solution of the reduced BCS model
(and other mean-field models).
Within the truncation described above, the bosonized flow yields
a reasonable solution without artificial features, but the gap
comes out a bit too small.
The reason for this is the truncation of $U_{\rm loc}^{\Lam}(\phi)$
at quartic order.
To recover the exact solution, one has to keep all orders in $\phi$
\cite{Strack08}.
The ansatz (\ref{ansatzGam_f}) for $\Gam_f^{\Lam}[\psi,\psib]$
neglects the generation of fermionic interactions by the flow.
In particular, quartic (two-fermion) interactions are
generated by box diagrams with four boson-fermion vertices.
These terms contain contributions from particle-hole fluctuations
which, among other effects, lead to a significant reduction of
the transition temperature.
The (re-)generated two-fermion interaction can be decoupled
at each step in the flow by a procedure called dynamical
bosonization \cite{Gies02,Gies04,Floerchinger09d}.
A general two-fermion interaction cannot be decoupled (exactly)
by a single Hubbard-Stratonovich field, such that several fields
may be needed to obtain accurate results.
Dynamical bosonization was used to include effects from
particle-hole fluctuations in attractively interacting Fermi
systems by \textcite{Floerchinger08a}.
Following the work of \textcite{Baier04} on the repulsive
Hubbard model at half-filling,
functional RG flow equations with Hubbard-Stratonovich fields
were also applied to the Hubbard model away from half-filling.
Commensurate and incommensurate antiferromagnetic fluctuations
were investigated \cite{Krahl09a}.
More importantly, it was clarified how the generation of
$d$-wave pairing from antiferromagnetic fluctuations can be
captured by a bosonized flow \cite{Krahl09b}, and the flow
was continued into the symmetry broken phase, with coupled
order parameter fields describing antiferromagnetism and
$d$-wave superconductivity \cite{Friederich10, Friederich11}.
Compared to the purely fermionic RG described in Sec.~IV.A,
the treatment of order parameter fluctuations is facilitated
considerably by the Hubbard-Stratonovich field.
On the other hand, fluctuation effects associated with other
channels (the particle-hole channel in case of superfluidity)
look more complicated.
For systems with competing instabilities the choice of an
adequate Hubbard-Stratonovich field becomes problematic,
since the fermionic interaction can be decoupled in different
ways, which, in combination with truncations, may lead to
ambiguities in the results.
In general, several Hubbard-Stratonovich fields must be used,
and the analysis done in Section \ref{sec:III} indicates which ones
are the most important. The decomposition of the interaction in
\cite{Husemann09a} allows to switch to Hubbard-Stratonovich fields
after the fermionic flow has been performed down to a certain scale,
and may thus be used to combine the two flow representations.
\section{QUANTUM CRITICALITY}
\label{sec:V}
Instabilities of the normal metallic state lead to a rich variety
of quantum phase transitions \cite{Sachdev99} in the ground state
of interacting electron systems, which can be tuned by a control
parameter such as pressure, doping, or a magnetic field.
Most interesting are {\em continuous} transitions which lead to
quantum critical fluctuations \cite{Belitz05}.
Near a quantum critical point (QCP) electronic excitations are
strongly scattered by order parameter fluctuations such that
Fermi liquid theory breaks down \cite{Vojta03,Loehneysen07}.
Quantum critical fluctuations are therefore frequently invoked
as a mechanism for non-Fermi liquid behavior observed in strongly
correlated electron compounds.
Quantum phase transitions in interacting Fermi systems are
traditionally described by an effective order parameter theory
pioneered by \textcite{Hertz76} and \textcite{Millis93}.
An order parameter field $\phi$ is introduced by a
Hubbard-Stratonovich decoupling of the fermionic interaction,
and the fermionic fields are subsequently integrated out.
The resulting effective action for the order parameter is
truncated at quartic order and analyzed by standard scaling
and RG techniques.
However, more recent studies revealed that the Hertz-Millis
approach is not always applicable, especially in low-dimensional
systems \cite{Belitz05,Loehneysen07}.
Since electronic excitations in a metal are gapless, integrating
out the electrons can lead to singular interactions in the
effective order parameter action which cannot be approximated
by a local quartic term.
The nature of the problem was identified and essential aspects
of its solution were presented first for disordered ferromagnets
by \textcite{Kirkpatrick96}, and later elaborated on by
\textcite{Belitz01a,Belitz01b}.
For clean ferromagnets, \textcite{Belitz97} showed that
Hertz-Millis theory breaks down, and no continuous quantum
phase transition can exist, in any dimension $d \leq 3$;
the transition is generically of first order \cite{Belitz99}.
The Hertz-Millis approach was also shown to be invalid for
the quantum antiferromagnetic transition in two dimensions
\cite{Abanov03,Abanov04,Metlitski10b}.
In that case a continuous transition survives, but the QCP
becomes non-Gaussian.
Applications of the functional RG to quantum phase transitions
and quantum criticality in interacting Fermi systems have
appeared only recently.
In Sec.~V.A we explain how the Hertz-Millis theory fits into
the functional RG framework and we review some extensions
relying on an effective order parameter action truncated at
quartic or hexatic order.
An application of a non-perturbative truncation, where all
orders in $\phi$ are (and must be) kept, is discussed in
Sec.~V.B.
Finally, in Sec.~V.C we briefly discuss the possibility to
study coupled flow equations for fermions and their
critical order parameter fluctuations in the functional RG
framework, and we refer to first steps in this direction.
\subsection{Hertz-Millis theory}
\label{sec:hertz}
In his seminal work on quantum phase transitions in metallic
electron systems, \textcite{Hertz76} proposed to decouple the
electron-electron interaction by introducing an order parameter
field $\phi$ via a Hubbard-Stratonovich transformation.
The resulting action is quadratic in the fermionic variables
$\psi$ and $\psib$, which can therefore be integrated out.
One thus obtains an effective action which depends only on $\phi$.
Truncating at quartic order in $\phi$, and discarding
irrelevant momentum and frequency dependences (in the sense
of standard power counting), leads to the Hertz action.
\begin{eqnarray} \label{S_hertz}
\cS[\phi] &=& \cS^{(0)} + \sum_q \phi(-q) \left[
A \bq^2 + Z \frac{|q_0|}{|\bq|^{z-2}} \right] \phi(q)
\nonumber \\
&+& \sum_x U_{\rm loc}(\phi(x)) \, ,
\end{eqnarray}
where $\cS^{(0)}$ is a field-independent term, and
\begin{equation}
U_{\rm loc}(\phi) = r \phi^2 + u \phi^4 \, .
\end{equation}
We write our equations for the case of a real scalar order
parameter for simplicity, using (again) the same letter
$\phi$ for the real and momentum space representations
of the field.
Except for the frequency dependence, the action has the
form of a $\phi^4$ theory for thermal phase transitions.
The frequency dependent term stems from low-energy particle-hole
excitations.
Here the dynamical exponent $z$ is an integer number $\geq 2$
depending on the type of transition.
Tuning the parameter $r$ one can approach the phase
transition, in particular the quantum phase transition at
$T = 0$.
The action (\ref{S_hertz}) has been analyzed by standard scaling
and RG techniques.
Due to the frequency dependence, the scaling behavior at the
QCP corresponds to a system with an effective dimensionality
$d_{\rm eff} = d + z$, where $d$ is the spatial dimension
\cite{Hertz76}.
As a consequence, the QCP appears to be Gaussian in three-
and even in two-dimensional systems.
An important insight by \textcite{Millis93} was that the
$\phi^4$ term in the action is nevertheless crucial to obtain
the correct temperature dependences near the QCP.
He derived the temperature dependence of the correlation
length $\xi$ and other quantities by using a perturbative
RG with a mixed momentum and frequency cutoff.
From a functional RG perspective, Millis' scaling theory
can be viewed as a simple truncation of the effective average
action $\Gam_R^{\Lam}[\phi]$ evolving from $\cS[\phi]$,
namely
\begin{eqnarray} \label{Gam_millis}
\Gam_R^{\Lam}[\phi] &=& \Gam^{(0)\Lam} +
\sum_q \phi(-q) \left[
A^{\Lam} \bq^2 + Z^{\Lam} \frac{|q_0|}{|\bq|^{z-2}} \right]
\phi(q) \nonumber \\
&+& \sum_x U_{\rm loc}^{\Lam}(\phi(x)) \, ,
\end{eqnarray}
where
\begin{equation}
U_{\rm loc}^{\Lam}(\phi) =
r^{\Lam} \phi^2 + u^{\Lam} \phi^4 \, ,
\end{equation}
and $\Lam$ parametrizes a mixed momentum and frequency cutoff.
The flow equations for the parameters in Eq.~(\ref{Gam_millis})
are obtained by inserting $\Gam_R^{\Lam}[\phi]$ in the exact
functional flow equation (\ref{floweqGamRb}) and comparing
coefficients.
Due to the local form of the $\phi^4$ interaction, the
self-energy is momentum and frequency independent such that
the parameters $A^{\Lam}$ and $Z^{\Lam}$ remain invariant.
The flow of $u^{\Lam}$, which is driven by a contribution of
order $(u^{\Lam})^2$, is important only in the marginal case
$d + z = 4$ and near the thermal phase transition at $T_c > 0$.
Hence, most of Millis' results on the region around the QCP
in the phase diagram are based on an analysis of the flow of
$r^{\Lam}$ and the thermodynamic potential
$\Omega^{\Lam} = T \Gam^{(0)\Lam}$
(for a review, see \textcite{Loehneysen07}).
Various extensions of Millis' analysis were derived within
the functional RG framework.
In particular, an extension to the symmetry-broken phase
was formulated, for cases where the broken symmetry is
{\em discrete} and does not gap out the fermionic excitations
\cite{Jakubczyk08}.
One such case is a nematic transition
driven by a Pomeranchuk instability of interacting electrons
on a lattice, where the discrete point-group symmetry of the
lattice is spontaneously broken \cite{Fradkin10}.
The symmetry-broken regime was described by the ansatz
(\ref{Gam_millis}) for $\Gam_R^{\Lam}[\phi]$, with a quartic
local potential which has a minimum away from zero:
\begin{equation}
U_{\rm loc}^{\Lam}(\phi) =
u^{\Lam} \left[ \phi^2 - (\phi_0^{\Lam})^2 \right]^2 \, .
\end{equation}
The resulting flow equations were used to compute $T_c$ and
the Ginzburg temperature $T_G^<$ below $T_c$ as a function
of the control parameter $r$.
To access the non-Gaussian thermal critical regime near $T_c$,
it is crucial to take the flow of the quartic coupling
$u^{\Lam}$ into account.
The parameters $A^{\Lam}$ and $Z^{\Lam}$ are now scale dependent, too.
While $Z^{\Lam}$ remains almost invariant, the flow of
$A^{\Lam}$ is important near $T_c$ and gives rise to an
anomalous scaling dimension.
A main result of the calculation was that the leading
$r$-dependence of $T_c$ is the same as that
of the Ginzburg temperatures below and above $T_c$ (the
latter was calculated by \textcite{Millis93}), but a fairly
large Ginzburg region opens in two dimensions
\cite{Jakubczyk08,Bauer11}.
In another extension a $\phi^6$-interaction was included
in $U_{\rm loc}^{\Lam}$ to study a possible change of the order
of the transition by fluctuations \cite{Jakubczyk09a},
as well as quantum tricritical points in metals
\cite{Jakubczyk10}.
Note that
the extensions mentioned above are based on perturbative
truncations resulting in flow equations with few running
couplings, which could have been obtained also by more
conventional RG methods.
\subsection{Full potential flow}
\label{sec:fullpot}
We now review an application to a problem where the effective
action cannot be truncated at any finite order in $\phi$,
such that the possibility to make non-perturbative truncations
becomes crucial \cite{Jakubczyk09b}.
The problem arises when asking how a nematic transition
caused by a $d$-wave Pomeranchuk instability in two dimensions
is affected by fluctuations.
Such a transition can be modelled by tight-binding electrons
on a square lattice with an attractive $d$-wave forward scattering
interaction \cite{Metzner03}:
\begin{equation} \label{f-model}
H = \sum_{\bk} \eps_{\bk} n_{\bk} +
\frac{1}{2L} \sum_{\bk,\bk',\bq} f_{\bk\bk'}(\bq) \,
n_{\bk}(\bq) \, n_{\bk'}(-\bq) \; ,
\end{equation}
where $n_{\bk}(\bq) = \sum_{\sg}
c^{\dag}_{\bk-\bq/2,\sg} c^{\phantom\dag}_{\bk+\bq/2,\sg}$
and $L$ is the number of lattice sites.
The interaction has the form
\begin{equation}
f_{\bk\bk'}(\bq) = - g(\bq) d_{\bk} d_{\bk'} \; ,
\end{equation}
where $d_{\bk} = \cos k_x - \cos k_y$ is a form factor with
$d_{x^2-y^2}$ symmetry. The coupling function $g(\bq) \geq 0$
has a maximum at $\bq={\bf 0}$ and is restricted to small momentum
transfers by a cutoff $\Lam_0$.
For sufficiently large $g = g({\bf 0})$ the interaction drives a
$d$-wave Pomeranchuk instability leading to a nematic state with
broken orientation symmetry, which can be described by the
order parameter
\begin{equation}
\phi = \frac{g}{L} \sum_{\bk} d_{\bk} \bra n_{\bk} \ket
\; .
\end{equation}
In the plane spanned by the chemical potential and temperature
a nematic phase is formed below a dome-shaped transition line
$T_c(\mu)$ with a maximal transition temperature near Van Hove
filling. In mean-field theory, the phase transition is usually
first order near the edges of the transition line, that is,
where $T_c$ is relatively low, and second order at the roof of
the dome \cite{Kee03,Khavkine04,Yamase05}.
Introducing an order parameter field via a Hubbard-Stratonovich
transformation, integrating out the fermions, and keeping only
the leading momentum and frequency dependences for small $\bq$
and small $q_0/|\bq|$ leads to a Hertz-type action $\cS[\phi]$
of the form (\ref{S_hertz}), with $z=3$ and a local potential
given by the mean-field potential
\begin{equation}
U_{\rm loc}(\phi) =
\frac{\phi^2}{2g} - \frac{2T}{L} \sum_{\bk}
\ln\left( 1 + e^{-(\eps_{\bk} - \phi d_{\bk} - \mu)/T}
\right) \; .
\end{equation}
At low temperatures,
the coefficients of a Landau expansion of $U_{\rm loc}(\phi)$ in
powers of the field,
$U(\phi) = a_0 + a_2 \phi^2 + a_4 \phi^4 + \dots$,
are typically negative for all exponents $2m \geq 4$.
Hence, $\cS[\phi]$ and consequently also the effective action
$\Gam_R^{\Lam}[\phi]$ cannot be truncated at any finite order
in $\phi$.
Fortunately, for bosonic fields the functional RG allows also
for non-perturbative approximations, where one expands only in
gradients, and not in powers of $\phi$ \cite{Berges02}.
In particular, one can use the ansatz (\ref{Gam_millis}) for
$\Gam_R^{\Lam}[\phi]$ without expanding the local potential
$U_{\rm loc}^{\Lam}(\phi)$.
The flow of $U_{\rm loc}^{\Lam}(\phi)$ is then determined by
a {\em partial} differential equation which contains a second
derivative of the potential with respect to $\phi$.
In Fig.~15 an exemplary plot of the evolution of the flowing
effective potential $U_{\rm loc}^{\Lam}(\phi)$ is shown for
$\Lam$ ranging from the ultraviolet cutoff
$\Lam_0 = e^{-1} \approx 0.37$ to the final value $\Lam = 0$.
The flow has been computed for electrons on a square lattice
with nearest neighbor hopping $t=1$, next-to-nearest neighbor
hopping $t'=-\frac{1}{6}$, and a coupling strength $g=0.8$.
The initial (mean-field) potential has a minimum at
$\phi_0 = 0.112$.
The final potential exhibits spontaneous symmetry breaking
with an order parameter $\phi_0 = 0.102$.
Fluctations shift $\phi_0$ toward a slightly smaller value
compared to the mean-field solutions.
The flat shape of $U_{\rm loc}^{\Lam}(\phi)$ for
$\phi \leq \phi_0$ at $\Lam = 0$ is imposed by the
convexity property of the grand canonical potential.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7cm]{fig15.eps}
\caption{(Color online)
Flowing effective potential $U_{\rm loc}^{\Lam}(\phi)$
for various values of $\Lam$ between $\Lam_0 = e^{-1}$
and $0$, at $\mu = -0.78$ and $T = 0.05$ \cite{Jakubczyk09b}.}
\end{center}
\label{fig:nematicflow}
\end{figure}
The transition line between normal and symmetry-broken phases
is shown in Fig.~16 for two choices of $\Lam_0$.
Compared to the corresponding mean-field result, the
transition temperature is suppressed, with a larger reduction
for larger $\Lambda_0$ (corresponding to a larger phase space
for fluctuations).
For $\Lambda_0 = 1$ the transition is continuous down to $T=0$,
leading to quantum critical points at the edges of the nematic
dome.
Increasing $\Lambda_0$ further (or reducing $g$), one may even
eliminate the nematic phase completely from the phase diagram
\cite{Yamase11}.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig16.eps}
\caption{(Color online) Critical temperatures versus chemical
potential for $\Lambda_0 = e^{-1}$ (larger dome with dots and
crosses) and $\Lambda_0 = 1$ (smaller dome with dots).
The mean-field transition line is also shown for comparison
\cite{Jakubczyk09b}.}
\end{center}
\label{fig:nematicphase}
\end{figure}
\vspace*{5mm}
\subsection{Coupled flow of fermions and order parameter fluctuations}
\label{sec:coupledflow}
There are various systems where integrating out the electrons leads
to an effective order parameter action with singular interactions
which cannot be approximated by a local coupling
\cite{Belitz05,Loehneysen07}.
In such cases it can be advantageous to keep the electrons in the
action, treating the coupled system consisting of electrons and their
order parameter fluctuations.
Several coupled boson-fermion systems exhibiting quantum criticality
have already been analyzed by various methods; see, for example,
\textcite{Vojta00,Belitz01a,Belitz01b,Abanov03,Rech06,Kaul08,Huh08,
Metlitski10a,Metlitski10b}.
The interplay of bosonic and fermionic infrared singularities
at the quantum critical point poses an interesting problem.
The functional RG for coupled bosons and fermions described in
Sec.~II.G provides a suitable framework to study such problems.
So far, it has not been applied to quantum phase transitions
in metallic electron systems.
However, encouraging works on relativistic field-theoretic models
with gapless bosons and fermions have already appeared.
For example, functional RG flow equations have been used to study
the Gross-Neveu model \cite{Rosa01},
quantum electrodynamics \cite{Gies04a},
and supersymmetric Wess-Zumino models \cite{Gies09}.
In the context of condensed matter physics, a toy model for a
semimetal-to-superfluid quantum phase transition has been studied
by coupled flow equations for the electrons and the superfluid
order parameter \cite{Strack10,Obert11}.
In dimensions $d<3$ the fermions and the order parameter
fluctuations acquire anomalous scaling dimensions at the QCP
of that model, leading to non-Fermi liquid behavior and non-Gaussian
criticality.
It will be interesting to devise suitable truncations of the
coupled boson-fermion flow equations for magnetic and nematic
quantum phase transitions in low-dimensional metallic systems,
where many open questions need to be clarified.
\section{CORRELATION EFFECTS IN QUANTUM WIRES AND QUANTUM DOTS}
\label{sec:VI}
As our last application of the functional RG to correlated fermion
systems we discuss many-body effects in quantum wires and dots.
The focus is on transport through such systems which are coupled
to two or more semi-infinite leads.
While in many of the above applications it was crucial
to devise an approximation scheme in which the flow of the two-particle
vertex (the effective two-particle interaction) was properly described,
in the ones reviewed in this section the physics is dominated
by the flow of the self-energy.
We start out with a
brief discussion of quantum transport through a region containing
correlations in Sec.~\ref{subsec:QT}. To study transport beyond the
linear response regime the functional RG was recently extended
to Fermi systems out of equilibrium. In Sec.~\ref{subsec:nonequfRG}
we review the main steps of this generalization.
After discussing the most elementary example of linear transport through an
inhomogeneous correlated quantum wire -- a chain with a single local
impurity -- in Secs.~\ref{subsec:LL1}-\ref{subsubnum}, we show how the
functional RG can be used (i) to describe transport on all energy scales
for more complex setups (Sec.~\ref{resotunsec}), (ii) to identify
unconventional low-energy fixed points of such systems
(Sec.~\ref{Yjunctionsec}), and (iii) to study finite bias
non-equilibrium transport (Sec.~\ref{subsecnonequiwire}).
As an example of the application of the functional RG to quantum dots, in
Sec.~\ref{subsubchargefluc} we consider an interacting chain of only
three lattice sites corresponding to a dot.
\subsection{Quantum transport}
\label{subsec:QT}
Experimental progress has made it possible to measure transport
through mesoscopic regions like one-dimensional (1D)
quantum wires of up to micrometer length and ``zero-dimensional''
quantum dots. The experiments provide evidence for correlation effects
\cite{Deshpande10,Hanson07}.\footnote{Correlation effects in effectively
1D electronic systems are also studied using photoemission. For a recent
review, see \textcite{Grioni09}.}
It is a great theoretical challenge to describe transport
when correlations in the mesoscopic system are important.
Usually the leads to which this correlated region is connected
are modeled as non-interacting.
A general formal expression in terms of Keldysh Green functions
for the current $I$ through an {\it interacting} system coupled
to two leads (indices $L,R$) in the stationary state
was presented by \textcite{Meir92}. Either for specific
models or applying certain approximations to the two-particle
interaction for each channel $\zeta$ it can be brought into a
Landauer-B\"uttiker type form \cite{Landauer57,Buettiker86}
\begin{equation}
\label{Landauer}
I_{\zeta} = \frac{1}{2 \pi} \int {\mathcal T}_{\zeta}(\epsilon,T,V_b)
\left[f_L(\epsilon)-f_R(\epsilon)\right] d\epsilon \; ,
\end{equation}
where we set $e=1=\hbar$ such that the conductance quantum per channel
is given by $1/(2 \pi)$.
Here ${\mathcal T}_{\zeta}$ is an {\it effective} transmission
probability, $V_b=\mu_L-\mu_R$ the bias voltage, and $f_{L/R}$
are Fermi functions with the chemical potentials of
the left and right leads $\mu_{L/R}$.
For the transport through a non-interacting system ${\mathcal T}_{\zeta}$
is the single-particle transmission probability.
The goal is to compute ${\mathcal T}_{\zeta}$ in the presence of correlations.
Here truncations of the functional RG flow equations which
lead to {\it frequency independent} self-energies are considered.
In this case using Eq.\ (\ref{Landauer}), with ${\mathcal T}_{\zeta}$ being
proportional to the ``contact-to-contact'' matrix element
of the (retarded) one-particle Green function (see below) does
not present an additional approximation as current vertex
corrections vanish \cite{Oguri01,Enss05a}.
In this approximation the two-particle interaction
affects the transport only via the renormalized self-energy which acts
as an additional, $T$ and $V_b$ dependent
scattering potential
on non-interacting electrons \cite{Oguri01,Langer61}.
For the {\it linear} conductance $g_{\zeta}(T)$ the transmission probability
enters only at zero bias $V_b = 0$,
\begin{equation}
\label{linearcond}
g_{\zeta}(T) = \frac{1}{2 \pi}\int {\mathcal T}_{\zeta}(\epsilon,T,0)
\left ( -\frac{\partial f}{\partial \epsilon}\right )d\epsilon \; ,
\end{equation}
i.e.\ $ {\mathcal T}_{\zeta}(\epsilon,T,0) $ is an {\it equilibrium} property.
At zero temperature Eq.\ (\ref{linearcond}) simplifies further to
$g_{\zeta}(0) = {\mathcal T}_{\zeta}(\mu,0,0)/(2 \pi) $.
\subsection{Functional RG in non-equilibrium}
\label{subsec:nonequfRG}
Recently, the functional RG approach was extended to study steady state
non-equilibrium transport through quantum wires and dots in the presence
of a finite bias voltage given by the difference of the chemical potentials
of the left and right leads $V_b = \mu_L-\mu_R$
\cite{Jakobs03,Gezzi07,Jakobs07a,Jakobs10a,Karrasch10a,Jakobs10b,Karrasch10b}.
The basic idea behind this extension is the use of real time or real frequency
Green functions on the Keldysh contour \cite{Rammer86} instead of Matsubara
Green functions.
As usual in diagrammatic approaches based on Keldysh Green functions one
assumes that the initial statistical operator does not contain any correlations
\cite{Rammer86}.
One can then either use a functional integral formulation \cite{Kamenev04}
of the non-equilibrium many-body problem \cite{Gezzi07} or a purely
diagrammatic approach \cite{Jakobs03,Jakobs07a} to derive the (same) flow
equations for the self-energy and higher order vertex functions.
Although the method allows to work with two-time Green functions and to study
transient dynamics, in the current implementation of the functional RG in
non-equilibrium for fermions the system is {\it assumed} to be in the steady
state. For interacting bosons functional RG was also used to study dynamics
\cite{Gasenzer08,Kloss11}.
On a technical level and compared to the equilibrium Matsubara functional RG
in the steady state the Keldysh structure only leads to an additional index
(the so-called Keldysh index $\pm$ referring to the upper and lower branch of
the Keldysh contour) to be added to the set of quantum numbers.
One of the main challenges of the non-equilibrium functional RG is to devise
cutoff schemes which do not violate causality and general Kubo-Martin-Schwinger
(KMS) relations \cite{Jakobs10b,Jakobs10c} after truncation of the infinite
hierarchy of flow equations. For a general cutoff fulfilling the requirements
discussed in Sect. \ref{sec:flowparameters} it is only guaranteed that
causality and KMS relations hold up to the truncation order, e.g.~first order
for the level-1 truncation. In fact, an
infrared (real) frequency cutoff similar to Eq.\ (\ref{cutfun}) violates
causality in second order \cite{Jakobs03,Gezzi07}. Its breaking constitutes
a severe problem as relations connecting the Keldysh contour matrix elements
of the Green function cannot be used. In particular,
one cannot rotate from the $G^{a,b}$ Green function, with Keldysh indices
$a,b = \pm$, to retarded, advanced, and Keldysh Green functions as it is usually
done \cite{Rammer86}. The momentum cutoff scheme used in other sections of this
review avoids this problem but is not suitable for models with broken translational
invariance. We will return to this issue and discuss appropriate cutoff schemes
when reviewing applications of non-equilibrium functional RG to transport through
wires in Sec.~\ref{subsecnonequiwire} and dots in Sec.~\ref{subsubchargefluc}.
Besides its ability to be applied to non-equilibrium problems the real time
(or real frequency) Keldysh functional RG has a distinct advantage even in
equilibrium: the analytic continuation from Matsubara to real frequencies
can be avoided. In fact, computing the real frequency dependence of observables
obtained by a numerical solution of the Matsubara RG flow equations
truncated on a level which includes a flowing two-particle vertex and self-energy
of initially unknown frequency structure presents a formidable task
\cite{Karrasch08a,Karrasch10d} as it is known from other imaginary time quantum
many-body methods such as quantum Monte-Carlo.
This advantage was utilized in the real time functional RG study of the
single-impurity Anderson model \cite{Jakobs10a,Jakobs10b} in which a frequency
dependent two-particle vertex and self-energy (complete level-2 truncation)
were kept (for the imaginary time analog of this study, see Sec. \ref{subsubspinfluc},
\textcite{Hedden04}, and \textcite{Karrasch08a}), and can be expected to be useful
also in functional RG studies of other models.
\subsection{Impurities in Luttinger liquids}
The metallic state of correlated fermions in one dimension is a non-Fermi liquid.
It falls into the Luttinger liquid (LL) class \cite{Haldane80}.
This state of matter is characterized by a power-law decay of space-time
correlation functions with interaction dependent exponents
\cite{Luttinger63,Mattis65,Luther74,Mattis74,Giamarchi04,Schoenhammer05} and
spin-charge separation \cite{Dzyaloshinskii74,Meden92,Voit93}.
In particular, after Fourier transforming,
the $2 k_F$-component of the density-density response function
shows a power-law divergence \cite{Mattis74,Apel82} for repulsive interactions,
instead of a logarithmic one for non-interacting electrons (Lindhard
function in 1D). This indicates that
any local inhomogeneity
with a non-vanishing $2 k_F$ component strongly affects the
low-energy physics and thus the transport
characteristics of a LL at low temperatures.
Perturbation theory is insufficient to describe the interaction effects
as it fails to capture the RG flow of the inhomogeneity appearing even to
first order in the interaction and higher order diagrams diverge.
As will become clear below the functional RG approach captures the
impurity flow and does {\it not} require a simplified modeling of the
inhomogeneity.
\subsubsection{A single local impurity--the local sine-Gordon model}
\label{subsec:LL1}
To understand the effect of a single impurity on the low-energy
physics of a {\it spinless, infinite} LL (absence of non-interacting leads),
bosonization was used first (for a review of this method see
e.g.~\textcite{vonDelft98}, \textcite{Giamarchi04}, and \textcite{Schoenhammer05}).
Within this approach the Fourier components of fermionic density
operators are split into their chiral parts which
obey Bose commutation relations in the low energy
subspace \cite{Tomonaga50}. In this limit the
kinetic energy and the two-particle interaction can be written as
bilinears in these operators \cite{Tomonaga50}, while a single-particle
scattering term (impurity) generally
takes a complicated form in the bosonic degrees of freedom. It
simplifies if only the pure forward scattering $\tilde V(0)$
and backward scattering $\tilde V(2k_F)$ contributions
are kept. Using bosonization to obtain results beyond this
approximate modelling of the impurity is rather involved.
The forward
scattering term is linear in the bosons and can easily be treated
leading to a phase shift. This is irrelevant for the physics described
in the following.
The backward scattering term consists of the cosine of a local
bosonic field and the resulting Hamiltonian is known
from field theory as the local sine-Gordon (LSG) model. One can
either use the exact Bethe ansatz solution \cite{Fendley95} of
this model or a bosonic RG which is perturbative in $\tilde V(2k_F)$
\cite{Kane92,Furusaki93a} to obtain
analytical results. Alternatively numerical methods can be
applied to the LSG model \cite{Moon93,Egger95}.
This led to a complete picture of the RG scaling of $\tilde V(2k_F)$
which has direct consequences for the linear conductance.
The RG flow connects two fixed points,
the perfect chain fixed point with conductance $g=K/(2 \pi)$
and the open chain fixed point with $g=0$.
In 1D spinless fermion systems only a single transport channel
exists and we thus suppress the channel index $\zeta$ from now on.
Here $K>0$ denotes the so-called
LL parameter which depends on the underlying model of the
quantum wire and its parameters such as the strength and range of
the two-particle interaction as well as
the band filling. Independently of
the model considered $K<1$ for repulsive interactions and $K>1$ for
attractive ones, while the
non-interacting case corresponds to $K=1$.
The corrections to the fixed point conductances are
power laws $s^{\gamma_{p/o}}$ with the infrared energy scale $s$
(e.g. the temperature $T$ or the energy cutoff $\Lambda$ in a
RG procedure).
The exponents are {\it independent}
of the bare impurity strength and given by $\gamma_p=2(K-1)$ and
$\gamma_o=2(1/K-1)$, respectively.
The sign of the scaling exponents implies that the open chain fixed point
is stable for repulsive interactions and unstable for attractive ones.
The opposite holds for the perfect chain fixed point.
These insights confirmed earlier indications that
impurities with a backscattering
component strongly alter the low-energy physics of LLs
with repulsive interactions \cite{Mattis74,Apel82}.
The behavior can be summarized by saying
that even a weak single impurity grows and eventually
cuts the chain into two parts with open boundary conditions at the
end points.
The exponent $\gamma_o$ characterizing (for repulsive
two-particle interactions) the suppression of $g$ on small scales is
twice the scaling exponent of the local single-particle
spectral function of a LL close to an open boundary.
This can be understood by viewing transport across the
impurity as an end-to-end tunneling between two semi-infinite
LLs (see e.g. \textcite{Kane92}).
Bosonization was not only used for an infinite LL wire
but also for the experimentally more relevant case in which an
interacting wire containing a single
impurity is contacted to two semi-infinite
non-interacting leads. Contacts generically lead to single-particle
backscattering and thus have an effect similar to the impurity (see below).
To disentangle the effect of the contacts and the impurity one often
models the contacts such that they do not lead to any backscattering.
In this case and in the absence of the impurity the conductance takes the
maximal value $1/(2 \pi)$ (instead of $K/(2 \pi)$ for an infinite LL).
Using bosonization this can either be achieved
within the so-called {\it local Luttinger liquid} picture
\cite{Safi95,Maslov95,Ponomarenko95,Janzen06}
or by appropriate boundary conditions for the bosonic
fields \cite{Egger97,Egger00}. The fixed points and scaling
exponents turned out to be the same as in the LSG model.
For weak two-particle interactions the problem of a single impurity in a
LL was also studied using a fermionic RG \cite{Matveev93,Yue94}. In this
approach a flow equation for the transmission coefficient at $k_F$ is
derived using poor man's RG. An extension of this method to interactions of
arbitrary strength was presented by \textcite{Aristov08} and \textcite{Aristov09}.
Remarkably, no intermediate fixed points appear within the LSG model and the
crossover between the perfect and open chain is characterized by a one-parameter
scaling function \cite{Kane92,Moon93,Fendley95,Egger00}. In one spatial
dimension a general impurity can be described by matrix elements $V_{k,k'}$
(only $ V_{k_F,-k_F}= \tilde V(2 k_F)$ is kept in the LSG model).
The RG analysis for such an impurity
involves the coupling of all matrix elements $V_{k,k'}$ in the flow and
one might wonder if this leads to an intermediate fixed point absent in
the LSG model. To lowest order in the impurity strength
the flow of $V_{k_F,-k_F}$ upon lowering the cutoff $\Lambda$ is driven by
$V_{k_F,-k_F}$ itself (see the next section). For repulsive two-particle
interactions $V_{k_F,-k_F}$ increases and the
perturbative RG breaks down. Now the other couplings, absent in the LSG model,
might become relevant and eventually cut off the flow of $V_{k_F,-k_F}$ to large
values, that is towards the open chain fixed point. That this does not happen
can nicely be revealed within a functional RG approach.
Before reviewing how this question was approached (see Sec.~\ref{subsubnum})
we first present the most elementary functional RG flow equation to tackle
inhomogeneous LLs and analytically show that it leads to the scaling behavior
known from bosonization in the limit of weak impurities.
\subsubsection{The functional RG approach to the single impurity problem}
\label{subsubsimp}
As in the previous applications because of the necessary truncations the
functional RG can (presently) only be used for small to intermediate
two-particle interactions. For the application to inhomogeneous LLs
it is crucial that it is non-perturbative in the single-particle
inhomogeneity. Here the focus is on a description
in which the RG flow and the interaction dependent exponents
characterizing the physics close to fixed points are kept at least to
leading order in the interaction. Following the discussion
in the last paragraph of the preceeding subsection
the feedback of the impurity into
the flow of the self-energy dominates the physics to be studied.
It is thus advantageous to use the one-particle irreducible functional RG
scheme in which the full propagator including the self-energy
appears on the right hand side of the flow equations. For spinless fermions
in a {\it homogeneous} wire the electron-electron interaction is renormalized
only by a finite amount of order interaction squared \cite{Solyom79}.
A single impurity does not alter this. To obtain the fixed point value
of the effective interaction to leading order one can
therefore replace the flowing two-particle
vertex by the antisymmetrized bare interaction corresponding to the
level-1 truncation introduced in Sec.\ \ref{sec:truncations}.
An improvement which includes the flow of the static two-particle
vertex is reviewed in Sec.~\ref{subsubnum}. After presenting the
most elementary functional RG flow equation for the self-energy we show
that it leads to the correct scaling properties for a weak impurity.
As one deals with systems in which translational
invariance is broken it is advantageous to introduce the infrared
cutoff $\Lambda$ in {\it frequency} space. To set up the
functional RG flow equations for the self-energy the propagator
$G_0$ of the non-interacting Hamiltonian $H_0$
containing only the kinetic energy
is replaced by $G_0^{\Lambda}(i \omega_n) = \Theta^{\Lambda}(\omega_n)
G_0(i \omega_n)$
with a function $\Theta^{\Lambda}$ which is
unity for $|\omega_n| \gg \Lambda$ and vanishes for
$|\omega_n| \ll \Lambda$.
More specifically
\begin{eqnarray}
\mbox{} \Theta^\Lambda(\omega_n) & =
\left\{ \begin{array}{ll}
0 \, ,& |\omega_n| \leq \Lambda- \pi T \\
\frac{1}{2} + \frac{|\omega_n|-\Lambda}{2\pi T}
\, , & \Lambda - \pi T \leq |\omega_n | \leq \Lambda +\pi T \\
1 \, ,& |\omega_n| \geq \Lambda + \pi T
\end{array} \right. \nonumber \\
\label{cutfun}
\end{eqnarray}
was used \cite{Enss05b}, where $\Lambda$ starts at $\infty$ and goes
down to $0$.\footnote{A significant speed-up of the numerical solution of the
differential flow equations can be achieved using the alternative cutoff
scheme discussed in the Appendix of \textcite{Andergassen06b}.}
For $T \to 0$,
$\Theta^{\Lambda}$ becomes a sharp $\Theta$-function and Matsubara frequencies
with $|\omega| < \Lambda$ are suppressed.
In this subsection a general continuum or lattice model of spinless fermions
is considered
with one-particle states $\left| \alpha \right>$, which in the
following will be either local states $\left| x \right>$, where $x=j$
is the site
index for the lattice model, with lattice constant $a=1$, or
momentum states $\left| k_n \right>$ with $k_n=2\pi n/L$.
A general two-particle interaction
Eq.\ (\ref{bareinteraction}) is assumed and an impurity term
$V_{\rm imp}= \sum_{\alpha,\beta} V_{\alpha,\beta} \, \psi^{\dag}_{\alpha} \psi^{}_{\beta} $.
The flow for the self-energy Eq.\ (\ref{floweqSigma}) in the
level-1 truncation reads \cite{Meden02b}
\begin{eqnarray}
&& \hspace{-.5cm} \frac{d}{d \Lambda}\Sigma^{\Lambda}_{\alpha,\beta} =
T \sum_{\omega_l} e^{i\omega_l 0^+} \!\!
\sum_{\gamma,\delta} \Bigg\{
\bigg[ 1- G_0^{\Lambda}(i \omega_l)
\Sigma^{\Lambda} \bigg]^{-1} \nonumber \\*
&& \times \frac{d G_0^{\Lambda}(i \omega_l)}{d \Lambda}
\bigg[ 1- \Sigma^{\Lambda}
G_0^{\Lambda}(i \omega_l) \bigg]^{-1}
\Bigg\}_{\delta,\gamma} U_{\alpha,\gamma,\beta,\delta} \; , \hskip 5mm
\label{volker:sflow}
\end{eqnarray}
where $G_0^{\Lambda}$ and $\Sigma^{\Lambda}$ are matrices.
The initial condition is given by
$\Sigma^{\Lambda=\infty}_{\alpha,\beta} = V_{\alpha,\beta}$ and
$U_{\alpha,\beta,\gamma,\delta}$ denotes the antisymmetrized bare
two-particle vertex.
At temperature $T=0$ and applying Eq.\ (\ref{morrislemma}) to products of $\Theta$-
and $\delta$-functions, Eq.\ (\ref{volker:sflow}) simplifies to
\begin{eqnarray}
\frac{d}{d \Lambda}\Sigma^{\Lambda}_{\alpha,\beta} =
- \frac{1}{2 \pi}
\sum_{\omega = \pm \Lambda} \sum_{\gamma,\delta}
U_{\alpha,\gamma,\beta,\delta} \,
\tilde G_{\delta,\gamma}^{\Lambda}(i \omega) \, e^{i \omega 0^+},
\label{volker:sigmalater}
\end{eqnarray}
where
\begin{eqnarray}
\tilde G^{\Lambda}(i\omega) = \left[ Q_0( i \omega) -
\Sigma^{\Lambda} \right]^{-1}
\label{volker:fullgdef}
\end{eqnarray}
is the full propagator for the cutoff dependent
self-energy and $Q_0=(G_0)^{-1}$.
The convergence factor $e^{i \omega 0^+}$
in Eq.\ (\ref{volker:sigmalater}) is relevant only for determining
the flow from $\Lambda=\infty$ down to some arbitrarily large
$\Lambda_0$. For $\Lambda_0$ much larger than the band
width this high energy part of the flow can be computed
analytically leading to the initial condition
$ \Sigma^{\Lambda_0}_{\alpha,\beta} = V_{\alpha,\beta} +
\sum_{\gamma} U_{\alpha,\gamma,\beta,\gamma}/2 $.
Within the present scheme $\Sigma^{\Lambda}_{\alpha,\beta}$ is
frequency independent and can be considered as
a flowing effective impurity potential.
To obtain an approximation for the Green function of the original
cutoff free problem for arbitrary impurity parameters
one has to determine the self-energy $\Sigma^{\Lambda}_{\alpha,\beta}$
at $\Lambda=0$ by numerically solving the set of differential equations
(\ref{volker:sflow}) or (\ref{volker:sigmalater}). To compute the
right hand side of the flow equations one has to invert the matrix
Eq.\ (\ref{volker:fullgdef}), i.e., to solve the problem of a single
particle moving in the effective scattering potential
$\Sigma^{\Lambda}_{\alpha,\beta}$.
Because of the matrix inversion involved in calculating
the right hand side of Eq.\ (\ref{volker:sigmalater}) the flow equations
can be solved analytically only in limiting cases, one being the situation
of a weak impurity. One then works in momentum space and considers
$\Sigma^{\Lambda}_{k,k'}$ for $k\ne k'$. The term linear in
$\Sigma^{\Lambda}$ presents the leading approximation in the expansion
of $\tilde G^{\Lambda}$ on the right hand side of Eq.\ (\ref{volker:sigmalater})
and one obtains \cite{Meden02b}
\begin{eqnarray}
\frac{d}{d \Lambda}\Sigma^{\Lambda}_{k,k'} &=& - \frac{1}{2 \pi}
\sum_{k_1,k_2} U_{k,k_1,k',k_2}
\nonumber \\*
&& \mbox{} \hspace{-2.cm}\times \left[ \frac{1}{i \Lambda - \xi_{k_2} }
\Sigma^{\Lambda}_{k_2,k_1}
\frac{1}{i \Lambda - \xi_{k_1} } + \left( \Lambda \to - \Lambda \right) \right] \; ,
\label{volker:expansion}
\end{eqnarray}
where $\xi_k = \epsilon_k - \mu$ with the one-particle dispersion
$\epsilon_k$. The antisymmetrized two-body matrix element
$U_{k,k_1,k',k_2}$ contains a momentum conserving Kronecker
delta, where $k+k_1=k'+k_2$ modulo the reciprocal lattice vector
$ 2\pi n$ for the lattice model
with $n=0,\pm 1$, when the four momenta are in the first Brillouin
zone. The umklapp processes $n=\pm 1$ involve low
energy excitations only for special fillings--half filling for the
nearest neighbor hopping model discussed later.
First models are considered for which
umklapp processes are absent. To determine the
backscattering properties of the self-energy one can put $k=k_F$
and $k'=-k_F-q$ with $|q| \ll k_F$. In order to read off the
dominant behavior for small $\Lambda$ the remaining
summation variable is shifted $k_1=-k_F+\tilde k_1$
\begin{eqnarray}
&& \frac{d}{d \Lambda}\Sigma^{\Lambda}_{k_F,-k_F-q} = - \frac{1}{2 \pi}
\sum_{\tilde k_1} U_{k_F,-k_F+\tilde k_1,-k_F-q,k_F+q+\tilde k_1}
\nonumber \\*
&& \times \Big[ \frac{1}{i \Lambda - \xi_{k_F+q+\tilde k_1} }
\Sigma^{\Lambda}_{k_F+q+\tilde k_1,-k_F+\tilde k_1}
\frac{1}{i \Lambda - \xi_{-k_F+\tilde k_1} } \nonumber \\*
&& +\left( \Lambda \to - \Lambda \right) \Big].
\label{volker:expansion2}
\end{eqnarray}
For $|\tilde k_1|\ll k_F$ one can linearize the dispersion
$\xi_{-k_F+\tilde k_1}\approx -v_F \tilde k_1$ and $\xi_{k_F+q+\tilde
k_1}\approx v_F (\tilde k_1+q)$,
where $v_F$ denotes the Fermi velocity.
In the thermodynamic limit the two $G_0$ factors for $q=0$ and
$\Lambda \to 0$ are
proportional to $\delta(\tilde k_1)$.
If only the singular contributions
are kept the differential equation for $\Sigma^{\Lambda}_{k_F,-k_F}$
reads
\begin{eqnarray}
\frac{d}{d \Lambda}\Sigma^{\Lambda}_{k_F,-k_F} =
-\frac{\hat U_{k_F,-k_F,k_F,-k_F }}{2 \pi v_F}
\frac{1}{\Lambda} \, \Sigma^{\Lambda}_{k_F,-k_F} \,
\label{volker:sigmakfmkf}
\end{eqnarray}
where $\hat U_{k_F,-k_F,k_F,-k_F } = L U_{k_F,-k_F,k_F,-k_F }$
is independent of the
system size. For the continuum model $\hat U_{k_F,-k_F,k_F,-k_F}=
\tilde U(0)-\tilde U(2k_F)$, where $\tilde U(k)$ is the Fourier
transform of the two-particle potential $U(x-x')$.
This leads to the scaling relation
\begin{eqnarray}
\Sigma^{\Lambda}_{k_F,-k_F} \sim \left( \frac{1}{\Lambda}
\right)^{ \hat U_{k_F,-k_F,k_F,-k_F } /(2 \pi v_F)} \; .
\label{volker:sigmascal}
\end{eqnarray}
As Eq.\ (\ref{volker:sigmakfmkf}) was derived by
expanding the Green function $G^{\Lambda}$ in powers of the self-energy,
the scaling behavior Eq.\ (\ref{volker:sigmascal}) can be trusted
only as long as $\Sigma^{\Lambda}$ stays small. Thus a single-particle
backscattering term is a relevant perturbation for repulsive interactions
and an irrelevant one for attractive ones consistent with
the bosonization result.
Equation (\ref{volker:sigmakfmkf}) holds even in the half-filled band
case for the nearest neighbor
hopping lattice model. The additional singular contribution due to
umklapp scattering is proportional to $\hat U_{k_F,k_F,-k_F,-k_F}$
which vanishes because of the antisymmetry of the matrix element
\cite{Meden02b}.
Neglecting all interaction effects beyond the renormalization of the
impurity potential and using the Born approximation the correction to
the perfect chain conductance is given by the self-energy squared.
The corresponding exponent $ \hat U_{k_F,-k_F,k_F,-k_F } /(\pi v_F)$
agrees to leading order in the interaction \cite{Schoenhammer05} with
$\gamma_p=2(K-1)$ obtained within bosonization.
The opposite limit of a weak link can be treated analytically as well
leading to results consistent to those of the bosonization
approach with an exponent characterizing the deviation from the open chain
fixed point conductance $g=0$ which agrees to leading order in the
interaction with $\gamma_o=2(1/K-1)$ \cite{Meden02b}.
Next the numerical solution of the
RG flow for a specific lattice model with
arbitrary impurity strength is discussed. This allows us to address the
question of additional fixed points.
\subsubsection{Basic wire model}
\label{basicmodel}
In the following the
tight-binding model of spinless fermions with nearest
neighbor interaction supplemented by an impurity is considered.
The Hamiltonian is given by $ H = H_{\rm kin} + H_{\rm int} + H_{\rm imp} $
with kinetic energy
\begin{equation}
\label{volker:kindef}
H_{\rm kin} = -\sum_{j=-\infty}^{\infty} \big( \,
c^{\dag}_{j+1} c_j^{\phantom\dag} + c^{\dag}_j \, c_{j+1}^{\phantom\dag}
\, \big) \; ,
\end{equation}
where $c^{\dag}_j$
and $c_j^{\phantom\dag}$ are the creation and annihilation operators on
site $j$, respectively. The
corresponding non-interacting dispersion is
$\epsilon_k=-2\cos k$.
The interaction is restricted to electrons on $N$ neighboring sites \cite{Enss05b}
\begin{eqnarray}
\label{volker:intdef}
\mbox{} \hspace{-.5cm} H_{\rm int} & = & \sum_{j=1}^{N-1} U_{j,j+1} \left[ n_j - \nu(n,U)
\right] \left[ n_{j+1} - \nu(n,U) \right] , \hskip 5mm
\end{eqnarray}
with the local density operator $n_j = c^{\dag}_j \,
c_j^{\phantom\dag}$. The two regions of the lattice with $j < 1$ and
$j>N$
constitute the semi-infinite non-interacting leads.
To model contacts which do not lead to single-particle
backscattering (see above) the interaction $U_{j,j+1}$
between electrons on sites $j$
and $j+1$ is allowed to depend on the position.
A conductance $g=1/(2 \pi)$ in the absence of single-particle
impurities is {\it only} achieved if $U_{j,j+1}$ is taken as a smoothly
increasing function of $j$ starting form zero at the bond $(1,2)$ and
approaching a constant bulk value $U$ over a sufficiently large number
of bonds. Equally, the $U_{j,j+1}$ are switched off
close to the bond $(N-1,N)$.
The results are independent of the shape of the envelope
function as long as it is sufficiently smooth.
An abrupt two-particle inhomogeneity acts similarly to a
single-particle impurity.
A detailed disussion on the effect of the spatial variation
of the two-particle interaction was presented by
\textcite{Meden03b} and \textcite{Janzen06}.
In Eq.\ (\ref{volker:intdef}) the density $n_j$ is shifted by
a parameter $\nu(n,U)$, which depends on the filling factor $n$
and the bulk interaction $U$. This is equivalent to introducing
an additional one-particle potential which can compensate
the Hartree potential in the bulk of the interacting wire.
In the half-filled band case $\nu(1/2,U)=1/2$ \cite{Enss05b}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.40\textwidth,clip]{fig17.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{Schematic plot of the quantum dot situation, where the
barriers are modeled by two site impurities \cite{Enss05b}.
\label{fig:doublebar}}
\end{figure}
The general form of the impurity part of the Hamiltonian is
\begin{equation}
\label{volker:impdef}
H_{\rm imp} = \sum_{j,j'}
V_{j,j'}^{\phantom\dag} \; c^{\dag}_{j} \, c_{j'}^{\phantom\dag} \; ,
\end{equation}
where $V_{j,j'}$ is a static potential.
Site impurities are given by
$V_{j,j'} = V_j \, \delta_{j,j'} $.
For a single site impurity $ V_j = V \, \delta_{j,j_0} $
$j_0$ is chosen to be far away from both leads.
Impurities close to the contacts were discussed by \textcite{Furusaki96} and
\textcite{Enss05b}.
Resonant tunneling can be studied considering two site
impurities of strengths $V_l$ and $V_r$ on the sites
$\tilde j_l=j_l-1$ and $\tilde j_r=j_r+1$.
The $N_D$ sites between $j_l$ and $j_r$ define
a quantum dot. The effect of a
gate voltage restricted to the dot region is described by a constant
$V_g$ on sites $j_l$ to $j_r$.
This situation is sketched in Fig.~\ref{fig:doublebar}.
Hopping impurities are achieved setting
$V_{j,j'} = V_{j',j} = - t_{j,j+1} \, \delta_{j',j+1} $.
For the special case of a single hopping impurity,
$t_{j,j+1} = (t'-1) \, \delta_{j,j_0} $,
the unit hopping amplitude is replaced by $t'$ on the bond linking
the sites $j_0$ and $j_0+1$. In the double-barrier problem
a hopping $t_l$ across the bond $(\tilde j_l,j_l)$ and $t_r$
across $(j_r,\tilde j_r)$ is considered.
The {\it homogeneous} model $H=H_{\rm kin} + H_{\rm int}$ with a constant
interaction $U$ across {\it all bonds}--not only the ones within $[1,N]$--can
be solved exactly by the Bethe ansatz
and $K$ is determined by a system
of coupled integral equations \cite{Haldane80}. In
the half-filled case they can be solved analytically leading to
\begin{equation}
\label{volker:BetheAnsatz}
K = \left[\frac{2}{\pi} \,
\arccos \left(-\frac{U}{2} \right) \right]^{-1}
\end{equation}
for $|U| \leq 2$. At other fillings the integral equations
can be solved numerically.
The model shows LL behavior for all
particle densities $n$ and any interaction strength except at
half filling for $|U| > 2$ were either phase separation sets in (for
$U< -2$) or the system orders into a charge-density-wave state
(for $U>2$).
Due to the presence of the semi-infinite
leads the direct calculation of the non-interacting
propagator related to $H_{\rm kin} + H_{\rm imp}$
requires the inversion of an infinite matrix. Using a
standard projection technique \cite{Taylor72} it can be
reduced to the inversion of an $N \times N$ matrix.
The leads then provide an additional $\omega_n$-dependent
diagonal one-particle potential on sites $1$ and $N$ \cite{Enss05b}
\begin{eqnarray}
\label{volker:leadpotdef}
V_{j,j'}^{\rm lead}(i\omega_n) & = &
\frac{i\omega_n+\mu}{2} \left( 1 -
\sqrt{1 - \frac{4}{(i\omega_n+\mu)^2}} \, \right) \nonumber \\
&& \times \delta_{j,j'} \left( \delta_{1,j} + \delta_{N,j} \right) \; .
\end{eqnarray}
Since the interaction is only non-vanishing
on the bonds between the sites $1$ to $N$ the problem including the
semi-infinite leads is this way reduced to the problem of an
$N$-site chain. In the functional RG it is then advantageous to
replace the {\it projected}
non-interacting propagator $G_0$ {\it including} the impurity
by a cutoff dependent one \cite{Enss05b}.
\subsubsection{Numerical solution of improved flow equations}
\label{subsubnum}
With a minor increase in the numerical effort one can go beyond
Eq.\ (\ref{volker:sflow}) for the flow of
the self-energy and include a $\Lambda$-dependent static
interaction $U^\Lambda$ \cite{Andergassen04}.
Its flow equation is derived from
the general one for the two-particle
vertex Eq.\ (\ref{floweqGamma4}) applying the following approximations:
(i) The three-particle vertex is set to zero.
(ii) All frequencies are set to zero. (iii) The feedback of the inhomogeneity
on the flow of the interaction is neglected. (iv) The interaction is assumed
to remain of nearest-neighbor form.
Then $U^\Lambda$ obeys a simple differential equation
\cite{Andergassen04},
\begin{eqnarray}
\label{RGU}
\frac{d}{d\Lambda} U^\Lambda = h(\Lambda) \left( U^\Lambda \right)^2 \; ,
\end{eqnarray}
where the function $h(\Lambda)$ depends only on the cutoff $\Lambda$ and
the Fermi momentum $k_F$.
The solution of the flow equation is lengthy for arbitrary fillings
\cite{Andergassen04} but has a simple form for half-filling
\begin{eqnarray}
\label{flussU}
U^{\Lambda} = \frac{U}
{1 +
\left(\Lambda - \frac{2 + \Lambda^2}{\sqrt{4 + \Lambda^2}} \right) \,
U/(2\pi)} \; .
\end{eqnarray}
These approximations are sufficient to correctly describe the RG flow
of the two-particle vertex on the Fermi surface of the homogeneous system
\cite{Andergassen04} to second order, as it is usually done in the
so-called g-ology method \cite{Solyom79}. For the inhomogeneous LLs
the flow of the effective interaction leads to improved results
for the scaling exponents (see below).
Within these approximations and using the projection of the leads,
the self-energy at $T \geq 0$ is a frequency-independent tridiagonal
matrix in real space
determined by the flow equation
($j,j\pm 1 \in [1,N]$)
\begin{eqnarray}
\frac{\partial}{\partial \Lambda} \Sigma_{j,j}^\Lambda =
-\frac{1}{2\pi}
\sum_{|\omega_n| \approx \Lambda} \sum_{r=\pm 1}
U_{j,j+r}^\Lambda \left[ \frac{1}{Q_0(i\omega_n)-\Theta^\Lambda(\omega_n)
\Sigma^\Lambda} \right. \nonumber \\* \left. \times
Q_0(i\omega_n) \, \frac{1}{Q_0(i\omega_n)-\Theta^\Lambda(\omega_n)
\Sigma^\Lambda } \right]_{j+r,j+r} \; , \nonumber\\
\frac{\partial}{\partial \Lambda} \Sigma_{j,j\pm 1}^\Lambda =
\frac{1}{2\pi}
\sum_{|\omega_n| \approx \Lambda} U_{j,j\pm 1}^\Lambda
\left[ \frac{1}{Q_0(i\omega_n)-\Theta^\Lambda(\omega_n) \Sigma^\Lambda }
\right. \nonumber \\*
\left. \times Q_0(i\omega_n) \, \frac{1}{Q_0(i\omega_n)-\Theta^\Lambda(\omega_n)
\Sigma^\Lambda} \right]_{j,j\pm 1} \; , \nonumber
\end{eqnarray}
\vspace{-.5cm}
\begin{equation}
\vspace{-.2cm}
\label{volker:sigmaflowtg0}
\end{equation}
where the matrix $Q_0=(G_0)^{-1}$ is the inverse of
the projected non-interacting propagator with impurity.
The symbol $|\omega_n| \approx \Lambda$ stands for taking the positive
as well as negative frequency with absolute value closest to
$\Lambda$.\footnote{To achieve this result the cutoff scheme discussed in the
Appendix of \textcite{Andergassen06b} was used.}
The initial conditions for $\Sigma$ at $\Lambda = \Lambda_0 \to \infty$
are independent of the precise realization of the inhomogeneity and read
$\Sigma^{\Lambda_0}_{j,j} =
\left[1/2- \nu(n,U)\right] \left( U_{j-1,j} +
U_{j,j+1} \right) $ and
$\Sigma^{\Lambda_0}_{j,j\pm 1} =0 $.
The frequency
dependence of the self-energy which appears in the exact solution
in order $U^2$ is {\it not} captured by this scheme.
Thus only the leading order is completely kept in the flow of $\Sigma$.
The coupled flow equations can be solved numerically by an algorithm
which approximately scales as $N$ \cite{Andergassen04}. Typically
systems of $10^4$ lattice sites
were considered, roughly corresponding to the length of quantum wires
accessible to transport experiments. For the interacting wire of finite
length the energy scale $ \delta_N = v_F / N$ forms a cutoff for any RG flow.
The flowing self-energy Eq.\ (\ref{volker:sigmaflowtg0})
depends on the three scales $T$, $\delta_N$, and $\Lambda$.
Saturation of $\Sigma^{\Lambda}$ for $\Lambda \lessapprox T$
or $\Lambda \lessapprox \delta_N$ sets in ``automatically''
in contrast to more intuitive RG schemes in
which the flowing couplings depend on $\Lambda$ only and
the flow is stopped ``by hand'' by replacing $\Lambda \to T$ or
$\Lambda \to \delta_N$, respectively \cite{Kane92,Yue94}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.40\textwidth,clip]{fig18.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{Self-energy near a site impurity of strength $V=1.5$
filling $n=1/4$, and interaction $U=1$; the impurity
is located at $j_0=513$ with $N = 1025$ sites. (Data
taken from \textcite{Andergassen04}.)
\label{fig:self1}}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.49\textwidth,clip]{fig19.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{ Decay of the oscillatory part of the off-diagonal matrix
element of the self-energy away from a single hopping impurity at
bond $j_0,j_0+1$. Results for $t'=0.1$, $j_0=5000$, $N=10^4$, $U=1$,
$n=1/2$, and different temperatures $T=10^{-1}$ (solid line),
$T=10^{-2}$ (dotted line), $T=10^{-3}$ (dashed line), and
$T=10^{-4}$ (dashed-dotted line) are presented. The left panel
shows the data on a log-log scale, the right panel on a linear-log scale.
For comparison the left panel contains a power-law $(j-j_0)^{-1}$
(thin solid line) \cite{Enss05b}. \label{fig:self2}}
\end{figure}
Figure \ref{fig:self1} shows the self-energy $\Sigma$ at the end of the RG
flow, for $T=0$ in the vicinity of a site impurity
of intermediate strength. Both
the onsite energy $\Sigma_{j,j}$ as well as the hopping $\Sigma_{j,j+1}$ become
oscillatory functions with wave number $2 k_F$ and a decaying amplitude.
The asymptotic value of $\Sigma_{j,j+1}$ away from
the impurity leads to a broadening of
the band due to the interaction. A more detailed analysis of the oscillatory
part $|\Delta \Sigma_{j,j+1}|= \left| \Sigma_{j,j+ 1} - \bar \Sigma_{\rm off} \right|$,
with the spatial average $\bar \Sigma_{\rm off}$,
for different $T>0$ is presented in Fig.\ \ref{fig:self2}. The left panel
shows that for $|j-j_0| \gtrapprox 10$ it decays as $1/|j-j_0|$ up to a thermal
length scale $\sim 1/T$ (provided $T > \delta_N$)
beyond which the decay becomes exponential; see the right panel.
For $U>0$ this is the generic behavior for large bare impurities or on
asymptotical large length scales.
It is the scattering off such a long-ranged oscillatory
potential--so-called Wigner-von Neumann potential \cite{Reed75}--which leads
to the power-law suppression of the conductance and the local spectral weight.
One can analytically show that the amplitude of the asymptotic $1/|j-j_0|$
decay determines the exponent \cite{Barnabe05a}.
By virtue of the RG flow this amplitude--and thus the
exponent--becomes independent of the impurity strength. For this reason
first order perturbation theory fails. It also leads to an
oscillatory self-energy which decays as $1/|j-j_0|$ but with an amplitude
which depends on the bare impurity strength \cite{Meden02b} incorrectly
leading to a power-law with an impurity dependent exponent.
The idea of an oscillatory decaying potential is
similarly inherent to a poor man's fermionic RG
approach \cite{Yue94}. Often these oscillations of the effective
renormalized potential are referred to as Friedel oscillations.
This is misleading as this term is reserved to the spatial
oscillations of the {\it electron density.} In particular, in
an inhomogeneous LL the effective potential decays as $|j-j_0|^{-1}$,
while the density oscillations asymptotically decay as $|j-j_0|^{-K}$ \cite{Egger95}.
The latter can also be shown within the functional RG
formalism presented here \cite{Andergassen04}.
The application of the {\it self-consistent} Hartree-Fock approximation
leads to an oscillatory self-energy with a constant amplitude and thus to a
charge density wave state \cite{Meden02b}. This is an unphysical artifact of
the approximation.
Using scattering theory \cite{Enss05b} one can show that the effective
transmission ${\mathcal T}(\epsilon,T)$ is given by the $(1,N)$ matrix
element of the
single-particle Green function $ {\mathcal T}(\epsilon_k,T)=4 \sin^2 k \,
|\langle N| G(\epsilon_k+i0)|1\rangle|^2$. Via the $T$-dependent self-energy
(see Fig.~\ref{fig:self2})
$G$ and thus $\mathcal T$ carries a temperature dependence. A typical example
for the $T$-dependence of the linear conductance $g(T)$ for a strong local impurity
is shown as the solid line in Fig.\ \ref{fig:reso}. It clearly follows the expected
power-law behavior for $\delta_N \lessapprox T \ll B$ with the band width $B=4$.
The $T^{-1}$ scaling at larger $T$ is a band effect.
For $-0.5 \leq U \leq 1.5$ and fillings $n=1/2$ as well as $1/4$ the exponent extracted
(see lower panel of Fig.\ \ref{fig:reso})
agrees well with the one of the LSG model $\gamma_o=2(1/K-1)$, with $K$
taken from the Bethe ansatz. Even for $U=1.5$ the
relative error is less than 5 percent (see Fig.\ 5 of \textcite{Enss05b}).
Higher order corrections in $U$ present in the numerical solution of the
flow equations (\ref{flussU}) and (\ref{volker:sigmaflowtg0})
clearly improve the result over the one of the perturbative (in the impurity strength)
analytical solution of Sec.\ \ref{subsubsimp} which yields a purely linear exponent.
We emphasize that this improvement is not systematic
as second and higher order terms are only partly kept in the RG.
A similar agreement can be found for $\gamma_p$ \cite{Enss05b}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{fig20.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{One-parameter scaling plot of the conductance. Open
symbols represent results obtained for $U=0.5$, $n=1/2$, and
different $T$ and $V$, while filled symbols were calculated for
$U=0.851$, $n=1/4$. Both pairs of $U$ and $n$ lead to the same
$K=0.85$ (within the present approximation). The solid line indicates
the non-interacting scaling function $(1+x^2)^{-1}$ and the dashed
one the LSG model power-law decay with exponent $-2/K$.
(Data taken from \textcite{Enss05b}.)
\label{fig:scaling}}
\end{figure}
Within the LSG model no intermediate fixed points appear
which is reflected by one-parameter scaling
$g = \tilde g_K(x)/2 \pi$ with $x=\left[ T/T_0\right]^{K-1} $ and
a non-universal scale $T_0$ \cite{Kane92,Moon93,Fendley95,Egger00}.
For appropriately chosen $T_0$ data for different
$T$ and $\tilde V(2k_F)$ but fixed $K$ can be collapsed onto
the $K$-dependent
scaling function $\tilde g_K(x)$. It has the limiting behavior
$\tilde g_K(x) \propto 1-x^2$ for $x \to 0$ and $\tilde g_K(x) \propto x^{-2/K}$
for $x \to \infty$.
One can perform a similar scaling with data from the numerical solution
of the flow equations for the microscopic lattice model
considering different $V$ and $T$ as well as
two sets of $(U,n)$ leading to the same LL parameter \cite{Enss05b}.
The perfect collaps of the data of Fig.\ \ref{fig:scaling}
shows that the improved description of the impurity flow
beyond the single amplitude approximation
inherent to the LSG model does not lead to additional
fixed points.
The functional RG scaling function shows a sensible $K$ dependence.
The exponent of the large $x$ power-law decay is smaller than the
non-interacting one $-2$ (solid line at large $x$) and very close to the
LSG model exponent $-2/K$ shown as the dashed line in Fig.\ \ref{fig:scaling}.
This has to be contrasted to the $K$ {\it independent} (non-interacting)
scaling function $\tilde g = (1+x^2)^{-1}$ resulting from the poor man's
fermionic RG \cite{Yue94}; solid line in Fig.\ \ref{fig:scaling}.
The functional RG results for a single local impurity in a LL show
that the LSG model describes the physics (two fixed points, exponents,
one-parameter scaling) of a broader class of models.
The same approach was also used to study the persistent current through
a LL ring with a local impurity prierced by a magnetic
flux \cite{Meden03a,Meden03b,Gendiar09}
Aspects resulting from the {\it spin} degree of freedom of electrons
were discussed by \textcite{Andergassen06a} and \textcite{Andergassen06b}.
\subsubsection{Resonant tunneling}
\label{resotunsec}
We next review the results on resonant transport through a double barrier--defining
an interacting quantum dot embedded in a LL \cite{Meden05,Enss05b}. The setup is
sketched in Fig.\ \ref{fig:doublebar}.
The linear conductance $g$ is characterized
by a hierarchy of energy scales. The functional RG is a unique tool to access this
problem as it provides reliable results on all scales.
For a fixed dot size $N_D$ and fixed barriers $V_{l/r}$
(or $t_{l/r}$) the dot can be tuned to
resonance varying $V_g$. Only for symmetric dots with $V_l=V_r$ (or $t_l=t_r$)
the peak conductance becomes ``perfect'' $g_p=1/(2\pi)$. For asymmetric
barriers a backscattering component of the single-particle inhomogeneity remains,
leading to a reduced $g_p$.
Due to the interaction backscattering grows during a RG procedure
and on asymptotic
scales the conductance vanishes with scaling exponent $\gamma_o$. The same
holds away from resonance regardless of the ratio $V_l/V_r$ (or $t_l/t_r$).
Thus the non-interacting resonance of finite width
either disappears (asymmetric barriers) or turns
into a resonance of {\it zero
width} (symmetric barriers). A rich $T$-dependence is found on
resonance and for symmetric barriers on which we now focus.
Without loss of generality we only consider site impurities as barriers.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{fig21.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{Upper panel: $g_p(T)$ for
$U=0.5$, $N=10^4$, $V_{l/r}=10$, $n=1/2$, and
$N=6$ (squares), $100$ (diamonds). The arrows indicate the relevant energy
scales $B$, $\delta_{N_D}$, $T^\ast_{N_D}$ and $\delta_N$.
The solid curve shows $g(T)/2$ for a single barrier with $V=10$ and $U=0.5$,
$N=10^4$. Lower panel: Logarithmic
derivative of the conductance. Dashed line: $\gamma_o$; dashed-dotted
line: $\gamma_o/2-1$. (Data taken from \textcite{Enss05b}.) \label{fig:reso}}
\end{figure}
The functional RG procedure can directly be applied to the double
barrier problem. The dot parameters only enter via the non-interacting
propagator. Figure \ref{fig:reso} shows the peak conductance $g_p(T)$ for
a dot with high barriers and two different dot sizes. The relevant
energy scales $B$, $\delta_{N_D}=v_F/N_D$,
$T^\ast_{N_D}$ (see below), and $\delta_N$ are indicated
by the arrows. For $\delta_{N_D} \lessapprox T $ the two barriers behave as
independent impurities. Using scattering theory one can show that in
this case $g_p$ is
obtained by adding the resistances of the two barriers
\cite{Enss05b,Jakobs07b}. This
explains why for $N_D=100$, for which this temperature regime is clearly developed,
$g_p(T)$ agrees to the solid line obtained by taking $g(T)/2$ of a single site
impurity of equal heigth as used for the double barrier. Note that it
is a non-trivial
fact that in this temperature regime the individual resistances can be added
to give the total resistance. In the presence of {\it inelastic processes}
one would of course expect this result (resistors in series) but they
are absent in the
mesoscopic setup--the ones resulting from the electron-electron interaction
are suppressed by the approximations. In fact, the case of three barriers
constitutes an example for which adding resistances does
no longer hold \cite{Jakobs07b}.
For $T \lessapprox \delta_{N_D}$ the width of
$-\partial f/\partial \epsilon$ is smaller than $\delta_{N_D}$
and only the resonance peak around $\epsilon=0$ of
${\mathcal T}(\epsilon,T)$ contributes to the integral in
Eq.\ (\ref{linearcond}). The width $w$ of this peak vanishes as
$T^{\gamma_o/2}/N_D$ \cite{Enss05b} leading to $g_p(T) \propto
T^{\gamma_o/2-1}$. The lower bound of this scaling regime,
first discussed using bosonization \cite{Furusaki93b,Furusaki98}, is reached
when $T$ equals
$w$, i.e.\ at $T_{N_D}^\ast \propto N_D^{-1/(1-\gamma_o/2)}$.
For $T < T_{N_D}^\ast$,
$2 \pi g_p$ approaches $1$. For $T$ reaching $\delta_N$ any power-law scaling in
$T$ with an interaction dependent exponent
is cut off by the finite size of the interacting part of the quantum wire.
In addition of identifying the different temperature regimes the functional RG
approach allows to (i) quantify the size of the crossover regime--typically
half an order of magnitude--and to (ii) obtain results for ``non-universal''
regimes as e.g.\ realized for $N_D=6$ and $\delta_{N_D} < T \ll B$. For dots with
weak barriers and sufficiently large $N_D$ only the regime with scaling exponent
$\gamma_o/2-1$ is realized and for weak barriers and small $N_D$ none of the
above power-law regimes emerges \cite{Meden05,Enss05b}.
Resonant transport in LLs was also studied by bosonization
\cite{Furusaki93b,Furusaki98}, poor man's fermionic RG \cite{Nazarov03,Polyakov03},
and numerically \cite{Huegle04}.
The temperature dependence of the peak conductance of resonant tunneling nicely
exemplifies that the functional RG approach provides sensible results on
{\it all energy scales} even for problems with a hierachy of scales.
Other examples from this class are situations in which the wire-lead contacts
are not modeled as being ``perfect'' \cite{Jakobs07b} and models in which the
leads and contacts are described in a more realistic way \cite{Waechter09}.
\subsubsection{Y-junctions}
\label{Yjunctionsec}
The power of the functional RG approach to uncover unconventional fixed points
and the related interesting physics was exemplified by discussing a specific
junction of three 1D wires, a so-called Y-junction. The three
LL wires (index $\nu=1,2,3$) each of length $N$ and coupled to a non-interacting
semi-infinite lead via a ``perfect'' contact are described by the basic model
discussed in Sec.\ \ref{basicmodel}.
The {\it symmetric} junction {\it pierced by a magnetic flux} $\phi$, is sketched in
Fig.\ \ref{fig:skizze} and given by
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.16\textwidth,clip]{fig22.eps}
\end{center}
\vspace{-0.6cm}
\caption[]{Sketch of the symmetric Y-junction of three quantum wires \cite{Barnabe05}.
\label{fig:skizze}}
\end{figure}
\begin{eqnarray}
\label{hamjunct}
H_{\rm Y} & = & - t_{\rm Y} \sum_{\nu=1}^{3}
\left( c_{1,\nu}^\dag c_{0,\nu} + \mbox{h.c.} \right) +
V \sum_{\nu=1}^{3} n_{0,\nu} \nonumber \\*
&& - t_{\triangle} \sum_{\nu=1}^{3} \left( e^{i \phi/3}
c_{0,\nu}^\dag c_{0,\nu+1} +
\mbox{h.c.} \right) \; ,
\end{eqnarray}
where the wire indices 4 and 1 are identified. The junction is
characterized by the three parameters $t_{\rm Y}$, $V$, and $ t_{\triangle}$.
Using scattering theory \cite{Barnabe05,Barnabe05a,Enss05b},
the $U=0$ conductance from wire $\nu$ to wire $\nu'$ can be written as
\begin{eqnarray}
\label{trans}
2 \pi g_{\nu,\nu'} = \frac{4 \left( \mbox{Im} \, \kappa \right)^2
\left| e^{-i \phi} - \kappa \right|^2}{\left|
\kappa^3-3 \kappa+ 2 \cos{\phi} \right|^2} \; ,
\end{eqnarray}
with a {\it single complex parameter}
$\kappa=(-V-t_{\rm Y}^2 \, \hat G^0_{1,1})/|t_{\triangle}|$. The Green
function $\hat G^0$ is obtained for one of the equivalent
disconnected ($t_{\rm Y}=0$) wires and $\hat G_{1,1}^0 \in
{\mathbb C}$ denotes its diagonal matrix element taken at the first
site $j=1$. It is evaluated at energy $\epsilon +i0$ with $\epsilon \to 0$.
Equation (\ref{trans}) holds if $\nu$ and $\nu'$ are in cyclic order and is
independent of the pair considered; $g_{\nu',\nu}$ follows by replacing
$\phi \to - \phi$.
If $\phi$ does not correspond to an integer multiple of $\pi$ and for
generic junction parameters the conductance from $\nu$ to $\nu'$
{\it differs} from the one with reversed indices indicating the breaking of
time-reversal symmetry. This constitues the most interesting
situation and we focus on such fluxes. In Fig.\ \ref{fig:U0cond} the
conductance from wire $\nu$ to $\nu'$ (cyclic) for $\phi = 0.4 \pi$ is
shown for the
upper half of the complex $\kappa$-plane. For restored time-reversal symmetry
the largest conductance allowed by the unitarity of the scattering matrix is
$2 \pi g_{\nu,\nu'}=4/9$ (denoted the ``perfect junction value'' in the following);
even for optimized parameters a reflection of $1/9$ is unavoidable.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth,clip]{fig23.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{(Color online) The non-interacting conductance $2 \pi g_{\nu,\nu'}$
(cyclic indices) as a function of the complex parameter $\kappa$ which in
turn is a function of the junction parameters $t_{\rm Y}$, $V$, and
$t_{\triangle}$. The flux is $\phi=0.4 \pi$.
\label{fig:U0cond}}
\end{figure}
For $U \neq 0$ the Y-junction can straightforwardly be treated within
the functional RG based approximation scheme \cite{Barnabe05}. We
here focus on $T=0$. To compute the conductance from
Eq.\ (\ref{trans}), $\hat G^0$ must be replaced by the
{\it auxiliary} Green function $\hat G$ obtained by
considering $\Sigma$ (at the end of the RG flow for the full system)
as an effective potential for a {\it single disconnected} wire
setting $t_{\rm Y}=0$ \cite{Barnabe05}.
Via the RG flow of $\Sigma$, $\hat G$
develops a dependence on ($t_{\rm Y},t_{\triangle},V$), $U$, and
$\delta_N=v_F/N$. The latter energy scale is a natural infrared
cutoff--in contrast to the flow parameter $\Lambda$ which is
artificial and sent to 0.
A comprehencive picture of the low-energy physics
is obtained from the dependence of $\kappa$ on
$\delta_N$. In Fig.\ \ref{fig:Yscal} each line is
for a fixed set of junction parameters and $\delta_N$ as a
variable. The flux is chosen as $\phi= 0.4 \pi$ and the arrows
indicate the direction of decreasing $\delta_N$.
As $\mbox{Im} \, \kappa$ has the opposite sign of $\mbox{Im} \,
\hat G_{1,1} <0$ it is restricted to positive values.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{fig24.eps}
\end{center}
\vspace{-0.3cm}
\caption[]{(Color online) Flow of $\kappa$ for $U=-1$, $n=1/2$, and $\phi=0.4 \pi$.
Arrows indicate the direction for $U<0$. For $U>0$ it is reversed.
For details see the text \cite{Barnabe05}.
\label{fig:Yscal}}
\end{figure}
Equation (\ref{trans}) allows for {\it four} distinguished conductance situations
(see Fig.\ \ref{fig:U0cond}): (i) on the line $\mbox{Im} \, \kappa = 0$,
$g_{\nu,\nu'}=g_{\nu',\nu}=0$ for almost all $\mbox{Re} \, \kappa$;
(ii) it is interrupted by three points having flux-dependent positions with
the conductance $2 \pi g_{\nu,\nu'}=2 \pi g_{\nu',\nu}=4/9$;
(iii) for a specific flux-dependent $\kappa$ one finds $2 \pi g_{\nu,\nu'}= 1$ and
$2 \pi g_{\nu',\nu}= 0$; (iv) $g_{\nu,\nu'}=g_{\nu',\nu}=0$ is also reached for
$|\kappa| \to \infty$.
These are the fixed points of the RG flow as is evident from Fig.\ \ref{fig:Yscal}.
(i) is an interupted line of decoupled chain fixed points with vanishing conductances
which is stable for $U>0$ and unstable in the opposite case. Analyzing the
dependence of $g_{\nu,\nu'}$ on $\delta_N$ in its vicinity
for different $U$ one finds that the scaling exponent is independent of $\phi$ given
by $\gamma_o$ as obtained for the single impurity. (ii) constitute three perfect
junction fixed points (circles in Fig.\ \ref{fig:Yscal}).
For $U>0$ each of the three fixed points has a {\it basin of attraction} given
by one of the three parts of the curve ${\mathcal C}(\phi)$ (curved thick line in
Fig.\ \ref{fig:Yscal} interrupted by the square)
on which the reflection $1-2 \pi g_{\nu,\nu'} - 2 \pi g_{\nu',\nu}$ takes a local minimum.
The $U$ dependence of the scaling exponent when approaching one of the fixed points
along its corresponding line is shown in Fig.\ \ref{fig:Yexp} (circles).
It is independent of $\phi$ and for small $|U|$ it can be fitted by
$U/(3 \pi)$. These fixed points have not been found by any method which is
based on bosonization and the exact dependence of their scaling exponent on $K$
is presently unknown. Because of the factor $1/3$ in the leading order it must be
different from the $K$-dependence of any of the exponents discussed so far.
(iii) The basins of attraction are separated by the maximal asymmetry fixed point
(maximal breaking of time-reversal symmetry; square in Fig.~\ref{fig:Yscal}).
For $\phi=\pi/2$ this fixed point was identified by a bosonization based approach
\cite{Chamon03}, and it was conjectured that the behavior found holds for all
fluxes different from integer multiples of $\pi$. The functional RG
results indeed confirm this--at least for small to intermediate $|U|$--as one
finds this fixed point for {\it all} such $\phi$ and obtains a {\it flux-independent}
scaling exponent which to leading order agrees to the bosonization result
$\gamma_{\rm Y}=2(\Delta-1)$ with $\Delta=4K/(3+K^2)$ (see Fig.\ \ref{fig:Yexp}).
The bosonization exponent shows a {\it non-monotonic} dependence on $K$
and thus $U$, which the approximate functional RG approach does not capture.
This implies that the maximal asymmetry fixed point is unstable for repulsive
interactions, and stable for sufficiently small attractive ones but turns
unstable again for larger attractive interactions.
(iv) In the mapping of the complex plane onto the Riemann sphere the
$g=\infty$ fixed point (north pole) is part of the projected line of
decoupled chain fixed points and shows the same stability properties
and scaling dimension.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.35\textwidth,clip]{fig25.eps}
\end{center}
\vspace{-.3cm}
\caption[]{Scaling exponents of the Y-junction close to the fixed
points \cite{Barnabe05}.\label{fig:Yexp}}
\end{figure}
The most interesting physics is associated with the perfect junction
fixed points which for $U>0$ each have one stable direction.
If the junction parameters of a non-interacting system at fixed $\phi \neq m \pi$,
$m \in {\mathbb N}_0$ are chosen such that the resulting $\kappa$ lies on
${\mathcal C}(\phi)$, but not on one of the three special points (ii),
$g_{\nu,\nu'} \neq g_{\nu',\nu}$ and the conductance
indicates the breaking of time-reversal symmetry as expected.
Turning on an interaction $U>0$ the ``fine-tuned'' system flows to one
of the perfect chain fixed points with equal ``perfect'' conductances
$2 \pi g_{\nu,\nu'}=4/9$ and $2\pi g_{\nu',\nu}=4/9$. Therefore, at small energy
scales the junction conductance does
{\it no longer indicate the explicit breaking of time-reversal symmetry.}
For generic junction parameters away from ${\mathcal C}(\phi)$
one finds related behavior.
Close to the line of decoupled chain fixed points the relative
difference $|g_{\nu,\nu'}-g_{\nu',\nu}|/(g_{\nu,\nu'}+g_{\nu',\nu})$ scales as a
power law in $\delta_N$ with an exponent given by
$\gamma_o/2$ and thus vanishes if $U>0$. This implies
that $g_{\nu,\nu'}$ and $g_{\nu',\nu}$ {\it become equal} faster than they go
to zero. In that sense for $U>0$ and up to the unstable maximal asymmetry
fixed point, on small scales the conductance does not show the breaking of
time-reversal symmetry--time reversal symmetry is ``restored'' by the
interaction.
Other types of junctions of an arbitrary number of LL wires were studied
using functional RG \cite{Barnabe05a} as well as by the poor man's
fermionic RG \cite{Lal02,Aristov10} and bosonization based approaches
\cite{Nayak99,Chen02}.
\subsubsection{Non-equilibrium transport through a contacted wire}
\label{subsecnonequiwire}
Non-equilibrium functional RG was used to study a finite bias transport
geometry with an impurity-free $N$ site interacting wire contacted to two
non-interacting semi-infinite leads by tunnel barriers modeled by reduced
hopping matrix elements as introduced in subsection \ref{basicmodel}:
$t_{0,1}=(t_L-1)$ and $t_{N,N+1} = (t_R-1)$ \cite{Jakobs07a}.
In equilibrium this model features a local single-particle spectral function
$\rho_j(\omega)$ which close to the chemical potential, in the vicinity of the
contacts, and for repulsive interactions is suppressed:
$\rho_j(\omega) \sim \omega^{\gamma_o}$ \cite{Enss05b}. The
linear conductance behaves as $g(T) \sim T^{\gamma_o}$ which can be understood
from viewing transport as an end-to-end tunneling between a LL and a Fermi
liquid lead and using the sum of two resistances as discussed in subsection
\ref{resotunsec}.
A cutoff scheme which conserves causality to any truncation order
\cite{Jakobs10b} is given by an imaginary frequency cutoff.
The Fermi function of the two leads which can be written as a Matsubara sum
\begin{eqnarray}
f_{L/R}(\omega) & = & \left[ e^{\beta(\omega-\mu_{L/R})} +1 \right]^{-1} \nonumber \\
\label{imagfermi}
& = & \beta^{-1} \sum_{\omega_n} \frac{e^{i \omega_n 0^+}}{i \omega_n - \omega +\mu_{L/R}}
\end{eqnarray}
is replaced by
\begin{eqnarray}
\label{imagcutoff}
f_{L/R}^\Lambda(\omega) = \beta^{-1} \sum_{\omega_n} \frac{\Theta(|\omega_n|-\Lambda)
e^{i \omega_n 0^+}}{i \omega_n - \omega +\mu_{L/R}} \; .
\end{eqnarray}
Details of this procedure including a discussion of the initial conditions
and its relation to the temperature flow scheme \cite{Honerkamp01c} are
presented by \textcite{Jakobs10b}.
Within the lowest-order truncation and after taking the equilibrium limit this
cutoff implemented for Keldysh Green functions yields the same flow equations as
the Matsubara functional RG with the frequency cutoff Eq.\ (\ref{cutfun})
\cite{Jakobs07a,Jakobs10b,Jakobs10c}.
In the presence of a finite bias voltage the level-1 truncation scheme (bare
two-particle vertex) with the cutoff procedure (\ref{imagcutoff}) was applied.
As discussed in Sec.~\ref{subsubsimp}, in equilibrium this is sufficient
to obtain scaling exponents correctly to leading order in $U$.
For weak tunneling
$\Gamma_{L/R} = \pi t_{L/R}^2 \rho_{0} \ll 1$,
with $\rho_{0}$ the
density of states of the disconnected, non-interacting leads taken at the last
lattice site, the flow of the retarded non-equilibrium self-energy matrix
$\Sigma^{{\rm ret},\Lambda}$ is given by a weighted sum of two equilibrium flows
\begin{eqnarray}
\label{nonequiflow}
\frac{d}{d \Lambda} \Sigma^{{\rm ret},\Lambda} =
\sum_{\lambda=L,R} \frac{\Gamma_\lambda}{\Gamma_L + \Gamma_R}
\left[ \frac{d}{d \Lambda} \Sigma^{{\rm eq},\Lambda} \right]_{\mu=\mu_\lambda} \; ,
\end{eqnarray}
where the terms inside the brackets on the right hand side are given by
Eq.\ (\ref{volker:sigmaflowtg0}) with the chemical potential set to $\mu_L$ or $\mu_R$
respectively (and $U^\Lambda \to U$). As discussed in subsection \ref{subsubnum} each
such term leads to an oscillatory slowly decaying self-energy originating at the
inhomogeneity--the tunnel barriers in the present case--and extending into the
interacting part of the wire.
The two chemical potentials $\mu_{L/R}$ imply two different wave numbers $2 k_F^{(L/R)}$.
Because of the weighting factor $\Gamma_\lambda/(\Gamma_L+\Gamma_R)$ the amplitudes
of the two superimposed decaying oscillations are generically different and depend
on the strength of the bare inhomogeneity.
The resulting non-equilibrium effect of two different and
$\Gamma_\lambda$-dependent exponents characterizing the scaling
of the spectral function close to $\mu_L$ and $\mu_R$
goes beyond the naive expectation that the bias voltage
plays the role of an infrared cutoff scale only (see e.g. \textcite{Schoeller09a}).
In Fig.~\ref{fig:nonequiwire} the local spectral function near the left contact
and for a restricted energy range around $\mu_{L/R}$ is shown. Due to the finite
temperature ($T=10^{-4}$) and the finite size of the interacting wire
($N=2 \times 10^4$) the suppression at $\mu_{L/R}$
is incomplete (cut off by $\mbox{max} \{ T,\delta_N \}$), but the difference in
the exponent is still apparent.
A detailed analysis shows that the exponents at $\mu_{L/R}$ are given by
$ \gamma_{L/R} = \Gamma_{L/R} \gamma_o(\mu_{L/R}) / (\Gamma_L + \Gamma_R)$
where the argument in the open boundary exponent $\gamma_o$ indicates that
it depends on the band filling and thus the chemical potential.
After adding a third probe lead these ``non-universal'' exponents can be
measured in a transport experiment \cite{Jakobs07a,Jakobs10b}.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{fig26.eps}
\end{center}
\vspace{-.3cm}
\caption[]{Suppression of the local one-particle spectral weight as a function
of energy near the left contact (at site $5$) of an interacting wire
driven out of equilibrium by a finite bias current. The parameters
are $T = 10^{-4}$,
$N=2 \time 10^4$, $U=0.5$, $t_L=0.075$, $t_R=0.15$, and
$\mu_{L/R}=\pm 0.05$. (Data taken from \textcite{Jakobs07a}.)
\label{fig:nonequiwire}}
\end{figure}
\subsection{Quantum dots}
A spatially confined system featuring a few energy levels is called a quantum dot.
In a transport geometry the dot is coupled to at least two leads.
Quantum dots show interesting physics if all relevant energy scales
(e.g. level-lead couplings and $T$)
are smaller than the level spacing of the isolated system. Due to the
strong confinement the two-particle interaction on the dot cannot be neglected
and leads to phenomena such as Coulomb blockade and the Kondo effect.
\subsubsection{Spin fluctuations}
\label{subsubspinfluc}
In the Kondo regime the physics is dominated by {\it spin
fluctuations.} The virtues and limitations of the functional RG approach to
describe aspects of Kondo physics in and out off equilibrium were extensively
studied within the single-impurity Anderson model and more complex
variants of the latter
\cite{Hedden04,Andergassen06a,Meden06,Karrasch06,Gezzi07,Karrasch07a,Karrasch07b,
Karrasch08a,Karrasch08b,Weyrauch08,Xu08,Karrasch09a,Kashcheyevs09,Eichler09,
Goldstein09,Bartosch09a,Karrasch10b,Jakobs10a,Jakobs10b,Xu10,Schmidt10,Isidori10}.
As the number of correlated degrees of freedom in quantum dots is
small the static truncation used for LLs was extended to contain
all second order
processes including a frequency dependent two-particle vertex and self-energy,
capturing the full real-space as well as spin structure
\cite{Hedden04,Karrasch08a,Jakobs10a,Jakobs10b,Karrasch10a,Karrasch10b}.
In fact, the studies of the single-impurity Anderson model constitute
one of the rare examples of the functional RG approach to correlated Fermi
systems using a complete level-2 truncation (even supplemented by parts of
the six-point vertex through the replacement discussed in the second part
of Sec.\ \ref{sec:truncations}).
Including the frequency dependence clearly improves the results beyond bare
perturbation theory of the same (that is second) order but it is presently not
possible to reach the strong coupling regime in a controlled way.
A discussion of the problems yet to be solved was presented by \textcite{Karrasch08a},
\textcite{Karrasch10a}, \textcite{Karrasch10b}, \textcite{Jakobs10a}, and
\textcite{Jakobs10b}.
An alternative way to include the full frequency dependence was recently
introduced for another impurity model by \textcite{Schmidt11}.
In the following a simple quantum dot model dominated by {\it charge fluctuations}
is discussed.
\subsubsection{Charge fluctuations in non-equilibrium}
\label{subsubchargefluc}
The quantum dot model belongs to the class of spinless models introduced
in subsection \ref{basicmodel}.
For a three site interacting chain ($N=3$) with $U_{1,2}=U_L$ and $U_{2,3}=U_R$
two hopping impurities are located at bonds $(1,2)$ and $(2,3)$: $t_{1,2}=t_L-1$,
$t_{2,3}=t_R-1$. Lattice site 2 constitutes a single-level dot which can be adjusted
in energy by a gate voltage $V_g$ (see Fig.\ \ref{fig:doublebar}).
An electron on this site interacts with lead electrons via a nearest-neighbor
coupling $U_{L/R}$ which are otherwise non-interacting.
Choosing $\nu=1/2$ in Eq.\ (\ref{volker:intdef}), $V_g$=0 corresponds to the
particle-hole symmetric point with dot occupation $\left< n_2 \right> =1/2$.
This model is a lattice realization of the
{\it interacting resonant level model} (IRLM).
The use of a variety of analytical as well as numerical methods
led to a rather complete understanding of the physics of this model in equilibrium
(see e.g. \textcite{Borda07} and references therein).
In addition, the current under a finite bias voltage $\mu_L=V_b/2$ and
$\mu_R = - V_b/2$ was investigated \cite{Borda07,Doyon07,Boulat08}.
Field theoretical methods were applied in the {\it scaling limit} in which all
energy scales are much smaller than the band width $B$.
In the following the focus is on this limit.
Functional RG results for the equilibrium and non-equilibrium properties beyond
the scaling limit including a favorable comparison with recent numerical
time-dependent density-matrix renormalization group data \cite{Boulat08} are
presented by \textcite{Karrasch10a}, \textcite{Karrasch10b}, and
\textcite{Karrasch10c}.
First order perturbation theory in $U_{L/R}$ leads to logarithmic terms in the
self-energy of the form $U_{L/R} \ln{ (t_{L/R}/B)}$
which in the scaling limit become large.
They indicate the appearance of power laws in $t_{L/R}$ with $U_{L/R}$
dependent exponents.
To uncover them requires a treatment which goes beyond perturbation theory.
In the limit of weak to intermediate two-particle interactions a Keldysh
functional RG approach to the IRLM in the level-1 truncation leads to a
comprehensive picture of the physics in and out of equilibrium.
In particular, it allows to identify the relevant energy scales.
For the present model instead of Eq.~(\ref{imagcutoff}) another cutoff
scheme suitable for non-equilibrium \cite{Jakobs10a,Jakobs10b} was implemented
and tested \cite{Karrasch10a,Karrasch10b}.
In this approach each of the three interacting sites is coupled to its own
auxiliary lead--in addition to the coupling of sites 1 and 3 to the physical leads.
The local density of states at the contact points of the auxiliary leads
is assumed to be energy independent (wide band limit) such that the
hybridization is energy independent and forms an additional onsite
``energy'' $i \Lambda$ on each of the three sites.
The auxiliary couplings are then considered as the cutoff and flow from
$\Lambda=\infty$, at which regularization is achieved, down to $\Lambda=0$,
at which the auxiliary leads are decoupled and the original problem is restored.
One can show that in the lowest order truncation and in the equilibrium limit
the Keldysh contour flow equations become equal to the equilibrium ones obtained
using the Matsubara formalism with the (at $T=0$) sharp energy cutoff
Eq.\ (\ref{cutfun}).
Similarly to the imaginary frequency cutoff of subsection \ref{subsecnonequiwire}
it conserves causality even after truncation of the functional RG flow equation
hierarchy.
In addition, in the equilibrium limit this so-called {\it reservoir cutoff}
scheme obeys the KMS relation in any truncation order
\cite{Jakobs10a,Jakobs10b,Jakobs10c}.
In the scaling limit and to lowest order in $U$ only flow equations
for the hydridizations $\Gamma^\Lambda_\lambda$ with initial values
$\Gamma_\lambda^{\rm ini} = \pi \rho_{0} t_\lambda^2$ appear ($\lambda=\mbox{L/R}$);
the flow of the level energies of the sites 1 to 3 is of order $U^2$.
The renormalized hybridizations set the width of the resonance at $V_g=0$.
For $\Lambda$ being smaller than the band width the flow equations for the
rates read ($\Gamma^\Lambda = \Gamma_L^\Lambda +\Gamma_R^\Lambda $)
\begin{eqnarray}
\label{flowIRLMgeneral}
\frac{d \Gamma^\Lambda_\lambda }{d \Lambda} =
-2 \rho_{0} U_\lambda \Gamma^\Lambda_\lambda
\frac{\Lambda + \Gamma^\Lambda}{(\mu_\lambda - V_g)^2 +
(\Lambda + \Gamma^\Lambda)^2} \; .
\end{eqnarray}
They have the approximate solutions
\begin{eqnarray}
\label{renorwidthLR}
\Gamma_\lambda \approx \Gamma^{\rm ini}_\lambda
\left( \frac{\Lambda_0}{\mbox{max} \{|\mu_\lambda-V_g|,\Gamma/2 \} }
\right)^{2 \rho_{0} U_\lambda} \; .
\end{eqnarray}
The scale $\Lambda_0$ is of the order of the band width.
Within the static approximation the current takes the form of the non-interacting
expression with the bare hybridizations $\Gamma_\lambda^{\rm ini}$ replaced by the
renormalized ones
\begin{equation}
\label{eq:curr}
I=\frac{1}{\pi}\frac{\Gamma_L\Gamma_R}{\Gamma}
\left[\text{arctan}\frac{V_b/2-V_g}{\Gamma}+
\text{arctan}\frac{V_b/2+ V_g}{\Gamma}\right] \; .
\end{equation}
It turns out to be useful \cite{Karrasch10c,Andergassen11} to introduce the two
scales $T_u^\lambda = \Gamma_\lambda^{\rm ini} \Lambda_0/T_u$, with $T_u = T_u^L + T_u^R$,
and the asymmetry parameter $c^2 = T_u^L/T_u^R$. The same flow equation
can be derived using the so-called real-time RG in frequency space
\cite{Schoeller09a,Karrasch10c,Andergassen11}.
Within this approach also the relaxation into the steady state was analyzed
in detail \cite{Karrasch10c,Andergassen11}.
We first review the results obtained for left-right symmetric model with
$t_L = t_R = t'$ and $U_L = U_R =U$ as well as particle-hole symmetry
$V_g=0$ \cite{Karrasch10b,Karrasch10c,Andergassen11}.
From Eq.\ (\ref{renorwidthLR}) it follows that in this case the maximum
of either $|\mu_\lambda|=|V_b|/2$ or $\Gamma$ itself cuts off the RG flow.
The charge susceptibility $\chi=-\left. d \left<n_2 \right>/d V_g\right|_{V_g=0}$ is
directly given by the renormalized width $\chi^{-1} = \pi \Gamma$, which at $V_b=0$ and
to leading order in $U$ ($\Gamma_\lambda \to \Gamma_\lambda^{\rm ini}$ on the right
hand side of Eq.\ (\ref{renorwidthLR})) gives the scaling relation
\begin{eqnarray}
\label{scalchi}
\chi \sim (\Gamma^{\rm ini})^{\alpha_\chi-1} \; , \;\;\;
\alpha_\chi=2\rho_{0} U + {\mathcal O}(U^2) \; .
\end{eqnarray}
In the non-interacting case $\chi \sim (\Gamma^{\rm ini})^{-1}$ as expected.
From Eq.\ (\ref{eq:curr}) it follows that the current for
$T_u^\lambda = \Gamma_\lambda \ll V_b \ll B$ is given by
\begin{eqnarray}
\label{scalI}
I \sim \Gamma \sim V_b^{- \alpha_I} \; , \;\;\;
\alpha_I=2\rho_{0} U + {\mathcal O}(U^2) \; .
\end{eqnarray}
Equations (\ref{scalchi}) and (\ref{scalI}) were also obtained using other
approaches \cite{Doyon07,Borda07,Boulat08} and suggest that the bias voltage
merely plays the role of an additional infrared cutoff, besides e.g. $\Gamma$
or temperature \cite{Borda10}.
That this is in general not the case is nicely shown by a functional RG treatment
{\it away} from particle-hole and/or left-right symmetry
\cite{Karrasch10b,Karrasch10c,Andergassen11}.
The differential conductance $g= dI / dV_b$ has a maximum when $V_g$ crosses
the chemical potential at $V_g = \pm V_b/2$ \cite{Karrasch10c,Andergassen11}.
In the on-resonance case the current for $V \gg \Gamma$ reads
\begin{equation}
\label{eq:IVon}
I(V_b) \approx \frac{\Gamma_L \Gamma_R}{2 \Gamma}
= T_u\frac{\left(\frac{T_u}{\Gamma}\right)^{2 \rho_{0} U_L}
\left(\frac{T_u}{V_b}\right)^{2 \rho_{0} U_R}}
{c\left(\frac{T_u}{\Gamma}\right)^{2 \rho_{0} U_L}
+\frac{1}{c}\left(\frac{T_u}{V_b}\right)^{2 \rho_{0} U_R}}\frac{c}{1+c^2} \; .
\end{equation}
The bias voltage dependence is clearly more involved than
in Eq.\ (\ref{scalI}).
In particular, simple power-law scaling with exponent $-2 \rho_{0}
U_R$ is only recovered in the
extreme limits of either $V_b \ggg T_u$ or $c\gg1$ \cite{Andergassen11}
as the exponent of the second term in the denominator $2 \rho_{0} U_R$ is small.
Off resonance (e.g.\ at $V_g=0$) and for $V_b \gg \Gamma$ the current is given by
\begin{equation}
\label{eq:IVoff}
I(V_b) \approx
T_u\frac{\left(\frac{T_u}{|V_b/2- V_g|}\right)^{2 \rho_{0} U_L}
\left(\frac{T_b}{|V_b/2 +V_g|}\right)^{2 \rho_{0} U_R}}
{c\left(\frac{T_u}{|V_b/2 - V_g|}\right)^{2 \rho_{0} U_L}
+ \frac{1}{c}\left(\frac{T_u}{|V_b/2 + V_g|}\right)^{2 \rho_{0} U_R}}
\frac{2 c}{1+c^2} \; .
\end{equation}
The more involved role of $V_b$ is again apparent.
A power law is obtained in the above studied left-right symmetric case or for very
strong left-right asymmtery ($c \ll 1$ or $c\gg1$) \cite{Andergassen11}.
This concludes the analysis of the IRLM which shows that the functional RG can be a
tool to obtain a comprehensive picture of the equilibrium and steady-state
non-equilibrium physics of a dot model dominated by charge fluctuations.
The approach allows for an unbiased analysis of the non-equilibrium rates and
cutoff scales.
\section{CONCLUSION}
\label{sec:VII}
\subsection{Summary}
The functional RG has proven to be a valuable source of new
approximation schemes for interacting fermion systems.
The heart of the method is an exact flow equation, which
describes the flow of the effective action as a function of
a suitable flow parameter.
The flow provides a smooth evolution from the bare action
to the final effective action from which all properties of
the systems can be obtained.
Approximations are obtained by truncating the effective action.
In many cases, rather simple truncations turned out to capture
rather complex many-body phenomena.
Compared to the traditional resummations of perturbation
theory these approximations have the advantage that infrared
singularities are treated properly due to the built-in RG
structure.
Approximations derived in the functional RG framework can be
applied directly to microscopic models, not only to
renormalizable effective field theories.
Remarkably, the functional RG reviewed here as a computational
tool is very similar to RG approaches used by mathematicians to
derive general rigorous results for interacting fermion systems.
We have dedicated a large portion of this review to general
features of the functional RG method for interacting Fermi
systems (Sec.~II).
After defining the relevant generating functionals, we have
presented a self-contained derivation of the exact functional
flow equation and its expansion leading to an exact hierarchy
of flow equations for vertex functions.
We have reviewed the different choices of flow parameters
used so far, along with their advantages and disadvantages.
Truncations and their justification by power-counting have
been discussed in detail for translation invariant bulk
systems, with links to the closely related mathematical
literature.
In Secs.~III-VI we have reviewed applications of the functional
RG to specific systems.
Most of the approximations used in these sections are based on
relatively simple truncations involving only the flow of the
two-particle vertex and/or the self-energy.
Nevertheless a rich variety of phenomena associated with low
energy singularities and instabilities is captured.
Instead of summarizing the content of each section, let us merely
highlight some distinctive features.
Sec.~III reviews functional RG work on the stability analysis of
two-dimensional electron systems with competing instabilities.
The main advantage of the functional RG based one-loop computation
of the two-particle vertex, compared to other weak-coupling
approximations, is that particle-particle and particle-hole channels
are treated on equal footing, such that there is no artificial bias
due to a selection or a different treatment of channels.
In the conventional many-body framework a summation of all parquet
diagrams would be required to achieve this, but a solution of the
parquet equations is extremely difficult.
Spontaneous symmetry breaking, the topic of Sec.~IV, can be treated
either by a purely fermionic flow, or by coupled flow equations
for the fermions and a Hubbard-Stratonovich field for the order
parameter. It seems that a comprehensive treatment of all relevant
fluctuation effects related to symmetry breaking can be achieved.
Applications of the fRG to quantum criticality, reviewed in Sec.~V,
have begun only recently. Approximations beyond Hertz-Millis
theory can be obtained from non-perturbative truncations of the
effective order parameter action, or by treating fermions and
order parameter fluctuations in a coupled flow instead of
integrating the fermionic degrees of freedom at once.
While the applications reviewed in Secs.~III-V address translation
invariant bulk systems, the purpose of Sec.~VI is to show how the
functional RG can be fruitfully applied to inhomogeneous systems
such as quantum wires and quantum dots -- in thermal equilibrium
and also in a non-equilibrium steady state.
A strikingly simple truncation of the flow equation hierarchy
turned out to describe a wealth of non-trivial quantum transport
properties characterized by low-energy power-laws and complex
crossover phenomena.
\subsection{Future directions}
The number of functional RG based works on interacting Fermi
systems has increased steadily over the last decade, but the
possibilities opened by this approach are far from being
exhausted.
There are many opportunities and challenges concerning both
fundamental developments of the method and the extension to
a broader range of systems.
On the methodological side there are a number of open issues.
In systems with an instability of the normal metallic state,
the flow of the effective interactions is not yet fully
understood, even on the level of truncations involving only
the two-particle vertex and the self-energy, since a
faithful parametrization of singular momentum and frequency
dependences of the vertex is not easy.
The most outstanding challenge is probably to identify
accurate and computable truncations of the exact flow equation
for strongly interacting systems such as systems close to a
Mott metal-insulator transition.
It is clear that three-particle and higher order vertices
cannot be discarded in a strongly interacting system.
However, they will usually not lead to qualitative changes
such as new singularities.
Hence, there is hope that the contribution from many-body
vertices can be absorbed in the structure appearing already
on the two-particle level.
After all, many strong coupling phenomena, including the Mott
transition, consist essentially in the formation of two-particle
bound states.
To capture effects related to strong local correlations, such
as the Mott transition, one may also try to treat higher order
vertices in a local approximation. This would make a link to
the dynamical mean-field theory (DMFT), where all vertices,
including the self-energy, are approximated by local
functions \cite{Georges96}.
For systems with strongly interacting order parameter
fluctuations there are already a number of non-perturbative
approximations for bosonic actions on the market.
The local potential approximation presented in Sec.~V is
only the simplest one. It can be extended by taking non-local
contributions into account, either in a derivative expansion
\cite{Berges02}, or by including the full momentum or
frequency dependence up to a certain level in the hierarchy
\cite{Blaizot05}.
Such approximations may be very useful for studying
incommensurate density wave instabilities in cases where the
modulation vector of the density wave can be determined only
after taking fluctuations into account.
Recently, the functional RG was extended to a real time (or
real frequency) Keldysh functional RG which can be used for
studying correlated Fermi systems in non-equilibrium
\cite{Jakobs03,Gezzi07,Jakobs10b,Karrasch10b}.
First applications, partly reviewed in Sec.~VI, indicate that
also for these type of problems the functional RG constitutes a
useful tool of outstanding flexibility.
So far only non-equilibrium steady states were studied.
To investigate a time evolution is technically straightforward
but requires a significantly increased computational effort or
additional approximations.
With few exceptions, applications of the functional RG to
interacting Fermi systems have so far been limited to purely
fermionic one-band systems.
There are many extensions of this restricted class of
systems, where the flexibility of the functional RG can be
fruitfully used in the future.
Multi-band models have been studied already for the pnictide
superconductors, but there are many more and qualitatively
different models for transition metal oxides with orbital
degrees of freedom.
One may include phonons and analyze the electron-phonon
interaction effects beyond the Eliashberg theory.
Allowing for disorder one may study the complex interplay
of interaction and disorder effects.
It is not hard to generalize the exact flow equations for
the extensions listed above. The interesting task is then
to devise suitable truncations.
Last but not least, the functional RG is an ideal many-body
tool to be combined with {\em ab initio}\/ band structure
calculations.
A lot of work in the last 15 years has been dedicated to
the ab initio calculation of correlated electron materials
with the DMFT \cite{Kotliar06,Anisimov10}.
As in DMFT, an arbitrary band structure can be used as
input for a functional RG calculation.
Furthermore, one can easily implement non-local potentials
and non-local two-particle interactions.
\begin{acknowledgments}
We are very grateful for fruitful collaborations and/or discussions
with S. Andergassen, J. Bauer, D. Baeriswyl, C. Castellani,
A. Chubukov, C. Di Castro, J. von Delft, A. Eberlein, T. Enss,
J. Feldman, R. Gersch, H. Gies, W. Hanke,
C. Husemann, S. Jakobs, P. Jakubczyk,
C. Karrasch, A. Katanin, H. Kn\"orrer, P. Kopietz, D.-H. Lee,
B. Obert, J. Pawlowski, C. Platt, M. Pletyukhov, M. Rice,
D. Rohe, A. Rosch, H. Schoeller, P. Strack, S. Takei, R. Thomale,
E. Trubowitz, C. Wetterich, P. W\"olfle, and H. Yamase.
All of us greatly benefitted from the DFG research group
{\em Functional renormalization group in correlated fermion systems}
(FOR 723).
\end{acknowledgments}
\begin{appendix}
\section{Wick-ordered flow equations}
\label{sec:wick}
In this appendix we present a derivation of Wick ordered
flow equations for fermions, which have been used for
calculations of instabilities and symmetry-breaking in
the two-dimensional Hubbard model.
{\em Wick ordered} $m$-particle functions $W_m^{\Lam}$ are
generated from the Wick-ordered effective interaction
\cite{Salmhofer99}
\begin{equation}
\cW^{\Lam}[\chi,\chib] =
e^{\Delta_{\bar G_0^{\Lam}}} \, \cV^{\Lam}[\chi,\chib] \; .
\end{equation}
The exponent in the Wick-ordering factor is the functional
Laplacian
$\Delta_{\bar G_0^{\Lam}} =
(\partial_{\chi}, \bar G_0^{\Lam} \partial_{\chib})$ with
$\bar G_0^{\Lam} = G_0 - G_0^{\Lam}$.
The Wick-ordered interaction converges to $\cV$ for
$\Lam \to 0$, since $\bar G_0^{\Lam}$ vanishes in that limit.
However, the flow equations for $\cW^{\Lam}$ and the corresponding
$m$-particle functions differ from those for $\cV^{\Lam}$.
The flow equation for the generating functional $\cW^{\Lam}$
reads \cite{Salmhofer99}
\begin{equation} \label{rgewick}
\frac{d}{d\Lam} \cW^{\Lam} =
\frac{1}{2} \, e^{\Delta^{\rm diff}_{\bar G_0^{\Lam}}} \,
\Delta_{\dot{\bar G}_0^{\Lam}}^{\rm diff} \,
\cW^{\Lam} \, \cW^{\Lam} \; ,
\end{equation}
where the superscript ''diff'' indicates that the Laplacian
takes one derivate on the first, and the other on the second
factor $\cW^{\Lam}$ on the right-hand side.
This equation is obtained as follows. Using the definition of
$\cW^{\Lam}$ and the flow equation for $\cV^{\Lam}$, one can
write
\begin{eqnarray}
\frac{d}{d\Lam} {\cW^{\Lam}} &=&
\frac{d}{d\Lam}
\big( e^{\Delta_{\bar G_0^{\Lam}}} \cV^{\Lam} \big)
\nonumber \\
&=& \Delta_{\dot{\bar G}_0^{\Lam}}
e^{\Delta_{\bar G_0^{\Lam}}} \cV^{\Lam}
\nonumber \\
&+& e^{\Delta_{\bar G_0^{\Lam}}}
\big( - \Delta_{\dot{\bar G}_0^{\Lam}} \cV^{\Lam} +
\frac{1}{2} \, {\Delta_{\dot{\bar G}_0^{\Lam}}^{\rm diff}} \,
\cV^{\Lam} \cV^{\Lam} \big)
\nonumber \\
&=& \frac{1}{2} \, e^{\Delta_{\bar G_0^{\Lam}}} \,
{\Delta_{\dot{\bar G}_0^{\Lam}}^{\rm diff}} \,
\cV^{\Lam} \cV^{\Lam} \; .
\nonumber
\end{eqnarray}
Using the decomposition
$\Delta_{\bar G_0^{\Lam}} =
\Delta_{\bar G_0^{\Lam}}^{\rm factor 1} +
\Delta_{\bar G_0^{\Lam}}^{\rm factor 2} +
{\Delta_{\bar G_0^{\Lam}}^{\rm diff}} \;$
(when acting on a product), this yields
$\partial_{\Lam} {\cW^{\Lam}} =
e^{\Delta_{\bar G_0^{\Lam}}^{\rm diff}} \,
\frac{1}{2} \, {\Delta_{\dot{\bar G}_0^{\Lam}}^{\rm diff}}
\big( e^{\Delta_{\bar G_0^{\Lam}}} \cV^{\Lam} \big) \,
\big( e^{\Delta_{\bar G_0^{\Lam}}} \cV^{\Lam} \big)$
and thus Eq.~(\ref{rgewick}).
\vskip 1mm
Expanding in powers of Grassmann fields and comparing
coefficients, one obtains a hierarchy of flow equations for
the $m$-particle functions $W^{(2m)\Lam}$, which is illustrated
diagrammatically in Fig.~\ref{fig:floweqwick}.
\begin{figure}[ht]
\centerline{\includegraphics[width = 7cm]{fig27.eps}}
\caption{Diagrammatic representation of the flow equations for
the effective $m$-particle interactions $W^{(2m)\Lam}$ in the
Wick-ordered version of the functional RG; the internal line with
a dash corresponds to $\partial_{\Lam} \bar G_0^{\Lam}$, the
others to $\bar G_0^{\Lam}$; all possible pairings leaving
$m$ ingoing and $m$ outgoing external legs have to be summed.}
\label{fig:floweqwick}
\end{figure}
The line with the dash is due to contractions generated by
$\Delta_{\dot{\bar G}_0^{\Lam}}^{\rm diff}$ in Eq.~(\ref{rgewick}),
the other lines are generated by the exponential of
$\Delta^{\rm diff}_{\bar G_0^{\Lam}}$.
Note that the right-hand side of the Wick-ordered flow equations
is bilinear in the effective interactions, and no tadpole
terms appear. Note also that the propagators connecting the
vertices have support for energies at and \emph{below} the
cutoff scale $\Lam$, such that the integration region shrinks
as $\Lam$ decreases.
One might worry that the low-energy propagators lead to
infrared divergences even for $\Lam > 0$. This is not the
case, as can be seen from the general infrared power-counting
analysis presented by \textcite{Salmhofer99}.
\section{Details of power counting}
\label{app:powercounting}
\subsection{Propagator bounds}\label{appssec:propbounds}
\newcommand{\eta}{\eta}
Here we show, using properties of the dispersion function and the Fermi
surface, that
\begin{equation}\label{propbounds}
\snorm{\Lambda} \le
a + b \log \sfrac{\Lambda_0}{\Lambda}
\quad \mbox{ and } \quad
\Vert G^\Lambda\Vert \le c \; \Lambda^{-1}\;
\end{equation}
where $a, b$ and $c$ are constants that do not depend on $\Lambda$.
In absence of Van Hove singularities, $b = 0$.
We first consider the case where the self-energy effects
are not taken into account (they are discussed in Subsection \ref{sssec:selfenergy}).
Then one can simply take
\chi^\Lambda (k) = \chi_> \left(
\sfrac{k_0^2+\xi_{\bk}^2}{\Lambda^2}
\right)
where $\chi_> (\eta)$ is a fixed (i.e.\ $\Lambda$-independent)
increasing function that vanishes at least linearly at $\eta=0$,
tends to $1$ as $\eta \to \infty$ and satisfies
$\chi'_>(\eta) \le \eta^{-2}$ for large $\eta$.
We can then verify (\ref{propbounds})
by scaling, as follows. The full propagator is $G^\Lambda (k) =(i k_0 -
\xi_{\bk})^{-1} \chi^\Lambda (k)$, so
|G^\Lambda (k)|
\le
\frac{c}{\Lambda}
$
where $c = \max \{ \eta^{-1} \chi_> (\eta^2): \eta > 0\}$ is finite.
The single-scale propagator is
\begin{equation}
S^\Lambda (k)
=
- \frac{2}{\Lambda^3} (ik_0 + \xi_{\bk}) \; \chi'_> \left(
\sfrac{k_0^2+\xi_{\bk}^2}{\Lambda^2}
\right) .
\end{equation}
Using that the Matsubara sum is a Riemann sum for the convergent integral
of $S^\Lambda$ over $k_0$
and introducing the density of states
$N(E) = \int d^dk \, \delta (\xi_{\bk} - E)$,
we get
\begin{eqnarray}
\snorm{\Lambda}
\le
\frac{4}{\Lambda^3}
\int dk_0 \int dE \, N(E) \sqrt{k_0^2 + E^2} \,
\chi'_> \left(
\sfrac{k_0^2+E^2}{\Lambda^2}
\right) , \hskip 3mm
\end{eqnarray}
where the $4$ instead of $2$ gives a (crude) bound for the change from the
Matsubara sum to the integral for large enough $\beta$.
Changing variables to $\rho =(k_0^2 + E^2)^{\frac12} $ and a polar angle
$\varphi$, we obtain
\begin{equation}
\snorm{\Lambda}
\le
\frac{4}{\Lambda^3}
\int_0^\infty \rho^2 d \rho\; \chi'_> \left(
\sfrac{\rho^2}{\Lambda^2}
\right)
\int_0^{2\pi} d\varphi\; N(\rho \cos \varphi).
\end{equation}
If the density of states $N$ is bounded, using $N(E) \le N_0$ and scaling out
$\Lambda$ implies $\snorm{\Lambda} \le a$, with
$a = 8 \pi N_0 \int \rho^2 \chi'_> (\rho^2) d\rho < \infty$.
In presence of a Van Hove point on the Fermi surface,
$N$ stays bounded in dimensions $d \ge 3$, but diverges logarithmically
for $d=2$. In this case, the $\varphi$-integral contributes an additional factor $\log
\Lambda$.
Thus (\ref{propbounds}) holds. The hypotheses on $\chi_>$ are satisfied
in particular for the standard strict cutoff functions that vanish identically
near $\eta=0$, and which are identically $1$ for $\eta \ge 1$.
For such cutoffs, the single-scale propagator is nonvanishing only in a
``momentum shell'' of thickness $\Lambda$ around the Fermi surface,
and the above bounds can also
be obtained by estimating the $\bk$-space volume of this shell
(see also Section \ref{ssec:imppoco}).
\subsection{Power counting}
Here we prove (\ref{poco}) to all orders in the running coupling expansion.
All terms on the right hand side of the flow equation for $\Gamma_r^{(2m)\Lambda}$
are of the form
\begin{equation}
\frac12 \, {\rm tr} \left(
\bS^\Lambda \,
{\bf P}^\Lambda
\right)({\underline{k}},{\underline{\sigma}})
=
\frac12
\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} l
\sum_{\alpha,\alpha'}
S^\Lambda_{\alpha,\alpha'} (l) \,
\hat P^\Lambda_{\alpha,\alpha'} ({\underline{k}},{\underline{\sigma}};l,-l) ,
\end{equation}
where $\hat P^\Lambda = \hat\Gamma_r^{(2m+2)\Lambda}$ in the first term of (\ref{mqexp})
and given by the other summands in (\ref{mqexp}) for the other terms,
and $\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} l = \frac1\beta \sum\limits_{l_0} \int \frac{d^d l}{(2\pi)^d}$ contains
both frequency and momentum summations.
Taking the absolute values inside the sum and estimating the factor
$\hat P^\Lambda$ by its maximum $\Vert P^\Lambda \Vert$,
we obtain
\begin{equation}
\Vert
\sfrac12 \; {\rm tr}\; (
\bS^\Lambda\;
{\bf P}^\Lambda)
\Vert
\le
\snorm{\Lambda}
\;
\Vert P^\Lambda \Vert
\end{equation}
with $\snorm{\Lambda} = \max\limits_\alpha \sum\limits_{\alpha'}\;
\int {\mkern2mu\mathchar'26\mkern-2mu\mkern-9mud} k \;
|\hat S^\Lambda_{\alpha,\alpha'} (k) |$.
The second simple inequality that we shall use is that
$\Vert P_1 \ldots P_n \Vert \le \Vert P_1 \Vert \ldots \Vert P_n \Vert$.
It implies bounds for all ${\cal L}^\Lambda_p$-contributions in terms of
$\Vert G^\Lambda\Vert$ and $\Vert \Gamma_{r_q}^{(2m_q)\Lambda} \Vert$,
so that
$\Vert
\sfrac{d}{d\Lambda} \Gamma_r^{(2m)\Lambda}
\Vert
$
is bounded by
\begin{eqnarray}\label{inductivebounds}
&&
\snorm{\Lambda} \;
\Big[
\Vert \Gamma_r^{(2m+2)\Lambda}\Vert
+
\Vert \Four{\Lambda} \Vert\;
\Vert G^\Lambda \Vert\;
\Vert \Gamma_{r-1}^{(2m)\Lambda}\Vert
\nonumber \\
&+&
\sum_{p \ge 2}
\Vert G^\Lambda \Vert^{p-1}
\sum {}'\;
\Vert \Gamma_{r_1}^{(2m_1)\Lambda}\Vert\;
\ldots
\Vert \Gamma_{r_p}^{(2m_p)\Lambda}\Vert
\Big] . \hskip 5mm
\end{eqnarray}
The power counting is now determined by $\snorm{\Lambda}$ and $\Vert G^\Lambda\Vert$.
Given (\ref{inductivebounds}) and (\ref{propbounds}), the proof of (\ref{poco})
is an effortless induction argument. The inductive scheme proceeds in the
standard way of \cite{Polchinski84}, namely upwards in $r \ge 1$ and at
fixed $r$, downwards in $m$, starting at $m=r$, where $\Gamma_r^{(2m+2)\Lambda}=0$.
The induction start $r=1$ is trivial.
Let $r \ge 2$ and assume (\ref{poco}) to hold for all $(r',m')$ with
$r' < r$ and for $r'=r$, $m' > m$.
The right hand side of (\ref{inductivebounds}) contains only terms to which
the inductive hypothesis (\ref{poco}) applies. Inserting it, using (\ref{propbounds}),
and collecting powers in the form $1 - p + \sum_q (2-m_q) = 1-m$
and
$\sum_{q=1}^p (r_q-m_q+1) = r-m$,
we obtain
\begin{equation}\label{Gadotbound}
\Vert
\sfrac{d}{d\Lambda} \Gamma_r^{(2m)\Lambda}
\Vert
\le
\tilde\gamma_r^{(2m)}
{\snorm{\Lambda}}^{r-m+1}
{f_\Lambda}^r \Lambda^{1-m}
\end{equation}
where the constant $\tilde\gamma_r^{(2m)}$ is a weighted
sum of products of the $\gamma_{r_q}^{(2m_q)}$.
We now use the initial condition $\Gamma_r^{(2m)\Lambda} = 0$ to write
$\Gamma_r^{(2m)\Lambda} = - \int_\Lambda^{\Lambda_0} d \ell \;
\frac{d}{d \ell}\Gamma_r^{(2m)\ell}\;$,
take the norm of this equation, and use (\ref{Gadotbound}).
This gives
\begin{equation}
\Vert
\Gamma_r^{(2m)\Lambda}
\Vert
\le
\tilde\gamma_r^{(2m)}
\int_\Lambda^{\Lambda_0} d \ell \;
{\snorm{\ell}}^{r-m+1}
{f_\ell}^r \ell^{1-m}
\end{equation}
By definition, $f_{\Lambda} \ge f_{\Lambda'}$
if $\Lambda \le \Lambda'$, so
$f_\ell \le f_\Lambda$ for all
$\ell $ in the integration interval.
Thus the last integral is bounded by
$\tilde\gamma_r^{(2m)}
{f_\Lambda}^r
\int_\Lambda^{\Lambda_0} d \ell \;
{\snorm{\ell}}^{r-m+1}
\ell^{1-m}$.
Because $\snorm{\ell}$ is at most logarithmic in $\ell$, and $m \ge 3$,
$\int_\Lambda^{\Lambda_0} d \ell \;
{\snorm{\ell}}^\alpha
\ell^{1-m}
\le
K \Lambda^{2-m} \snorm{\Lambda}^\alpha $
with a constant $K$ that depends on $\alpha$ and $m$.
This, together with an appropriate choice of $\gamma_r^{(2m)}$,
completes the induction step.
For $m=2$, doing the last integral increases the power of the logarithm
by one. This case is discussed in more detail in Subsection \ref{ssec:imppoco}.
For $m=1$, the self-energy term, the same simple bound gives
$\Vert \frac{d}{d\Lambda} \Sigma^\Lambda \Vert
\le
\snorm{\Lambda}\; f_\Lambda$, so the integral gives a contribution
of order $f_\Lambda$. When a counterterm is used to keep the Fermi surface fixed,
the initial condition for $\Sigma^\Lambda$ at $\Lambda=\Lambda_0$
is given by the counterterm, which needs to be adjusted such that
at low scales $\Lambda$, $\Sigma^\Lambda (0, \bk) = O(\Lambda)$
whenever $\xi_{\bk} = 0$. This leads to the self-consistency relation mentioned
in section \ref{sssec:selfenergy}.
A similar proof can be given in the Wick ordered scheme \cite{Salmhofer98b};
it is even simpler because the double induction used here is replaced by
single induction on $r$.
A crucial point in obtaining (\ref{Gadotbound}) is that all the
dependence on $p$ and on the $m_q$ drops out when the power
of $\Lambda$ is collected. It is this property that makes
many-fermion models with short-range interactions
{\em renormalizable} in the strict quantum-field-theoretical sense.
The classification in relevant, marginal and irrelevant
terms now also becomes apparent
because for $m \ge 3$, the $\Gamma^{(2m)\Lambda}_r$
grow as $\Lambda$ decreases:
suppose we add an additional $(2m \ge 6)$-point interaction vertex
$\tilde V^{(2m)\Lambda_0}$ of order $1$ to the initial interaction
at $\Lambda_0$. Its insertion at a lower scale $\Lambda$ is
a factor $(\frac{\Lambda}{\Lambda_0})^{m-2}$ smaller than that
of the effective $2m$-point-vertex created by the two-particle
interaction. Thus the influence of $\tilde V^{(2m)\Lambda_0}$
wanes at low scales -- it is irrelevant.
A simple adaptation of the above inductive argument indeed shows
that the inclusion of such additional terms
with $m \ge 3$ in the interaction at $\Lambda_0$
changes only prefactors in the power counting bounds.
For $m=2$, this suppressing factor is absent, so that these terms
are marginal (the more detailed analysis of Section
\ref{ssec:imppoco} shows how to separate the marginally relevant
from the marginally irrelevant terms).
Moreover, it is clear that this power counting breaks down when
$\Gamma^{(4)\Lambda}$ develops singularities as a function
of $\bk$ and $\omega$, because then $f_\Lambda = \infty$.
Finally, for $m=1$, the scale derivative of the self-energy obtained
by the above argument is of order $\Lambda^{1-m}$,
as in (\ref{Gadotbound}),
but since $m=1$, this integrates to $O(1)$ instead of $O(\Lambda^{2-m})$
-- this term is relevant.
To get its size back to $O(\Lambda^{2-m})$ in the momentum shell
where $|\xi(\bk)| \sim \Lambda$, one needs a cancellation by
a counterterm, as described briefly in Section \ref{sssec:selfenergy}.
In the Taylor expansion
required to do the cancellation, the derivative of the self-energy
appears. By the above power counting, this is a marginal term.
In the Luttinger model, it is really marginal and
causes the anomalous exponents.
For curved Fermi surfaces in $d \ge 2$ dimensions, it is seen to be
irrelevant by the arguments discussed in
Section \ref{sssec:OLloops}.
There is a hard problem hidden in the recursion of the constants
$\gamma_r^{(2m)}$. In the recursion described above, the number of terms
that gets added corresponds to the number of Feynman graphs with $r$
vertices, which grows factorially in $r$, so that the bound obtained in
this way
for $\gamma_r^{(2m)}$ and $c_r^{(2m)}$ is of order $r!$. If saturated,
it would lead to a convergence problem.
However, due to the fermionic antisymmetry, sign cancellations in the sum
over Feynman diagrams prevent this factorial from arising. For proofs,
we refer interested readers to the literature (see
\textcite{FMRT92,FKT98,Disertori00,Salmhofer00,FKT02,Pedra08}
and references therein). In their application to propagators with Fermi
surfaces, these proofs also provide a rigorous basis
for the use of Fermi surface patches (first used in \cite{FMRT92}
and there called ``sectors''). Patching the Fermi surface has become
an essential tool also in applications, see Section \ref{sec:III}.
In a typical lattice model, the kinetic energy per particle is bounded, so that
the flow is usually started at the highest value (the bandwidth) of the
kinetic energy, $\Lambda_0$.
The $\chi^\Lambda$ we used here also cuts off large frequencies.
Thus the starting interaction is
in this case one where the degrees of freedom with frequencies $|k_0|$
above $\Lambda_0$ have already been integrated over.
This starting action can be obtained by convergent perturbation theory,
see \textcite{Pedra08}.
\subsection{Improved power counting}\label{ssec:imppoco}
This is a refinement of power counting, valid in a large class of bulk
fermion systems in $d \ge 2$ \cite{FT1,Shankar94,FST1,FST2,FST3,FST4}.
It is the deeper reason behind the emergence of Fermi liquid behaviour
and of dominant Cooper pairing tencencies in weakly coupled standard
fermion systems, and it provides a precise link between Fermi surface
geometry and scaling properties of the effective $m$-particle vertices
in general.
We discuss this in the absence of self-energy effects,
to bring out the main effects as clearly as possible.
(The self-energy changes the Fermi surface; if the interacting
Fermi surface is regular, the following analysis remains essentially
unchanged.) We also assume a strict cutoff function,
i.e.\ $\chi_> (\eta ) = 0 $ for $\eta \le (1-\delta)^2$, where
$0 < \delta < 1/2$ is fixed, and $\chi_> (\eta)=1$ for $\eta \ge 1$.
Again, this choice is not essential; it just simplifies the discussion.
\subsubsection{Effects of curvature on power counting}
The integral
$I_\Lambda (k) = \int d p_0 d^dp |S^\Lambda (p)| \, |G^\Lambda (\pm p+k)|$
arises from the trace on the right hand side of the RG equation
when all effective vertices and all but one of the propagators $G^\Lambda$
have been estimated by their maximal values. It
thus determines the maximal possible value of a term on the right hand
side of the RG equation, where the dependence on one external momentum
is kept.
In particular, $I_\Lambda$ is directly relevant for the one-loop contributions
to the flowing four-point vertex. The power counting done above corresponds
to the estimate $I_\Lambda (k) \le \Vert G^\Lambda\Vert \, \snorm{\Lambda}
\le \Lambda^{-1} \snorm{\Lambda}$, so that $\int_\Lambda I_\lambda d\lambda$
grows logarithmically in $\Lambda$ for small $\Lambda$.
Since $\Sigma^\lambda =0$,
$S^\Lambda (k) = (ik_0 - \xi_{\bk})^{-1} \, \partial_\Lambda \chi^\Lambda (k)$
and
\begin{equation}\label{Gint}
G^\Lambda (k)
=
\frac{\chi^\Lambda (k)}%
{i k_0 - \xi_{\bk}}
=
G^{\Lambda_0} (k)
-
\int_\Lambda^{\Lambda_0} d \lambda\;
S^\lambda (k) .
\end{equation}
The term $G^{\Lambda_0}$ is nonvanishing at large frequencies, but
not important
($\Vert G^{\Lambda_0}\Vert \le \Lambda_0^{-1}$, hence a factor
$\Lambda/\Lambda_0$ smaller than $\Vert G^{\Lambda}\Vert $
when $\Lambda$ gets small. Thus
$\int d p_0 d^dp |S^\Lambda (p)| \, |G^{\Lambda_0} (p+k)|
\le \snorm{\Lambda} \Lambda_0^{-1}$, hence its integral over $\Lambda$
is bounded by a constant).
The derivative of the strict cutoff function vanishes unless
$\Lambda (1-\delta) \le |ik_0 - \xi_{\bk}| \le \Lambda$, so
$S^\Lambda (p)$ vanishes unless $|p_0| \le \Lambda$ and
$\bp $ is in the momentum space shell
$\mshell{\Lambda} = \left\{ \bk: |\xi_{\bk}| \le \Lambda \right\}$,
and there, $|S^\Lambda (p) |\le \frac{1}{\Lambda^2}$.
The $p_0$-sum in $I_\Lambda$ gives at most $2 \Lambda$, and the
$\bp$-integral gives the $d$-dimensional volume of the intersection
$\mshell{\Lambda} \cap (\bk \pm \mshell{\lambda})$
of two momentum space shells, where one is shifted by $\bk$.
It follows that
\begin{equation}\label{Iint}
I_\Lambda (k)
\le
O(\Lambda_0^{-1})
+
\frac{2}{\Lambda}
\int_\Lambda^{\Lambda_0} \frac{d\lambda}{\lambda^2} \;
\mbox{vol}_d \left(\mshell{\Lambda} \cap (\bk \pm \mshell{\lambda})\right)
.
\end{equation}
This links the scaling behaviour of terms in the RG equation to
the geometric properties of the Fermi surface.
Obviously, the volume of the intersection is at most
as large as the volume of $\mshell{\Lambda}$ itself:
$\mbox{vol}_d (\mshell{\Lambda} \cap (\bk \pm \mshell{\lambda}))
\le \mbox{vol}_d \mshell{\Lambda} \le \mbox{const.} \Lambda$.
Using this in (\ref{Iint}) gives
the general power counting bound mentioned at the beginning of
this section, $I_\Lambda (k) \le $ const.$\Lambda^{-1}$.
Assuming that $\xi_{-\bk} = \xi_{\bk}$,
this bound is always saturated for $\bk = 0$,
and also for those $\bk$ for which
the shift by $\bk$ makes the two shells overlap over a significant
region of the Fermi surface, that is, when $\bk$ is a nesting vector of
the Fermi surface.
For other values of $\bk$,
the intersection volume can be much smaller than that of $\mshell{\Lambda}$.
A general definition of non-nesting was given,
and power counting bounds were derived when it is satisfied, by \textcite{FST1},
and extended to the case with Van Hove singularities in \cite{FS1,FS2}.
Here we only cite the result for the case of a strictly convex and positively
curved Fermi surface without Van Hove singularities, discussed also in
the Appendix of \cite{Salmhofer99}.
In that case, and for $\Lambda \le \lambda \le v_{\rm F,min} |\bk| $,
one can show that the volume ratio
$\frac{{\rm vol}_d\, (\mshell{\Lambda} \cap (\bk \pm \mshell{\lambda}))}%
{{\rm vol}_d\,\mshell{\Lambda}}$
is proportional to
\begin{eqnarray}
\sfrac{\lambda}{|\bk|v_{\rm F,min} \; \kappa}
& \mbox{ if } &
\bk \not\in {\cal F}^{(2)}_\lambda
\label{transversal}\\
\left(\sfrac{\lambda}{\kappa}\right)^{\frac{d-1}{2}}
& \mbox{ if } &
\bk \in{\cal F}^{(2)}_\lambda.
\label{2kF}
\end{eqnarray}
Here $v_{\rm F,min}$ is the smallest value of $|\nabla e|$ on the Fermi
surface, ${\cal F}^{(2)}_\lambda$ is a $O(\lambda)$-neighbourhood
of the set $\left\{2 \bk: \xi_{\bk} = 0 \right\}$ (note that $2\bk$ is
taken modulo reciprocal lattice vectors), and $\kappa$ denotes the
minimal curvature on the Fermi surface.
This is illustrated for $\lambda = \Lambda$ in Fig.~\ref{fig:schnitte}.
In the first case, the intersection is transversal,
which decreases the intersection volume by the factor in (\ref{transversal}).
The second case corresponds to a $2k_F$-intersection,
where the curvature in a region of size $\sqrt{\lambda}$ determines
the intersection volume, corresponding to (\ref{2kF}).
In the third case, $|\bk|$ is so small that the volume of the intersection
is essentially equal to that of $\mshell{\Lambda}$.
\begin{figure}[ht]
\centerline{\includegraphics[width = 7cm]{fig28.eps}}
\caption{Intersections of a momentum shell around the Fermi surface with
its translate, as arising in loop integrals on the right hand side of the
RG equation. When the Fermi surface is curved, the intersection volume
decreases strongly unless the translating momentum is small.}
\label{fig:schnitte}
\end{figure}
The scale in the flow where the improvements set in is
determined by the curvature of the Fermi surface,
because there is really only an improvement if the
additional factors are smaller than one.
In cases where the curvature is small on large parts of the Fermi surface,
as in the Hubbard model near to half-filling and at small next-to-nearest hopping,
one thus has an effective nesting at those scales and at those $\bk$ where the quotients
in (\ref{transversal}) and (\ref{2kF}) are so big that they give a bound
that is bigger than the trivial bound $1$ for the volume ratio.
Eqs.\ (\ref{transversal}) and (\ref{2kF}) imply that
for small $|\bk|$,
\begin{equation}
\int_{0}^{\Lambda_0} I_\Lambda (k) d\Lambda
\le
\mbox{const. } \log \frac{\Lambda_0}{|\bk| v_{\rm F,min}}
\end{equation}
(where the constant depends on the curvature of the Fermi surface)
and that the function remains bounded for $|\bk|$ not close to zero
(for details, see \textcite{FST1,Salmhofer98a,Salmhofer99}).
Thus, for convex curved Fermi surfaces, the four-point function can diverge only
at $\bk = 0$ and there, only logarithmically
(by a similar argument, one can see that it can diverge only at $k_0 =0$).
The particle-particle correction
to the vertex function has exactly this behaviour. In the particle-hole term,
there is an additional sign cancellation that removes the logarithm.
The same argument shows that in general, divergences can occur
only at nesting vectors of the Fermi surface.
\subsubsection{Uniform improvement from overlapping loops}
\label{sssec:OLloops}
An extension of these geometric estimates to two-loop integrals
of the type $\int d p\; \int d q\; S^\Lambda (p) \, S^{\Lambda'} (q)
S^{\Lambda''} (p\pm q\pm k)$ is very useful for $d \ge 2$:
it is shown in \cite{FST1} that
in absence of nesting and Van Hove singularities,
such integrals contain a scaling improvement independently of $\bk$.
Such two-loop integrals associated to graphs with overlapping loops
arise when the RG equation gets iterated;
the graph classification of \cite{FST1,FST3} shows that in a precise sense,
the overwhelming majority of graphs in the Feynman graph expansion
contains one or even two such subintegrals, hence becomes subleading
at low scales.
As is explained in detail in \cite{FST1,Salmhofer98a},
in absence of nesting and Van Hove singularities, this justifies
the particle-particle-ladder aproximation,
it singles out the
Hartree-Fock type contributions to the self-energy by scaling arguments,
and it allows to show that the derivative of the self-energy is
RG-irrelevant instead of marginal.
\end{appendix}
\bibliographystyle{apsrmp}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,741 |
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With this virtually unlimited bankroll, Soros has become something of a puppet master in both international markets and political arenas, using his influence to enhance his riches and further his leftist political agenda.
Of all those who had their hopes for a progressive utopia dashed on the rocks of November's presidential election, perhaps none had more to lose with the defeat of Hillary Clinton than Soros. Yet, disappointment and petulance is no excuse for the harsh and highly hypocritical words Soros has levelled at our next president. After all, it was Soros who helped build his empire on a foundation of flimflam and manipulation; even at the expense of entire nations.
Seeing how profitable it was to manipulate currencies, Soros would use his talents again in Thailand a few years later. As he did in the U.K., Soros and his hedge fund bet against the Thai baht in 1997, shorting their investments, and prompting an overreaction by Thailand's central bank to save the currency – a move that even Soros himself admitted was "catastrophic." Of course, Soros (who fancies himself a financial prophet to be ignored at one's peril) blamed Thai authorities for how they responded to his bets. Never mind the impact on Thai citizens held at the mercy of the games played between Soros and their country's officials; profit was to be had.
Trump is a successful businessman, worth billions of dollars; but even he, at his most grandiose, would likely balk at the claim to possess the clout in a single breath to topple governments and reduce global currencies to Monopoly money, as Soros in the past has done. Nor has Trump shown any of Soros' zeal for influencing the political process with shadow money and "AstroTurf" grassroots organizations. In fact, Trump's genuinely organic draw among everyday Americans, coupled with receiving just a fraction of super PAC money during the primaries compared to Clinton, shows him to be the polar opposite of Soros when it comes to expending private dollars for public influence.
Furthermore, Trump's cabinet selections thus far show that he has more interest in reducing the government's grip over the nation, than solidifying it -- as Soros would no doubt prefer. In his post-election whining, Soros has proven himself to be nothing more than a disgruntled emperor who cries "dictator!" when his own reign is threatened by an outsider. | {
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} | 1,616 |
Fox News Guest On Sandra Bland: "If She Had Been My Daughter, I'd Have Been Embarrassed"
Media, Recent News
Fox News has a rotation of black conservatives that they throw on their programs to help push their particular narratives on race-based issues. One of their favorites is Milwaukee Sheriff David Clarke, who they turned to quite a bit in the wake of the Eric Garner and Michael Brown killings. On Thursday night, Clarke appeared on The Kelly File to discuss the Sandra Bland case. And, as you'd probably guess, he found that Officer Brian Encinia did absolutely nothing wrong when he threatened to light up Bland with a taser over a minor traffic infraction.
In a segment discussing Bland's autopsy report and any potential fallout from it, Kelly spoke with Clarke and essentially let him sully the dead Bland's name and character while describing Encinia as a model of law enforcement. At the same time, while Kelly seemed mildly disapproving of Encinia's actions during the traffic stop and subsequent arrest, she also told her viewers that they need to listen to the cops at all times and merely complain later. Nice to see Kelly's penchant for victim blaming hasn't subsided.
When Kelly asked Clarke what he thought about the dashcam video of the arrest, Clarke insisted that the officer did everything by the book, and there was absolutely nothing wrong with his actions. He did say that he probably would have said "I'll light you up!" like Encinia did, but he saw no problem with him threatening to use a taser to remove Bland from her vehicle. The sheriff also didn't see any issues with Encinia escalating the situation the way he did, claiming that Bland was the only one at fault, and she should have followed every single one of Encinia's commands. Because, ya know, the police are infallible.
Now, no one should be shocked that Clarke, with his history of 100% support of anything law enforcement does, would claim that the officer was in the right and Bland was in the wrong when viewing the tape. However, it was what Clarke said personally about Bland that is drawing wide condemnation. After talking about how Encinia did a fucking amazing job in the arrest, Clarke vomited out the following crap:
"Look, what we train officers to say when we're going to use a taser is 'Do X or I'm going to tase you. But I'm not going to get hung up on that. I was more appalled by the language she was using with an authority figure. If she was my daughter I would have been embarrassed at the kind of language she was using on the scene.
She did some things that caused an officer to have to move up in terms of his response to keep her safe and to keep himself safe. That's unfortunate that the autopsy report– umm you know — the manner of death is suicide."
Fuck you, David Clarke. Fuck you very much.
Below is video of the segment, courtesy of Fox News:
Cable NewsDavid ClarkeFox NewsMegyn KellySandra BlandTicker
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Man Seen Carrying Pelosi's Lectern During Capitol Riot Arrested
Tucker Carlson Says Trump 'Recklessly Encouraged' Capitol Hill Riot | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/codegolf.stackexchange.com\/questions\/92834\/counting-goats-to-sleep\/92842","text":"# Counting Goats to Sleep\n\nSome people count sheep to get to sleep. Others count goats.\n\nWrite a program or function that takes in a positive integer N and outputs N-1 awake goats followed by one sleeping goat, as if someone was counting N goats and at the very last one they fell asleep.\n\nAwake goats look like this:\n\n \\\n___\/o>\n-(___)\"\n'' ''\n\n\nSleeping goats look like this:\n\n \\\n___\/->\n,(___)\"\n \n\n\nThey are chained together with a single space between beard and tail of adjacent goats:\n\n \\ \\ \\\n___\/o> ___\/o> ___\/->\n-(___)\" -(___)\" ,(___)\"\n'' '' '' '' \n\n\nThe output is allowed to have trailing spaces and a single trailing newline.\n\nThe shortest code in bytes wins.\n\n## Examples\n\nN = 1:\n\n \\\n___\/->\n,(___)\"\n \n\n\nN = 2:\n\n \\ \\\n___\/o> ___\/->\n-(___)\" ,(___)\"\n'' '' \n\n\nN = 3:\n\n \\ \\ \\\n___\/o> ___\/o> ___\/->\n-(___)\" -(___)\" ,(___)\"\n'' '' '' '' \n\n\nN = 4:\n\n \\ \\ \\ \\\n___\/o> ___\/o> ___\/o> ___\/->\n-(___)\" -(___)\" -(___)\" ,(___)\"\n'' '' '' '' '' '' \n\n\nLarger N should work just as well.\n\n\u2022 I think your \"goats\" look more like 4-footed angry birds ;-) Sep 9 '16 at 21:19\n\u2022 Aww I was hoping to count some goats, not the other way round Sep 9 '16 at 21:45\n\u2022 I think I know who counts goats to sleep Sep 9 '16 at 21:57\n\u2022 I don't think you can count goat to sleep unless \"bleeeeeeet\" makes you feel sleepy :P +1 great challenge Sep 9 '16 at 22:19\n\u2022 Psychopaths fall asleep by counting screaming goats. Sep 12 '16 at 16:36\n\n# MATL, 56 53 bytes\n\n:\"'!!((!((!!#*).?p0!!!]'8eP!P]'p(.' '.a-'XE&hqc\n\n\nTry it online!\n\n### Explanation\n\nAwake goat\n\nThe awake goat can be packed into the string\n\n '' '' \")___(->o\/___ \\\n\n\nand unpacked as will be explained shortly. However, the single-quote symbols would need to be duplicated in order to escape them, so the string literal would have to be defined as (note the enclosing single-quote symbols and the duplication of the original ones):\n\n' '''' '''' \")___(->o\/___ \\'\n\n\nTo save bytes, we define the string using characters one code point above that, thus avoiding duplication. The string literal becomes\n\n'!!((!((!!#*).?p0!!!]'\n\n\nAt the end of the code we will subtract 1 and convert to char. (We could do it now, right after the string literal; but leaving it for the end will save another single-quote duplication, as we will see).\n\nTo explain how the string is unpacked, we will work with the original characters (that are produced at the end of the code by subtacting 1), so the explanation is easier to follow. We first reshape the string\n\n '' '' \")___(->o\/___ \\\n\n\ninto an 8-row 2D char array, in column-major order (down, then across). This automatically pads the last column with char 0 (at the end of the code, subtracting 1 will transform it into number \u22121, which converted to char gives again char 0). Char 0 is displayed as a space. So effectively we are padding with spaces. The result of reshaping is\n\n >\n\"o\\\n')\/\n'__\n__\n'__\n'(\n-\n\n\nWe now flip vertically:\n\n -\n'(\n'__\n__\n'__\n')\/\n\"o\\\n>\n\n\nand then transpose and flip vertically again to produce the awake goat:\n\n \\\n___\/o>\n-(___)\"\n'' ''\n\n\nThe two flip operations are needed because the original packed string is \"in reverse\". This is to exploit the fact that the actual 2D char array representing the goat has 6 initial spaces in its first row, which get automatically filled by padding when the string is reshaped into an 8-row 2D array. But the padding is done at the end (not beginning) of the last column (not row), hence the flips and transpose.\n\nSleeping goat\n\nA sleeping goat is generated from an awake goat by transliterating chars o, ', - into -, , , respectively. Actually, because of the one-code-point-above transformation, we transliterate chars p, (, ' into ., a, -, which again saves us from having to duplicate the single-quote symbol. This is why the subtract-one operation was left for the end of the program.\n\nCode structure\n\n1. Generate an awake goat N times, working with code points increased by 1.\n2. Transform the last goat into a sleeping goat.\n3. Concatenate all goats horizontally. Subtract 1 to code points and cast to char.\n\nCommented code\n\n: % (Step 1) Implicitly input N. Push range [1 2 ... N]\n\" % For each (i.e. repeat N times)\n'!!((!((!!#*).?p0!!!]' % Push this string. Quotes are escaped by duplicating\n8e % Reshape into an 8-row 2D array of char, in\n% column-major order, padding last column with\n% char 0\nP % Flip vertically\n!P % Transpose and flip vertically\n] % End\n'p(.' % (Step 2) Push this string: source for transliteration\n'.a-' % Push this string: target for transliteration\nXE % Transliterate. Transforms last goat into sleeping\n&h % (Step 3) Horizontally concat all 2D char arrays\nqc % Subtract 1 and convert to char. 0 becomes \u22121, which\n% is converted to char 0, which is displayed as a space\n% Implicitly display\n\n\u2022 This is some serious goat theory ;) Sep 9 '16 at 23:05\n\n## Javascript, 122 bytes\n\nf=(n,r='repeat')=>' \\\\ '[r](n--)+\n${' ___\/o>'[r](n)} ___\/->${'-(___)\" '[r](n)},(___)\"\n+ '' '' [r](n)+' '\n\n\nSide note\nIn the following code (91 bytes) the goats are aligned verticaly. It does not comply with the output format but i though it could be interesting to note that the horizontal alignment required in the output format needs more bytes:\n\nf=n=>\n\\\\\n___\/${--n?'o':'-'}>${n?'-':','}(___)\"\n{n?' ':'' ''}+(n?f(n):'') \u2022 Why include the vertical submission? The challenge asks for horizontal alignment. \u2013 user45941 Sep 9 '16 at 22:47 \u2022 @Mego What's wrong with showing how much more golfable it would have been? \u2013 Neil Sep 9 '16 at 22:54 \u2022 @Neil Because it's entirely tangential to the challenge. \u2013 user45941 Sep 9 '16 at 22:54 \u2022 @Mego I think it's interesting to note. Sep 9 '16 at 23:06 \u2022 @Mego I thought it could be interesting. I edited the post to make it more obvious that the vertical alignment is not a valid answer. \u2013 Hedi Sep 10 '16 at 8:42 ## Batch, 234 bytes @echo off set\/pn= call:l \" \\ \" \" \\\" call:l \" ___\/o]\" \" ___\/-]\" call:l \"-(___)@ \" \",(___)@\" call:l \" '' '' \" \" \" exit\/b :l set s=%~2 for \/l %%i in (2,1,%n%)do call set s=%~1%%s%% set s=%s:@=\"% echo %s:]=^>% Takes input from stdin. Batch has trouble with \" and > for various reasons so I have to use placeholders and then switch them at the end. \u2022 I had no idea set\/pn works ._. Sep 9 '16 at 23:07 \u2022 The ^ escapes characters. \u2013 Krii Sep 14 '16 at 14:03 \u2022 @Krii Doesn't work when I need it to. \u2013 Neil Sep 14 '16 at 14:50 ## Pyke, 56 54 bytes Fhqd6*\"\\ ___\/o> -(___) '' ''\"+23\\\":RI\"-o'\"\",-\".:(P Try it here! 4 bytes too many because Pyke doesn't allow double quotes in strings :( ## JavaScript (ES6), 110 109 bytes f= n=> \\\\ \\\\ ___\/o> ___\/-> -(___)\" ,(___)\" '' '' .replace(\/^.{8}\/gm,\"&\".repeat(n-1))+\" \"\n;\n<input type=number min=1 oninput=o.textContent=f(this.value)><pre id=o>\n\nHaving to support all three kinds of quote characters was annoying, but fortunately @pinkfloydx33's comment gave me the flash of inspiration that I could add the backquotes at the end thus saving me 1 byte.\n\n\u2022 Can you save a byte by switching the quote type in the middle and concatenating two strings '+\"'' ''\" (assume single quotes are back ticks since I've no idea how to get a backtick into a code block in comments) Sep 10 '16 at 9:35\n\u2022 @pinkfloydx33 I thought I'd already tried that but then I realised that I could add those back ticks at the end which does save me a byte. Also to get a back tick in a comment code block just prefix it with a backslash.\n\u2013\u00a0Neil\nSep 10 '16 at 10:22\n\u2022 You can remove the semicolon Sep 12 '16 at 1:07\n\u2022 @howderek I didn't include it or the f= in my byte count, it's just there for completeness.\n\u2013\u00a0Neil\nSep 12 '16 at 7:52\n\n# GolfScript, 91 bytes\n\n~:a 1-:b;\" \\\\ \"a*n\" ___\/o>\"b*\" ___\/->\"n\"-(___)\\\" \"b*\",(___)\\\"\"n\" '' '' \"b*\" \"n\n\n\nInput: 3\n\nOutput:\n\n \\ \\ \\\n___\/o> ___\/o> ___\/->\n-(___)\" -(___)\" ,(___)\"\n'' '' '' '' \n\n\n### Explanation\n\n~:a 1-:b; # Parse and save the input\n\" \\\\ \"a*n # Repeat the first line 'a' times\n\" ___\/o>\"b* # Repeat the head 'b' times\n\" ___\/->\"n # Then add the sleeping goat's head\n\"-(___)\\\" \"b* # Idem\n\",(___)\\\"\"n #\n\" '' '' \"b* # Idem\n\" \"n #\n\n\nTry it online!\n\n\u2022 Nearly misread as GoatScript Sep 10 '16 at 19:03\n\n# Jelly, 62 56 bytes\n\n\u2076\u1e8b6;\u201c\\ ___\/o>-(___)\" '' '' \u201ds8\n\u00a2\u201c-,o-'\u201dy\u00d0\u20ac\n\u20191\u00a3\u1e8b\u20ac\u017c\u00a2Y\n\n\nTest it at TryItOnline\n\nHow?\n\n\u2076\u1e8b6;\u201c\\ ___\/o>-(___)\" '' '' \u201ds8 - Link 1: make a goat, niladic\n\u2076\u1e8b6 - space character, \u2076, repeated 6 times\n\u201c\\ ___\/o>-(___)\" '' '' \u201d - rest of the awake goat text\n; - concatenate\ns8 - split into length 8 parts\n\n\u00a2\u201c-,o-'\u201dy\u00d0\u20ac - Link 2: put a goat to sleep, niladic\n\u00a2 - last link (make a goat)\n\u201c-,o-'\u201d - characters to remap\ny\u00d0\u20ac - map for each (change \"-\" into \",\", \"o\" into \"-\", and \"-\" into \"\"\n\n\u20191\u00a3\u1e8b\u20ac\u017c\u00a2Y - Main link: n\n\u2019 - decrement (nAwakeGoats)\n1\u00a3 - call link 1 as a nilad (make an awake goat)\n\u1e8b\u20ac - repeat nAwakeGoats times\n\u00a2 - last link (make a sleeping goat)\n\u017c - zip\nY - join with line feeds\n- implicit print\n\n\n## Knight, 96 bytes\n\n;O*\" \\ \"=n+0P;O+*+=t' ___\/''->'=l-nT+t'o>';O+*+','=t'(___)\" 'l+'-'tO+*' 'l\" '' ''\"\n\n\nPretty simple, nothing super fancy\u2014You just print out n-1 versions of the sleeping goat and then the woke goat. Repeatable elements are saved in t so they aren't typed out again.\n\nExpanded:\n\n# Print out the ears. They're the same for both goat types.\n; OUTPUT (* \" \\ \" (= n (+ 0 PROMPT)))\n; OUTPUT\n(+\n(* (+ (= tmp ' ___\/')'->') (= m - n TRUE))\n(+ tmp 'o>'))\n; OUTPUT\n(+\n(* (+ ',' (= tmp '(___)\" ')) m)\n(+ '-' tmp))\nOUTPUT (+ (* ' ' m) (\" '' ''\"))\n\n\u2022 -2 bytes: *\" ''\"2 Jun 15 at 21:59\n\n# PHP , 200 Bytes\n\n$a=[\" \\ \",\" ___\/o>\",'-(___)\" ',\" '' '' \",\" \"];$z=8*$n=$argv[1];for($i=0;$i<4;)$o.=str_repeat($a[$i],$i++==3?$n-1:$n);$o[$z*2-2]=\"-\";$o[$z*3-8]=\",\";$o.=$a[4];echo chunk_split($o,$z,\"\\n\");\n\n\u2022 You are coding way too clean, J\u00f6rg. I could golf off 32 bytes from that in 11 steps. Want hints? Sep 10 '16 at 16:31\n\u2022 Thanks I want only to solve this challenge in any way. Sometimes is clean better then a wrong solution. You can paste your way. Sep 10 '16 at 17:05\n\u2022 I took a different approach; but if you want tips for yours, just ask. 24 bytes in the first 5 steps. Sep 10 '16 at 19:33\n\u2022 @J\u00f6rgH\u00fclsermann Modifying other people's answers seems to be heavily frowned upon in this site. Sep 10 '16 at 21:57\n\u2022 @Carcigenicate Did You mean I should modifying answers from other People or viceversa? I tend more to solve a problem clean if I am only interesting like in this case. Ascii Art is normally not my priority Sep 10 '16 at 23:18\n\n## C++, 180 bytes\n\nauto f(int n)\n{\nstring a,b,c,d;\nwhile(n--)\n{\na+=\" \\\\ \";\nb+=\" ___\/\";b+=n?\"o>\":\"->\\n\";\nc+=n?\"-(___)\\\" \":\",(___)\\\" \\n\";\nd+=n?R\"( '' '' )\":\" \\n\";\n}\nreturn a+'\\n'+b+c+d;\n}\n\n\u2022 Welcome to PPCG! Please include the one line version so that you can actually count it. You can always include a readable version separately so people don't have to read the one-liner. :) Sep 10 '16 at 18:21\n\u2022 Martin, thanks for the link. I originally measured the size by file size and now I fix it. Sep 10 '16 at 18:40\n\u2022 Answers should indeed be measured by file size. My point was that your code does work without the linefeeds, so the answer should include that version. Sep 10 '16 at 18:46\n\u2022 Okay, I wrote the size by file size. About how this code works - there is no difference between readable and one line version. Sep 10 '16 at 19:01\n\u2022 I don't think it's valid to not include the #include <string> and either using namespace std; or using std::string; in your byte count if your function cannot be compiled without them.\n\u2013\u00a0hvd\nSep 11 '16 at 12:42\n\n## Pip, 60 + 1 = 61 bytes\n\nOne byte added for the n flag.\n\nYsX6.\\\"\\ ___\/o>-(___)\" '' '' \\\"<>8yXa-1.YyR^\"-o'\"Y^\",-\"\n\n\nConstructs an awake goat as a list of lines and yanks it into y. String-multiplies to get a-1 awake goats. Replaces -o' with ,- in y and concatenates it to the end. Prints, newline-separated.\n\nTry it online!\n\n(I think this is my first time using Pip's escaped-string syntax \\\"...\\\", which allows for literal double quotes in the string.)\n\n# IBM\/Lotus Notes Formula, 187174 188 bytes (not competing)\n\nEDIT Found a space that shouldn't have been there and removed an unneeded @Implode\n\n188 as I had missed the fact that the tail of the sleeping goat is different :-(\n\nB:=@Repeat(\" \\\\ \";a);C:=@Repeat(\" \/o> \";a-1)+\" \/->\";D:=@Repeat(\" --- \";a);E:=@Repeat(\",(___)\\\" \";a);F:=@Repeat(\" \";a);@Implode(B:C:D:E:F;@NewLine)\n\n\nUngolfed:\n\nB:=@Repeat(\" \\\\ \";a);\nC:=@Repeat(\" \/o> \";a-1)+\" \/->\";\nD:=@Repeat(\" --- \";a);\nE:=@Repeat(\"(___)\\\" \";a-1)+\",(___)\\\" \";\nF:=@Repeat(\" \";a);\n@Implode(B:C:D:E:F;@NewLine)\n\n\nUsage:\n\nCreate a Notes form with two fields named a and g.\n\na=editable, number, g=computed, text.\n\nPaste the above formula into g and give a a default value of 0.\n\nSet the form font to Terminal.\n\nCreate a new document with the form, enter a number in a and press F9 to update the goats.\n\nSamples:\n\nNot competing as the format messes up when the number of goats reaches the width of the page.\n\nGiven an infinitely wide screen it should will work for any number of goats though. This is what it looks like when the page is not wide enough.\n\n\u2022 Why is it non-competing? Every answer is like that. That's called wrapping. Sep 12 '16 at 19:04\n\u2022 Thanks for the clarification @mbomb007. Was trying to be honest and not disrespect better golfers than me. I'm new to this. Ok it competes. It won't win but I bet there won't be too many Lotus Notes golfers to beat me with formula language \u263a Sep 12 '16 at 19:45\n\u2022 To be honest, I bet there won't be too many Lotus Notes golfers. Sep 12 '16 at 20:26\n\n# CJam, 58 bytes\n\nri{S6*\"\\ ___\/o>,(___)\\\" '' '' \"+\\{'o\"-\"er}|8\/}%W%zN*\n\n\nTry it online!\n\nExplanation\n\nri e# Read an integer from input\n{ e# Map the following block to the range 0..input-1\nS6* e# Push 6 space characters\n\"\\ ___\/o>,(___)\\\" '' '' \"+ e# Push this string and concatenate with the spaces\n\\ e# Bring the number being mapped to the top\n{ e# If it's 0, execute this block:\n'o e# Push the string \"'o\"\n\"-\" e# Push the string \"-\"\ner e# Transliterate the large string by replacing characters\ne# from \"'o\" with respective characters from \"-\"; this\ne# makes the sleeping goat.\n}| e# (end if)\n8\/ e# Split the string into chunks of length 8\n}% e# (end map)\nW% e# Reverse the array, since the sleeping goat was made at\ne# the beginning\nz e# Transpose\nN* e# Join with newlines\n\n\n# Python 2.7, 101 113 bytes\n\nEdit: Added function definition\n\ndef f(n):\nm=n-1\nprint \" \\ \"*n+\"\\n\"+\" ___\/o>\"*m+\" ___\/->\\n\"+'-(___)\" '*n+\"\\n\"+\" '' '' \"*m+\" \"*2\n\n\nde-golfified:\n\nm=n-1 # Replacement variable. Saves 6 bytes\n\" \\ \"*n+\"\\n\"+ # Print ears, same for all goats!\n\" ___\/o>\"*m+ # Print eyes of n-1 awake goat\n\" ___\/->\\n\"+ # Print eye of sleeping goat\n'-(___)\" '*m+ # Print body of n-1 awake goat\n',(___)\"\\n'+ # Print body of sleeping goat\n+\" '' '' \"*m+ # Print the legs of n-1 awake goat\n\" \"*2 # Print legs of sleeping goat using *2 operator to save 1 byte\n\n\nNote Python2.7 is one byte shorter than Python3 due to that it doesn't need parentesis when printing.\n\n\u2022 Needs to receive input n, and you've missed the tail change for the sleeping goat (also did you see the Py 3.6 answer?). Sep 11 '16 at 3:40\n\u2022 Hi! The tail change is there, wasn't sure about if you needed to handle input. Had a look a the Python3.6 answer after writing my own. Does it that recieve input though?\n\u2013\u00a0tigr\nSep 11 '16 at 10:55\n\u2022 Ah, ok. It needs to be either a program or a function. Will update with worse solution, for now :(\n\u2013\u00a0tigr\nSep 11 '16 at 10:56\n\u2022 Yep, function or program, you got it! You can remove the space in print \"..., and place everything on one line, using 1 ; to separate the two statements. The tail still not in the golfed code shown, but looks like you've counted it, all in that should make it 112 bytes. Sep 11 '16 at 15:28\n\n# 05AB1E, 66 bytes\n\n\u2019 \\ 0 ___\/1>02(___)\" 0 33 33 \u20190\u00a1v123SDys\u2026o-'S:I<\u00d7?ys\u2026-,S:,\n\n\nTry it online!\n\n### Explanation\n\n\u2019 \\ 0 ___\/1>02(___)\" 0 33 33 \u20190\u00a1v123SDys\u2026o-'S:I<\u00d7?ys\u2026-,S:, Argument n\n\u2019 \\ 0 ___\/1>02(___)\" 0 33 33 \u2019 The goat, newline replaced by 0 and the eye replaced by 1\n0\u00a1 Split on 0\nv For each y in array, do:\n123SD Push the array [1,2,3] twice\nys\u2026o-'S: Replace [1,2,3] with ['o','-','\\'']\nI<\u00d7? Print that n-1 times without newline\nys\u2026-,S:, Replace [1,2,3] with ['-',',',''] and print\n\n\n# Perl 5-n, 105 bytes\n\nsay for' \\\\ 'x(--$_+1), ' ___\/o>'x$_.' ___\/->',\n'-(___)\" 'x$_.',(___)\" ', \" '' '' \"x$_.' '\n\n\nTry it online!\n\n# Bash + GNU Coreutils, 165 155 bytes\n\na=\" \\\n>o\/___\n\\\")___(-\n'' '' \"\neval paste -d \\'\\' $(seq$1|while read;do\nprintf '<(echo \"$a\") ' done) | sed \"s\/-\/,\/;s\/o\/-\/;s\/'' ''\/\"' \/'|rev Run with: bash my_pgm.bash N Basically the program prints N times of the same goat (reversed), and substitutes the first -, for ,, the first o for - and the first '' '' for the backticks. Then reverses the lines. # PHP, 133 131 bytes for(;$y<32;$y+=8)for($x=$argv[1];$x--;)echo substr(\" \\ ___\/\".($x?\"o>-(___)\\\" '' '' \":\"->,(___)\\\" \"),$y,8),\"\n\"[$x]; I found two bytes to golf away from one of the version without curlys. ## PowerShell v2+, 96 bytes param($n)' \\ '*$n-- ' ___\/o>'*$n+' ___\/->'\n'-(___)\" '*$n+',(___)\"' \" '' '' \"*$n+' '\n\n\n(ab)uses the default Write-Output formatting to include a newline between elements. Leverages string concatenation and multiplication to construct the goats line by line. The only real trick is the first line $n-- to output the correct number of ears and then post-decrement $n so it's correct for the rest of the lines.\n\nPS C:\\Tools\\Scripts\\golfing> 1..4|%{.\\counting-goats-to-sleep.ps1 $_} \\ ___\/-> ,(___)\" \\ \\ ___\/o> ___\/-> -(___)\" ,(___)\" '' '' \\ \\ \\ ___\/o> ___\/o> ___\/-> -(___)\" -(___)\" ,(___)\" '' '' '' '' \\ \\ \\ \\ ___\/o> ___\/o> ___\/o> ___\/-> -(___)\" -(___)\" -(___)\" ,(___)\" '' '' '' '' '' '' ## Ruby, 102 bytes m=-1+n=gets.to_i puts' \\ '*n,' ___\/o>'*m+' ___\/->',(?-+a='(___)\" ')*m+?,+a,\" '' '' \"*m+\" \"*2 # Python 3. 170 bytes lambda n:'\\n'.join(map(lambda*l:''.join(l),*map(lambda w:(' '*6+'\\ ',' ___\/'+(w and'o'or'-')+'>',(w and'-'or',')+'(___)\" ',w and\" '' '' \"or' '),range(n)[::-1]))) hmm, apparently constructing the string without doing list manipulation yield shorter code ## Emacs Lisp, 241 bytes (defvar s'(\" \\\\\"\" ___\/->\"\",(___)\\\"\"\" \"))(defun a()(dotimes(n 4 g)(setf(nth n g)(format\"%s%s\"(nth n'(\" \\\\ \"\" ___\/o>\"\"-(___)\\\" \"\" '' '' \"))(nth n g)))))(defun g(n)(let((g(copy-seq s)))(mapcar'message(dotimes(i(- n 1)g)(a))))) \"Slightly ungolfed\" (defvar s'(\" \\\\\"\" ___\/->\"\",(___)\\\"\"\" \")) (defun a()(dotimes(n 4 g)(setf(nth n g)(format\"%s%s\"(nth n'(\" \\\\ \"\" ___\/o>\"\"-(___)\\\" \"\" '' '' \"))(nth n g))))) (defun g(n)(let((g(copy-seq s)))(mapcar'message(dotimes(i(- n 1)g)(a))))) where s is one sleeping goat, a adds an awake goat and g(n) is the counting function. # Java 8, 236222218 173 bytes n->{String x=\"\\n\",a=\"\",b=a,c=a,d=a;for(;n-->0;a+=\" \\\\ \",b+=\" ___\/\"+(n<1?\"-\":\"o\")+\">\",c+=(n<1?\",\":\"-\")+\"( )\\\" \")d+=(n<1?\" \":\" '' ''\")+\" \";return a+x+b+x+c+x+d;} Explanation: Try it online. n->{ \/\/ Method with integer parameter and String return-type String x=\"\\n\", \/\/ New-line String to reduce bytes a=\"\", \/\/ Row 1 String, starting empty b=a, \/\/ Row 2 String, starting empty c=a, \/\/ Row 3 String, starting empty d=a; \/\/ Row 4 String, starting empty for(;n-->0; \/\/ Loop n times: a+=\" \\\\ \", \/\/ Append the horns to row 1 b+=\" ___\/\"+(n<1?\"-\":\"o\")+\">\", \/\/ Append the back and head to row 2 c+=(n<1?\",\":\"-\")+\"( )\\\" \") \/\/ Append the tail, body, and beard to row 3 d+=(n<1?\" \":\" '' ''\")+\" \";\/\/ Append the legs to row 4 return a+x+b+x+c+x+d;} \/\/ Return the four rows, separated by new-lines # Canvas, 58 39 bytes \uff01\uff0d\u2265\uff19A\u00b2\uff3d]\uff4b\uff11\uff5b\uff5bK\uff14\uff45d\uff52\u2199FX\u201f\u00d7-\uff12\uff12\u254b,\uff18\uff13\u254b \uff13\uff14\u254b\u2194 Try it here! Constructs the awake goat facing left from a compressed string, repeats it $$\\n\\$$ times horizontally, modifies the left-most goat to the sleeping goat, then flips the whole string horizontally. # Brainfuck 420 bytes Its not very golfed yet... but it works... whit numbers from 0 to 255 :) >+[[-]>,[+[-----------[>[-]++++++[<------>-]<--<<[->>++++++++++<<]>>[-<<+>>]<+>]]]<]<[->+>+>+>+<<<<]>>>>>>++++[<++++++++>-]>-[<+>---]<+++++++>>-[<+>-----]<---->>-[<-->-------]<+>>----[<+>----]<-><<<<<<[>......>.<.<-]++++++++++.>>+++<<<-[>>..>...>.>.>.<<<<<<-]>>..>...>.--.>>.[<]>.<<-[>>>>>.-----.<...>+.++++<<++.--.<<<-]>>>>>-.----.<...>+.<<++.--.>>[-<<<<+>>>>]<[<]>-->.<<<-[>>>>.<<..>>.<<..>>..<<<<-]>>>>>+<.>..<.>..<.. Try it online Explained version: Read number from 0 to 255 \"thanks to esolangs ;)\" >+[[-]>,[+[-----------[>[-]++++++[<------>-]<--<<[->>++++++++++<<]>>[-<<+>>]<+>]]]<]< Its a little bit hard explain this part because it is creating letters and printing lines at the same time [->+>+>+>+<<<<] Make 4 copies of the number (one for every line) >>>>>>++++[<++++++++>-]>-[<+>---]<+++++++> Make space and '\\' >-[<+>-----]<---- Make '\/' >>-[<-->-------]<+ Make 'o' >>----[<+>----]<-> Make mayor sign <<<<<<[>......>.<.<-] Print first line ++++++++++.>>+++<< Make line jump and '_' <-[>>..>...>.>.>.<<<<<<-] Print second line >>..>...>.--.>>.[<] Make minus sign and print the line of last sheep >.<<-[>>>>>.-----.<...>+.++++<<++.--.<<<-] Print thirth line >>>>>-.----.<...>+.<<++.--.>>[-<<<<+>>>>] Print the line of last sheep <[<]>-->.<<<-[>>>>.<<..>>.<<..>>..<<<<-] Print fourth line >>>>>+<.>..<.>..<.. Print the line of last sheep # Zsh, 97 bytes $1()O+=${s\/o\/-} for s (' \\ ' \" ___\/o>\" '-(___)\" ' \" '' '' \")(eval {1..$1}'||O+=$s;';<<<$O)\n\n\nTry it online!\n\nFeels way too long.\n\n# 05AB1E, 63 61 bytes\n\n\u2022D:\u00a8M\u00df\u00b2\u043d\u0100\u00f9\u00dc<3\u00b9\u00ba\u2022\u201c\\\n_\/o>-()\"'\u201c\u00c5\u0432J.B\u00d7\u00bbR\"-, o-\"#vy.;2F\u201e'.;]R\n\n\nTry it online!\n\n\u2022...\u2022\u201c...\u201c\u00c5\u0432J.B\u00d7\u00bbR\"...\"#vy.;2F\u201e'.;]R # trimmed program\nR # push reversed...\n.B # list of lines in...\nJ # joined...\n\u00c5\u0432 # list of characters in...\n\u201c...\u201c # literal...\n\u00c5\u0432 # with indices in base-10 values of base length of...\n\u201c...\u201c # literal...\n\u00c5\u0432 # digits of...\n\u2022...\u2022 # 258955905826915235144641900393055...\n.B # with length of longest line minus length of each line spaces appended to each line...\n# (implicit) with each element...\n\u00d7 # repeated...\n# implicit input...\n\u00d7 # times...\n\u00bb # joined by newlines...\nR # reversed...\n.; # with first occurence of...\n# second character of...\ny # variable...\n.; # replaced by...\n # first character of...\ny # variable...\nv # for y in...\n\"...\" # literal...\n# # split by spaces...\n.; # with first occurence of...\n# second character of...\n\u201e' # literal...\n.; # replaced by...\n# first character of...\n\u201e' # literal...\nF # for n in [0, 1, 2, ...,\n2 # ..., literal...\nF # ]\n] # exit loop\n] # exit loop\n# implicit output\n`","date":"2021-10-16 09:55:05","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\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.581536591053009, \"perplexity\": 8458.136219303862}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323584554.98\/warc\/CC-MAIN-20211016074500-20211016104500-00609.warc.gz\"}"} | null | null |
\section{Tools}
In this section we state two main tools that we use in our analysis, namely the cactus representation of minimum cuts and strongly Rayleigh probability distributions. We will conclude this section by giving an overview of our proof.
\subsection{Min-cuts and the cactus}\label{subsec:mincuts}
To understand our algorithm and analysis it is useful to recall the {cactus representation~\cite{FF09}} of the min-cuts of a graph. We briefly recall some basic definitions and the recursive construction of the cactus.
We will rely on a number of basic facts about min-cuts. {For proofs, see~\cite{FF09}}. Suppose $G$ is a $k$-edge connected graph.
\begin{fact}
\label{fact:min-cuts1}
If two tight sets $S$ and $S'$ cross, then each of $S \smallsetminus S'$,
$S' \smallsetminus S$, $S \cap S'$ and $\overline{S \cup S'}$ are tight. Moreover, there are no edges from $S \smallsetminus S'$ to
$S' \smallsetminus S$, and there are no edges from
$S \cap S'$ to $\overline{S \cup S'}$.
Therefore, if two distinct tight sets $S$ and $S'$ cross each other, then $\delta (S) \cap \delta (S') = \emptyset$.
\end{fact}
The following fact is especially useful to us, since the support graph of $x$ is 4-edge-connected.
\begin{fact}
\label{fact:min-cuts2}
Suppose that every proper mincut is crossed by some other proper mincut. Then $k$ is even and $G$ is a cycle, with $k/2$ parallel edges between each adjacent pair of vertices.
\end{fact}
\begin{fact}\label{fact:lastcycle}
In step \ref{alg:step:cycle} of the algorithm the remaining graph $G$ is a cycle of length at least 3 such that there are exactly two parallel edges between each pair of consecutive vertices.
\end{fact}
\begin{proof}
Let $G$ be the graph which remains after the while loop in the algorithm terminates. By the algorithm, $e^+=\{u,v\}$ is not contracted yet. $G$ has at least 3 vertices, as otherwise in the last of the while we contracted a set $S$ where $e^+\in \delta(S)$ which contradicts \cref{fact:e+notindS}. If $G$ has 3 vertices then it must be a cycle. Otherwise, $\{u,v\}$ is a proper tight set in $G$, and it must be crossed. In this case by \cref{fact:min-cuts2} $G$ is a cycle of length at least $4$.
\end{proof}
\begin{definition}[Cactus Graph] A loopless and 2-edge connected graph $C = (U,F)$
is a {\em cactus} if each edge belongs to exactly one cycle.
\end{definition}
\begin{theorem}[Dinits, Karzanov, Lomonosov] Let $G = (V, E)$ be a loopless graph with min-cut size $k \ge 1$. There is a cactus $C = (U,F)$
and a mapping $\phi: V \rightarrow U$ such that the 2-element cuts of
$C$ are in one to one correspondence with the min-cuts of $G$.
Equivalently, $S$ is at tight set of $G$ if and only if $\phi(X)$ is a tight set
of $C$.
\end{theorem}
Our algorithm can be viewed as essentially constructing a cactus representation of the min-cuts. More precisely, the critical cuts of our algorithm (defined below) are in one to one correspondence with the cycles of the cactus.
In the rest of this section we will more fully explore the interaction between our algorithm and the structure of the cactus representation of minimum cuts. From now on, assume $G$ is the 4-regular support graph of the half-integral LP solution $x$ and that we have executed our algorithm on $G$.
\paragraph{Critical sets and cuts:}
A tight set $S$ selected in step \ref{alg:step:mintightset} of the algorithm is called a \textbf{critical set} and the corresponding cut $\delta (S):= E(S, \overline S)$ is called a \textbf{critical cut}.
Vertices of $G$ are degenerate critical sets.
There is a natural {\em hierarchy of critical sets} associated with the execution
of the algorithm. The leaves of the hierarchy are vertices of the original graph.
If $S$ and $S'$ are critical sets such that $S$ or a contracted version of $S$ is a vertex in $S'$, then $S$ is a child of $S'$ (respectively $S'$ is the parent of $S$). If $\tilde S$ is an ancestor of $S$ in the hierarchy of critical sets, then we say that $\tilde S$ is a \textbf{higher} critical set
than $S$ (resp. $S$ is a \textbf{lower} critical set than $\tilde S $).
For example, in \cref{fig:exectree}, critical set $F$ is the parent of and is higher in the hierarchy than critical sets $A$, $B$ and $C$.
The root of
the hierarchy is the graph $G$ once we get to step \ref{alg:step:cycle} of the algorithm.
\begin{definition}[Going higher]\label{def:gohigher}
An edge $e$ in $\delta(S)$ {\em goes higher} if the lowest critical set $S'$ such that $S\subsetneq S'$ satisfies $e\in \delta(S')$.
\end{definition}
Note that by \cref{fact:independencecritical} any edge going higher is independent of all edges which do not.
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\caption{Example execution on a half integral graph.
In the first figure, we visualize five tree operations in parallel, which we may do since all these tight sets have size 4 and are not crossed by other tight sets. Similarly in the second figure we do two operations in parallel. In the final step, a cycle is chosen by picking two edges at random. $A,B,C,D,E$ are all ``degree cuts" whereas $F$ and $G$ are both ``cycle cuts." $t$ is an example top edge (as are all edges picked in the first graph). $a,g$ and $b,h$ are cycle partners with respect to the cut $F$. $e,f$ are companions.
} \label{fig:execution}
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\caption{An execution tree on this graph.}
\label{fig:exectree}
\end{figure}
\paragraph{Structure of critical cuts:} Consider a critical set $S$ chosen
in step \ref{alg:step:mintightset} in the algorithm. We will abuse notation and, at any time during the execution of the algorithm, refer to $G$ with vertex set $V$ as the graph remaining at that time, after contraction of all trees that have been sampled before $S$ is considered. Consider the graph $G' := G/ V \smallsetminus S$ and let $w$ be the contracted vertex representing
$V \smallsetminus S$.
There are two possibilities for the structure of $G'$:
\begin{itemize}
\item Case 1: There are no proper min-cuts inside $S$.
In this case, we call $\delta (S)$ a \textbf{degree cut}. In \cref{fig:execution}, $A,B,C,D,E$ are all degree cuts.\footnote{ These cuts correspond to cycles of length two in the cactus.}
\item Case 2: There is a proper min-cut $(S_0, \overline S_0)$ such that
$S_0 \subsetneq S$. In this case,
it (and every other proper min-cut inside $S$) is crossed by some other
min-cut (or would be more minimal than $S$).
It follows that
in $G'$, every proper mincut is crossed by some other proper mincut and therefore, by \cref{fact:min-cuts2}, the graph is a cycle with two edges between each pair of adjacent vertices in the cycle. In this case, we call $\delta(S)$ a {\em cycle cut}. For example, $F$ and $G$ in \cref{fig:execution} are cycle cuts.
We divide the 4 edges from $w$ into two pairs, such that each pair share an endpoint inside $S$. We call each such pair {\em cycle partners} with respect to $\delta(S)$.
Every other pair of edges between two adjacent vertices in the cycle are called {\em companions}.
For example, in \cref{fig:execution}, $\delta(F)$ is a cycle cut
and $a,g$ and $b,h$ are cycle partners with respect to $\delta(F)$. $e$ and $f$ are companions.
Cycle cuts correspond to cycles of length 3 or more in the cactus.
\end{itemize}
Note that every edge has at most one companion but possibly many partners depending on the underlying cactus.
\begin{definition}[Highest critical cuts]
For a vertex $u$ and an edge $e=\{u,v\}$, let $S_{u,e}$ be the highest critical set $S$ such that $u \in S$ and $v \not \in S$, and let $S_{e}$ be the lowest critical set such that both $S_{u,e}$ and $S_{v,e}$ are (contracted) nodes in $S_{e}$. Then $\delta(S_{u,e})$ and $\delta(S_{v,e})$ are the highest critical cuts containing $e$.
If the edge $e$ is clear from context, we may drop $e$ in the notation $S_{u,e}$.
\end{definition}
\begin{definition}[Bottom Edge and Top Edges]
For an edge $e$, if $S_{e}$ is a cycle cut, we say that $e$ is a {\em bottom edge} and otherwise it is a {\em top edge}.
\end{definition}
For example, in \cref{fig:execution}, $e,f,a,g,b,h$ are bottom edges (among the labeled edges) and $t$ is a top edge.
The following fact is immediate:
\begin{fact}
Companion bottom edges $e,f$ are in or out of $T$ independently of every other edge of $T$.
\label{fact:efinout}
\end{fact}
\paragraph{Min-cuts containing a particular edge:}
The set of min-cuts an edge $e=(u,v)$ is on are the following:
\begin{enumerate}
\item[(a)] all critical degree cuts $\delta(S)$ such that $e \in \delta(S)$. (This
includes the cuts $(u, V\smallsetminus u)$ and $(v, V\smallsetminus v)$.)
\item[(b)]
For any set $S$ such that $\delta(S)$ is a critical cycle cut, and
$e$ is either in $S$ or on $\delta(S)$, every cut of the cycle that includes
the edge $e$ is a min-cut $e$ is on.
\end{enumerate}
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\begin{tikzpicture}[scale=0.8,inner sep=1.8]
\tikzstyle{every node}=[draw,fill=red,circle];
\begin{scope}[shift={(-4,0)}]
\foreach \i in {0,...,5}{
\node at (\i*60:1.5) (a_\i) {};
}
\foreach \i/\j in {0/1, 1/2, 2/3, 3/4, 4/5, 5/0}{
\path (a_\i) edge (a_\j);
}
\end{scope}
\begin{scope}[shift={(4,0)}]
\foreach \i in {0,...,5}{
\node at (\i*60:1.5) (a_\i) {};
}
\foreach \i/\j in {0/1, 1/2, 2/3, 3/4, 5/0}{
\path (a_\i) edge (a_\j);
}
\draw [color=blue,dashed,line width=1.3] (a_5)+(-.3,-.3) arc (180:55:1);
\draw [color=blue,dashed,line width=1.3] (a_5)+(-.5,-.3) arc (180:120:3);
\draw [color=blue,dashed,line width=1.3,shift=(a_5)] (a_5)+(-.7,-.3) -- (-.7,3);
\draw [color=blue,dashed,line width=1.3] (a_5)+(-.9,-.3) arc (0:60:3);
\draw [color=blue,dashed,line width=1.3] (a_5)+(-1.1,-.3) arc (0:120:1);
\end{scope}
\end{tikzpicture}
\caption[An Example of a Graph with no Good Edge]{Consider the Hamiltonian cycle shown at the left. In any spanning tree of this graph
all edges are contained in at least one odd minimum cut. The dashed blue arcs in the right shows the odd near minimum cuts for one spanning tree.}
\label{fig:cyclegoodedges}
\end{figure}
}
It is easy to see that each of the above is a min-cut. To see that there are no others, it suffices to observe by induction that whenever a set $S$ is contracted,
we have accounted for all min-cuts in which nodes inside $S$ are partitioned between the two sides of the cut.\old{ this follows immediately from the fact that when a set $S$ is selected, either there are no proper
min-cuts inside $S$, or the remaining graph with $V \smallsetminus S$ contracted is a cycle with two edges between each pair of edges (and case (b) above accounts for all such cuts).}
\paragraph{Other facts:}
We end this part by recording the following basic facts about structure of min cuts, and we will use them throughout our proofs.
\begin{fact} \label{fact:cyclecut} Suppose that $S$ is a critical set.
If some (contracted) vertex $v \in S$ has two edges to $w := V \smallsetminus S$, then $S$ is a cycle cut.
\end{fact}
\begin{proof} This is immediate if $|S|=2$, so suppose that $|S|> 2$.
Then
$v$ has two edges to $w$, which has two edges to $S \smallsetminus v$ which has two edges to $w$. Since $S \smallsetminus v$ is therefore a proper min-cut but was not selected in step \ref{alg:step:mintightset}, it must be crossed by some other set, which, by the earlier discussion of the structure of critical cuts , means that $\delta(S)$ is a cycle cut.
\end{proof}
\begin{fact} \label{fact:2cutinter} Suppose that $S$ and $S'$ are two distinct tight sets.
Then $|\delta (S) \cap \delta (S')| \le 2$.
\end{fact}
\begin{proof}
By contradiction. Suppose that $S$ and $S'$ are both proper min-cuts and have
$\delta (S) \cap \delta (S') \ge 3 $. Then by \cref{fact:min-cuts1}, they do not cross.
Therefore it must be that, say, $S \subset S'$. But in this case, if $\delta (S) \cap \delta (S') \ge 3$, since $\delta(S)$ and $\delta(S')$ are both min-cuts, there can only be one edge from $S$ to $S'\smallsetminus S$ and at most one edge from $ S'\smallsetminus S$ to $V \smallsetminus (S \cup S')$ which contradicts $\delta (S' \smallsetminus S) \ge 4$.
\end{proof}
\begin{fact}
\label{fact:2cycle} Suppose that $S$ and $S'$ are two critical sets such that $S\subset S'$.
Then if $\delta (S) \cap \delta (S') = 2$, then $S'$ is a cycle cut.
\end{fact}
\begin{proof}
Once $S$ is contracted, it has two edges to $V \smallsetminus S'$, and therefore by
\cref{fact:cyclecut} is a cycle cut.
\end{proof}
\begin{fact}
\label{fact:cyclepartners}
Suppose that $S \subset S'$ are two critical cycle cuts. Then any two edges are cycle partners on at most one of these (cycle) cuts.
\end{fact}
\begin{proof}
Suppose not. Then there is a pair of edges
$e$ and $f$ that are cycle partners on both. Suppose that
$g$ and $h$ are the other pair of cycle partners on $\delta(S)$ and that their endpoint inside $S$ is node $u$.
Then $(S'\smallsetminus S) \cup u$ is a min-cut that crosses $S$, which is a contradiction to the selection of $S$. [Essentially this means that in fact there is a larger cycle here.]
\end{proof}
\begin{fact}\label{fact:2gohigherbottom}
Say $S$ is a critical set and exactly two edges of $\delta(S)$ are bottom edges that do not go higher. Then the other two edges of $\delta(S)$ must go higher.
\end{fact}
\begin{proof}
Say $\delta(S)=\{a,b,c,d\}$ and suppose $a,b$ are bottom edges that do not go higher. Say $S'$ is the parent of $S$ in the hierarchy of critical cuts. This implies that $\delta(S')$ is a cycle cut. So, $a,b$ are companions in this cycle. This implies that either $c,d$ are also companions or they are cycle partners in $\delta(S')$.
\end{proof}
\subsection{Strongly Rayleigh Distributions}
\label{sec:SRprops}
Let ${\cal B}_E$ be the set of all probability measures on the Boolean algebra $2^E$.
Let $\mu\in{\cal B}_E$. The generating polynomial $g_\mu: \mathbb{R}[\{y_{e}\}_{e\in E}]$ of $\mu$ is defined as follows:
$$ g_\mu(y):=\sum_S \mu(S) \prod_{e\in S} y_e.$$
We say $\mu$ is a strongly Rayleigh distribution if $g_\mu\neq 0$ over all $\{y_e\}_{e\in E} \in \mathbb{C}^E$ where $\text{Im}(y_e)>0$ for all $e\in E$. Strongly Rayleigh (SR) distributions were defined in \cite{BBL09} where it was shown any $\lambda$-uniform spanning tree distribution is strongly Rayleigh. In this subsection we recall several properties of SR distributions proved in \cite{BBL09,OSS11} which will be useful to us.
\paragraph{Closure Operations.}
Strongly Rayleigh distributions are closed under the following operations.
\begin{itemize}
\item {\bf Projection.} For any $\mu\in {\cal B}_E$, and any $F\subseteq E$, the projection of $\mu$ onto $F$ is the measure $\mu_F$ where for any $A\subseteq F$,
$$ \mu|_F(A)=\sum_{S: S\cap F=A} \mu(S).$$
\item {\bf Conditioning.} For any $e\in E$, $\{\mu | e \text{ out}\}$ and $\{\mu | e \text{ in}\}$.
\end{itemize}
\paragraph{Negative Dependence Properties.}
An {\em increasing function} $f:2^E\rightarrow \mathbb{R}$, is a function where for any $A\subseteq B\subseteq E$, we have $f(A)\leq f(B)$.
For example, if $E$ is the set of edges of a graph $G$, then the existence of a Hamiltonian cycle is an increasing function, and the $3$-colorability of $G$ is a decreasing function.
\begin{definition}[Negative Association]
\label{def:negativeassociation}
A measure $\mu \in {\cal B}_E$ is {\em negatively associated} or NA if
for any increasing functions $f,g: 2^E\to \mathbb{R}$, that depend on {\em disjoint} sets of edges,
$$ \EE{\mu}{f}\cdot \EE{\mu}{g} \geq \EE{\mu}{f\cdot g} $$
\end{definition}
\noindent
It is shown in \cite{BBL09,FM92} that strongly Rayleigh measures
are negatively associated.
\vspace{0.1in}
\noindent
Let $\mu$ be a strongly Rayleigh measure on edges of $G$. For a set $A$, let $$X_A=|A\cap S|$$ be the random variable indicating the number of edges in $A$ chosen in a random sample $S$. The following facts immediately follow from the negative association property and the fact that any tree has exactly $n-1$ edges, see \cite{OSS11} for more details.
\begin{fact}
\label{fact:updown}
If $\mu$ is a $\lambda$-uniform spanning tree distribution on $G=(V,E)$, then
for any $S\subset E$, $p\in \mathbb{R}$
\begin{enumerate}
\item If $e\notin S$, then $\EE{\mu}{X_{e} \big| X_S \geq p} \leq \EE{\mu}{X_{e}}$ and
$\EE{\mu}{X_{e} \big| X_S \leq p} \geq \EE{\mu}{X_{e}} $
\item If $e\in S$, then $\EE{\mu}{X_e | X_S \geq p} \geq \EE{\mu}{X_e}$ and $\EE{\mu}{X_e | X_S\leq p} \leq \EE{\mu}{X_e}$.
\end{enumerate}
\end{fact}
\begin{fact}
For any set of edges $S$ and $e\not\in S$,
\begin{equation}
\EE{\mu}{X_S} \le \EE{\mu}{ X_S | X_e = 0} \le \EE{\mu}{X_S} + x_e
\label{fact:e0}
\end{equation}
and
\begin{equation}
\EE{\mu}{X_S} - x_e \le\EE{\mu}{ X_S | X_e = 1} \le \EE{\mu}{X_S}.
\label{fact:e1}
\end{equation}
\end{fact}
\begin{lemma} \label{lem:correlation}
Let $S = \{e_1, \ldots, e_k\}$ be a set of $k$ edges and suppose that $e \not\in S$.
Then there are $k-1$ edges in $S$ such that for each edge $e_i$ among these $k-1$
$$0.25 \le \P{e_i \in T | e \in T} \le 0.5.$$
\end{lemma}
\begin{proof}
By \cref{fact:e1}, $\P{e_i | e\in T} \leq 0.5$ for all $e_i$.
Suppose that (after renaming) $\P{e_1 | e\in T}$ and $\P{e_2 | e\in T}$ are the smallest among all $1\leq i\leq k$. Observe that by \cref{fact:e1} $\E{X_{e_1}+X_{e_2} | e\in T}\geq 0.5$. Therefore, the bigger one is at least $0.25$.
\end{proof}
\paragraph{Rank Sequence.}
The {\em rank sequence} of $\mu$ is the sequence
$$\P{X_E=0}, \P{X_E=1}, \ldots,\P{X_E=m}.$$
Let $g_\mu(y)$ be the generating polynomial of $\mu$.
The {\em diagonal specialization} of $\mu$, $\bar{g}(.)$ is a univariate polynomial obtained by treating $g(.)$ as a univariate polynomial (i.e., considering $g(y,y,\ldots,y)$). Observe that $\bar{g}(.)$ is the generating polynomial of the rank sequence of $\mu$. If $\bar{g}(c)=0$ for $c\in \mathbb{C}$, then $g(c,c,\ldots,c)=0$. So, if $g(.)$ is a real stable polynomial then so is $\bar{g}$. Since a univariate polynomial with real coefficients is stable if and only if all of its roots are real, $\bar{g}(.)$ is a polynomial with real roots.
Generating polynomials of probability distributions with real roots are very well studied in the literature (see \cite{Pit97} and references therein). If $\bar{g}(.)$ is a real rooted univariate polynomial of degree $m$ with nonnegative coefficients, then coefficients of $\bar{g}(.)$ correspond to the probability density function of the convolution of a set of $m$ independent Bernoulli random variables (up to a normalization). In other words,
there are $m$ independent Bernoulli random variables $B_1,\ldots,B_m$ with success probabilities $p_1,\ldots,p_m\in[0,1]$ such that the probability that exactly $k$ variables succeed
is the coefficient of $y^k$ in $\bar{g}(.)$.
\begin{fact}[{\cite{BBL09,Pit97}}]
\label{fact:SRsumindependentBernoulli}
The rank sequence of a strongly Rayleigh measure is the probability distribution of the number
of successes in $m$ independent trials for some sequence of success probabilities $p_1,\ldots,p_m\in [0,1]$.
\end{fact}
Given this, we can apply the following theorem by Hoeffding \cite{Hoe56}, following the approach of~\cite{OSS11}.
\begin{theorem}[{\cite[Corollary 2.1]{Hoe56}}]\label{thm:hoeffding}
Let $g:\{0,1,\dots,m\}\to \mathbb{R}$ and $0\leq p\leq m$ for some integer $m\geq 0$. Let $B_1,\dots,B_m$ be $m$ independent Bernoulli random variables with success probabilities $p_1,\dots,p_m$ that minimizes (or maximizes)
$$ \E{g(B_1+\dots+B_m)}$$
over all distributions in ${\cal B}_m(p)$. Then, $p_1,\dots,p_m\in\{0,x,1\}$ for some $0<x<1$.
\end{theorem}
\begin{lemma}\label{lem:420} Let $S \subseteq E$ with $|S| = 3$. Furthermore, assume that $\P{|S \cap T| \ge 1} = 1$. Then, $\P{|S \cap T| = 1} \ge \frac{1}{2}$ and $\P{|S \cap T| = 2} \ge \frac{3}{8}$.\end{lemma}
\begin{proof}
By \autoref{fact:SRsumindependentBernoulli}, we can write the rank sequence of $|S \cap T|$ as a sum of 3 independent Bernoullis $B_1,B_2,B_3$, and since $\P{|S \cap T| \ge 1} = 1$ we know that for one Bernoulli $p=1$. Without loss of generality let $p_1 = 1$. Then by \autoref{thm:hoeffding} we know that $\P{|S \cap T| = 1}$ and $\P{|S \cap T| = 2}$ are minimized when $p_2 = p_3 = \frac{1}{4}$ or $p_2 = \frac{1}{2}$ and $p_3 = 0$. Therefore:
$$\P{|S \cap T| = 1} \ge \min\bigg\{\bigg(\frac{3}{4}\bigg)^2,\frac{1}{2}\bigg\} = \frac{1}{2}$$
$$\P{|S \cap T| = 2} \ge \min\bigg\{2\bigg(\frac{1}{4}\bigg)\bigg(\frac{3}{4}\bigg),\frac{1}{2}\bigg\} = \frac{3}{8}$$
\end{proof}
The following two lemmas are proved using a similar analysis.
\begin{lemma}\label{lem:420-2} Let $S \subseteq E$ with $|S| = 2$. Let $\frac{1}{2} \le \E{|S \cap T|} \le \frac{3}{2}$. Then $\P{|S \cap T| = 1} \ge \frac{3}{8}$.\end{lemma}
\begin{lemma} \label{lem:cut_even}
For a min-cut $C$, $\P{|T\cap C|\text{ even}}\geq 13/27$.
\end{lemma}
\begin{lemma}\label{lem:427} Let $S_1,S_2 \subseteq E$ with $|S_1 \cap S_2| = \varnothing$. Let $|S_1|=|S_2|=2$, or equivalently $\E{|S_1 \cap T|} = \E{|S_2 \cap T|} = 1$. Then $\P{|S_1 \cap T| = 1 \land |S_2 \cap T| = 1} \ge \frac{3}{16}$.\end{lemma}
\begin{proof}
Let $S_1 = \{e,f\}$. Then condition on $e \in T$: this occurs with probability $\frac{1}{2}$. By \autoref{fact:updown} we have
$$\E{|f \cap T| \mid e \in T} \le \frac{1}{2}$$
Then condition on $f \not\in T$. Given the above, this happens with probability at least $\frac{1}{2}$. Similarly consider the event $e \not\in T$ and $f \in T$. One of these occurs with probability $\frac{1}{2}$. Therefore, in either event we have:
$$\frac{1}{2} \le \E{|S_2 \cap T|} \le \frac{3}{2}$$
And now by \autoref{lem:420-2} both events occur simultaneously with probability at least $\frac{3}{16}$.
\end{proof}
\section{Overview of Analysis}
As already mentioned, our algorithm consists of two steps: sample a 1-tree $T$, and then construct an optimal $O$-join for the odd degree vertices in the 1-tree.
Given a feasible LP solution $x$, the choice $y_e = x_e/2$ for each edge $e\in E$ (which gives $y_e := 1/4$ in the half integral case), yields an $O$-join solution of total cost at most $OPT/2$. However, this is essentially Christofides' algorithm and
guarantees only a 3/2 approximation.
The key to improving on this is the observation that constraint
\eqref{eq:tjoinlp} in the $O$-join LP is not binding if the intersection of the cut $\delta(S)$ with the tree is even.
\begin{definition}[Even cuts]
A cut $\delta(S)$ is \textbf{even} in $T$ (or simply ``even'' when T is understood) if $|T \cap \delta(S)|$ is even.
\end{definition}
Thus, for every edge $e$ with the property that every min-cut that $e$ is on is even, we can reduce $y_e$ to 1/6, since every non-min-cut has at least 6 edges, and therefore this guarantees that constraint \eqref{eq:tjoinlp} remains satisfied everywhere. This is the gist of the approach taken in \cite{OSS11}.
Suppose that there are multiple ``good" edges $e$ with the property that every min-cut they are on is even, say with probability at least $p$ (over the randomness in the selection of $T$). Then for those outcomes $T$ in which $e$ has this property, we could set $y_e := 1/6$ and satisfy the $O$-join constraints. This would save us $\frac{1}{12}c(e)$ on every such edge $e$ (the reduction from $1/4$ to $1/6$) and thereby
guarantee a reduction in the cost of the $O$-join solution
of $\sum_{e\text{ "good"}} \frac{p}{12} c(e)$.
Unfortunately, in general, it is not possible to
argue that every min-cut an edge is on is even simultaneously in $T$ with constant probability. So, we will use a careful charging scheme.
\begin{definition}[Last Cuts]
For an edge $e$, the last cuts of $e$ are the only ({\em two}) min-cuts containing $e$ and edges going higher in the graph right before contracting $S_e$.
\end{definition}
Observe that the last cuts of a top edge are critical cuts, but the last cuts of bottom edges are not critical.
\begin{definition}[Even at Last]
For an edge $e$ we say $e$ is even at last if the two last cuts of $e$ are even.
Equivalently, if $e$ is a bottom edge, we say $e$ is even at last if all the min cuts containing $e$ on the cycle defined by the graph consisting of $S_e$ with $V \smallsetminus S_e$ contracted are even.
Otherwise, if $e=\{u,v\}$ is a top edge, then it is even at last if the critical cuts $S_u,S_v$ are even simultaneously.
\end{definition}
\begin{fact}
\label{fact:criticalsat}
If a bottom edge $e$ is even at last, then all (bottom) edges $f$ where $S_f=S_e$ are even at last.
\end{fact}
\begin{proof}
Since $\delta(S_e)$ is a cycle cut, the edges inside $S_e$ form a path,
and thus, exactly one edge between each pair of (possibly contracted) vertices inside $S_e$ is selected as part of the tree on $S_e$ chosen in step 6 of the algorithm.
If, $e$ is even at last, we must have exactly one of each pair of cycle partners on $\delta(S)$
is in $T$; therefore, every pair of adjacent nodes in the cycle have one edge connecting them in the tree. So, all cuts on the cycle have exactly two edges in $T$. This implies every bottom edge of this cycle is even at last.
\end{proof}
\begin{remark}
By \cref{fact:min-cuts2}, the companion of every bottom edge $e$ has exactly the same pair of last cuts as $e$.
\end{remark}
\begin{definition}[Good edges]\label{defn:good}
An edge $e=(u,v)$ is \textbf{good} if it is even at last with probability at least $p$ for some constant $p > 0$.
\end{definition}
Instead of proving that all min-cuts that a single edge is on are even, we will instead prove that every minimum cut contains at least one good edge. Each good edge $e$ will then be responsible for its last two cuts. This will allow edges to be reduced when they are even at last, as all cuts lower in the hierarchy are handled by other edges.
\begin{theorem}\label{thm:mainprobabilistic}
There is a universal constant $p\geq 1/27$ such that every every min-cut has at least one good edge.
\end{theorem}
The proof of the above theorem together with some strengthened statements will be in \cref{sec:probFacts}. The proof mainly exploits properties of strongly Rayleigh distributions.
As we hinted at, we will reduce the value of $y_e$ to 1/6 whenever an edge $e$ is even at last. However, since $e$ may also be on many other lower min-cuts, if we reduce $y_e$, the solution may not be feasible (\eqref{eq:tjoinlp} may be violated) as the lower min-cuts may be odd. To handle any lower min-cut $C$ that $e$ is on, we show that, conditioned on $e$ being even at last, the probability that $C$ is also even is at least $q$ for some $q\geq \Omega(1)$. Therefore, we only need to worry about the lower cuts with probability $1-q$ each. In the bad event that a lower cut $C$ is odd, we will need to fix the solution to guarantee that \eqref{eq:tjoinlp} still holds: our approach is to split the deficit introduced in the $O$-Join constraint for $C$ among the good edges that do not go higher (see \cref{def:gohigher}). We then simply show that in expectation each edge gains.
This part of the proof heavily exploits the properties of cactus representation of the min-cuts that we discussed above.
We note that \cref{thm:mainprobabilistic} on its own is not enough to run our charging argument; so, we need a slightly stronger version. In particular, in some cuts we may need to have two or three good edges.
\section{Analysis}
\section{Probabilistic lemmas}\label{sec:probFacts}
In this section, we present three probabilistic lemmas which show that in every min-cut there is at least one good edge, (and in some there are even more).
This immediately proves \cref{thm:mainprobabilistic}.
Note the last cycle that we choose in step \ref{alg:step:cycle} of the algorithm has all edges even at last so we don't need to address it in this section. Furthermore, by \cref{fact:e+notindS}, $e^+$ does not belong to any critical cut.
\begin{restatable}[Bottom edge lemma]{lemma}{bottom}
\label{lem:bottom-highest}
Suppose that $e=(u,v)$ is a bottom edge. Then $e$ is good (where $p \ge 3/16$). \end{restatable}
\begin{figure}[!htbp]
\centering
\begin{tikzpicture}
\foreach \a/\b in {-3/1,0/0,3/1}{\draw[bleudefrance,fill=blizzardblue,thick] (\a,\b) circle (12pt);}
\node at (0,-0.7) {$S_{u,e}$};
\node at (3,0.3) {$S_{v,e}$};
\node [draw=none] at (-3.6,-0.5) () {$S_e$};
\draw[gray,xshift=-0.05cm,yshift=-0.15cm] (-3,1) to (-0.15,0.05);
\draw[gray,xshift=0.05cm,yshift=--0.15cm] (-3,1) to (-0.15,0.05);
\draw[gray,xshift=-0.05cm,yshift=0.15cm] (0.15,0.05) to (3,1);
\path[-] (0.15,0.05) edge[below,midway,thick,orange,xshift=0.05cm,yshift=-0.15cm] node {$e=(u,v)$} (3,1);
\path[-] (-3,1) edge[below,midway,thick,purple,xshift=-0.2cm,yshift=-0.05cm] node {$a$} (-4,2.5);
\path[-] (-3,1) edge[right,midway,thick,purple,xshift=-0.05cm,yshift=0.15cm] node {$b$} (-3,2.75);
\path[-] (3,1) edge[below,midway,thick,purple,xshift=0.2cm,yshift=0.05cm] node {$c$} (4,2.5);
\path[-] (3,1) edge[right,midway,thick,purple,xshift=0.05cm,yshift=0.15cm] node {$d$} (3,2.75);
\draw [black,line width=1.2pt, dashed] (0,1) ellipse (4.5 and 2);
\end{tikzpicture}
\caption{Illustration of edges in Bottom Edge Lemma.}
\label{fig:bottom}
\end{figure}
\begin{proof}
If $e$ is a bottom edge then $S_e$ ($= S_{(u,v)}$) is a cycle cut. By construction, when a tree on $S_e$ is
selected in step 8 of the algorithm, exactly one edge is chosen between every pair of adjacent
nodes in $E(S_e)$. So it suffices to consider the edges in $\delta(S_e)$.
These divide up into two pairs of cycle partners connecting $S_e$ to $V \smallsetminus S_e$, say $\{a, b\}$ and $\{c,d\}$. (See \cref{fig:bottom}.)
Then by \cref{lem:427}, setting $S_1:= \{a, b\}$ and $S_2:= \{c,d\}$,
we have $\P{|S_1 \cap T|= 1\text{ and } |S_2 \cap T| = 1} \ge 3/16$.
\end{proof}
\begin{restatable}[Top edge lemma]{lemma}{top}
\label{lem:top-good}
In a critical cut $\delta(S)$ with one edge that goes higher, of the remaining three edges in the cut, at least two are good with $p \ge \frac{1}{16}$.
\end{restatable}
\begin{proof}
\begin{figure}[!htbp]
\centering
\begin{tikzpicture}
\foreach \a/\b/\c in {-3/e/3,-1/a/2,1/b/2,3/c/2}{
\draw[bleudefrance,fill=blizzardblue,thick] (\a,0) circle (12pt);
\path[-] (\a,0) edge[right,midway,thick,purple,yshift=0.5em] node {$\b$} (\a,\c);
}
\path[-] (-3,0) edge[above,midway,thick,goodgreen] node {$f$} (-1,0);
\path[-] (-3,0) edge[below,midway,thick,goodgreen,bend right=25] node {$g$} (1,0);
\path[-] (-3,0) edge[below,midway,thick,goodgreen,bend right=38] node {$h$} (3,0);
\draw [black,line width=1.2pt, dashed] (0,-0.4) ellipse (4.2 and 1.3);
\node[black] at (-3.8,0.6) {$\delta(S)$};
\draw [black,line width=1.2pt, dashed] (0,0) ellipse (5.2 and 3);
\end{tikzpicture}
\caption{Illustration of top lemma. The figure shows the case where all three of $S_f$, $S_g$ and $S_h$ have an edge that goes higher ($a,b,c$ respectively). It may be that some number of these nodes do not have an edge going higher.}
\label{fig:top}
\end{figure}
First, suppose that $e$ is the edge that goes higher from $\delta(S)$, and $f$, $g$ and $h$ are the other edges
in $\delta(S)$.
If the other endpoint of one of these three edges, say $S_h$, has no edge that goes higher then $h$ is good. (See left side of \cref{fig:top}.) To see this, observe that we can condition on $\delta (S_h)$ being
even which by \cref{lem:cut_even} has probability at least 13/27. Given this event,
$|\{f,g,h\}\cap T|$ is either even or odd. In either case,
the event $e \in T$ is an independent event that occurs with probability 1/2. Therefore,
$$\P{ |\delta(S_h) \cap T|\text{ even}} \cdot\P {e\text{ makes }\delta(S) \text{ even} ~|~|\delta(S_h) \cap T |\text{ even}} \ge \frac{13}{27}\cdot\frac{1}{2}.$$
Therefore, at least two of $f$,$g$ and $h$ are good if at least two of $S_f$, $S_g$ and $S_h$ do not have an edge that goes higher.
Consider next the case that, say, $S_f$ and $S_g$ both have an edge that goes higher, but $S_h$ doesn't. As before, $h$ is good. We claim that one of $f$ and $g$ is also good.
To see this, since $e\in T$ is independent of $f,g,h \in T$, we
observe using \cref{lem:420} that $$\P{e \in T \text{ and } |\{f,g,h\}\cap T|
= 1} \ge \frac{1}{2} \cdot \frac{1}{2}.$$
Next, apply \cref{lem:correlation} to $a$ and $b$ to conclude that for one of $a$ and $b$, say $a$,
$$\P{a\in T | e \in T} \ge \frac{1}{4}.$$
Therefore, as before, regardless of the even/odd status of $X\cup f$ after conditioning $e$ in and $f,g,h$ to 1, the cut can be fixed by $a$, meaning we have $p \ge \frac{1}{16}$.
Finally, if all three of $S_f$, $S_g$ and $S_h$
have an edge that goes higher, then again, condition $e$ in and $\{f,g,h\}\cap T|
= 1$. Then
apply \cref{lem:correlation} to $a$, $b$ and $c$ to conclude that for two of them, say $a$ and $b$, their probability of being in $T$ given that $e \in T$ is at least 1/4.
Therefore, each can fix their corresponding cut ($\delta(S_f)$ for $a$ and $\delta(S_g)$ for $b$) and both $f$ and $g$ are good.
\end{proof}
\begin{lemma}
For every critical cut $S$, there is an edge $e\in\delta(S)$ such that $\P{e\text{ even at last}} \geq 1/27$.
\end{lemma}
\begin{proof}
First, if there is a critical set $S'$ ($S'\neq S)$ such that $|\delta(S)\cap\delta(S')|=2$, then by \cref{fact:2gohigherbottom}, there exist at least two bottom edges in $\delta(S)$ and by \cref{lem:bottom-highest} they are good and we are done. Furthermore, if there is an edge in $\delta(S)$ that goes higher then we are done by \cref{lem:top-good}.
Otherwise, assume $S$ has at most one edge to any other critical set and no edge goes higher. Recall that $T^-=T- e^+$ is always a spanning tree.
Let $\mu'$ be the conditional measure where $|\delta(S)\cap T^-|=1$. Observe that in $\mu'$, first we sample a tree\footnote{Note $T$ is not a spanning tree in $G\smallsetminus S$, that is why we need to look at $T^-$.} in $G\smallsetminus S$, and then we {\em independently} add an edge in $\delta(S)$. Therefore, by \cref{fact:updown}, $\PP{\mu'}{e\in T}\leq 1/2$.
Thus, for at least one edge $e\in \delta(S)$, we have $1/4\leq \PP{\mu'}{e\in T}\leq 1/2$. In the special case that $\P{|\delta(S)\cap T|=1}=0$, we just let $e$ be an arbitrary edge in $\delta(S)$. Say, $\{e\}= \delta(S)\cap \delta(S')$.
It remains to prove that $\PP{\mu}{\delta(S),\delta(S') \text{ even}}>1/27$ since $S,S'$ are the last cuts of $e$.
\begin{eqnarray*}
\P{\delta(S),\delta(S') \text{ even}} &=& 1- \P{\text{$\delta(S)$ or $\delta(S')$ odd}}\\
&=& 1- \P{\delta(S) \text{ odd}} - \P{\delta(S') \text{ odd}} + \P{\delta(S),\delta(S') \text{ odd}}\\
&\geq & 13/27 - \P{\delta(S) \text{ odd}} + \P{|T\cap \delta(S)|=1 \wedge \delta(S')\text{ odd}},
\end{eqnarray*}
by \cref{lem:cut_even}. First, note if $\P{|T\cap \delta(S)|=1}=0$, then we get that $\P{|T\cap \delta(S)|=2}=1$ (since $\P{|\delta(S)\cap T|\geq 1}=1$, and $\E{|\delta(S)\cap T|}=2$). So, the RHS is $13/27$ and we are done.
Otherwise,
\begin{eqnarray*} \P{|T\cap \delta(S)|=1 \wedge \delta(S')\text{ odd}} &=& \P{\delta(S')\text{ odd} \Big| |T\cap\delta(S)|=1}\cdot \P{|T\cap \delta(S)|=1}\\
&\geq& \frac14\cdot\P{|T\cap \delta(S)|=1}.
\end{eqnarray*}
The inequality is because edge $e$ can make $\delta(S')$ odd by being in/out of the tree and that has probability at least $1/4$.
Therefore,
\begin{eqnarray*}
\P{\delta(S),\delta(S') \text{ even}}&\geq & 13/27 - \P{\delta(S)\text{ odd}} + \frac14 \P{|T\cap \delta(S)|=1}\\
&=& 13/27 - \frac34\P{|T\cap \delta(S)|=1} - \P{|T\cap \delta(S)|=3}.
\end{eqnarray*}
Finally, by \cref{fact:e1}, the RHS attains its minimum value when $|T\cap \delta(S)|$ is sum of 4 Bernoullis with success probabilities $1,1/3,1/3,1/3$.
\end{proof}
\section{Proof of Main Theorem}
Recall that every good edge is even on top with probability at least $p$. The following statement is the main technical result of this section.
\begin{lemma}\label{prop:goodojoin}
There is a (random) feasible $O$-join solution such that for every good edge $e$,
$$\E{y_e} \leq 1/4-p/240,$$
and for every bad edge $y_e=1/4$ with probability 1.
\end{lemma}
Before, proving the above statement we use it to prove \cref{thm:main}.
\begin{proof}[Proof of \cref{thm:main}]
Consider the trivial $O$-join solution $y'$ where $y'_e=1/2$ if $e$ is good and $y'_e=1/6$ otherwise. Note that this is a valid $O$-join by \cref{thm:mainprobabilistic}.
Now, define $z=\alpha y + (1-\alpha)y'$ for some $\alpha$ that we choose later. It follows that for any good edge $e$,
$$\E{z_e} \le \alpha(\frac{1}{4} - \frac{p}{240}) + (1-\alpha)\frac{1}{2},$$
and for a bad edge $f$:
$$\E{z_f} = \alpha\frac{1}{4} + (1-\alpha)\frac{1}{6}$$
So, for $p=\frac{1}{27}$ and $\alpha = \frac{2160}{2161}$ we obtain $\E{z_e}\leq 0.249962$.
Since any edge $e$ is chosen in $T$ with probability $1/2$ (up to a $2^{-n})$ error), we pay at most $1/2+\E{z_e}$ for any edge $e$ whereas $x$ pays $1/2$. Therefore, we get a $0.749962/0.5$ approximation algorithm.
\end{proof}
So, in the rest of this section we prove \cref{prop:goodojoin}.
\paragraph{O-join construction for good edges:}
For each good edge $e$, define $B_e$ to be an independent Bernoulli random variable which is 1 with probability $p/p_e$, where $p$ is the {\em lower bound} on the probability that {\em any} good edge is even on top, and $p_e$ is the {\em actual} probability that $e$ is even on top.
If $e,f$ are bottom edge companions, then we let $B_f=B_e$ (with probability 1).
Note that this still makes selection of $e,f$ independent of $B_e$ and any other edge of the graph.
We then construct an $O$-join solution for each 1-tree $T$ using the following three step process:
\begin{enumerate}
\item Initialize $y_e := 1/4$ for each edge $e \in E$.
\item Next, {if $e$ is even at last in $T$ and $B_e=1$}, reduce $y_e$ by $r_e(T)$ where:
$$
r_e(T):= \begin{cases}
\beta &\text{ if $e$ is a bottom edge.}\\
\tau_2 & \text{if $e=\{u,v\}$ is a good top edge and there are exactly 2 good top edges}\\ & \text{ in both $\delta(S_u)$ and $\delta(S_v)$ that do not go higher.}\\
\tau_3 & \text{if $e$ is a good top edge that does not meet the previous criteria.}
\end{cases}
$$
$\beta,\tau_2,\tau_3$ are parameters we will set later. For now, we just assume $\tau_3\leq \tau_2\leq \beta \leq 1/12$. When $r_e(T) > 0$, we say that $e$ is \textbf{reduced}.
\item On each cut $C$ that is odd, let $\Delta(C):= \sum_{e \in C} r_e (T)$ be the amount by which edges on that cut were reduced in step 2, and let $G_C$ be the set of good edges on $C$ such that $C$ is one of their last cuts.
Now, for an edge $e\in G_C$, let $C'$ (and $C$) be the last cuts of $e$.
Then increase $y_e$ by $\max\left\{\frac{\Delta(C)}{|G_C|}, \frac{\Delta(C')}{|G_{C'}|}\right\}$.
Notice that in this case, since $C$ is one of $e$'s top cuts, $e$ is not even on top in $T$ and therefore is not reduced in step 2.
\end{enumerate}
By construction, this is a valid $O$-join solution since on every min-cut we began with at least 4 edges crossing every cut with $y_e = 1/4$ and then guaranteed that every reduction on an odd cut in step 2 was compensated for by a matching increase on that cut in step 3. (The ultimate gain of course will come from the fact that many cuts will be even and hence, there will not need to be an increase.)
All non-min-cuts have at least 6 edges on them, each of value at least 1/6 (after reduction) and are therefore satisfied in \eqref{eq:tjoinlp}.
In the rest of the proof, it is enough to show that for any good edge $e$,
$\E{y_e}\leq 1/4-p/240.$
We complete the proof using the following two lemmas.
\begin{lemma}\label{lem:topedgejoin}
If $\beta \geq 5\tau_2/4$, then for any good top edge $e=\{u,v\}$,
$$\E{y_e}\leq \frac14 - p\min\{\tau_2-\beta/2,\tau_3-5\beta/12\}.$$
\end{lemma}
\begin{proof}
Let $C:= \delta (S_u)$ and $C':= \delta (S_v)$. Recall that, since $e$ is a top edge, $C$ and $C'$ are critical cuts.
Let $H_C$ (resp. $H_{C'}$) be the good edges in $C$ (resp. $C'$) that go higher.
Note that $|H_C|$ and $|H_{C'}|$ is either 0 or 1 by \cref{fact:2cycle}. We consider 3 cases:
\begin{description}
\item [Case (i): $|G_C| = |G_{C'}| = 2$.]
In the worst case, there is an edge, say $f\in H_C$ and an edge $g\in H_{C'}$.
If $f$ is a bottom edge, then
$$\E{r_f(T)} = \beta\cdot p\quad\text{ and }\quad
\P{C\text{ is odd} | f \text{ reduced}} = 1/2$$
since $$\P{\text{parity of }|f \cap T| = \text{ parity of }|(\delta(S_u) \smallsetminus f) \cap T|} = 1/2,$$
by \cref{fact:efinout}.
Therefore, the expected reduction in step 2 on $C$ is at most
$$ \E{r_f(T)}\cdot\frac{1}{|G_C|}\cdot \P{C\text{ is odd} | f \text{ reduced}} =
\beta p\cdot \frac{1}{2}\cdot\frac{1}{2} = \frac{\beta\cdot p}{4}.$$
On the other hand, if $f$ is a top edge,
then the expected reduction in step 2 on $C$ is at most
$$ \E{r_f(T)}\cdot\frac{1}{|G_C|}\cdot \P{C\text{ is odd} | f \text{ reduced}} \le
\tau_2 p\cdot \frac{1}{2}\cdot\frac{5}{8} \leq \frac{\beta p}{4},$$
where we use \cref{lem:420} and the fact that edge $f$ is independent of the rest of edges in $C$ to infer that , $\P{C\text{ is odd} | f \text{ reduced}}\leq 5/8$.
The same reasoning applies to $g$, so we get
$$\E{y_e} \le \frac{1}{4} - \tau_2p + 2\cdot \frac{\beta p}{4}.$$
\begin{figure}[!htbp]
\centering
\begin{tikzpicture}
\foreach \a/\b/\c in {-1/f/2,1/g/2}{
\draw[bleudefrance,fill=blizzardblue,thick] (\a,0) circle (12pt);
\path[-] (\a,0) edge[right,midway,thick,purple,yshift=0.5em] node {$\b$} (\a,\c);
}
\path[-] (-1,0) edge[above,midway,thick,blue,xshift=-0.3em,yshift=0.1em] node {} (-2,0.25);
\path[-] (-1,0) edge[below,midway,thick,orange,xshift=-0.3em,yshift=-0.1em] node {} (-2,-0.25);
\path[-] (1,0) edge[above,midway,thick,blue,xshift=0.3em,yshift=0.1em] node {} (2,0.25);
\path[-] (1,0) edge[below,midway,thick,orange,xshift=0.3em,yshift=-0.1em] node {} (2,-0.25);
\path[-] (-1,0) edge[below,midway,thick,orange] node {$e$} (1,0);
\draw [black,line width=1.2pt, dashed] (0,-0.2) ellipse (2.5 and 1.1);
\node[black] at (-2.7,0.6) {$\delta(S_{e})$};
\end{tikzpicture}\qquad
\begin{tikzpicture}
\foreach \a/\b/\c in {-1/f/2,1/g/2}{
\draw[bleudefrance,fill=blizzardblue,thick] (\a,0) circle (12pt);
\path[-] (\a,0) edge[right,midway,thick,purple,yshift=0.5em] node {$\b$} (\a,\c);
}
\path[-] (-1,0) edge[above,midway,thick,orange,xshift=-0.3em,yshift=0.1em] node {} (-2,0.25);
\path[-] (-1,0) edge[below,midway,thick,orange,xshift=-0.3em,yshift=-0.1em] node {} (-2,-0.25);
\path[-] (1,0) edge[above,midway,thick,blue,xshift=0.3em,yshift=0.1em] node {} (2,0.25);
\path[-] (1,0) edge[below,midway,thick,orange,xshift=0.3em,yshift=-0.1em] node {} (2,-0.25);
\path[-] (-1,0) edge[below,midway,thick,orange] node {$e$} (1,0);
\draw [black,line width=1.2pt, dashed] (0,-0.2) ellipse (2.5 and 1.1);
\node[black] at (-2.7,0.6) {$\delta(S_{e})$};
\end{tikzpicture}
\caption{Cases (i) is on the left and case (ii) is on the right. Orange edges are good. On the left, $y_e$ is reduced by $\tau_2$ with probability $p$ and pays half the burden on both sides; on the right $y_e$ is reduced by $\tau_3$ with probability $p$ and pays half the burden on one side and one third of the burden on the other.}
\label{fig:casebc}
\end{figure}
\item [Case (ii): $|G_C|\geq 3$ or $|G_{C'}|\geq 3$ (or both).]
Again, in the worst case, there is an edge, say $f\in H_C$ and an edge $g\in H_{C'}$.
By the same reasoning as above, $\E{y_e}$ is largest if $f$ and $g$ are bottom
edges. In this case, the same calculation as above gives,
$$ \E{r_f(T)}\cdot\frac{1}{|G_C|}\cdot \P{C\text{ is odd} | f \text{ reduced}} \le
\beta p\cdot \frac{1}{|G_C|}\cdot\frac{1}{2},$$
and similarly for $g$,
so
$$\E{y_e} \le \frac{1}{4} - \tau_3 p + \frac{\beta p}{2}\left(\frac{1}{2} + \frac{1}{3}\right).$$
\item [Case (iii): $|H_C|+|H_{C'}|\leq 1$.]
In the worst case, $|H_C|=1$ and $|H_{C'}|=0$ and $f\in H_C$ is a bottom edge.
Note that in this case we may have $G_{C'}=\{e\}$. But the advantage is that we never increase $y_e$ to fix $C'$ since $H_{C'}=\emptyset$.
Then,
$$\E{y_e} = \frac{1}{4} - \tau_3 p + \beta p\cdot \frac1{|G_C|}\cdot \frac12 \leq 1/4 -\tau_3 p +\frac{\beta p}{4}.$$
\end{description}
Note that if case (iii) does not happen, then by \cref{lem:top-good} we have $|G_C|,|G_{C'}|\geq 2$. So, either (i) or (ii) will happen.
\end{proof}
\begin{lemma}\label{lem:bottomedgeojoin}
If $3\tau_3\leq 2\tau_2$, then for any (good) bottom edge $e$,
$$\E{y_e}\leq 1/4 - p\min\{\beta/4, 3\beta/4-\tau_2, \beta-4\tau_2/3\}.$$
\end{lemma}
\begin{proof}
Say $f$ is the companion of $e$.
Let $S=S_e$ and $S'$ be the parent of $S$ in the hierarchy of critical cuts.
Say the last cuts of $e$ (and $f$) are $C=\{e,f,a,b\}$ and $C'=\{e,f,g,h\}$.
In other words $a,b$ are partners and $g,h$ are partners.
Note that $|G_C|=|G_{C'}|=2$ because all edges $\{a,b,g,h\}$ go to the higher ciritical cut $S_e$.
\begin{description}
\item [{Case (i)}: $a$ and $g$ go higher than $S$.]
We have $a,g\in \delta(S')$. So, by \cref{fact:2cycle}, $S'$ is also a cycle cut.
This means that $b$ and $h$ are companions and $a$ and $g$ are cycle partner pairs on $\delta(S')$. (See \cref{fig:casebc}).
Edge $e$ has to increase to fix the cuts $C,C'$ whenever $a,b,g$ or $h$ are decreased and the corresponding cut is odd. The expected increase in $y_e$ due to reductions on $a,b,g,h$ divides into two types.
\begin{figure}[!htbp]
\centering
\begin{tikzpicture}
\foreach \a/\b in {-3/1,0/0,3/1}{\draw[bleudefrance,fill=blizzardblue,thick] (\a,\b) circle (12pt);}
\node at (0,-0.7) {$S_{u,e}$};
\node at (3.5,0.4) {$S_{v,e}$};
\node [draw=none] at (-2.5,-1) () {$S$};
\node [draw=none] at (-4.5,-1.6) (){$S'$};
\draw[gray,xshift=-0.05cm,yshift=-0.15cm] (-3,1) to (-0.15,0.05);
\draw[gray,xshift=0.05cm,yshift=--0.15cm] (-3,1) to (-0.15,0.05);
\draw [dashed,red, line width=1.1pt] (0.5,-0.5) arc (-20:170:2.2);
\node [draw=none,red] at (-0.6,1.9) () {$C$};
\draw [dashed,red,line width=1.1pt] (3.7,1) arc (0:250:0.7);
\node [draw=none,red] at (3.9,1.2) () {$C'$};
\draw[gray,xshift=-0.05cm,yshift=0.15cm] (0.15,0.05) to node [above] {$f$} (3,1);
\path[-] (0.15,0.05) edge[below,midway,thick,orange,xshift=0.05cm,yshift=-0.15cm] node {$e=(u,v)$} (3,1);
\path[-] (-3,1) edge[below,midway,thick,purple,xshift=-0.2cm,yshift=-0.05cm] node {$a$} (-5,4);
\path[-] (-3,1) edge[right,midway,thick,purple,xshift=-0.05cm,yshift=0.15cm] node {$b$} (-3,2.75);
\path[-] (3,1) edge[below,midway,thick,purple,xshift=0.2cm,yshift=0.05cm] node {$g$} (5,4);
\path[-] (3,1) edge[right,midway,thick,purple,xshift=0.05cm,yshift=0.15cm] node {$h$} (3,2.75);
\draw [black,line width=1.2pt, dashed] (0,1) ellipse (4.5 and 2);
\draw [black,line width=1.2pt, dashed] (0,1.2) ellipse (5.5 and 3.5);
\end{tikzpicture}
\caption{Case (i) of \cref{lem:bottomedgeojoin}.}
\label{fig:casebc}
\end{figure}
We start by $b,h$: By \cref{fact:efinout} and that $B_b=B_h$, we know that $b$ and $h$ are always reduced at the same time. Furthermore, conditioned on $b$ (and $h$) being reduced, we have
$$ \text{parity of } |T\cap C| = \text{parity of }|T\cap C'|. $$
This is simply because $b$ (and $h$) are reduced only when they are even at last which implies $|T\cap \{a,g\}|=1$.
So, we can fix the reduction of $b,h$ simultaneously when we increase $e$ (or $f$). In other words, it is enough to only take into account the expected increase of $y_e$ due to $b$, i.e.,
$$ \E{r_b(T)}\cdot \frac1{|G_C|} \cdot \P{|C\cap T| \text{ is odd} | b \text{ reduced}} =
\beta p\cdot \frac{1}{2}\cdot\frac{1}{2} = \frac{\beta p}{4}.$$
Now, we calculate the expected increase due to $a,g$: We compute the charge due to $a$ and the same will hold for $g$. Again, by \cref{fact:efinout}, $b$ and $h$ are chosen independently of $a,g$, i.e., $\P{|T\cap C|\text{ odd} | a \text{ reduced}} \leq 1/2$. Therefore, the expected increase due to $a$ is
$$ \E{r_a(T)} \cdot \frac1{|G_C|} \cdot\P{|C\cap T|\text{ odd} | a\text{ reduced}}\leq \beta p\cdot \frac12\cdot \frac12=\frac{\beta p}{4}$$
using $\tau_3,\tau_2\leq \beta$.
Therefore, altogether,
$$\E{y_e} \le \frac{1}{4} - \beta p + 3\cdot \frac{\beta p}{4}.$$
\item [Case (ii): Only one edge, say $a$, goes higher than $S$.]
In this case, by the same reasoning as above, we have
$$ \E{r_a(T)}\cdot\frac{1}{|G_C|}\cdot \P{|C\cap T| \text{ is odd} | a \text{ reduced}} \le
\beta p\cdot \frac{1}{2}\cdot\frac{1}{2},$$
since $b$ is independent of $a$ and it can be chosen to correct the parity.
For the remaining three edges $\{b,g,h\}$, by \cref{lem:top-good} either two or three of $b,g$ and $h$ are good.
If two are good, then each has an expected reduction of at most $\tau_2 p$ or
three are good and each has an expected reduction of at most $\tau_3 p$.
Therefore,
altogether,
$$\E{y_e} \le \frac{1}{4} - \beta p + \frac{\beta p}{4} + \frac{p}{2}\cdot\max\{2\tau_2, 3\tau_3\} \leq \frac14 -\frac{3\beta p}{4} + p\tau_2,$$
by the assumption of the lemma.
\item [Case (iii): $a,g$ are companions and $b,h$ are companions (on the next critical cut).]
This case follows the same analysis as case (i) but gains because in this case $a,g$ are also reduced simultaneously.
\item [Case (iv): No edge goes higher than $S$ and all $\{a,b,g,h\}$ are top edges.]
Some number of these edges are good; if more than two are good we pay $4\tau_3$ (at most) with probability $p$ and otherwise we pay $2\tau_2$ (at most) with probability $p$ . Then:
$$\E{y_e} \le \frac{1}{4} - \beta p + \frac{p}{|G_C|}\max\{2\tau_2,4\tau_3\} \leq \frac14 - \beta p + \frac{4\tau_2 p}{3}.$$
\end{description}
To finish the proof we just need to argue that we exhausted all cases. By \cref{fact:2cutinter}, among $\{a,b,g,h\}$ at most two go higher. By \cref{fact:cyclepartners}, from each pair of cycle partners, i.e., $\{a,b\}$ or $\{g,h\}$, at most one goes higher. Therefore, if case (i) does not happen, we have at most one that goes higher. If (i), (ii) do not happen, then no edge goes higher. So, by \cref{fact:2gohigherbottom} either all four edges in $\{a,b,g,h\}$ are bottom edges, i.e., case (iii), or none are bottom edges, i.e., case (iv).
\end{proof}
To finish the proof of \cref{prop:goodojoin}, let $\beta=1/12$, $\tau_2=7/120$ and $\tau_3=7/180$ chosen to satisfy $\tau_3\leq\tau_2\leq\beta$, $\beta\geq 5\tau_2/4$ and $3\tau_3\leq2\tau_2$. Plugging in these numbers into \cref{lem:topedgejoin} and \cref{lem:bottomedgeojoin} we obtain that $\E{y_e}\leq 1/4-p/240$ for any good edge $e$ as desired.
\bibliographystyle{alpha}
\section{Introduction} \label{sec:intro}
In an instance of the traveling salesperson problem (TSP) we are given a set of $n$ cities along with their pairwise symmetric distances. The goal is to find a Hamiltonian cycle of minimum cost. In the metric TSP problem, which we study here, the distances satisfy the triangle inequality. Therefore the problem is equivalent to finding a closed Eulerian connected walk of minimum cost. It is NP-hard to approximate TSP with a factor better than $\frac{185}{184}$ \cite{Lam12}. A classical algorithm of Christofides~\cite{Chr76} from 1976 gives a $\frac32$-approximation algorithm for TSP and remains the best known approximation algorithm for the general version of the problem despite significant work~\cite{Wol80,SW90,BP91,Goe95,CV00,GLS05,BM10,BC11,SWV12}.
Polynomial-time approximation schemes (PTAS) have been found for Euclidean \cite{Aro96,Mitchell99}, planar \cite{GKP95, AGKKW98, Kle05}, and low-genus metric \cite{DHM07} instances.
The case of graph metrics has received significant attention. In 2011, the third author, Saberi, and Singh~\cite{OSS11} found a $\frac{3}{2} - \epsilon_0$ approximation for this case. M\"omke and Svensson \cite{MS11} then
obtained a combinatorial algorithm for graphic TSP with an approximation ratio of 1.461. This approximation ratio was later improved by Mucha \cite{Muc12} to $\frac{13}{9} \approx 1.444$, and by Seb\"o and Vygen \cite{SV12} to $1.4$.
In this paper we study metric TSP for instances in which an optimal solution to the subtour linear programming relaxation is half-integral, i.e., when the optimal solution $x$ satisfies $x_e\in \{0,1/2,1\}$ for all edges $e$.
These instances are conjectured to be ``the hardest'' instances of TSP by Schalekamp, Williamson and van Zuylen.
\begin{conjecture}[\cite{SWvZ13}]\label{conj:SWZ}
The integrality gap for the subtour LP is attained on half-integral vertices of the polytope.
\end{conjecture}
The above conjecture is motivated by the fact that the worst known integrality gap examples of TSP (and TSP-path) are half-integral. Furthermore, as shown in \cite{SWvZ13} the worst case ratio of 2-matchings to optimal solution of the subtour-LP is attained by half-integral instances.
Very little progress has been made on half integral instances even though they have been a subject of study for decades, \cite{CR98, BC11, BS17, HNR17, HN19}.
Our main result is the following theorem:
\begin{theorem}\label{thm:main} There is a randomized polynomial time algorithm which when given any half-integral fractional solution $x$ of the subtour LP produces a tour with expected cost at most $1.49993$ times the cost of $x$.
\end{theorem}
So, if \cref{conj:SWZ} holds affirmatively, the above theorem implies that the integrality gap of the subtour-LP is at most $1.49993$.
Our result also has a direct consequence to the minimum cost 2-edge connected subgraph problem. About 20 years ago, Carr and Ravi \cite{CR98} showed that the integrality gap of the half-integral LP solutions of the min cost 2-edge connected subgraph problem is at most $4/3$. But, to the best of our knowledge, no polynomial time algorithm with an approximate factor better than $3/2$ is known. Our theorem also implies a $1.49993$ approximation algorithm for half-integral LP solutions of the minimum cost 2-edge connected subgraph problem.
\subsection{Overview of algorithm}
There are two well known lines of attack to metric TSP: (i) Start from an optimal Eulerian subgraph and make it connected by adding new edges while preserving the parity of the degrees, or (ii) Start with an optimal connected subgraph, then correct the parities of vertex degrees by adding the minimum cost Eulerian augmentation.
Here, we take the second approach.
It turns out that in approach (ii) the minimum cost Eulerian augmentation of any connected graph is simply the min cost matching on odd degree vertices which can be computed in polynomial time. So, the main question is how to choose a spanning tree of cost at most OPT such that the cost of the minimum Eulerian augmentation is bounded away from $\text{OPT}/2$. Here we follow the approach initiated in \cite{OSS11}.
We sample a random spanning tree that does not cost more than the optimum in expectation. More precisely, we sample from the maximum entropy distribution of spanning trees with marginals equal to the given LP solution, $x$, and then add the minimum cost matching on odd degree vertices.
It was conjectured in \cite{OSS11} that this algorithm beats Christofides for general metric TSP, but the authors could only justify a variant of this conjecture for graph metrics.
To bound the approximation factor for graph metrics, \cite{OSS11} showed that this random spanning tree
``locally'' looks like a Hamiltonian cycle, e.g., each vertex has degree 2 with constant probability, and, except in some special cases, each pair of vertices have degree 2 simultaneously with a constant probability.
Roughly speaking, the analysis of \cite{OSS11} is ``local'' in the sense that it shows that there is a set $F$ of edges with $x(F)\geq \Omega(n)$, such that each $e\in F$ is only contained in a constant number of ``local'' (near) min cuts and all these (near) min cuts have even number of edges in the random spanning tree with constant probability.
Such a method provably fails for the problem on general metrics since most of the cost of the LP may be concentrated on edges which show up in many (near) min cuts. So, one needs a more ``global'' analysis technique.
In this paper we take the first step towards a global {\em amortized} analysis.
The hard instances of TSP are those where the cost of the LP is dominated by the edges which show up in many (near) min cuts; in this case, there is no hope to show that all such cuts are even simultaneously in a random spanning tree with constant probability. Our high level framework is to build a hierarchy over edges. Roughly speaking, a more ``global'' edge shows up higher in the hierarchy. When an edge $e$ is even in its highest cuts in the hierarchy, we gain from $e$ at the expense of using descendants of $e$ in the hierarchy to pay for lower cuts of $e$ which may be odd. We then show that the amount $e$ gains when it reduces exceeds by a constant factor the amount it may have to pay to fix cuts containing edges going higher in the hierarchy.
Putting this together, we show that a variant of the max entropy sampling algorithm beats Christofides if the underlying LP solution is half-integral.
We expect that many of our techniques can be generalized to apply to LP solutions that are not half-integral, however the most difficult barrier to overcome seems to be that the structure of near minimum cuts (all cuts in the LP of value within 2 and $2+\epsilon$ for a fixed $\epsilon > 0$) is more complex than the structure of minimum cuts.
\section{Our Algorithm} \label{sec:prelim}
Before we discuss our algorithm, we need a few definitions and tools.
Where $V$ is the set of vertices, let $c:V\times V\to\mathbb{R}_+$ denote the cost of going from $u$ to $v$ for any $u,v\in V$.
For a graph $G=(V,E)$ and a set $S\subseteq V$, we write
\begin{eqnarray*}E(S):=\{\{u,v\}\in E: u,v\in S\},\\
\delta(S):=\{\{u,v\}\in E: u\in S,v\notin S\}.
\end{eqnarray*}
For a vector $x:E\to\mathbb{R}$, and a set $F\subseteq E$, we write $x(F)=\sum_{e\in F} x_e$.
For a graph $G=(V,E)$ and $S\subseteq V$, we write $G\smallsetminus S$ to be the graph where $S$ and all edges incident to $S$ are removed, and we write $G/S$ to denote the graph in which $S$ is contracted. For two sets $S,T$, we write $S\smallsetminus T$ to denote the set difference.
\subsection{Held-Karp Relaxation}
\label{sec:HK}
The following linear program was first formulated by Dantzig, Fulkerson and Johnson \cite{DFJ54} and is
known as the subtour elimination polytope or the Held-Karp LP relaxation (see also \cite{HK70}).
\begin{equation}
\begin{aligned}
\min & \sum\limits_{u,v} c(u,v) x_{\{u,v\}} \\
\text{s.t.} & \sum_{u \in S, v \in \overline{S}} x_{\{u,v\}} \ge 2 & \forall \, S\subsetneq V\\
& \sum_{v \in V} x_{\{u,v\}} = 2~\hspace{6ex} & \forall \, u\in V\\
& x_{\{u,v\}}\geq 0 & \forall \, u,v \in V.
\end{aligned}
\label{eq:tsplp}
\end{equation}
\begin{assumption}
Throughout the paper, we assume that we are given a feasible half-integral solution of the Held-Karp LP, that is, for each $\{u,v\}$, $x_{\{u,v\}} \in \{0, 0.5, 1\}$.
\end{assumption}
\begin{remark}
We will often talk about the support graph $G=(V,E)$ of $x$, replacing any edge of value 1 with two parallel edges. Therefore the number of edges crossing any minimum cut is 4 (corresponding to fractional value 2),
and the graph is Eulerian. \textbf{Henceforth, any reference to the graph $G$ refers to this support graph.}
\end{remark}
In our algorithm we will repeatedly consider subsets $S$ of vertices such that $\delta (S)$ is a min-cut of $x$.
\subsubsection{Spanning Tree Polytope}
For any graph $G=(V,E)$,
Edmonds \cite{Edm70} gave the following description for the convex hull of spanning tree of a graph $\tilde G$, known as the {\em spanning tree polytope}.
\begin{equation}
\begin{aligned}
& z(E) = n-1 & \\
& z(E(S)) \leq |S|-1 & \forall S\subseteq V\\
& z_e \geq 0 & \hspace{6ex} \forall e\in E.
\end{aligned}
\label{eq:spanningtreelp}
\end{equation}
Edmonds \cite{Edm70} proved that the extreme point solutions of this polytope are the characteristic vectors of the spanning trees of $G$.
We formally define tight sets in \cref{subsec:algorithm} but for now assume $S$ is tight if $x(\delta(S))=2$. In the half-integral case this corresponds to $|\delta(S)|=4$.
\begin{fact} If in the fractional solution $x$ of the Held-Karp LP
the set $S$ is a tight set, then the restriction of $x$ to edges in $E(S)$, that is, the fractional solution $\{x_e\}_{e\in E (S)}$,
is in the spanning tree polytope on $(S, E(S))$.
\end{fact}
\begin{proof}
Since every min-cut has fractional value 2, and every vertex has fractional degree 2 in any Held-Karp solution, we have
$$\sum_{e \in E(S)} x_e = \frac{2|S|-2}{2} = |S|-1.$$
For the same reason, the constraint \eqref{eq:spanningtreelp} of the spanning tree polytope holds for each $S' \subset S$.
\end{proof}
It follows immediately that:
\begin{fact} \label{fact:expcostT}If $S$ is a tight set w.r.t. $x$, then
a random spanning tree on $S$ selected from a distribution whose marginals are $x_e$ for each $e\in E(S)$ has expected cost $\sum_{e\in E(S)} x_e c_e$.
\end{fact}
\subsection{Maximum entropy distribution}
We say a distribution $\mu$ over spanning trees is
{\em $\lambda$-uniform or maximum entropy} if there are nonnegative
weights $\lambda: E \rightarrow \mathbb{R}_+$ such that for any tree $T$,
$$ \P{T} \propto \prod_{e \in T} \lambda_e.$$
Given a point $z$ in the spanning tree polytope, for a connected graph $G = (V, E)$, Asadpour et al. \cite{AGMOS17} show that there is an efficient algorithm
that
finds non-negative $\lambda_e$'s in a such a way that for every edge
$e \in E$ and tree $T$ sampled from $\mu$, $\P{e \in T}$ is
(approximately) equal to $z_e$.
To sample from a distribution on spanning trees, we follow \cite{AGMOS17,OSS11} and
sample spanning trees using a distribution
\begin{theorem}[\cite{AGMOS17}]
\label{thm:maxentropycomp}
Let ${\bf z}$ be a point in the spanning tree polytope of the graph $\tilde G=(\tilde V, \tilde E)$.
For any $0 < {\epsilon}$, values $\lambda_e$ for all $e\in \tilde E$ can be found such
that
the corresponding $\lambda$-uniform spanning tree distribution, $\mu$, satisfies
$
\sum_{T\in {\cal T}: T \ni e} \PP{\mu}{T} \leq (1+\varepsilon)z_e,\hspace{3ex}\forall e\in E,$$
i.e., the marginals are approximately preserved. Furthermore, the running
time is polynomial in $n=|\tilde V|$, $- \log \min_{e\in E} z_e$ and $\log(1/{\epsilon})$.
\end{theorem}
We can now briefly explain the main algorithm of \cite{OSS11}: Given a feasible solution $x$ of subtour-LP, define $z=(1-1/n)x$. Then, sample $T$ from a $\lambda$-uniform distribution with marginals $z$ and add a min-cost matching on the odd degree vertices of $T$.
\subsection{Min-cost Eulerian augmentation}
\label{sec:Ojoin}
Once we have sampled a tree (or, as we shall see later, a tree plus an edge), we will be finding the minimum cost Eulerian augmentation. For this purpose, we use the following characterization of the $O$-join polytope due to Edmonds and Johnson \cite{EJ73}.
\begin{proposition}
\label{prop:tjoin}
For any graph $G=(V,E)$, cost function $c: E \to \mathbb{R}_+$, and a set $O\subseteq V$ with an even number of vertices, the minimum weight of an $O$-join equals the optimum value of the following integral linear program.
\begin{equation}
\begin{aligned}
\min \hspace{4ex} & c(y) \\
\text{s.t.} \hspace{3ex} & y(\delta(S)) \geq 1 & \forall S \subseteq V, |S\cap O| \text{ odd}\\
& y_e \geq 0 & \forall e\in E
\end{aligned}
\label{eq:tjoinlp}
\end{equation}
\end{proposition}
\subsection{Description of Algorithm}\label{subsec:algorithm}
Our algorithm is a slight modification of the one studied in \cite{OSS11}.
Given a feasible solution $x$ of the subtour-LP, without loss of generality, we assume that there exists an edge $e^+$ such that $x_{e^+}=1$. If such an edge does not exist, we split a node $v$ into two nodes $v_1,v_2$; connect 2 of the edges out of $v$ to $v_1$ and the other two to $v_2$. Then, we connect $v_1$ to $v_2$ with edge $e^+$ of cost $c(e^+)=0$ and $x_{e^+}=1$.
Given such a solution $x$ our algorithm is as follows: Define a fractional spanning tree $z$ where $z_{e^+}=0$ and $z_e=x_e$ for any $e\neq e^+$. Then, we sample $T$ from the $\lambda$-uniform spanning tree distribution $\mu$ with marginals $z$ for some $\epsilon=2^{-n}$ using \cref{thm:maxentropycomp}. Define $T=T\cup\{e^+\}$; this gives a {\bf 1-tree}.
A 1-tree is a union of a spanning tree and an edge. Finally, we add a minimum cost Eulerian augmentation on the odd degree vertices of $T$. Throughout we let $T^-=T\smallsetminus {e^+}$. $T^-$ is an actual spanning tree.
\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}[inner sep=1.7pt,scale=.8,pre/.style={<-,shorten <=2pt,>=stealth,thick}, post/.style={->,shorten >=1pt,>=stealth,thick}]
\draw [rotate=20,line width=1.1](0,0) ellipse (2cm and 1cm);
\draw [rotate=-20,line width=1.2](-1.35,-0.48) ellipse (2cm and 1cm);
\draw (-3.8, 0.5) node {$X$};
\draw (2.3, 0.5) node {$Y$};
\tikzstyle{every node} = [draw, circle,fill=red];
\node at (1.1,0.75) (){};
\node at (0.6,0.3) (){};
\path (1.3,0.1) node (){};
\path (-1.9,0.35) node (){};
\path (-2.5,0.5) node (){};
\path (-0.5,0.2) node (){};
\path (-1.2,-.7) node (){};
\path (0,-0.7) node (){};
\path (-1,1.5) node (){};
\end{tikzpicture}
\end{center}
\caption{An example of two crossing sets.}
\label{fig:crossingsets}
\end{figure}
There is an equivalent description of the above algorithm. Before discussing this, we need to define three more concepts.
\begin{definition}
Consider a graph $G= (V,E)$ with min-cuts of value $k$.
\begin{itemize}
\item Any set $S\subseteq V$ such that $|\delta (S) |= k$ (i.e., its boundary is a min-cut) is called a \textbf{tight} set.
\item A cut $(S, \overline S)$ is \textbf{proper} if
$|S| \ge 2$ and $|\overline S| \ge 2$.
\item Two sets $S$ and $S'$ \textbf{cross} if all of $S \smallsetminus S'$,
$S' \smallsetminus S$, $S \cap S'$ and $V \smallsetminus (S \cup S')$ are non-empty.
\end{itemize}
\end{definition}
See \cref{alg} for an equivalent description of our algorithm, which we will work with throughout the paper. As the equivalence is not fundamental to our proof, we omit the (simple) proof here.
\begin{algorithm}[h]
\caption{Algorithm for half-integral TSP}
\begin{algorithmic}[1]
\State Given a half-integral solution $x$ of the subtour LP, with an edge $e^+$ with $x_{e^+}=1$.
\State Let $G$ be the support graph of $x$.
\State Set $T = \emptyset$ \Comment{$T$ will be a 1-tree}
\While {there exists a proper tight set of $G$ that is not crossed (by a tight set)}
\State Let $S$ be a minimal such set such that $e^+\notin E(S)$ \Comment{Note such a set always exists, as $S,\overline{S}$ are both proper tight sets, so one does not have $e^+$. In \cref{fact:e+notindS} we show that $e^+\notin\delta(S)$.}\label{alg:step:mintightset}
\State Compute the maximum entropy distribution $\mu$ of $E(S)$
\State Sample a tree from $\mu$ and add its edges to $T$
\State Set $G = G/S$ \Comment{Note we never contract $e^+$.}
\EndWhile
\State Randomly sample a cycle from $G$ (including $e^+$) and add it to $T$ \Comment{In \cref{fact:lastcycle} we show $G$ itself is a cycle} \label{alg:step:cycle}
\State Compute the minimum O-Join on the odd nodes of $T$. Shortcut it and output the resulting Hamiltonian cycle.
\end{algorithmic}
\label{alg}
\end{algorithm}
\begin{fact}\label{fact:e+notindS}
In step \ref{alg:step:mintightset} of the algorithm, we have $e^+\notin\delta(S)$.
\end{fact}
\begin{proof}
Say $e^+\in\delta(S)$, and let $e^+=\{u,v\}$. Then, since $x_{e^+}=1$, $\{u,v\}$ is a tight set. It also crosses $S$ (as $S$ is a proper set). That is a contradiction.
\end{proof}
A few remarks are in order.
By \cref{fact:expcostT}, $\E{c(T)}=c(x)$ (up to an error of $2^{-n}$). Therefore, to prove \cref{thm:main} all we need to do is to bound the expected cost of the $O$-join by $c(x)(1/2-{\epsilon}_0)$ for some ${\epsilon}_0>0$.
Also, crucial to our analysis are the independence properties we get from our algorithm, see the following and \cref{fact:efinout}.
\begin{fact}\label{fact:independencecritical}
Any tree chosen from a max-entropy distribution corresponding to a proper tight set $S$ which is not crossed is independent of all other edges of $T$ that we choose in different iterations of the while loop in our algorithm.
\end{fact}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,441 |
Wendela Hebbe (9 de septiembre de 1808, Jönköping - 27 de agosto de 1899, Estocolmo), fue una periodista y escritora sueca. Podría decirse que fue la primera periodista contratada de forma permanente por un periódico sueco. Tuvo un lugar importante en los círculos literarios radicales de mediados del siglo XIX en Suecia y fue una referente controvertida para la mujer emancipada.
Biografía
Wendela Hebbe era la mayor de las tres hijas del párroco Anders Samuel Åstrand y Maria Lund. Su padre era aficionado a la literatura y tenía interés por la cultura y cultivó en sus hijas en las artes. De niña, a Wendela se le incentivaba a leer y aprender música, arte y literatura. Se la describió como talentosa en la música y la literatura y se la apodó "Fröken Frågvis" ("Señorita curiosa"). Esaias Tegnér era un conocido de su padre y un invitado habitual en su casa. Al parecer, él la cortejó sin éxito desde muy joven, incluso después de casada, dedicándole muchos de sus poemas. Ella lo rechazó pero le brindó su amistad.
En 1832, se casó con el abogado y escritor Clemens Hebbe (1804-1893), con quien tuvo tres hijas. En 1839, su esposo se declaró en bancarrota y huyó del país: primero a Inglaterra y luego emigró a los Estados Unidos. Wendela quedó abandonada con sus hijas y tuvo que procurarse el sustento por sí sola. Se instaló en Jönköping y comenzó a trabajar en la única profesión que se consideraba socialmente aceptable para una mujer educada en ese momento. Se desempeñó como profesora impartiendo clases de música, canto y dibujo, lo que apenas le era suficiente para mantenerse.
Carrera como periodista
En 1841, su primera novela, Arabella, fue publicada por Lars Johan Hierta, el jefe de redacción del periódico radical Aftonbladet, y ese mismo año fue contratada en Aftonbladet (se le dio un puesto permanente en 1844).
Wendela Hebbe es la primera periodista mujer en Suecia. Aunque las mujeres empezaron a escribir artículos en la prensa sueca aproximadamente desde 1738, como Margareta Momma por ejemplo, a la mayoría de ellas no se la ha podido identificar ya que escribieron bajo seudónimos. Es por esto que Wendela Hebbe fue, probablemente, la primera mujer reportera en ocupar un puesto permanente en un periódico sueco y, en ese sentido, la pionera de su profesión. No fue hasta el siglo XIX que la prensa sueca empezó a contratar personal permanente, y Wendela Hebbe es la primera mujer que figura en el registro de empleados de cualquier periódico sueco. Ella está inscrita en el registro de trabajadores de Aftonbladet entre 1844 y 1851, seguida de Marie Sophie Schwartz del Svenska Tidningen Dagligt Allehanda en 1851-1859.
Hebbe fue nombrada traductora y editora de la sección de cultura, siendo responsable de la difusión de temas relacionados con la cultura, la música y la literatura. Realizó reseñas de literatura y novelas, conciertos, espectáculos de ópera y obras teatrales, y dirigió la sección de series. Es conocida por haber utilizado su sección para promocionar a escritores nóveles publicando sus novelas en forma de series.
Fuera del ámbito cultural, se desempeñó activamente como reportera social y, de hecho, fue quizás la primera reportera en Suecia que introdujo el reportaje social en Suecia. Wendela compartía la visión liberal y humanitaria de Aftonbladet. Por su condición de mujer fue considerada idónea para inquirir sobre temas tales como la miseria social entre los pobres, y tuvo gran notoriedad con su primer reportaje social Biskopens besök (Visita del Obispo) en 1843, una publicación que contribuyó al debate social que se había iniciado en torno a las diferencias de clases en Suecia en aquella época. Gracias a su reportaje sobre la injusticia social, logró en varias ocasiones atraer la atención sobre áreas que necesitaban ser reformadas y ayudar a las personas que necesitaban ayuda.
Carrera literaria
Wendela se retiró del periodismo en 1851 para dedicarse a una carrera como novelista. Su novela debut, Arabella, fue una novela de amor convencional, pero sus novelas posteriores tienen un estilo más realista. Sus novelas se centran en la intriga como tal y no en los personajes, y están fuertemente asociadas a su época. Incluyó la crítica social como mensaje en sus novelas y es posible se haya inspirado en Dickens y en la literatura británica del siglo XVIII. Su novela Brudarne ha sido descrita como su novela más notable y es considerada la primera "novela para chicas" en Suecia. Como novelista ha sido considerada talentosa pero no original, consiguiendo solo un éxito moderado.
Tuvo más éxito como escritora de canciones y poemas para niños y adolescentes. Sus poemas infantiles fueron influenciados por su infancia idílica en Småland y representan juegos y rimas infantiles; además, del folclore tradicional. En particular, sus cuentos de hadas sobre animales fueron muy admirados por Bj Björnson y S. H. Grundtvig. Entre sus canciones, las composiciones Högt deruppe mellan fjällen ('En lo alto entre las cimas de las montañas') y Linnean ('Linnea') se hicieron muy populares.
Además de su producción personal, ella hizo una valiosa contribución histórica al escribir antiguas historias y canciones tradicionales del folclore popular.
Vida privada
Hebbe tuvo una larga relación sentimental con Lars Johan Hierta. Esta era de conocimiento público generando caricaturas en la prensa y rumores de que se le había dado su puesto producto del nepotismo. Hebbe y Hierta no podían casarse, ya que ambos estaban casados. Su matrimonio con su exiliado esposo no se disolvió hasta 1864. Wendela Hebbe y Lars Johan Hierta tuvieron un hijo, Edvard, en 1852. Incluso una mujer tan independiente como Hebbe, no quiso reconocer que tenía un hijo extramatrimonial. Edvard nació en secreto durante un viaje a Francia. Hebbe nunca lo reconoció, pero se lo consideraba como un hijo adoptivo de Hierta bajo un nombre falso, y también lo visitaba Hebbe de vez en cuando, hasta que se le dio un hogar permanente en Alemania. Su hijo se convirtió más tarde en el padre de la artista Mollie Faustman.
Wendela fue una figura central en la élite radical de Estocolmo, especialmente durante los años 1840 y 1850, y organizó un salón literario y musical que se convirtió en punto de encuentro para el mundo literario y artístico liberal, que se reunía para recitar, tocar música y debatir. A su círculo pertenecieron Johan Jolin, Gunnar Olof Hyltén-Cavallius y también Magnus Jacob Crusenstolpe, a quien apoyó en su lucha por la libertad de expresión. Una amistad particular fue la que mantuvo con Carl Jonas Amor Almquist, cuyo trabajo como escritor admiraba, ya que compartían un interés por la crítica social. También desempeñó un papel importante como asesora y secretaria en la creación de las composiciones de Almquist, en particular de sus canciones, según su hija Signe Hebbe, que recordó a su madre y a Almquist sentados junto al piano durante sus composiciones: "A principios de los años 40, cuando muchas de las canciones de Almquists fueron terminadas, Almqvist señalaba con un dedo en el teclado musical qué tono deseaba. Fue también H quien con su voz cálida y bella presentó las nuevas creaciones al círculo de amigos". Almqvist la festejó con la composición para piano "Vendelas mörka lockar" (los oscuros rizos de Wendela). Según Signe Hebbe, su madre y Almqvist nunca tuvieron una relación romántica, pero la correspondencia y el comportamiento de Almqvist sugieren que eran muy probablemente más que amigos. Wendela Hebbe demostró su leal apoyo a Carl Jonas Love Almquist durante el escándalo de 1851.
Su salón era una parte importante de la vida literaria de Estocolmo y era considerado un destino clave para un escritor que visitaba Estocolmo. Johan Ludvig Runeberg lo hizo durante su breve visita en 1851. Su casa siguió siendo un lugar de encuentro durante décadas, incluso después de que una enfermedad la dejara incapaz de caminar en 1878. Luego conoció a Ellen Key y Herman Sätherberg, para cuyos poemas ella compuso música. También acompañó a su hija, la famosa cantante de ópera Signe Hebbe, en sus giras por Europa.
Wendela nunca alcanzó fama como escritora, pero desempeñó un papel importante como anfitriona de su salón, y aunque nunca estuvo involucrada por la labor emancipación de las mujeres, fue un modelo a seguir para la mujer emancipada por su estilo de vida independiente y controvertido. Gösta Lundström dijo de ella: "Como escritora, Hebbe no puede tener un lugar destacado en nuestra historia. Pero como una fuerza de encuentro e inspiración en la vida cultural de la Suecia del siglo XIX, vale la pena recordarla. También como una de las primeras representantes de la emancipación de la mujer en nuestra nación, defiende su lugar como una de las mujeres suecas más famosas de su siglo. A pesar de que en muchos aspectos ilustró el 'alma' ideal de las mujeres de la época romántica, lo equilibra con su inteligencia y realismo." La escritora y periodista Jane Gernandt- Claine dijo de ella: "Alrededor de todo su ser etéreo había un aire indescriptible de refinamiento espiritual, esta nobleza del alma, que pertenece a las cosas más admirables de la vida. Nunca te acercaste realmente a ella y nunca quisiste realmente, estuviste demasiado feliz de estar a una distancia de tanta nobleza sentimental dentro de esta frágil y refinada coraza.
Wendela Hebbe fue muy cortejada por los artistas contemporáneos masculinos, pero se la conoce como una mujer sencilla. Ella se describía como alguien "con mucho anhelo y mucha frustración".
Legado
En 1983 se fundó la asociación Wendelas Vänner (Amigos de Wendela Hebbe) para preservar la memoria de Wendela Hebbe. La asociación conserva su casa de verano en Södertälje, que le fue cedida por Hierta en 1863, y la convirtió en museo.
El gimnasio (escuela) Wendela Hebbegymnasiet en Södertälje lleva su nombre.
Obras
Arabella (novela, 1841)
Svenska skaldestycken för ungdom (libro de poemas "para jóvenes", 1845)
Arbetkarlens hustru (La esposa de un trabajador) (reportaje, 1846)
Brudarne (Las novias) (novela, 1846). Su obra más famosa.
En fattig familj (Una familia pobre) (reportaje, 1850)
Tvillingbrodern (El hermano gemelo) (novela, 1851)
Lycksökarna (Los cazadores de fortunas) (novela, 1852)
Dalkullan (canción, 1858)
Yo Skogen (En el bosque) (libro infantil, 1871)
Bland trollen (Entre los ogros) (libro infantil, 1877)
Bajo hängranarne (Bajo los árboles colgantes) (novela, 1877)
Véase también
Catharina Ahlgren
Referencias
Enlaces externos
http://www.dagensvisa.com/minata/abl_wendela.html (Biografía en sueco, con imágenes).
http://www.wendela.se
Österberg, Carin et al., Svenska kvinnor: föregångare, nyskapare. Lund: Signum 1990. ()
Wendela Hebbe, urn:sbl:12676, Svenskt biografiskt lexikon (art av Gösta Lundström), hämtad 2015-11-12.
Escritores de Suecia del siglo XIX
Traductores de Suecia del siglo XIX
Escritoras de Suecia
Mujeres periodistas de Suecia
Traductoras de Suecia
Suecas del siglo XIX | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,315 |
Payne & Redemption – An Independent Film Inspired By The Works Of Sam Lake
Film Trivia
• The film will be distributed for free online, and an exclusive screening party will be held in London's Notting Hill.
• Between January 1st and December 31st, 2008, Payneandredemption.com had received eighty-four million, fifty-three thousand, and one-hundred and twenty seven unique hits.
• On April 22nd, 2008, 20th Century Fox demanded writer / director Fergle Gibson cease and desist work on "Max Payne: Payne & Redemption", a non-profit production, for unsolicited use of the Max Payne property. This news was met with infuriation by millions of Max Payne fans around the globe, who threatened to boycott Fox's "Max Payne" production if they didn't reconsider their position. For two weeks, Fergle and producer Luke Morgan-Rowe, tried to negotiate with Fox, but their conciliations fell on deaf ears. Undeterred, Fergle dropped the "Max Payne" name and continued working on his project. Although a Box Office success, Fox's Max Payne film was critially panned, scoring 18% at Rotten Tomatoes, and shot down by Scott Miller, Producer of the Max Payne videogames.
• In July of 2008, Payneandredemption.com hit an all-time-high, receiving over twelve MILLION unique hits in that month alone.
• Despite the lead character being portrayed as a heavy smoker in the trailers to Payne & Redemption, Writer / Director Fergle Gibson is staunchly AGAINST smoking of any kind.
• With pre-production taking place over the course of a year, principle photography begun on October 4th 2006, with a crew of over twenty industry professionals. Lasting four long rainy nights of gruelling guerilla film-making in sub-zero temperatures, the production wrapped on October 8th, with cast & crew quite happy to leave the sodden, depressing world of Max Payne behind them… for a hot bath and the promise of a summer shoot for the remaining story.
• In its first week of release, the teaser trailer to Payne & Redemption gained over twenty-three thousand views on Youtube.com alone.
• Payne & Redemption was one of the first films in the UK (and the first independent film) to be shot on Arri Media's THEN prototype Arriflex D-20 – A revolutionary high-definition film-style digital motion picture camera with a CMOS sensor the size of a Super 35mm film gate aperture.
• Before working on Payne & Redemption, John Attwood, the film's Chief Lighting Technician, had completed work on The Bourne Ultimatum. John has also worked on many other major releases, including Die Another Day, Tomb Raider 2, Alexander, Hitchhiker's Guide to the Galaxy and Casino Royale.
• Initially, the story of Payne & Redemption was written as a twenty minute short, but writer / director Fergle Gibson felt the storyline could be greatly expanded upon, and proposed the idea of making six twenty minute short films, with the idea that they would come together to form one feature length piece. This idea has since been scrapped due to financial difficulties.
• John Mangan's arrival necessitated a total re-write of the antagonist's role to accommodate the character's new, and very different direction.
• John Mangan is in fact the second actor to be cast as the antagonist, as John Ramirez, who was initially selected to play the part, had to pull out due to personal reasons.
• Approximately 1260 hours were spent on the storyboards for Payne & Redemption, consisting of around 350 shots within an estimated runtime of 20 minutes. These storyboards were then condensed down to around 150 shots whilst still keeping the estimated runtime.
• The Director, Fergle Gibson, and replacement Storyboard Artist, Ben Dean, worked on the storyboards, consistently, for an average of 18 hours a day, over the course of around 70 days, on a diet comprised of mostly cookies, energy drinks and toasted sandwiches.
• None of the storyboards were ever used for the final shoot.
• The original Storyboard Artist (Roger Williams) dropped out after re-reading the script, deeming it "sick and perverted".
Miscellaneous Trivia
• Writer & Director, Fergle Gibson, is a trained & qualified special effects makeup artist.
• Producer, Luke Morgan-Rowe, when not producing films, is in fact a Doctor at a well-known London hospital.
• Kylie Cushman, who plays the femme fatale in Payne & Redemption, hosts a popular quiz show on Sky Digital. She also appeared briefly in the British film "Love Actually".
• The location for some of the publicity stills is in fact one of the schools that the Director and Producer both used to attend.
• For one of the yet unreleased publicity stills, the production team flooded a church with 20 SWAT team members, all carrying M16 assault rifles, tactical knives and Sig Sauer handguns – This was met with some concern with the local Women's Institute.
• The Writer / Director's favourite film is Terminator 2: Judgment Day. His least favourite film is Terminator 3: Rise of the Machines.
Official IMDB
Official MySpace Page
Fergle Gibson On Twitter
Luke M-Rowe On Twitter
Ben Dean On Twitter
Justin R. Durban
Visual Fizzle
Payne Reactor
MaxPayne.hu
Remedy Games
NEW Announcements…
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Copyright © 2005-2013 Payne & Redemption - An Independent Film. All Rights Reserved.
This film is NOT associated with Take-Two Interactive, Rockstar Games or Twentieth Century Fox.
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Q: How to trigger python file through html Button - Flask I am quite new to programming and I need your kind support.
I am trying to build a simple WebApp with flask webforms and I'd like it to work like this:
*
*Insert data
*Press Button
*Trigger another python script and use the input
(As for what the .py script does, for now it is ok just to print(info))
@app.route("/form", methods=["POST"])
def form():`
tenant=request.form.get("t_tenant")
VRF=request.form.get("t_vrf")
BD=request.form.get("t_bd")
Subnet=request.form.get("t_subnet")
App=request.form.get("t_app")
global info
info=f"Tenant={tenant},\nVRF={VRF},\nBD={BD},\nSubnet={Subnet},\n, App={App}"
file = open(r'exec.py', 'r').read()
return render_template("forms.html", title=title, info=info)
Needless to say, it doesn't work as the "exec.py" doesn't see the "info" variable
Both .py scripts are in the same directory and the forms.html is in templates dir
Any advice are more than welcome
Thanks!
A: I have to write it here as answer as it is long;
let's say you have the following 'parse_file' function in your exec.py:
def parse_file():
print("running parse_file from exec.py")
Now import that exec.py file and use in form as ex.parse_file() the function. Instead of directly running the file, you can use the functions in this file. These functions may be some other Flask apps that may open other html files, etc.
import exec as ex
@app.route("/form", methods=["POST"])
def form():`
tenant=request.form.get("t_tenant")
VRF=request.form.get("t_vrf")
BD=request.form.get("t_bd")
Subnet=request.form.get("t_subnet")
App=request.form.get("t_app")
global info
info=f"Tenant={tenant},\nVRF={VRF},\nBD={BD},\nSubnet={Subnet},\n, App={App}"
ex.parse_file() # do what ever you want to do with the functions in exec
# file = open(r'exec.py', 'r').read()
return render_template("forms.html", title=title, info=info)
| {
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{"url":"https:\/\/ncatlab.org\/nlab\/show\/foundations+and+philosophy","text":"# nLab foundations and philosophy\n\nThe boundary between philosophical and logical foundational studies of mathematics is very unclear indeed \u2013 one has only to think of Russell\u2019s part in Principia Mathematica. For a discussion of more obviously logical foundational aspects of category theory see foundations.\n\nTo be done\n\n## Foundations and the philosophy of language in mathematics\n\nDifferent foundations of mathematics provide different languages to use for presenting concepts in mathematics, and the symbols and words used in various foundations have different meanings? from each other.\n\nWe work in a deductive system with hypothetical judgments, and give examples of how different foundations of mathematics written in natural deduction result in different meanings of symbols used in the judgments used in each foundation.\n\nFor example, take the symbol $\\in$ denoting membership in set theory and set-level dependent type theories. In unsorted set theory and simply sorted set theory, $\\in$ is a proposition in the context of an element variable and a set variable. The formal judgment $a \\in A \\; \\mathrm{prop}$ means that \u201c\u2018the element $a$ is in the set $A$\u2019 is a proposition\u201d, and the judgment $a \\in A \\; \\mathrm{true}$ is commonly abbreviated as $a \\in A$ and means that \u201cthe proposition that the element $a$ is in the set $A$ is true\u201d. In dependently sorted set theory, and in unlayered set-level dependent type theories, $\\in$ is a metatheoretic judgment, rather than being part of the actual structure of the theory. The formal judgment $a \\in A$ simply means that \u201c$a$ is in the set $A$\u201d, rather than \u201cthe proposition that the element $a$ is in the set $A$ is true\u201d; there is no propositional truth value attached to $a \\in A$. Thus, the foundational notion of membership in unsorted and simply sorted set theories is semantically different from the notion of membership in dependently sorted set theories and unlayered set-level dependent type theory.\n\nSimilarly, the symbol $=$ denoting equality have different meanings in different theories.\n\n## Philosophical papers\n\nJean-Pierre Marquis, 1995, Category Theory and the Foundations of Mathematics: Philosophical Excavations. Synthese 103 (3).\n\nColin McLarty, 2004, \u2018Exploring Categorical Structuralism\u2019, Philosophia Mathematica, 12, 37\u201353.\n\nLast revised on November 18, 2022 at 19:01:58. See the history of this page for a list of all contributions to it.","date":"2022-12-08 06:44:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 17, \"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\": 0.5537781119346619, \"perplexity\": 867.9955202940783}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711278.74\/warc\/CC-MAIN-20221208050236-20221208080236-00198.warc.gz\"}"} | null | null |
Giwakwa, Giwakwa ili Kee-wakw su zli ledeni divovi ljudožderi iz legendi kod južnoabenačkih plemena Abenaki, Penobscot, Malecite, Passamaquoddy. Prema većini legendi, Giwakwa je nekoć bio ljudsko biće koje je ili bilo opsjednuto zlim duhom ili je počinilo užasan zločin (osobito kanibalizam ili uskraćivanje hrane izgladnjeloj osobi), zbog čega mu se srce pretvorilo u led. U nekim legendama kod Abenakija, kameni divovi (Asinikiwakw) nisu bili transformirani ljudi, već iskonska čudovišta ljudožderi, koje je porazio kulturni heroj Gluskabi.
Ostalw varijante imena: Kiwakw, Kewahqu, Kee-wakw, Kewok, Kiwahq, Kewoqu, Kewawkqu', Kewawkgu, Kiwakwe, Kiwákwe, Kiwahkw, Kiwa'kw, Keewaqu', Kee-wowk, Kiawahq', Keewahkwee, Asinikiwakw, A-senee-ki-wakw. Isto i Giwakweska, Giwakweskwa, ili Kiwakweskwa (ženski oblik), Kiwahqiyik (plural)
Izvori
Indijanska mitologija (ljudožderi)
Abenaki mitologija
Penobscot mitologija
Passamaquoddy mitologija
Maliseet mitologija | {
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SET gp_create_table_random_default_distribution=off;
-- end_ignore
DROP TABLE IF EXISTS foo;
CREATE TABLE foo (a INT, b INT) WITH (appendonly=true);
INSERT INTO foo SELECT i as a, i as b FROM generate_series(1,20) AS i;
UPDATE foo SET b = 0 WHERE a = 1;
DELETE FROM foo WHERE a = 2;
DROP TABLE IF EXISTS foocs;
CREATE TABLE foocs (a INT, b INT) WITH (appendonly=true, orientation=column);
INSERT INTO foocs SELECT i as a, i as b FROM generate_series(1,20) AS i;
UPDATE foocs SET b = 0 WHERE a = 1;
DELETE FROM foocs WHERE a = 2;
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/physics.aps.org\/articles\/large_image\/f1\/10.1103\/Physics.5.66","text":"M. M\u00fcser\/Saarland Univ.\n\nSpring a leak. The simulated seal shown here has only $20$ percent of its surface in contact with the substrate (dark gray regions), yet the flow of liquid (in color) across the seal is largely restricted to a few channels, leading to less leakage than previous models would predict.","date":"2013-05-22 04:00:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5742955207824707, \"perplexity\": 2152.387436751704}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368701281163\/warc\/CC-MAIN-20130516104801-00068-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
The Yangians, defined by Drinfeld \cite{Dr1, Dr2}, are certain non-commutative Hopf algebras that are important examples of quantum groups. They were studied to generate rational solutions of the {\em Yang-Baxter equation} and there were many applications in statistical mechanics and mathematical physics. Nowadays, the study of Yangians gives many new points of view and important applications to classical Lie theory; see the book \cite{Mo} and references therein.
Consider $Y_N=Y(\mathfrak{gl}_N)$, the Yangian associated to the general Lie algebra $\mathfrak{gl}_N$.
Associated to each composition $\mu$ of $N$, Brundan and Kleshchev established a parabolic presentation for $Y_N$ in \cite{BK1}. Roughly speaking, this new presentation of $Y_N$ corresponds to the Levi decomposition of $\mathfrak{gl}_N$ with respect to $\mu$. In the special case when $\mu=(1,\ldots,1)$, the corresponding presentation is equivalent to Drinfeld's presentation (\cite[Remark 5.12]{BK1}). On the other extreme case when $\mu=(N)$, the corresponding presentation is called the {\em RTT presentation}; see \cite{MNO, Mo}.
The parabolic presentations play a fundamental role in their subsequent works. In \cite{BK2}, they established a concrete realization of finite $W$-algebras associated to {\em any} nilpotent element of type A in terms of Yangians, and a key step is to define a subalgebra of $Y_N$, called the {\em shifted Yangian}. Such a subalgebra can only be defined in terms of the new presentation found in \cite{BK1} except some special cases. The connection between Yangians and finite $W$-algebras was observed earlier in \cite{RS} for some particular nilpotent elements (called {\em rectangular} elements) with a different approach. Moreover, by means of such a realization, one may study the representation theory of finite $W$-algebras by studying the representation theory of Yangians; see \cite{BK3}.
The main goal of this article is to obtain the generalization of \cite{BK1} to the super Yangian $Y_{M|N}=Y(\mathfrak{gl}_{M|N})$, the super Yangian associated to the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$. It is defined by Nazarov \cite{Na} in terms of the RTT presentation as a super analogue of $Y_{N}$.
One of the major differences between $\mathfrak{gl}_N$ and $\mathfrak{gl}_{M|N}$ is that, in the case of $\mathfrak{gl}_N$, we may always choose a canonical Borel subalgebra, since all of the Borel subalgebras are conjugated by the action of the Lie group $GL_N$. This Borel subalgebra gives a root system, and for this given root system we have multiple choices of simple systems. It suffices to choose a canonical simple system since they are all conjugate by the Weyl group action. However, the above argument is no longer true in the case of $\mathfrak{gl}_{M|N}$. Therefore, in the study of $\mathfrak{gl}_{M|N}$ and its representation theory, we may want to specifically mention which simple system we are using, and the notion of {\em 01-sequence} (called $\epsilon\delta$-sequences in \cite[Section 1.3]{CW}) is introduced as a parameterizing set of the conjugacy classes of simple systems of $\mathfrak{gl}_{M|N}$ (also for some other types of Lie superalgebras) under the Weyl group action.
For example, if we identify $\mathfrak{gl}_{M|N}$ with the set of $(M+N)\times (M+N)$ matrices and choose the Cartan subalgebra $\mathfrak{h}$ to be the set of diagonal matrices, then the most common choice for $\mathfrak{s}$ is
$$\mathfrak{s}=\mathfrak{s}^{st}=\stackrel{M}{\overbrace{0\ldots0}}\,\stackrel{N}{\overbrace{1\ldots1}},$$
where a representative simple system of $\mathfrak{gl}_{M|N}$ in the corresponding Weyl group orbit is given by
\[
\Pi^{st}=\{ \delta_{i}-\delta_{i+1}, \epsilon_{j}-\epsilon_{j+1}, \delta_{M}-\epsilon_{1}\,|\, 1\leq i\leq M-1, 1\leq j\leq N-1\}.
\]
Here the notation $\delta_i$ and $\epsilon_j$ denote elements in $\mathfrak{h}^*$ such that
$\delta_{i}(X)$ equals to the $i$-th diagonal entry of $X\in\mathfrak{h}$, and $\epsilon_{j}(X)$ equals to the $(M+j)$-th diagonal entry of $X\in\mathfrak{h}$. With this standard choice, there is only one odd simple root and the behavior of $\mathfrak{gl}_{M|N}$ is ``closest" to the classical $\mathfrak{gl}_N$.
Note that the Weyl group is isomorphic to $S_M\times S_N$, which permutes those $\delta_i$'s and those $\epsilon_j$'s, respectively. Therefore, there is exactly one odd simple root, which is of the form $\delta_i-\epsilon_j$ for some $1\leq i\leq M$, $1\leq j\leq N$, in any other simple system in the Weyl group orbit of $\Pi^{st}$. It implies that the following two simple systems of $\mathfrak{gl}_{3|2}$ are in different Weyl group orbits, since $\Pi_1$ contains only one simple odd root and $\Pi_2$ contains 4 odd simple roots:
\begin{align*}
\Pi_1&=\{\delta_1-\delta_2, \delta_2-\delta_3,\delta_3-\epsilon_1,\epsilon_1-\epsilon_2\} \longleftrightarrow \mathfrak{s}_1=00011,\\
\Pi_2&=\{\delta_1-\epsilon_1, \epsilon_1-\delta_2,\delta_2-\epsilon_2,\epsilon_2-\delta_3\} \longleftrightarrow \mathfrak{s}_2=01010.
\end{align*}
We refer the reader to \cite[Chapter 1.]{CW} for more details and further applications of 01-sequences.
It is noticed in \cite{Pe2} that the notion of $01$-sequence can be perfectly equipped to the RTT presentation of $Y_{M|N}$. It turns out that, up to an isomorphism, the definition of $Y_{M|N}$ is independent of the choices of the 01-sequence $\mathfrak{s}$, and Nazarov's definition corresponds to the case when $\mathfrak{s}=\mathfrak{s}^{st}$ is the canonical one. Since the RTT presentation can be thought as merely a special case of the parabolic presentation by taking the composition $\mu=(M+N)$, the above observation suggests that it should be possible to obtain a corresponding parabolic presentation for {\em any} $\mu$, which triggered this work.
To be precise, the main result of this article (Theorem \ref{Pg}) is that for an {\em arbitrary} fixed 01-sequence $\mathfrak{s}$ of $\mathfrak{gl}_{M|N}$ and an {\em arbitrary} fixed composition $\mu$ of $M+N$, a presentation of $Y_{M|N}$ is obtained.
We quickly explain the idea, which is basically generalizing the argument in \cite{BK1} and adapting some techniques in \cite{Go, Pe1} dealing with the sign factors.
Fix a composition $\mu$ of $M+N$ and fix an arbitrary $01$-sequence $\mathfrak{s}$ of $\mathfrak{gl}_{M|N}$. We first define some distinguished elements in $Y_{M|N}$ associated to $\mu$, denoted by $D$'s, $E$'s and $F$'s, by {\em Gauss decomposition} (or {\em quasideterminants}).
Roughly speaking, the elements $D$'s are those elements in the diagonal blocks of the block matrix decomposition of $Y_{M|N}$ with respect to $\mu$, while the $E$'s and the $F$'s are those elements in the upper and lower diagonal blocks, respectively. Note that these elements depend on the shape $\mu$, where their parities are determined by the given $\mathfrak{s}$. These elements form a generating set for $Y_{M|N}$ (Theorem \ref{gendef}), so the next step is to find enough relations to achieve a presentation.
In the case of \cite{BK1}, if the generators are from two different blocks and the blocks are not ``close", then they commute. Fortunately, this phenomenon remains to be true under our general setting (Lemma~\ref{corcommute}) and it enormously reduces the number of the non-vanishing relations. As a consequence, we only have to focus on the supercommutation relations of the elements that are either in the same block, or their belonging blocks are ``close enough". Let $n$ be the length of $\mu$. When $n=2,3,4$, the situations are less complicated so that we may derive various relations among those generators by direct computation.
Next, we take advantage of the homomorphisms $\psi_L$ and $\zeta_{M|N}$ between super Yangians (see Section 4). These maps carry the relations in the special cases (with $n\leq 4$) to the general case, so that we obtain many relations in $Y_{M|N}$. Finally we prove that we have found enough relations for our presentation.
As a matter of fact, there are already a few results \cite{Go, Pe1} on such a generalization focusing on the canonical case when $\mathfrak{s}=\mathfrak{s}^{st}$.
In \cite{Go}, a presentation of $Y_{M|N}$ when $\mu=(1^{M+N})$ is obtained, which is a generalization of Drinfeld's presentation. In \cite{Pe1}, a similar result when $\mu$ is of the form $\mu=(\lambda,\nu)$, where $\lambda$ is a composition of $M$ and $\nu$ is a composition of $N$, is obtained.
However, the results only explained the case when $\mathfrak{s}=\mathfrak{s}^{st}$. Moreover, the compositions $\mu$ therein are very special so that the elements in one block must have the same parity so the super phenomenon only happens at a few specific places.
Under our setting, $\mu$ and $\mathfrak{s}$ are {\em both arbitrary} so that an even element and an odd element could exist in the same block. As a result, the super phenomenon could happen everywhere. Hence in our current consideration, the signs arising from the $\mathbb{Z}_2$-grading are much more involved than \cite{Go, Pe1} and one needs more elaborate notation and extra care when treating the sign issues. Roughly speaking, we need to correctly insert the necessary sign factors in almost every formula. Certainly, our main theorem covers the above results as special cases.
Finally we mention one undergoing application of our result, which can also be thought as the true motivation of this work. Following the classical $\mathfrak{gl}_N$ case, one may try to generalize the argument in \cite{BK2} so that a realization of finite $W$-superalgebras of type A in terms of the super Yangian $Y_{M|N}$ can be obtained. Such a connection was observed in \cite{BR} for {\em rectangular} nilpotent elements, but this is still open for the general nilpotent case.
In fact, based on \cite{Go, Pe1}, there are already some partial results \cite{BBG, Pe2, Pe3} about the the realization of finite $W$-superalgebras when the nilpotent element is {\em principal} or satisfying certain restrictions. As noticed in \cite{BBG, Pe3}, if we want to generalize the argument in \cite{BK2} to the case of $Y_{M|N}$ in full generality, then a more general presentation of $Y_{M|N}$ is required. One of the reasons is that the {\em shifted super Yangian}, which is a subalgebra of $Y_{M|N}$, can be defined only under some nice assumptions as in \cite{BBG, Pe3}. With our new presentations, the shifted super Yangian can be defined for {\em any} given nilpotent element in $\mathfrak{gl}_{M|N}$ so that it is possible to establish the connection in full generality, and this is currently in progress by the author.
This article is organized in the following fashion. In Section 2, we recall some basic properties of $Y_{M|N}$.
In Section 3, we explicitly define the parabolic generators by means of Gauss decomposition and show that they indeed form a generating set.
In Section 4, we define some homomorphisms between super Yangians so that we may reduce the general case to some less complicated special cases. Section 5 and 6 are devoted to further study about these special cases. Our main theorem is formulated and proved in Section 7.
\section{Prelimilaries}
Let $\mathfrak{s}$ be a $0^M1^N$-sequence (or 01-sequence for short) of $\mathfrak{gl}_{M|N}$, which is a sequence consisting of $M$ $0$'s and $N$ $1$'s, arranged in a row with respect to a certain order. It is well-known \cite[Proposition 1.27]{CW} that there is a bijection between the set of $0^M1^N$-sequence and the Weyl group orbits of simple systems of $\mathfrak{gl}_{M|N}$.
For homogeneous elements $A$ and $B$ in a $ \mathbb Z _2$-graded algebra $L$, the {\em supercommutator of $A$ and $B$} is defined by
\[
\big[ A,B \big] = AB-(-1)^{\pa{A}\pa{B}}BA,
\]
where $\pa{A}$ is the $ \mathbb Z _2$-grading of $A$ in $L$, or called the {\em parity} of $A$. By convention, a homogeneous element $A$ is called $even$ if $\pa{A}=0$, and called $odd$ if $\pa{A}=1$. For each $1\leq i\leq M+N$, let $\pa{i}$ denote the $i$-th digit of the fixed $0^M1^N$-sequence $\mathfrak{s}$.
\begin{definition}
For a given $\mathfrak{s}$, the super Yangian associated to the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$, denoted by $Y_{M|N}$ hereafter, is the associative $\mathbb{Z}_2$-graded algebra (i.e., superalgebra) over $\mathbb{C}$ generated by the {\em RTT generators} \cite{Na}
\begin{equation}\label{RTTgen}
\left\lbrace t_{i,j}^{(r)}\,| \; 1\le i,j \le M+N; r\ge 1\right\rbrace,
\end{equation}
subject to following relations:
\begin{equation}\label{RTT}
\big[ t_{i,j}^{(r)}, t_{h,k}^{(s)} \big] = (-1)^{\pa{i}\,\pa{j} + \pa{i}\,\pa{h} + \pa{j}\,\pa{h}}
\sum_{g=0}^{\mathrm{min}(r,s) -1} \Big( t_{h,j}^{(g)}\, t_{i,k}^{(r+s-1-g)} - t_{h,j}^{(r+s-1-g)}\, t_{i,k}^{(g)} \Big),
\end{equation}
where the parity of $t_{i,j}^{(r)}$ for $r>0$ is defined by $\pa{i}+\pa{j}$ (mod 2), and the bracket is understood as the supercommutator. By convention, we set $t_{i,j}^{(0)}:=\delta_{ij}$.
\end{definition}
Similar to the $\mathfrak{gl}_{M|N}$ case, for $r>0$, the element $t_{i,j}^{(r)}$ is called an $even$ ($odd$, respectively) element if its parity is $0$ ($1$, respectively).
The original definition in \cite{Na} corresponds to the case when $\mathfrak{s}$ is the canonical one; that is, $\mathfrak{s}$ is of the form
\[
\mathfrak{s}=\mathfrak{s}^{st}:=\stackrel{M}{\overbrace{0\ldots0}}\,\stackrel{N}{\overbrace{1\ldots1}}.
\]
As observed in \cite{Pe2}, the definition of $Y_{M|N}$ is independent of the choices of $\mathfrak{s}$, up to an isomorphism, so we often omit it in the notation.
For each $1\leq i,j\leq M+N$, define the formal power series
\[
t_{i,j}(u):= \sum_{r\geq 0} t_{i,j}^{(r)}u^{-r} \in Y_{M|N}[[u^{-1}]].
\]
It is well-known \cite[p.125]{Na} that $Y_{M|N}$ is a Hopf-superalgebra, where the comultiplication
$\Delta:Y_{M|N}\rightarrow Y_{M|N}\otimes Y_{M|N}$ is given by
\begin{equation}\label{Del}
\Delta(t_{i,j}^{(r)})=\sum_{s=0}^r \sum_{k=1}^{M+N} t_{i,k}^{(r-s)}\otimes t_{k,j}^{(s)}.
\end{equation}
Moreover, there exists a surjective homomorphism $$\operatorname{ev}:Y_{M|N}\rightarrow U(\mathfrak{gl}_{M|N})$$ called the {\em evaluation homomorphism}, defined by
\begin{equation}\label{ev}
\operatorname{ev}\big(t_{i,j}(u)\big):= \delta_{ij} + (-1)^{|i|} e_{ij}u^{-1},
\end{equation}
where $e_{ij}\in\mathfrak{gl}_{M|N}$ is the elementary matrix.
For homogeneous elements $x_1,\ldots,x_s$ in a superalgebra $A$, a {\em supermonomial} in $x_1,\ldots,x_s$ means a monomial of the form $x_1^{i_1}\cdots x_s^{i_s}$ for some $i_1,\ldots,i_s\in \mathbb{Z}_{\geq 0}$ and $i_j\leq 1$ if $x_j$ is odd. The following proposition is a PBW theorem for $Y_{M|N}$, where the proof in \cite{Go} works perfectly for any fixed $\mathfrak{s}$.
\begin{proposition}\cite[Theorem 1]{Go}\label{PBWSY}
The set of supermonomials in the following elements of $Y_{M|N}$
\[
\left\lbrace t_{i,j}^{(r)}\, |\, 1\leq i,j\leq M+N, r\geq 1 \right\rbrace
\] taken in some fixed order
forms a linear basis for $Y_{M|N}$.
\end{proposition}
Define the $loop$ $filtration$ on $Y_{M|N}$
\begin{equation}\label{filt2}\notag
L_0 Y_{M|N} \subseteq L_1 Y_{M|N} \subseteq L_2 Y_{M|N} \subseteq \cdots
\end{equation}
by setting $\deg t_{ij}^{(r)}=r-1$ for each $r\geq 1$ and let $L_kY_{M|N}$ be the span of all supermonomials of the form
$$t_{i_1j_1}^{(r_1)}t_{i_2j_2}^{(r_2)}\cdots t_{i_sj_s}^{(r_s)}$$
with total degree not greater than $k$. The associated graded superalgebra is denoted by $\operatorname{gr} Y_{M|N}.$
Let $\mathfrak{gl}_{M|N}[x]$ denote the {\em loop superalgebra} $\mathfrak{gl}_{M|N}\otimes \mathbb{C}[x]$ with the standard basis
$$\lbrace e_{ij}x^r \,|\, 1\leq i,j\leq M+N, r\geq 0\rbrace$$
and let $U(\mathfrak{gl}_{M|N}[x])$ denote its universal enveloping algebra. The next corollary follows from Proposition~\ref{PBWSY}.
\begin{corollary}\cite[Corollary 1]{Go}\label{Yloop}
The graded superalgebra $\operatorname{gr} Y_{M|N}$ is isomorphic to $U(\mathfrak{gl}_{M|N}[x])$ by the map
\begin{center}
$\operatorname{gr} Y_{M|N}\rightarrow U(\mathfrak{gl}_{M|N}[x])$\\
$\operatorname{gr}_{r-1}t_{ij}^{(r)}\mapsto (-1)^{\pa{i}}e_{ij}x^{r-1}.$
\end{center}
\end{corollary}
\section{Parabolic generators}
Let $\mu=(\mu_1,\ldots,\mu_n)$ be a given composition of $M+N$ with length $n$ and fix a $0^M1^N$-sequence $\mathfrak{s}$. We break $\mathfrak{s}$ into $n$ subsequences according to $\mu$; that is,
\[\mathfrak{s}=\mathfrak{s}_1\mathfrak{s}_2\ldots\mathfrak{s}_n,\]
where $\mathfrak{s}_1$ is the subsequence consisting of the first $\mu_1$ digits of $\mathfrak{s}$, $\mathfrak{s}_2$ is the subsequence consisting of the next $\mu_2$ digits of $\mathfrak{s}$, and so on.
For example, if we have $\mathfrak{s}=011100011$ and $\mu=(2,4,3)$, then
\[
\mathfrak{s}=\overbrace{01}^{\mathfrak{s}_1} \, \overbrace{1100}^{\mathfrak{s}_2} \, \overbrace{011}^{\mathfrak{s}_3}.
\]
For each $1\leq a\leq n$, let $p_a$ and $q_a$ denote the number of $0$'s and $1$'s in $\mathfrak{s}_a$, respectively. By definition, each $\mathfrak{s}_a$ is a $0^{p_a}1^{q_a}$-sequence of $\mathfrak{gl}_{p_a|q_a}$. Moreover, for all $1\leq i\leq \mu_a$, define the {\em restricted parity} $\pa{i}_a$ by
\begin{center} $\pa{i}_a$:= the $i$-th digits of $\mathfrak{s}_a$. \end{center}
By definition, for each $1\leq a\leq n$ and $1\leq i\leq \mu_a$, we have
\begin{equation}\label{respa}
\pa{i}_a=\pa{\sum_{j=1}^{a-1}\mu_j+i}.
\end{equation}
Basically, the techniques in \cite{BK1, Pe1} work perfectly with the notion of an arbitrary $\mathfrak{s}$ that we just introduced earlier. In order to make this article self-contained, we spend some time explaining the notation precisely.
Define the $(M+N)\times (M+N)$ matrix with entries in $Y_{M|N}[[u^{-1}]]$ by
\[
T(u):=\Big( t_{i,j}(u) \Big)_{1\leq i,j\leq M+N}
\]
Note that for any fixed $\mathfrak{s}$, the leading minors of the matrix $T(u)$ are invertible and hence it possesses a $Gauss$ $decomposition$ \cite{GR} with respect to $\mu$; that is,
\begin{equation}\label{T=FDE}
T(u) = F(u) D(u) E(u)
\end{equation}
for unique {\em block matrices} $D(u)$, $E(u)$ and $F(u)$ of the form
$$
D(u) = \left(
\begin{array}{cccc}
D_{1}(u) & 0&\cdots&0\\
0 & D_{2}(u) &\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0 &\cdots&D_{n}(u)
\end{array}
\right),
$$
$$
E(u) =
\left(
\begin{array}{cccc}
I_{\mu_1} & E_{1,2}(u) &\cdots&E_{1,n}(u)\\
0 & I_{\mu_2} &\cdots&E_{2,n}(u)\\
\vdots&\vdots&\ddots&\vdots\\
0&0 &\cdots&I_{\mu_{n}}
\end{array}
\right),\:
$$
$$
F(u) = \left(
\begin{array}{cccc}
I_{\mu_1} & 0 &\cdots&0\\
F_{2,1}(u) & I_{\mu_2} &\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
F_{n,1}(u)&F_{n,2}(u) &\cdots&I_{\mu_{n}}
\end{array}
\right),
$$
where
\begin{align}
D_a(u) &=\big(D_{a;i,j}(u)\big)_{1 \leq i,j \leq \mu_a},\\
E_{a,b}(u)&=\big(E_{a,b;i,j}(u)\big)_{1 \leq i \leq \mu_a, 1 \leq j \leq \mu_b},\\
F_{b,a}(u)&=\big(F_{b,a;i,j}(u)\big)_{1 \leq i \leq \mu_b, 1 \leq j \leq \mu_a},
\end{align}
are $\mu_a \times \mu_a$,
$\mu_a \times \mu_b$
and $\mu_b \times\mu_a$ matrices, respectively, for all $1\le a\le n$ in (3.3)
and all $1\le a<b\le n$ in (3.4) and (3.5).
Also note that all the submatrices $D_{a}(u)$'s are invertible, and we define the $\mu_a\times\mu_a$ matrix
$D_a^{\prime}(u)=\big(D_{a;i,j}^{\prime}(u)\big)_{1\leq i,j\leq \mu_a}$ by
\begin{equation*}
D_a^{\prime}(u):=\big(D_a(u)\big)^{-1}.
\end{equation*}
The entries of these matrices are certain power series
\begin{eqnarray*}
&D_{a;i,j}(u) =& \sum_{r \geq 0} D_{a;i,j}^{(r)} u^{-r},\\
&D_{a;i,j}^{\prime}(u) =& \sum_{r \geq 0}D^{\prime(r)}_{a;i,j} u^{-r},\\
&E_{a,b;i,j}(u) =& \sum_{r \geq 1} E_{a,b;i,j}^{(r)} u^{-r},\\
&F_{b,a;i,j}(u) =& \sum_{r \geq 1} F_{b,a;i,j}^{(r)} u^{-r}.
\end{eqnarray*}
In addition, for $1\leq a\leq n-1$, we set
\begin{eqnarray*}
&E_{a;i,j}(u) :=& E_{a,a+1;i,j}(u)=\sum_{r \geq 1} E_{a;i,j}^{(r)} u^{-r},\\
&F_{a;i,j}(u) :=& F_{a+1,a;i,j}(u)=\sum_{r \geq 1} F_{a;i,j}^{(r)} u^{-r}.
\end{eqnarray*}
We will show that the coefficients of these series forms a generating set for $Y_{M|N}$.
As a matter of fact, one may describe all these series in terms of the RTT generators explicitly by \textit{quasideterminants} \cite{GR}. Here we follow the notation in \cite[(4.3)]{BK1}. Suppose that $A, B, C$ and $D$ are $a \times a$, $a \times b$, $b \times a$ and $b \times b$ matrices respectively with entries in some ring.
Assuming that the matrix $A$ is invertible, we define
\begin{equation*}
\left|
\begin{array}{cc}
A&B\\
C&
\hbox{\begin{tabular}{|c|}\hline$D$\\\hline\end{tabular}}
\end{array}
\right| := D - C A^{-1} B.
\end{equation*}
Write the matrix $T(u)$ in block form according to $\mu$ as
$$
T(u) = \left(
\begin{array}{lll}
{^\mu}T_{1,1}(u)&\cdots&{^\mu}T_{1,n}(u)\\
\vdots&\ddots&\cdots\\
{^\mu}T_{n,1}(u)&\cdots&{^\mu}T_{n,n}(u)\\
\end{array}
\right),
$$ where each ${^\mu}T_{a,b}(u)$ is a $\mu_a \times \mu_b$ matrix.
\begin{proposition}\cite{GR}\label{quasi} We have
\begin{align}\label{quasid}
&D_a(u) =
\left|
\begin{array}{cccc}
{^\mu}T_{1,1}(u) & \cdots & {^\mu}T_{1,a-1}(u)&{^\mu}T_{1,a}(u)\\
\vdots & \ddots &\vdots&\vdots\\
{^\mu}T_{a-1,1}(u)&\cdots&{^\mu}T_{a-1,a-1}(u)&{^\mu}T_{a-1,a}(u)\\
{^\mu}T_{a,1}(u) & \cdots & {^\mu}T_{a,a-1}(u)&
\hbox{\begin{tabular}{|c|}\hline${^\mu}T_{a,a}(u)$\\\hline\end{tabular}}
\end{array}
\right|,\\[4mm]
&E_{a,b}(u) =\label{quasie}
D^{\prime}_a(u)
\left|\begin{array}{cccc}
{^\mu}T_{1,1}(u) & \cdots &{^\mu}T_{1,a-1}(u)& {^\mu}T_{1,b}(u)\\
\vdots & \ddots &\vdots&\vdots\\
{^\mu}T_{a-1,1}(u) & \cdots & {^\mu}T_{a-1,a-1}(u)&{^\mu}T_{a-1,b}(u)\\
{^\mu}T_{a,1}(u) & \cdots & {^\mu}T_{a,a-1}(u)&
\hbox{\begin{tabular}{|c|}\hline${^\mu}T_{a,b}(u)$\\\hline\end{tabular}}
\end{array}
\right|,\\[4mm]
&F_{b,a}(u) =\label{quasif}
\left|
\begin{array}{cccc}
{^\mu}T_{1,1}(u) & \cdots &{^\mu}T_{1,a-1}(u)& {^\mu}T_{1,a}(u)\\
\vdots & \ddots &\vdots&\vdots\\
{^\mu}T_{a-1,1}(u) & \cdots & {^\mu}T_{a-1,a-1}(u)&{^\mu}T_{a-1,a}(u)\\
{^\mu}T_{b,1}(u) & \cdots & {^\mu}T_{b,a-1}(u)&
\hbox{\begin{tabular}{|c|}\hline${^\mu}T_{b,a}(u)$\\\hline\end{tabular}}
\end{array}
\right|D^{\prime}_a(u),
\end{align}
for all $1\leq a\leq n$ in (\ref{quasid}) and $1\leq a<b\leq n$ in (\ref{quasie}), (\ref{quasif}).
\end{proposition}
Let $T_{a,b;i,j}(u)$ be the $(i,j)$-th entry of the $\mu_a\times\mu_b$ matrix $^\mu T_{a,b}(u)$ and let $T_{a,b;i,j}^{(r)}$ denote the coefficient of $u^{-r}$ in $T_{a,b;i,j}(u)$. As a consequence of Proposition \ref{quasi}, we have
\begin{equation}\label{Jacobi1}
E_{b-1;i,j}^{(1)}=T_{b-1,b;i,j}^{(1)}, \qquad F_{b-1;j,i}^{(1)}=T_{b,b-1;j,i}^{(1)},
\end{equation}
for all $2\leq b\leq n$, all $1\leq i\leq \mu_{b-1}, 1\leq j\leq \mu_{b}$.
In particular,
\begin{equation}\label{Dtident}
D_{1;i,j}^{(r)}=T_{1,1;i,j}^{(r)}=t_{i,j}^{(r)},\quad \text{for all}\quad 1\leq i,j\leq \mu_1, r\geq 0.
\end{equation}
Note that the matrix $T(u)$ is always invertible and we define the entries of its inverse by
\[
\big(T(u)\big)^{-1}:=\big(t_{ij}^{\prime}(u)\big)_{i,j=1}^{M+N}.
\]
Taking inverse to the matrix equation (\ref{T=FDE}), we have
\begin{equation}\label{Tp=FDE}
t_{i,j}^{\prime}(u) = \big( F(u)^{-1} D(u)^{-1} E(u)^{-1} \big)_{i,j}
\end{equation}
Applying the technique in \cite[Section 1]{MNO}, we may rewrite equation (\ref{RTT}) into the following series form
$$
(u-v)[t_{ij}(u),t_{hk}(v)]
=(-1)^{\pa{i}\,\pa{j}+\pa{i}\,\pa{h}+\pa{j}\,\pa{h}}
\Big(t_{hj}(u)t_{ik}(v)-t_{hj}(v)t_{ik}(u)\Big).
$$
The next lemma, which will be used frequently later, can be deduced from the above equation by a similar calculation as in \cite[Section 2]{Pe1}.
\begin{lemma}
\begin{multline}\label{usefull}
(u-v)[t_{ij}(u),t'_{hk}(v)]
=(-1)^{\pa{i}\,\pa{j}+\pa{i}\,\pa{h}+\pa{j}\,\pa{h}}\times\\
\Big(\delta_{h,j}
\sum_{g=1}^{M+N}
t_{ig}(u)t'_{gk}(v)-\delta_{i,k}\sum_{g=1}^{M+N}t'_{hg}(v)t_{gj}(u)\Big),
\end{multline}
for all $1\leq i,j,h,k\leq M+N$.
\end{lemma}
\begin{lemma}
For each pair $a$, $b$ such that $1<a+1<b\leq n-1$ and $1\leq i\leq\mu_a$, $1 \leq j \leq \mu_b$, we have
\begin{equation}
E_{a,b;i,j}^{(r)} = (-1)^{\pa{k}_{b-1}}[E_{a,b-1;i,k}^{(r)}, E_{b-1;k,j}^{(1)}],
\,\,\,\,\,
F_{b,a;j,i}^{(r)} = (-1)^{\pa{k}_{b-1}}[F_{b-1;j,k}^{(1)}, F_{b-1,a;k,i}^{(r)}],\label{ter}
\end{equation}
for any fixed $1 \leq k \leq \mu_{b-1}$.
\end{lemma}
\begin{proof}
This can be proved by induction on $b-a>1$. We perform the initial step when $a=1$ and $b=3$ for the $E$'s here, while the general case and the statement for the $F$'s can be established in a similar way.
By (\ref{quasie}) and (\ref{Jacobi1}), we have
$$[E_{1,2;i,h}^{(r)}, E_{2,3;k,j}^{(1)}]=[\sum_{p=1}^{\mu_1}\sum_{s=0}^{r}D_{1;i,p}^{\prime(s)} T_{1,2;p,h}^{(r-s)} , T_{2,3;k,j}^{(1)}].$$
Note that we may express $D_{1;i,p}^{\prime(r)}$ in terms of $t_{\alpha,\beta}^{\prime(r)}$, and the subscriptions $1\leq \alpha,\beta\leq \mu_1$ will never overlap with the subscriptions of $T_{2,3;k,j}^{(1)}=t_{\mu_1+k,\mu_1+\mu_2+j}^{(1)}$, where $1\leq k\leq \mu_2, 1\leq j\leq \mu_3$. Thus they supercommute by (\ref{usefull}), and the right-hand side of the equation equals to
$$\sum_{p=1}^{\mu_1}\sum_{s=0}^{r}D_{1;i,p}^{\prime(s)} [ T_{1,2;p,h}^{(r-s)}, T_{2,3;k,j}^{(1)}].$$
The bracket can be computed by (\ref{RTT}):
$$[ T_{1,2;p,h}^{(r-s)}, T_{2,3;k,j}^{(1)}]=[t_{p,\mu_1+h}^{(r-s)}, t_{\mu_1+k,\mu_1+\mu_2+j}^{(1)}]=(-1)^{\pa{k}_2} \delta_{h,k} t_{p,\mu_1+\mu_2+j}^{(r-s)}.$$
Thus, for any $1\leq k\leq \mu_2$, we have
$$(-1)^{\pa{k}_2}[E_{1,2;i,k}^{(r)}, E_{2,3;k,j}^{(1)}]=\sum_{p=1}^{\mu_1}\sum_{s=0}^{r}D_{1;i,p}^{\prime(s)}t_{p,\mu_1+\mu_2+j}^{(r-s)}=\sum_{p=1}^{\mu_1}\sum_{s=0}^{r}D_{1;i,p}^{\prime(s)}T_{1,3;p,j}^{(r-s)}.$$
By (\ref{quasie}), the right-hand side of the above equation is exactly $E_{1,3;i,j}^{(r)}$. The general case is similar, except that the expression of $E_{a,b-1;i,h}^{(r)}$ by (\ref{quasie}) is more complicated.
\end{proof}
\begin{theorem}\label{gendef}
The superalgebra $Y_{M|N}$ is generated by the following elements
\begin{align*}
&\big\lbrace D_{a;i,j}^{(r)}, D_{a;i,j}^{\prime(r)} \,|\, {1\leq a\leq n,\; 1\leq i,j\leq \mu_a,\; r\geq 0}\big\rbrace,\\
&\big\lbrace E_{a;i,j}^{(r)} \,|\, {1\leq a< n,\; 1\leq i\leq \mu_a, 1\leq j\leq\mu_{a+1},\; r\geq 1}\big\rbrace,\\
&\big\lbrace F_{a;i,j}^{(r)} \,|\, {1\leq a< n,\; 1\leq i\leq\mu_{a+1}, 1\leq j\leq \mu_a,\; r\geq 1}\big\rbrace.
\end{align*}
\end{theorem}
\begin{proof}
Multiplying the matrix equation (\ref{T=FDE}), we see that each $t_{ij}^{(r)}$ can be expressed as a sum of supermonomials in $D_{a;i,j}^{(r)}$, $E_{a,b;i,j}^{(r)}$ and $F_{b,a;i,j}^{(r)}$, in a certain order that all the $F$'s appear before the $D$'s and all the $D$'s appear before the $E$'s. By (\ref{ter}), it is enough to use $D_{a;i,j}^{(r)}$, $E_{a;i,j}^{(r)}$ and $ F_{a;i,j}^{(r)}$ only rather than use all of the $E$'s and the $F$'s.
\end{proof}
\begin{remark}
Actually we don't need those $D_{a;i,j}^{\prime(r)}$ to obtain a generating set.
They only appear in (\ref{p702}) and (\ref{p706}). One can in fact remove (\ref{p702}) from the defining relations and rewrite (\ref{p706}) into a new expression which is free from those $D^{\prime(r)}_{a;i,j}$.
The reason to include them is to make the presentation look concise.
\end{remark}
The generators of $Y_{M|N}$ in Theorem \ref{gendef} above are called {\em parabolic generators}. Note that these generators depend on the shape $\mu$ and their parities depend on the fixed sequence $\mathfrak{s}$ (see (\ref{pad})-(\ref{paf}) later). We will use the notation $Y_\mu$ or $Y_{M|N}(\mathfrak{s})$ or $Y_\mu(\mathfrak{s})$ to emphasize the choice of $\mu$ or $\mathfrak{s}$ or both when necessary. The goal of this article is to write down explicitly a set of defining relations of $Y_\mu(\mathfrak{s})$ with respect to the parabolic generators for {\em any} fixed $\mu$ and {\em any} fixed $\mathfrak{s}$.
\section{Homomorphisms between super Yangians}
To explicitly write down the relations among the parabolic generators in Theorem \ref{gendef}, we start with the special cases when $n$ are either 2 or 3, that are relatively less complicated. The other relations in full generality can be deduced from these special ones by applying certain nice injective homomorphisms that we are about to introduce.
We start with some notation. Let $\mathfrak{s}$ be a fixed $0^M1^N$-sequence. We define
\begin{itemize}
\item $\check{\mathfrak{s}}$:= the $0^N1^M$-sequence obtained by interchanging the 0's and 1's of $\mathfrak{s}$.
\item $\mathfrak{s}^r$:= the reverse of $\mathfrak{s}$.
\item $\mathfrak{s}^\dagger$:= $(\check{\mathfrak{s}})^r$, the reverse of $\check{\mathfrak{s}}$.
\end{itemize}
For instance, if $\mathfrak{s}=0011010$, then $\check{\mathfrak{s}}=1100101$, $\mathfrak{s}^r=0101100$, and $\mathfrak{s}^\dagger=1010011$.
Moreover, if $\mathfrak{s}_1$ and $\mathfrak{s}_2$ are two 01-sequences, then $\mathfrak{s}_1\mathfrak{s}_2$ simply means the concatenation of $\mathfrak{s}_1$ and $\mathfrak{s}_2$.
The following proposition is a generalization of \cite[Proposition 4.1]{Pe1} to an arbitrary 01-sequence $\mathfrak{s}$, where the proof is similar so we omit; see also \cite[Section~4]{Go}
\begin{proposition}
\begin{enumerate}
\item [1.]The map $\rho_{M|N}:Y_{M|N}(\mathfrak{s})\rightarrow Y_{N|M}(\mathfrak{s}^\dagger)$ defined by
\[
\rho_{M|N}\big(t_{ij}(u)\big)=t_{M+N+1-i,M+N+1-j}(-u)
\]
is an isomorphism.
\item [2.]The map $\omega_{M|N}:Y_{M|N}(\mathfrak{s})\rightarrow Y_{M|N}(\mathfrak{s})$ defined by
\[
\omega_{M|N}\big(T(u)\big)=\big(T(-u)\big)^{-1}
\]
is an automorphism.
\item [3.]The map $\zeta_{M|N}:Y_{M|N}(\mathfrak{s})\rightarrow Y_{N|M}(\mathfrak{s}^\dagger)$ defined by
\[
\zeta_{M|N}=\rho_{M|N}\circ\omega_{M|N}
\]
is an isomorphism.
\item [4.] Let $p,q\in\mathbb{Z}_{\ge0}$ be given and let $\mathfrak{s}_1$ be an arbitrary $0^p1^q$-sequence.
Let
$$\varphi_{p|q}:Y_{M|N}(\mathfrak{s})\rightarrow Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$$
be the injective homomorphism sending each $t_{i,j}^{(r)}$ in $Y_{M|N}(\mathfrak{s})$ to $t_{p+q+i,p+q+j}^{(r)}$ in $Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$. Then
the map $\psi_{p|q}:Y_{M|N}(\mathfrak{s})\rightarrow Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$ defined by
\[
\psi_{p|q}=\omega_{p+M|q+N}\circ\varphi_{p|q}\circ\omega_{M|N},
\]
is an injective homomorphism.
\end{enumerate}
\end{proposition}
In fact, only the maps $\zeta_{M|N}$ and $\psi_{p|q}$ will be used later so we write down their images explicitly.
\begin{lemma}
Let $1\leq i,j \leq M+N$.
\begin{enumerate}
\item [1.]For any $p,q\in\mathbb{Z}_{\geq 0}$, we have \begin{equation}\label{psit}
\psi_{p|q}\big(t_{ij}(u)\big)=
\left| \begin{array}{cccc} t_{1,1}(u) &\cdots &t_{1,p+q}(u) &t_{1, p+q+j}(u)\\
\vdots &\ddots &\vdots &\vdots \\
t_{p+q,1}(u) &\cdots &t_{p+q,p+q}(u) &t_{p+q, p+q+j}(u)\\
t_{p+q+i, 1}(u) &\cdots &t_{p+q+i,p+q}(u) &\boxed{t_{p+q+i, p+q+j}(u)}
\end{array} \right|.
\end{equation}
\item [2.] \begin{equation}\label{zetat}
\zeta_{M|N}\big(t_{ij}(u)\big)=t_{M+N+1-i,M+N+1-j}^{\prime}(u).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
The first one is exactly the same with \cite[Lemma 4.2]{BK1}, while the second one is immediate from the definition.\end{proof}
Set $L=p+q$ for convenience. Note that (\ref{psit}) depends only on $L$ so we may simply write $\psi_{p|q}=\psi_L$ when appropriate. As a consequence of Proposition \ref{quasi}, we have
\begin{eqnarray}\label{psid}
D_{a;i,j}(u)&=&\psi_{\mu_1+\mu_2+\ldots +\mu_{a-1}}\big(D_{1;i,j}(u)\big),\\ \label{psie}
E_{a;i,j}(u)&=&\psi_{\mu_1+\mu_2+\ldots +\mu_{a-1}}\big(E_{1;i,j}(u)\big),\\ \label{psif}
F_{a;i,j}(u)&=&\psi_{\mu_1+\mu_2+\ldots +\mu_{a-1}}\big(F_{1;i,j}(u)\big).
\end{eqnarray}
In addition, the map $\psi_{L}$ sends $t_{i,j}^{\prime}(u)\in Y_{M|N}(\mathfrak{s})$ to $t_{L+i,L+j}^{\prime}(u)\in Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$, which implies that the image of $\psi_{L}\big(Y_{M|N}(\mathfrak{s})\big)$ in $Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$ is generated by the following elements
\[
\{t_{L+i,L+j}^{\prime(r)}\in Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})\,|\,1\leq i,j\leq M+N, r\geq 0\}.
\]
If we pick any element $t_{i,j}^{(r)}$ in the northwestern $L\times L$ corner of $T(u)$, an $(L+M+N)\times(L+M+N)$ matrix with entries in $Y_{p+M|q+N}[[u^{-1}]]$, then the indices will never overlap with those of $\psi_{L}\big(Y_{M|N}\big)$, that are in the southeastern $(M+N)\times(M+N)$ corner of the same $T(u)$. As a result of equation (\ref{usefull}), they supercommute. Clearly, the elements in the northwestern $L\times L$ corner of $T(u)$ (of $Y_{p+M|q+N}$) generate a subalgebra which is isomorphic to $Y_{p|q}(\mathfrak{s}_1)$ by (\ref{RTT}), so we have obtained the following lemma. Roughly speaking, the map $\psi_{L}$ embeds the matrix $T(u)$ into the southeastern corner of a larger matrix $T(u)$.
\begin{lemma}\label{corcommute}
In $Y_{p+M|q+N}(\mathfrak{s}_1\mathfrak{s})$, the subalgebras $Y_{p|q}(\mathfrak{s}_1)$ and $\psi_{L}\big(Y_{M|N}(\mathfrak{s})\big)$ supercommute with each other.
\end{lemma}
Moreover, by equations (\ref{psit}), (\ref{psid}), (\ref{psie}) and (\ref{psif}), the parities of the parabolic generators can be easily obtained as follows:
\begin{eqnarray}
\label{pad}\text{ parity of } D_{a;i,j}^{(r)}&=&\pa{i}_a+\pa{j}_a \,\,\text{(mod 2)},\\
\label{pae}\text{ parity of } E_{b;h,k}^{(s)}&=&\pa{h}_{b}+\pa{k}_{b+1} \,\,\text{(mod 2)},\\
\label{paf}\text{ parity of } F_{b;f,g}^{(s)}&=&\pa{f}_{b+1}+\pa{g}_{b} \,\,\text{(mod 2)},
\end{eqnarray}
for all $r,s\geq 1$, $1\leq a\leq n$, $1\leq b\leq n-1$, $1\leq i,j\leq \mu_a$, $1\leq h,g\leq \mu_b$, $1\leq k,f\leq \mu_{b+1}$.
Next we analyze $\zeta_{M|N}$. Associate to $\mu$, we may define the elements \{$D_{a;i,j}^{(r)};D^{\prime(r)}_{a;i,j}$\}, \{$E_{a;i,j}^{(r)}$\}, \{$F_{a;i,j}^{(r)}$\} in $Y_{M|N}=Y_\mu(\mathfrak{s})$ by Gauss decomposition. Consider
\[\overleftarrow{\mu}:=(\mu_n,\mu_{n-1},\ldots,\mu_{1}),\]
the reverse of $\mu$.
We may similarly define elements \{$\overleftarrow{D}_{a;i,j}^{(r)};\overleftarrow{D}^{\prime(r)}_{a;i,j}$\}, \{$\overleftarrow{E}_{a;i,j}^{(r)}$\}, \{$\overleftarrow{F}_{a;i,j}^{(r)}$\} in $Y_{N|M}=Y_{\overleftarrow{\mu}}(\mathfrak{s}^\dagger)$ with respect to $\overleftarrow{\mu}$ in the same way. Their relations are described in the following proposition, which is a generalization of \cite[Proposition~1]{Go} and \cite[Proposition~4.4]{Pe1}, where the proof is very similar. Roughly speaking, the map $\zeta_{M|N}$ turns the matrix $T(u)$ up side down, so we take $\mathfrak{s}^\dagger$ in the image space in order to keep the parities unchanged.
\begin{proposition}\label{zetadef} For all $1\leq i,j,h\leq \mu_a$, $1\leq k\leq \mu_{a+1}$, we have
\begin{eqnarray}
\zeta_{M|N}\big(D_{a;i,j}(u)\big)&=&\overleftarrow{D}^{\prime}_{n+1-a;\mu_{a}+1-i,\mu_{a}+1-j}(u), \qquad\forall 1\leq a\leq n, \label{zd} \\
\zeta_{M|N}\big(E_{a;h,k}(u)\big)&=&-\overleftarrow{F}_{n-a;\mu_{a}+1-h,\mu_{a+1}+1-k}(u), \quad\forall 1\leq a\leq n-1, \label{sze}\\
\zeta_{M|N}\big(F_{a;k,h}(u)\big)&=&-\overleftarrow{E}_{n-a;\mu_{a+1}+1-k,\mu_{a}+1-h}(u), \quad\forall 1\leq a\leq n-1\label{szf}.
\end{eqnarray}
\end{proposition}
The following proposition follows directly from (\ref{RTT}), (\ref{psid}), and Lemma \ref{corcommute}. In consequence, the relations among the $D$'s are obtained. One should notice that these $D$'s could be even or odd according to (\ref{pad}). This is different with the cases in \cite{Go,Pe1}, in which they are always even.
\begin{proposition}\label{dd0}
The relations among the elements
$\{D_{a;i,j}^{(r)},D_{a;i,j}^{\prime (r)}\}$ for all $r\geq 0$, ${1\leq i,j\leq \mu_a}$, $1\leq a\leq n$
are given by
\[
D_{a;i,j}^{(0)}=\delta_{ij},\]
\[
{\displaystyle \sum_{t=0}^{r}D_{a;i,p}^{(t)}D_{a;p,j}^{\prime (r-t)}=\delta_{r0}\delta_{ij} },
\]
\begin{multline*}
[D_{a;i,j}^{(r)},D_{b;h,k}^{(s)}]=
\delta_{ab}(-1)^{\pa{i}_a\pa{j}_a+\pa{i}_a\pa{h}_a+\pa{j}_a\pa{h}_a}\times\\
\sum_{t=0}^{min(r,s)-1}\big(D_{a;h,j}^{(t)}D_{a;i,k}^{(r+s-1-t)}
-D_{a;h,j}^{(r+s-1-t)}D_{a;i,k}^{(t)}\big).
\end{multline*}
\end{proposition}
It can be observed from the relations that the elements $\{D_{a;i,j}^{(r)},D_{a;i,j}^{\prime (r)}\}$ generate a subalgebra of $Y_{\mu}$, called the $Levi$ $subalgebra$ of $Y_{\mu}$, and denote it by $Y^0_{\mu}(\mathfrak{s})$.
By Lemma \ref{corcommute}, we have
\begin{eqnarray*}
Y^0_{\mu}(\mathfrak{s})&=&Y_{\mu_1}(\mathfrak{s}_1)\psi_{\mu_1}(Y_{\mu_2}(\mathfrak{s}_2))\psi_{\mu_1+\mu_2}(Y_{\mu_3}(\mathfrak{s}_3))\cdots\psi_{\mu_1+\cdots+\mu_{n-1}}(Y_{\mu_n}(\mathfrak{s}_n))\\
&\cong &Y_{\mu_1}(\mathfrak{s}_1)\otimes Y_{\mu_2}(\mathfrak{s}_2)\otimes\cdots \otimes Y_{\mu_n}(\mathfrak{s}_n),
\end{eqnarray*}
where $\mu=(\mu_1,\mu_2,\ldots ,\mu_n)$ and $\mathfrak{s}=\mathfrak{s}_1\mathfrak{s}_2\cdots\mathfrak{s}_n$.
Note that in the special case when all $\mu_i=1$, the subalgebra $Y^0_{(1,\ldots,1)}(\mathfrak{s})$ is purely even and commutative. One may think $Y^0_{(1,\ldots,1)}(\mathfrak{s})$ in $Y_{M|N}$ as an analogue of the Cartan subalgebra consisting of all diagonal matrices in $\mathfrak{gl}_{M|N}$.
\section{Special Case: $n=2$}
In this section, we focus on the very first non-trivial case under our consideration; that is, $\mu=(\mu_1,\mu_2)$ with a fixed 01-sequence $\mathfrak{s}=\mathfrak{s}_1\mathfrak{s}_2$.
We list our parabolic generators as follow:
\begin{align*}
&\big\lbrace D_{a;i,j}^{(r)}, D_{a;i,j}^{\prime(r)} \,|\, {a=1,2;\; 1\leq i,j\leq \mu_a;\; r\geq 0}\big\rbrace,\\
&\big\lbrace E_{1;i,j}^{(r)} \,|\, 1\leq i\leq \mu_1, 1\leq j\leq\mu_{2};\; r\geq 1\big\rbrace,\\
&\big\lbrace F_{1;i,j}^{(r)} \,|\, 1\leq i\leq\mu_{2}, 1\leq j\leq \mu_1;\; r\geq 1\big\rbrace.
\end{align*}
The following proposition gives explicitly the relations among the generators other than those relations already obtained in Proposition \ref{dd0}.
{\allowdisplaybreaks
\begin{proposition}\label{n=2}
Let $\mu=(\mu_1,\mu_2)$ be a composition of $M+N$. The following identities hold in
$Y_{\mu}((u^{-1},v^{-1}))$:
\begin{align}
(u-v)[D_{1;i,j}(u), E_{1;h,k}(v)]\label{p511}
&=(-1)^{\pa{h}_1\pa{j}_1}\delta_{hj}\sum_{p=1}^{\mu_1}D_{1;i,p}(u)\big(E_{1;p,k}(v)-E_{1;p,k}(u)\big),\\[2mm]
(u-v)[E_{1;i,j}(u), D^{\prime}_{2;h,k}(v)]
&=(-1)^{\pa{h}_2\pa{j}_2}\delta_{hj} \sum_{q=1}^{\mu_2}\big(E_{1;i,q}(u)-E_{1;i,q}(v)\big)D^{\prime}_{2;q,k}(v),\label{p512}\\[2mm]
(u-v)[D_{2;i,j}(u), E_{1;h,k}(v)]
&\notag=(-1)^{\pa{h}_1\pa{k}_2+\pa{h}_1\pa{j}_2+\pa{j}_2\pa{k}_2}\times\\
&\qquad\qquad\qquad D_{2;i,k}(u)\big(E_{1;h,j}(u)-E_{1;h,j}(v)\big),\label{p513}\\[2mm]
(u-v)[D_{1;i,j}(u), F_{1;h,k}(v)]
&\notag=(-1)^{\pa{i}_1\pa{j}_1+\pa{h}_2\pa{i}_1+\pa{h}_2\pa{j}_1}\delta_{ik}\times\\
&\qquad\qquad\qquad \sum_{p=1}^{\mu_1}\big(F_{1;h,p}(u)-F_{1;h,p}(v)\big)D_{1;p,j}(u),\label{p514}\\[2mm]
(u-v)[F_{1;i,j}(u), D^{\prime}_{2;h,k}(v)]
&\notag=(-1)^{\pa{h}_2\pa{i}_2+\pa{h}_2\pa{j}_1+\pa{j}_1\pa{k}_2}\delta_{ik}\times\\
&\qquad\qquad\qquad \sum_{q=1}^{\mu_2}D^{\prime}_{2;h,q}(v)\big(F_{1;q,j}(v)-F_{1;q,j}(u)\big),\label{p515}\\[2mm]
(u-v)[D_{2;i,j}(u), F_{1;h,k}(v)]
&\notag=(-1)^{\pa{h}_2\pa{k}_1+\pa{h}_2\pa{j}_2+\pa{j}_2\pa{k}_1}\times\\
&\qquad\qquad\qquad \big(F_{1;i,k}(v)-F_{1;i,k}(u)\big)D_{2;h,j}(u),\label{p516}\\[2mm]
(u-v)[E_{1;i,j}(u), F_{1;h,k}(v)]
&\notag=(-1)^{\pa{h}_2\pa{i}_1+\pa{i}_1\pa{j}_2+\pa{h}_2\pa{j}_2}D_{2;h,j}(u) D^{\prime}_{1;i,k}(u)\\
&\qquad -(-1)^{\pa{h}_2\pa{k}_1+\pa{j}_2\pa{k}_1+\pa{h}_2\pa{j}_2}
D^{\prime}_{1;i,k}(v) D_{2;h,j}(v),\label{p517}\\[2mm]
(u-v)[E_{1;i,j}(u), E_{1;h,k}(v)]
&\notag=(-1)^{\pa{h}_1\pa{j}_2+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2}\times\\
&\qquad \big(E_{1;i,k}(u)-E_{1;i,k}(v)\big)\big(E_{1;h,j}(u)-E_{1;h,j}(v)\big),\label{p518}\\[2mm]
(u-v)[F_{1;i,j}(u), F_{1;h,k}(v)]
&\notag=(-1)^{\pa{i}_2\pa{j}_1+\pa{h}_2\pa{i}_2+\pa{h}_2\pa{j}_1}\times\\
&\qquad \big(F_{1;h,j}(v)-F_{1;h,j}(u)\big)\big(F_{1;i,k}(u)-F_{1;i,k}(v)\big).\label{p519}
\end{align}
The identities hold for all $1\leq i,j\leq \mu_1$ if $D_{1;i.j}(u)$ appears on the left-hand side of the equation,
for all $1\leq h,k\leq \mu_2$ if $D_{2;h,k}^\prime(u)$ appears on the left-hand side of the equation,
for all $1\leq i^\prime \leq \mu_1$, $1\leq j^\prime \leq \mu_2$ if $E_{1;i^\prime,j^\prime}(u)$ appears on the left-hand side of the equation, and
for all $1\leq h^\prime\leq \mu_2$, $1\leq k^\prime\leq \mu_1$ if $F_{1;h^\prime,k^\prime}(u)$ appears on the left-hand side of the equation.
\end{proposition}
}
\begin{proof}
Our approach is similar to those in \cite[Section 6]{BK1} and \cite[Section 5]{Pe1}.
We compute the matrix products (\ref{T=FDE}) and (\ref{Tp=FDE})
with respect to the composition $\mu=(\mu_1,\mu_2)$ and get the following identities.
\begin{eqnarray}
\label{511}t_{i,j}(u)&=&D_{1;i,j}(u), \qquad\qquad\qquad\quad\;\forall 1\le i,j\le \mu_1,\\
\label{512}t_{i,\mu_1+j}(u)&=&D_{1;i,p}E_{1;p,j}(u),\qquad\qquad\quad\;\;\forall 1\le i\le \mu_1, 1\le j\le \mu_2,\\
\label{513}t_{\mu_1+i,j}(u)&=&F_{1;i,p}(u)D_{1;p,j}(u),\qquad\quad\quad\;\forall 1\le i\le \mu_2, 1\le j\le \mu_1,\\
\label{514}t_{\mu_1+i,\mu_1+j}(u)&=&F_{1;i,p}(u)D_{1;p,q}(u)E_{1;q,j}(u)+D_{2;i,j}(u),\,\,\,\forall 1\le i,j\le \mu_2,\\
\label{515}t^{\prime}_{i,j}(u)&=&D^{\prime}_{1;i,j}(u)+E_{1;i,p'}(u)D^{\prime}_{2;p',q'}(u)F_{1;q',j}(u),\forall 1\le i,j\le \mu_1,\\
\label{516}t^{\prime}_{i,\mu_1+j}(u)&=&-E_{1;i,p'}(u)D^{\prime}_{2;p',j}(u),\qquad\quad\forall 1\le i\le \mu_1, 1\le j\le \mu_2,\\
\label{517}t^{\prime}_{\mu_1+i,j}(u)&=&-D^{\prime}_{2;i,p'}(u)F_{1;p',j}(u),\qquad\quad\,\forall 1\le i\le \mu_2, 1\le j\le \mu_1,\\
\label{518}t^{\prime}_{\mu_1+i,\mu_1+j}(u)&=&D^{\prime}_{2;i,j}(u),\qquad\qquad\qquad\quad\;\,\forall 1\le i,j\le \mu_2,
\end{eqnarray}
where the indices $p,q$ (respectively, $p',q'$) are summed over $1,\ldots,\mu_1$ (respectively, $1,\ldots,\mu_2$).
(\ref{p511})--(\ref{p513}) can be proved similar to \cite[Lemma~6.3]{BK1} and \cite[Proposition~5.1]{Pe1}, except for some issues about the sign factors that have to be carefully treated. (\ref{p514})--(\ref{p516}) and (\ref{p519}) follow from applying the map $\zeta_{M|N}$ to (\ref{p511})--(\ref{p513}) and (\ref{p518}) with suitable choices of indices.
We prove (\ref{p517}) and (\ref{p518}) in detail here as illustrating examples about some new phenomenons and how we deal with them.
To show (\ref{p517}), we need other identities. Computing the brackets in (\ref{p512}) and (\ref{p514}) by definition, we have
\begin{multline}\label{2ef1}
(u-v)E_{1;\alpha ,j}(u)D^{\prime}_{2;h,\beta}(v)
=(-1)^{\pa{h}_2\pa{j}_2}\delta_{hj}\big(E_{1;\alpha ,q}(u)-E_{1;\alpha ,q}(v)\big)D^{\prime}_{2;q,\beta}(v)\\
+(-1)^{(\pa{\alpha}_1+\pa{j}_2)(\pa{h}_2+\pa{\beta}_2)}(u-v)D^{\prime}_{2;h,\beta}(v)E_{1;\alpha ,j}(u),
\end{multline}
\begin{multline}\label{2ef2}
(u-v)F_{1;\beta ,k}(v)D_{1;i,\alpha}(u)=
(-1)^{(\pa{i}_1+\pa{\alpha}_1)(\pa{\beta}_1+\pa{k}_1)}(u-v)D_{1;i,\alpha}(u)F_{1;\beta ,k}(v)\\
-(-1)^{\pa{i}_1\pa{k}_1}\delta_{ki}\big(F_{1;\beta ,p}(u)-F_{1;\beta ,p}(v)\big)D_{1;p,\alpha}(u),
\end{multline}
where $\alpha$, $p$ (respectively, $\beta$, $q$) are summed over $1,\ldots,\mu_1$ (respectively, $1,\ldots, \mu_2$).
By (\ref{usefull}), we have
\begin{multline*}
(u-v)[t_{i,\mu_1+j}(u),t^{\prime}_{\mu_1+h,k}(v)]=(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{h}_2+\pa{j}_2\pa{h}_2}\times \\
\big(\delta_{hj}\sum_{g=1}^{M+N}t_{ig}(u)t^{\prime}_{gk}(v)
-\delta_{ki}\sum_{s=1}^{M+N}t^{\prime}_{\mu_1+h,s}(v)t_{s,\mu_1+j}(u)\big).
\end{multline*}
Substituting by (\ref{511})$-$(\ref{518}), we may rewrite the above identity as the following
\begin{align}\notag
&D_{1;i,\alpha}(u)(u-v)E_{1;\alpha ,j}(u)D^{\prime}_{2;h,\beta}(v)F_{1;\beta,k}(v)+
(-1)^{\pa{j}_2\pa{h}_2}\delta_{hj}D_{1;i,\alpha}(u)D^{\prime}_{1;\alpha ,k}(v)\\\notag
&+(-1)^{\pa{j}_2\pa{h}_2}\delta_{hj}D_{1;i,\alpha}(u)\big(E_{1;\alpha ,q}(v)-E_{1;\alpha ,q}(u)\big)D^{\prime}_{2;q,\beta}(v) F_{\beta ,k}(v)\\\notag
&=(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_2+\pa{k}_1)}(u-v)D^{\prime}_{2;h,\beta}(v) F_{1;\beta ,k}(v)D_{1;i,\alpha}(u)E_{1;\alpha,j}(u)\\\notag
&+(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_2+\pa{k}_1)}\delta_{ki}D_{2;h,\beta}^\prime(v)\big(F_{1;\beta ,p}(u)-F_{1;\beta ,p}(v)\big)D_{1;p,\alpha}(u) E_{1;\alpha ,j}(u)\\ \label{2ef3}
&+(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_2+\pa{k}_1)}\delta_{ki}D^{\prime}_{2;h,\beta}(v)D_{2;\beta ,j}(u),
\end{align}
where $\alpha$, $p$ (respectively, $\beta$, $q$) are summed over $1,\ldots,\mu_1$(resp. $1,\ldots,\mu_2$). Substituting (\ref{2ef1}) and (\ref{2ef2}) into (\ref{2ef3}) and simplifying the result, we obtain
\begin{multline*}
D_{1;i,\alpha}(u)D^\prime_{2;h\beta}(v)D_{2;\beta,j}(v)D^{\prime}_{1;\alpha ,k}(v)\\
+(-1)^{\pa{\alpha}_1\pa{h}_2+\pa{\alpha}_1\pa{k}_1+\pa{j}_2\pa{\beta}_2}(u-v)D_{1;i,\alpha}(u) D^{\prime}_{2;h,\beta}(v)E_{1;\alpha ,j}(u)F_{1;\beta ,k}(v)\\
=
(-1)^{\pa{\alpha}_1\pa{h}_2+\pa{\alpha}_1\pa{k}_1+\pa{j}_2\pa{k}_1}(u-v)D_{1;i,\alpha}(u) D^{\prime}_{2;h,\beta}(v) F_{1;\beta ,k}(v)E_{1;\alpha ,j}(u)\\
+(-1)^{\pa{k}_1\pa{j}_2+\pa{h}_2\pa{\alpha}_1+\pa{\beta}_2\pa{\alpha}_1+\pa{\beta}_2\pa{k}_1}D_{1;i,\alpha}(u) D^{\prime}_{2;h,\beta}(v) D_{1;\alpha,k}^\prime(u) D_{2;\beta ,j}(u).
\end{multline*}
Note that in the above equality, the index $i$ is not involved in those sign factors. We may multiply the matrix $D_1^\prime(u)$ from the left to both sides of the equality above so that we have:
\begin{multline*}
D^\prime_{2;h\beta}(v)D_{2;\beta,j}(v)D^{\prime}_{1;i ,k}(v)\\
+(-1)^{\pa{i}_1\pa{h}_2+\pa{i}_1\pa{k}_1+\pa{j}_2\pa{\beta}_2}(u-v)D^{\prime}_{2;h,\beta}(v)E_{1;i ,j}(u)F_{1;\beta ,k}(v)\\
=
(-1)^{\pa{i}_1\pa{h}_2+\pa{i}_1\pa{k}_1+\pa{j}_2\pa{k}_1}(u-v) D^{\prime}_{2;h,\beta}(v) F_{1;\beta ,k}(v)E_{1;i ,j}(u)\\
+(-1)^{\pa{k}_1\pa{j}_2+\pa{h}_2\pa{i}_1+\pa{\beta}_2\pa{i}_1+\pa{\beta}_2\pa{k}_1}D^{\prime}_{2;h,\beta}(v) D_{1;i,k}^\prime(u) D_{2;\beta ,j}(u).
\end{multline*}
Similar to the above computation, we want to multiply $D_2(v)$ from the left to the above identity. However, we can {\em not} do this directly since the index $h$ is involved in some sign factors; such a phenomenon didn't appear in \cite{BK1, Go, Pe1}.
It turns out that we may multiply a suitable sign factor $(-1)^{\pa{i}_1\pa{h}_2}$ to the above identity so that
\begin{multline*}
(-1)^{\pa{i}_1\pa{j}_2}D^\prime_{2;h,\beta}(v)D_{2;\beta,j}(v)D^{\prime}_{1;i ,k}(v)\\
+(-1)^{\pa{i}_1\pa{k}_1+\pa{j}_2\pa{\beta}_2}(u-v)D^{\prime}_{2;h,\beta}(v)E_{1;i ,j}(u)F_{1;\beta ,k}(v)\\
=
(-1)^{\pa{i}_1\pa{k}_1+\pa{j}_2\pa{k}_1}(u-v) D^{\prime}_{2;h,\beta}(v) F_{1;\beta ,k}(v)E_{1;i ,j}(u)\\
+(-1)^{\pa{k}_1\pa{j}_2+\pa{\beta}_2\pa{i}_1+\pa{\beta}_2\pa{k}_1}D^{\prime}_{2;h,\beta}(v) D_{1;i,k}^\prime(u) D_{2;\beta ,j}(u).
\end{multline*}
Observe that in the very first term we have $D^\prime_{2;h\beta}(v)D_{2;\beta,j}(v)$, which is $\delta_{hj}$, so we may replace $\pa{h}_2$ by $\pa{j}_2$ in the sign factor. Now those sign factors in the above result are free from the index $h$ so we may multiply $D_2(v)$ from the left to obtain
\begin{multline*}
(-1)^{\pa{i}_1\pa{j}_2}D_{2;h,j}(v)D^{\prime}_{1;i ,k}(v)
+(-1)^{\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_2}(u-v)E_{1;i ,j}(u)F_{1;h ,k}(v)\\
=
(-1)^{\pa{i}_1\pa{k}_1+\pa{j}_2\pa{k}_1}(u-v) F_{1;h ,k}(v)E_{1;i ,j}(u)\\
+(-1)^{\pa{k}_1\pa{j}_2+\pa{h}_2\pa{i}_1+\pa{h}_2\pa{k}_1} D_{1;i,k}^\prime(u) D_{2;h ,j}(u).
\end{multline*}
Collecting the corresponding terms and using the fact that $D_{2;h,j}(u)$, $D_{1;i,k}^\prime(u)$ supercommute, we derive (\ref{p517}).
Such a technique appears very often in the remaining part of this article.
For (\ref{p518}), we start with $[t_{i,\mu_1+j}(u), t^{\prime}_{h,\mu_1+k}(v)]=0.$
Multiplying $(u-v)^2$ and computing the bracket after substituting by (\ref{512}) and (\ref{516}), we have
\begin{align}\notag
&(u-v)^2D_{1;i,p}(u)E_{1;p,j}(u)E_{1;h,q}(v)D^{\prime}_{2;q,k}(v)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
&-(-1)^{\pa{p}_1\pa{h}_1+\pa{p}_1\pa{q}_2+\pa{j}_2\pa{q}_2}(u-v)E_{1;h,q}(v)D_{1;i,p}(u)(u-v)D^{\prime}_{2;q,k}(v)E_{1;p,j}(u)=0,\label{2ee1}
\end{align}
where the indices $p$ and $q$ are summed from 1 to $\mu_1$ and $\mu_2$, respectively.
Computing the brackets in (5.1) and (\ref{p512}), we obtain the following two identities
\begin{multline*}
(u-v)(-1)^{(\pa{i}_1+\pa{p}_1)(\pa{h}_1+\pa{q}_2)}E_{1;h,q}(v)D_{1;i,p}(u)\\
=(u-v)D_{1;i,p}(u)E_{1;h,q}(v)
-\delta_{hp}(-1)^{\pa{h}_1\pa{p}_1}D_{1;i,g_1}(u)\big(E_{1;g_1,q}(v)-E_{1;g_1,q}(u)\big),
\end{multline*}
\begin{multline*}
(u-v)(-1)^{(\pa{p}_1+\pa{j}_2)(\pa{q}_2+\pa{k}_2)}D^{\prime}_{2;q,k}(v)E_{1;p,j}(u)\\
=(u-v)E_{1;p,j}(u)D^{\prime}_{2;q,k}(v)
+ \delta_{jq}(-1)^{\pa{q}_2\pa{j}_2}\big(E_{1;p,g_2}(u)-E_{1;p,g_2}(v)\big)D^{\prime}_{2;g_2,k}(v),
\end{multline*}
where the indices $p, g_1$ (respectively, $q, g_2$) are summed from 1 to $\mu_1$ (respectively, $\mu_2$).
Substituting these two into the second term of (\ref{2ee1}), multiplying some suitable choices of sign factors as in the proof of (\ref{p517}) so that we may multiply $D_1(u)$ from the left and $D_2(v)$ from the right simultaneously, we derive that
\begin{multline}
(u-v)^2[E_{1;i,j}(u),E_{1;h,k}(v)]=\\
(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1}(u-v)E_{1;h,j}(v)\big(E_{1;i,k}(v)-E_{1;i,k}(u)\big)\\
+(-1)^{\pa{j}_2\pa{h}_1+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2}(u-v)\big(E_{1;i,k}(u)-E_{1;i,k}(v)\big)E_{;1h,j}(u)\\
+\big(E_{1;i,j}(v)-E_{1;i,j}(u)\big)\big(E_{1;h,k}(u)-E_{1;h,k}(v)\big).\label{2ee2}
\end{multline}
For a power series $P$ in $Y_{\mu}[[u^{-1}, v^{-1}]]$, we write $\big\lbrace P\big\rbrace_{d}$ for the homogeneous component of $P$ of total degree $d$ in the variables $u^{-1}$ and $v^{-1}$. Then (\ref{p518}) is a consequence of the following claim.
{\bf{Claim:}} For $d\ge 1$, we have
\begin{multline*}
(u-v)\big\lbrace [E_{1;i,j}(u), E_{1;h,k}(v)]\big\rbrace_{d+1}=\\
\big\lbrace (-1)^{\pa{j}_2\pa{h}_1+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2}\big(E_{1;i,k}(u)-E_{1;i,k}(v)\big)\big(E_{1;h,j}(u)-E_{1;h,j}(v)\big)\big\rbrace_d
\end{multline*}
We prove the claim by induction on $d$.
For $d=1$, take $\big\lbrace\:\big\rbrace_0$ on (\ref{2ee2}) so that
\[
(u-v)^2\big\lbrace [E_{1;i,j}(u),E_{1;h,k}(v)]\big\rbrace_2=0.
\]
Note that the right-hand side of (\ref{2ee2}) is zero when $u=v$, hence we may divide both sides by $(u-v)$ and therefore $(u-v)\big\lbrace [E_{1;i,j}(u),E_{1;h,k}(v)]\big\rbrace_2=0$, as desired.
\\
Assuming that the claim is true for some $d>1$, so we have
\begin{multline}
(-1)^{\pa{k}_2\pa{j}_2+\pa{k}_2\pa{h}_1+\pa{j}_2\pa{h}_1+1}\big\lbrace[E_{1;i,k}(u), E_{1;h,j}(v)]\big\rbrace_{d+1}=\\
\Big\lbrace \frac{\big(E_{1;i,j}(v)-E_{1;i,j}(u)\big)\big(E_{1;h,k}(u)-E_{1;h,k}(v)\big)}{u-v}\Big\rbrace_d\label{2ee3}
\end{multline}
Note that the right-hand side of (\ref{2ee3}) is zero when $u=v$, which implies
\begin{equation}
E_{1;i,k}(v)E_{1;h,j}(v)=(-1)^{\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1+\pa{i}_1\pa{j}_2+1}E_{1;h,j}(v)E_{1;i,k}(v).\label{2ee4}
\end{equation}
Take $\big\lbrace\:\big\rbrace_d$ on (\ref{2ee2}) and replace the last term by (\ref{2ee3}):
\begin{align*}
(&u-v)^2\big\lbrace [E_{1;i,j}(u), E_{1;h,k}(v)]\big\rbrace_{d+2}\\
&=(u-v)(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1}\big\lbrace E_{1;h,j}(v)\big(E_{1;i,k}(v)-E_{1;i,k}(u)\big)\big\rbrace_{d+1}\\
& +(u-v)(-1)^{\pa{j}_2\pa{h}_1+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2}\big\lbrace \big(E_{1;i,k}(u)-E_{1;i,k}(v)\big)E_{1;h,j}(u)\big\rbrace_{d+1}\\
& +(u-v)\big\lbrace (-1)^{\pa{k}_2\pa{j}_2+\pa{k}_2\pa{h}_1+\pa{j}_2\pa{h}_1}E_{1;i,k}(u)E_{1;h,j}(v)-\\
&\qquad\qquad\qquad\qquad\qquad\qquad
(-1)^{\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1+\pa{i}_1\pa{j}_2+1}E_{1;h,j}(v)E_{1;i,k}(u)\big\rbrace_d\\
&=(u-v)\big\lbrace (-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1}E_{1;h,j}(v) E_{1;i,k}(v)\big\rbrace_{d+1}\\
& +(u-v)(-1)^{\pa{j}_2\pa{h}_1+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2}\big\lbrace E_{1;i,k}(u)E_{1;h,j}(u)-E_{1;i,k}(v)E_{1;h,j}(u) \big\rbrace_{d+1}\\
& +(u-v)\big\lbrace (-1)^{\pa{k}_2\pa{j}_2+\pa{k}_2\pa{h}_1+\pa{j}_2\pa{h}_1}E_{1;i,k}(u)E_{1;h,j}(v)\big\rbrace_d
\end{align*}
Substituting the term $(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{h}_1+\pa{j}_2\pa{h}_1}E_{1;h,j}(v) E_{1;i,k}(v)$ by (\ref{2ee4}) and simplifying the result, we have
\begin{align*}
&(u-v)^2\big\lbrace [E_{1;i,j}(u), E_{1;h,k}(v)]\big\rbrace_{d+2}=\\
&(u-v)\big\lbrace (-1)^{\pa{j}_2\pa{h}_1+\pa{j}_2\pa{k}_2+\pa{h}_1\pa{k}_2} \big(E_{1;i,k}(u)-E_{1;i,k}(v)\big)\big(E_{1;h,j}(u)-E_{1;h,j}(v)\big)\big\rbrace_{d+1}.
\end{align*}
Dividing both sides by $u-v$ establishes the claim.
\end{proof}
\section{Special Cases: $n=3$ and the super Serre relations}
In this section, we will consider the generators $D$'s, $E$'s and $F$'s in different super Yangians at the same time but using the same notation. It should be clear from the context which super Yangian we are dealing with.
Similar to the proof of Proposition \ref{n=2}, we compute the matrix products
(\ref{T=FDE}) and (\ref{Tp=FDE}) with respect to the composition $\mu=(\mu_1,\mu_2,\mu_3)$ and derive the following identities.
\begin{eqnarray}
\label{610}t_{i,j}(u)&=&D_{1;i,j},\\
\label{611}t_{i,\mu_1+j}(u)&=&\sum_{p=1}^{\mu_1}D_{1;i,p}E_{1;p,j}(u),\\
\label{612}t_{i,\mu_1+\mu_2+j}(u)&=&\sum_{p=1}^{\mu_1}D_{1;i,p}E_{1,3;p,j}(u),\\
\label{613}t^{\prime}_{i,\mu_1+\mu_2+j}(u)&=&\sum_{p^\prime=1}^{\mu_3}\sum_{q=1}^{\mu_2}\big(E_{1;i,q}(u)E_{2;q,p^\prime}(u)-E_{1,3;i,p^\prime}(u)\big)D^{\prime}_{3;p^\prime,j}(u),\\
\label{614}t^{\prime}_{\mu_1+i,\mu_1+\mu_2+j}(u)&=&-\sum_{p^\prime=1}^{\mu_3}E_{2;i,p'}(u)D^{\prime}_{3;p^\prime,j}(u),\\
\label{615}t^{\prime}_{\mu_1+\mu_2+i,\mu_1+j}(u)&=&-\sum_{p^\prime=1}^{\mu_3}D^{\prime}_{3;i,p^\prime}(u)F_{2;p^\prime,j}(u),
\end{eqnarray}
where (\ref{610}) holds for all $1\leq i,j\leq \mu_1$,
(\ref{611}) holds for all $1\leq i\leq \mu_1$, $1\leq j\leq \mu_2$,
(\ref{612}) and (\ref{613}) hold for all $1\leq i\leq \mu_1$, $1\leq j\leq \mu_3$,
(\ref{614}) holds for all $1\leq i\leq \mu_2$, $1\leq j\leq \mu_3$,
and (\ref{615}) holds for all $1\leq i\leq \mu_3$, $1\leq j\leq \mu_2$.
\begin{lemma}\label{3be}
The following identities hold in $Y_{(\mu_1,\mu_2,\mu_3)}((u^{-1},v^{-1}))$:
\begin{equation}\label{61a}
[E_{1;i,j}(u), F_{2;h,k}(v)] = 0,
\end{equation}
\begin{multline}\label{61b}
[E_{1;i,j}(u), E_{2;h,k}(v)] = \\
\dfrac{(-1)^{\pa{j}_2\pa{h}_2}}{u-v}\delta_{hj}\sum_{q=1}^{\mu_2}\big\lbrace\big(E_{1;i,q}(u)-E_{1;i,q}(v)\big)E_{2;q,k}(v)
+ E_{1,3;i,k}(v) - E_{1,3;i,k}(u)\big\rbrace,
\end{multline}
\begin{multline}\label{61c}
[E_{1,3;i,j}(u), E_{2;h,k}(v)] =\\
(-1)^{\pa{i}_1\pa{j}_3+\pa{i}_1\pa{h}_2+\pa{h}_2\pa{j}_3+\pa{g}_2}
E_{2;h,j}(v) [E_{1;i,g}(u), E_{2;g,k}(v)],
\end{multline}
\begin{multline}\label{61d}
\sum_{q=1}^{\mu_2}[E_{1;i,j}(u), E_{1,3;h,k}(v) - E_{1;h,q}(v) E_{2;q,k}(v)] =\\
(-1)^{\pa{h}_1\pa{j}_2+\pa{j}_2\pa{k}_3+\pa{h}_1\pa{k}_3+\pa{g}_2+1}
[E_{1;i,g}(u), E_{2;g,k}(v)] E_{1;h,j}(u).
\end{multline}
Here (\ref{61a}) holds for all $1\leq i\leq \mu_1$, $1\leq j,k\leq \mu_2$, $1\leq h\leq \mu_3$, (\ref{61b}) holds for all $1\leq i\leq \mu_1$, $1\leq j,h\leq \mu_2$, $1\leq k\leq \mu_3$, (\ref{61c}) holds for all $1\leq i\leq \mu_1$, $1\leq h\leq \mu_2$, $1\leq j,k\leq \mu_3$, and (\ref{61d}) holds for all $1\leq i,h\leq \mu_1$, $1\leq j,g\leq \mu_2$, $1\leq k\leq \mu_3$.
\end{lemma}
\begin{proof}
By (\ref{usefull}), we have $[t_{i,\mu_1+j}(u),t^{\prime}_{\mu_1+\mu_2+h,\mu_1+k}(v)]=0$. Substituting by (\ref{611}) and (\ref{615}), we have
\[
[D_{1;i,p}(u)E_{1;p,j}(u),-D^{\prime}_{3;h,q}(v)F_{2;q,k}(v)]=0.
\]
Computing the bracket, we obtain
\begin{multline}\label{3ee1}
D_{1;i,p}(u)E_{1;p,j}(u)D^{\prime}_{3;h,q}(v)F_{2;q,k}(v)-\\
(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_3+\pa{k}_2)}D^{\prime}_{3;h,q}(v)F_{2;q,k}(v)D_{1;i,p}(u)E_{1;p,j}(u)=0,
\end{multline}
where $p$ and $q$ are summed over $1,\ldots,\mu_1$ and $1,\ldots,\mu_3$, respectively. Similarly, by (\ref{usefull}), we have
\[
[t_{ij}(u),t^{\prime}_{\mu_1+\mu_2+h,\mu_1+k}(v)]=[t_{i,\mu_1+j}(u),t^{\prime}_{\mu_1+\mu_2+h,\mu_1+\mu_2+k}(v)]=0,
\]
which implies that
\[
[D_{1;i,j}(u),F_{2;h,k}(v)]=[E_{1;i,j}(u),D^{\prime}_{3;h,k}(v)]=0.
\]
Substituting these into (\ref{3ee1}) and using the fact that $D_{1;i,j}(u)$, $D^{\prime}_{3;h,k}(v)$ supercommute, we have
\begin{multline*}
(-1)^{\pa{j}_2\pa{q}_3}D_{1;i,p}(u)D^{\prime}_{3;h,q}(v)E_{1;p,j}(u)F_{2;q,k}(v)-\\
(-1)^{\pa{j}_2\pa{k}_2+\pa{p}_1\pa{q}_3+\pa{p}_1\pa{k}_2}D_{1;i,p}(u)D^{\prime}_{3;h,q}(v)F_{2;q,k}(v)E_{1;p,j}(u)=0.
\end{multline*}
The sign factors are free from the indices $i$ and $h$. Multiplying $D_{3}(v)D^{\prime}_{1}(u)$ from the left, we obtain (\ref{61a})
By (\ref{usefull}) again, we have
\[
(u-v)[t_{i,\mu_1+j}(u),t^{\prime}_{\mu_1+h,\mu_1+\mu_2+k}(v)]=(-1)^{\pa{j}_2\pa{h}_2}\delta_{jh}\sum_{s=1}^{M+N}t_{is}(u)t_{s,\mu_1+\mu_2+k}(v).
\]
Substituting by (\ref{610})--(\ref{615}), we have
\begin{multline}\label{3ee2}
(u-v)[D_{1;i,p}(u)E_{1;p,j}(u),-E_{2;h,q}(v)D^{\prime}_{3;q,k}(v)]=\\
(-1)^{\pa{j}_2\pa{h}_2}\delta_{jh}D_{1;i,p}(u)\big\lbrace \big(E_{1;p,r}(v)E_{2;r,q}(v)-E_{1,3;p,q}(v)\big)\\
-E_{1;p,r}(u)E_{2;r,q}(v)+E_{1,3;p,q}(u)\big\rbrace D^{\prime}_{3;q,k}(v),
\end{multline}
where the indices $p,q,r$ are summed from 1 to $\mu_1,\mu_3,\mu_2$, respectively.
Using the facts that
\begin{eqnarray*}
\big[E_{1;i,j}(v),D^{\prime}_{3;h,k}(u)\big]=0,& \qquad\big(\text{proved in the proof of (\ref{61a})}\big)\\
\big[E_{2;i,j}(v),D_{1;h,k}(u)\big]=0,& \qquad \big(\text{obtained from}\;\;[t_{i,j}(u),t^{\prime}_{\mu_1+h,\mu_1+\mu_2+k}(v)]=0\big)
\end{eqnarray*}
we may rewrite (\ref{3ee2}) as the following
\begin{multline}\label{3ee3}
(u-v) D_{1;i,p}(u) \lbrace E_{1;p,j}(u)E_{2;h,q}(v)-\\
(-1)^{(\pa{j}_2+\pa{p}_1)(\pa{h}_2+\pa{q}_3)}E_{2;h,q}(v)E_{1;p,j}(u)\rbrace D^{\prime}_{3;q,k}(v)\\
=(-1)^{\pa{j}_2\pa{h}_2+1}\delta_{jh}D_{1;i,p}(u)\big\lbrace \big(E_{1;p,r}(v)E_{2;r,q}(v)-E_{1,3;p,q}(v)\big)\\
-E_{1;p,r}(u)E_{2;r,q}(v)+E_{1,3;p,q}(u)\big\rbrace D^{\prime}_{3;q,k}(v).
\end{multline}
The sign factors are free from the indices $i$ and $k$. Canceling $D_{1}(u)$ from the left and $D^{\prime}_3(v)$ from the right on both sides of (\ref{3ee3}) and dividing both sides by $u-v$, we have deduced (\ref{61b}).
To show (\ref{61c}), the identity (\ref{p512}) in $Y_{(\mu_2,\mu_3)}((u^{-1},v^{-1}))$ reads as
\begin{equation*}
(u-v)[E_{1;h,k}(u),D^{\prime}_{2;i,j}(v)]=(-1)^{\pa{i}_2\pa{k}_2}\delta_{ki}\big(E_{1;h,p}(u)-E_{1;h,p}(v)\big)D^{\prime}_{2;p,j}(v).
\end{equation*}
Applying the map $\psi_{\mu_1}$ to this identity and using (4.3)$-$(4.5), we have the following identity in $Y_{(\mu_1,\mu_2,\mu_3)}((u^{-1},v^{-1}))$
\begin{equation*}
(u-v)[E_{2;h,k}(u),D^{\prime}_{3;i,j}(v)]=(-1)^{\pa{i}_3\pa{k}_3}\delta_{ki}\big(E_{2;h,p}(u)-E_{2;h,p}(v)\big)D^{\prime}_{3;p,j}(v).
\end{equation*}
Taking the coefficient of $u^0$ in the above identity, we obtain
\begin{equation*}
[E_{2;h,k}^{(1)},D^{\prime}_{3;i,j}(v)]=-(-1)^{\pa{i}_3\pa{k}_3}\delta_{ki}E_{2;h,p}(v)D^{\prime}_{3;p,j}(v).
\end{equation*}
Also by (\ref{ter}), we have
\begin{equation*}
E_{1,3;i,j}(u)=(-1)^{\pa{g}_2}[E_{1;i,g}(u),E^{(1)}_{2;g,j}],\; \text{for any} \; 1\leq g\leq\mu_2.
\end{equation*}
By (\ref{612}), (\ref{613}), the super Jacobi identity and the fact that $[E_{1;i,g}(u),D^{\prime}_{3;h,k}(v)]=0$, we have
\begin{align}\notag\label{3ee4}
[E_{1,3;i,j}(u),D^{\prime}_{3;h,k}(v)]&=\big[(-1)^{\pa{g}_2}[E_{1;i,g}(u),E^{(1)}_{2;g,j}],D^{\prime}_{3;h,k}(v)\big]\\\notag
&=(-1)^{\pa{g}_2}\big[E_{1;i,g}(u),[E^{(1)}_{2;g,j},D^{\prime}_{3;h,k}(v)]\big]+0\\\notag
&=(-1)^{\pa{g}_2}\big[E_{1;i,g}(u),-(-1)^{\pa{j}_3\pa{h}_3}\delta_{hj}E_{2;g,p}(v)D^{\prime}_{3;p,k}(v)\big]\\
&=(-1)^{1+\pa{g}_2+\pa{j}_3\pa{h}_3}\delta_{hj}\big[E_{1;i,g}(u),E_{2;g,p}(v)\big]D^{\prime}_{3;p,k}(v).
\end{align}
By (\ref{usefull}) and (\ref{610})--(\ref{615}), we have
\[
[t_{i,\mu_1+\mu_2+j}(u),t^{\prime}_{\mu_1+h,\mu_1+\mu_2+k}(v)]=[D_{1;i,p}(u)E_{1,3;p,j}(u),-E_{2;h,q}(v)D^{\prime}_{3;q,k}(v)]=0,
\]
where $p$ and $q$ are summed from 1 to $\mu_1$ and $\mu_3$, respectively. Multiplying $D_1^{\prime}(u)$ from the left, we have $[E_{1,3;i,j}(u),E_{2;h,q}(v)D^{\prime}_{3;q,k}(v)]=0,$ which may be written as
\begin{multline*}
[E_{1,3;i,j}(u),E_{2;h,q}(v)]D^{\prime}_{3;q,k}(v)+\\
(-1)^{(\pa{i}_1+\pa{j}_3)(\pa{h}_2+\pa{q}_3)}E_{2;h,q}(v)[E_{1,3;i,j}(u),D^{\prime}_{3;q,k}(v)]=0.
\end{multline*}
Substituting the last bracket by (\ref{3ee4}), we have
\begin{multline*}
[E_{1,3;i,j}(u),E_{2;h,q}(v)]D^{\prime}_{3;q,k}(v)=\\
(-1)^{(\pa{i}_1+\pa{j}_3)(\pa{h}_2+\pa{j}_3)+\pa{g}_2+\pa{j}_3\pa{j}_3}E_{2;h,j}(v)[E_{1;i,g}(u),E_{2;g,p}(v)]D^{\prime}_{3;p,k}(v).
\end{multline*}
Multiplying $D_{3}(v)$ from the right to the above equality, we acquire (\ref{61c}).
Taking the coefficient of $u^0$ in (\ref{61b}), we have
\begin{equation}\label{3ee5}
[E_{1;i,j}^{(1)},E_{2;h,k}(v)]=(-1)^{\pa{j}_2\pa{h}_2}\delta_{hj}\big(-E_{1;i,q}(v)E_{2;q,k}(v)+E_{1,3;i,k}(v)\big).
\end{equation}
Taking the coefficient of $v^0$ in (\ref{p511}), we have
\begin{equation}\label{3ee6}
[D_{1;i,j}(u),E_{1;h,k}^{(1)}]=(-1)^{\pa{j}_1\pa{h}_1}\delta_{hj}D_{1;i,p}(u)E_{1;p,k}(u).
\end{equation}
Together with the super Jacobi identity and the fact that $[D_{1;i,j}(u),E_{2;g,k}(v)]=0$, (\ref{3ee5}) and (\ref{3ee6}) imply that
\begin{align*}
[D_{1;i,j}(u),E_{1,3;h,k}(v)&-E_{1;h,q}(v)E_{2;q,k}(v)]=[D_{1;i,j}(u),(-1)^{\pa{g_2}}\big[E_{1;h,g}^{(1)},E_{2;g,k}(v)]\big]\notag\\
&=(-1)^{\pa{g_2}}\big[[D_{1;i,j}(u),E_{1;h,g}^{(1)}],E_{2;g,k}(v)\big]+0\\
&=(-1)^{\pa{g_2}}[(-1)^{\pa{j}_1\pa{h}_1}\delta_{hj}D_{1;i,p}(u)E_{1;p,g}(u),E_{2;g,k}(v)]\\
&=(-1)^{\pa{g_2}+\pa{j}_1\pa{h}_1}\delta_{hj}D_{1;i,p}(u)[E_{1;p,g}(u),E_{2;g,k}(v)].
\end{align*}
Summing $j$ from 1 to $\mu_1$ in the above identity, we derive
\begin{multline}\label{3ee7}
(-1)^{\pa{i}_1(\pa{h}_1+\pa{k}_3)}\big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big)D_{1;i,p}(u)\\
-(-1)^{\pa{p}_1(\pa{h}_1+\pa{k}_3)}D_{1;i,p}(u)\big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big),\\
=(-1)^{\pa{g}_2+\pa{p}_1\pa{k}_3}\delta_{hp}D_{1;i,p'}(u)[E_{1;p',g}(u),E_{2;g,k}(v)]
\end{multline}
where $r,p,p'$ are summed over $\mu_2, \mu_1 ,\mu_1$, respectively.
On the other hand, by (\ref{usefull}) and (\ref{610})--(\ref{615}), we have
\begin{multline}\label{3ee8}
[t_{i,\mu_1+j}(u),t^{\prime}_{h,\mu_1+\mu_2+k}(v)]=\\
[D_{1;i,p}(u)E_{1;p,j}(u),\big(E_{1;h,r}(v)E_{2;r,q}(v)-E_{1,3;h,q}(v)\big)D^{\prime}_{3;q,k}(v)]=0,
\end{multline}
where $p$ and $q$ are summed from 1 to $\mu_1$ and $\mu_3$, respectively. Multiplying $D_{3}(v)$ from the right, note that $D_3$ supercommutes with $E_1$ and $D_1$, and computing the bracket, we obtain
\begin{multline}\label{3ee9}
D_{1;i,p}(u)E_{1;p,j}(u)\big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big)\\
-(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_1+\pa{k}_3)} \big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big)D_{1;i,p}(u)E_{1;p,j}(u) =0
\end{multline}
where $p$ and $r$ are summed from 1 to $\mu_1$ and $\mu_2$, respectively. Substituting (\ref{3ee7}) into the second term of (\ref{3ee9}), we have
\begin{multline*}
D_{1;i,p}(u)E_{1;p,j}(u)\big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big)\\
-(-1)^{(\pa{p}_1+\pa{j}_2)(\pa{h}_1+\pa{k}_3)} \big\lbrace D_{1;i,p}(u)\big(E_{1;h,r}(v)E_{2;r,k}(v)-E_{1,3;h,k}(v)\big)\big\rbrace E_{1;p,j}(u) \\
-(-1)^{(\pa{g}_2+\pa{j}_2)(\pa{h}_1+\pa{k}_3)} \big\lbrace D_{1;i,p}(u)\big[E_{1;p,g}(u),E_{2;g,k}(v)\big] \big\rbrace E_{1;h,j}(u)=0.
\end{multline*}
Multiplying $D^{\prime}_{1}(u)$ from the left, we deduce that
\begin{multline*}
E_{1;i,j}(u)\big(E_{1;h,q}(v)E_{2;q,k}(v)-E_{1,3;h,k}(v)\big)\\
-(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{h}_1+\pa{k}_3)}\big(E_{1;h,q}(v)E_{2;q,k}(v)-E_{1,3;h,k}(v)\big)E_{1;i,j}(u)\\
=(-1)^{\pa{j}_2(\pa{h}_1+\pa{k}_3)+\pa{g}_2+\pa{h}_1\pa{k}_3}\big[E_{1;i,g}(u),E_{2;g,k}(v)\big]E_{1;h,j}(u).
\end{multline*}
Simplifying the above, we obtain (\ref{61d}).
\end{proof}
The following lemma can be deduced by applying the automorphism $\zeta_{N|M}$ to the corresponding identities of Lemma \ref{3be} in $Y_{\overleftarrow{\mu}}=Y_{(\mu_3,\mu_2,\mu_1)}=Y_{N|M}$.
\begin{lemma}\label{3bf}
The following identities hold in $Y_{(\mu_1,\mu_2,\mu_3)}((u^{-1},v^{-1}))$:
\begin{equation}\label{62a}
[F_{1;i,j}(u),E_{2;h,k}(v)]=0,
\end{equation}
\begin{multline}\label{62b}
[F_{1;i,j}(u),F_{2;h,k}(v)]=
\dfrac{(-1)^{\pa{i}_2\pa{j}_1+\pa{i}_2\pa{h}_3+\pa{j}_1\pa{h}_3}}{u-v}\delta_{ik}\times\\
\big\lbrace \sum_{q=1}^{\mu_2}F_{2;h,q}(v)\big(F_{1;q,j}(v)-F_{1;q,j}(u)\big)-F_{3,1;h,j}(v)+F_{3,1;h,j}(u)\big\rbrace,
\end{multline}
\begin{multline}\label{62c}
[F_{3,1;i,j}(u),F_{2;h,k}(v)]=
(-1)^{\pa{i}_3\pa{j}_1+\pa{i}_3\pa{h}_3+\pa{j}_1\pa{h}_3+\pa{g}_2+1}
[F_{2;h,g}(v),F_{1;g,j}(u)]F_{2;i,k}(v),
\end{multline}
\begin{multline}\label{62d}
\sum_{q=1}^{\mu_2}[F_{1;i,j}(u) , F_{2;h,q}(v)F_{1;q,k}(v)-F_{3,1;h,k}(v)]=\\
(-1)^{(\pa{h}_3+\pa{j}_1)(\pa{k}_1+\pa{g}_2)} F_{1;i,k}(u)[F_{1;g,j}(u),F_{2;h,g}(v)].
\end{multline}
Here (\ref{62a}) holds for all $1\leq i,h\leq \mu_2$, $1\leq j\leq \mu_1$, $1\leq k\leq \mu_3$, (\ref{62b}) holds for all $1\leq i,k\leq \mu_2$, $1\leq j\leq \mu_1$, $1\leq h\leq \mu_3$, (\ref{62c}) holds for all $1\leq i,h\leq \mu_3$, $1\leq j\leq \mu_1$, $1\leq j,k\leq \mu_2$, and (\ref{62d}) holds for all $1\leq i,g\leq \mu_2$, $1\leq j,k\leq \mu_1$, $1\leq h\leq \mu_3$.
\end{lemma}
Our next lemma is a generalization of \cite[Lemma 6.3]{Pe1}. It is surprising that there are no sign factors appearing in the resulting identities.
\begin{lemma}\label{3ef0}
The following identities hold in $Y_{(\mu_1,\mu_2,\mu_3)}[[u^{-1},v^{-1},w^{-1}]]$:
\begin{eqnarray}
\label{63a}&\big[[E_{1;i,j}(u),E_{2;h,k}(v)],E_{2;f,g}(v)\big]=0,\\[1mm]
\notag&\big[E_{1;i,j}(u),[E_{1;h,k}(u),E_{2;f,g}(v)]\big]=0,\\[1mm]
\label{63c}&\big[[E_{1;i,j}(u),E_{2;h,k}(v)],E_{2;f,g}(w)\big]+
\big[[E_{1;i,j}(u),E_{2;h,k}(w)],E_{2;f,g}(v)\big]=0,\\[1mm]
\notag&\big[E_{1;i,j}(u),[E_{1;h,k}(v),E_{2;f,g}(w)]\big]+
\big[E_{1;i,j}(v),[E_{1;h,k}(u),E_{2;f,g}(w)]\big]=0,\\[1mm]
\notag&\big[[F_{1;i,j}(u),F_{2;h,k}(v)],F_{2;f,g}(v)\big]=0,\\[1mm]
\notag&\big[F_{1;i,j}(u),[F_{1;h,k}(u),F_{2;f,g}(v)]\big]=0,\\[1mm]
\notag&\big[[F_{1;i,j}(u),F_{2;h,k}(v)],F_{2;f,g}(w)\big]+\big[[F_{1;i,j}(u),F_{2;h,k}(w)],F_{2;f,g}(v)\big]=0,\\[1mm]
\notag&\big[F_{1;i,j}(u),[F_{1;h,k}(v),F_{2;f,g}(w)]\big]+\big[F_{1;i,j}(v),[F_{1;h,k}(u),F_{2;f,g}(w)]\big]=0.
\end{eqnarray}
The identities hold for all $1\leq i\leq \mu_a, 1\leq j\leq \mu_{a+1}$ if $E_{a;i.j}(u)$ appears and
hold for all $1\leq h \leq \mu_{a+1}$, $1\leq k \leq \mu_a$ if $F_{a;h,k}(u)$ appears, where $a$=1 or 2.
\end{lemma}
\begin{proof}
We prove (\ref{63a}) and (\ref{63c}) in detail while the others are similar.
To show (\ref{63a}), we first claim that
\[
[E_{a;i,j}(v),E_{a;h,k}(v)]=0 \quad \text{for}\; a=1,2 \quad\text{in}\quad Y_{(\mu_1,\mu_2,\mu_3)}[[v^{-1}]].
\]
Indeed, the case $a=1$ follows from (\ref{p518}) and $a=2$ follows from applying the map $\psi_{\mu_1}$ to (\ref{p518}) in $Y_{(\mu_2,\mu_3)}[[v^{-1}]]$.
By the super Jacobi identity, together with the above claim and (\ref{61b}), it suffices to prove the case when $j=h=f$. In this case, we compute the following bracket by Lemma~\ref{3be} as below.
{\allowdisplaybreaks
\begin{align*}
(&u-v)\big[[E_{1;i,j}(u),E_{2;j,k}(v)],E_{2;j,g}(v)\big]\\
&=(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{j}_2+\pa{k}_3)}(u-v)\big[E_{2;j,k}(v),[E_{1;i,j}(u),E_{2;j,g}(v)]\big]\\
&=(-1)^{(\pa{i}_1+\pa{j}_2)(\pa{j}_2+\pa{k}_3)+(\pa{i}_1+\pa{g}_3)(\pa{j}_2+\pa{k}_3)}(u-v)\big[\,[E_{1;i,j}(u),E_{2;j,g}(v)] \,, E_{2;j,k}(v)\big]\\
&=(-1)^{(\pa{j}_2+\pa{g}_3)(\pa{j}_2+\pa{k}_3)}\big[(-1)^{\pa{j}_2} E_{1;i,q}(u) E_{2;q,g}(v)-E_{1;i,q}(v) E_{2;q,g}(v)\\
&\qquad\qquad + E_{1,3;i,g}(v) - E_{1,3;i,g}(u)\, , \,E_{2;j,k}(v)\big]\\
&=(-1)^{\pa{j}_2\pa{k}_3+\pa{j}_2\pa{g}_3+\pa{g}_3\pa{k}_3}\big\lbrace[E_{1;i,q}(u)E_{2;q,g}(v),E_{2;j,k}(v)]+[E_{1,3;i,g}(v),E_{2;j,k}(v)]\\
&\qquad -[E_{1;i,q}(v)E_{2;q,g}(v),E_{2;j,k}(v)]-[E_{1,3;i,g}(u),E_{2;j,k}(v)]\big\rbrace\\
&=(-1)^{\pa{j}_2\pa{k}_3+\pa{j}_2\pa{g}_3+\pa{g}_3\pa{k}_3}\big\lbrace
E_{1;i,q}(u)[E_{2;q,g}(v) , E_{2;j,k}(v)] \\
&\,+(-1)^{(\pa{q}_2+\pa{g}_3)(\pa{j}_2+\pa{k}_3)}[ E_{1;i,q}(u), E_{2;j,k}(v)] E_{2;q,g}(v)
-E_{1;i,q}(v)[E_{2;q,g}(v) , E_{2;j,k}(v)]\\
&\,-(-1)^{(\pa{q}_2+\pa{g}_3)(\pa{j}_2+\pa{k}_3)}[ E_{1;i,q}(v), E_{2;j,k}(v)] E_{2;q,g}(v)\\
&\,+(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{g}_3+\pa{j}_2\pa{g}_3+\pa{j}_2} E_{2;j,g}(v) [E_{1;i,j}(v) , E_{2;j,k}(v)]\\
&\,-(-1)^{\pa{i}_1\pa{j}_2+\pa{i}_1\pa{g}_3+\pa{j}_2\pa{g}_3+\pa{j}_2} E_{2;j,g}(v) [E_{1;i,j}(u) , E_{2;j,k}(v)]\big\rbrace\\
&=(-1)^{\pa{j}_2\pa{k}_3+\pa{q}_2\pa{j}_2+\pa{q}_2\pa{k}_3} [E_{1;i,q}(u), E_{2;j,k}(v)] E_{2;q,g}(v) \\
&\,+(-1)^{\pa{j}_2+\pa{i}_1\pa{j}_2+\pa{i}_1\pa{g}_3+\pa{j}_2\pa{k}_3+\pa{g}_3\pa{k}_3} E_{2;j,g}(v)[ E_{1;i,j}(v), E_{2;j,k}(v)]\\
&\,-(-1)^{\pa{j}_2\pa{k}_3+\pa{q}_2\pa{j}_2+\pa{q}_2\pa{k}_3} [E_{1;i,q}(u), E_{2;j,k}(v)] E_{2;q,g}(v) \\
&\,-(-1)^{\pa{j}_2+\pa{i}_1\pa{j}_2+\pa{i}_1\pa{g}_3+\pa{j}_2\pa{k}_3+\pa{g}_3\pa{k}_3} E_{2;j,g}(v)[ E_{1;i,j}(u), E_{2;j,k}(v)]\\
&=(-1)^{\pa{j}_2}\big[ [E_{1;i,j}(u) , E_{2;j,k}(v)] , E_{2;j,g}(v) \big]
-(-1)^{\pa{j}_2}\big[ [E_{1;i,j}(v) , E_{2;j,k}(v)] , E_{2;j,g}(v) \big]
\end{align*}
}
Thus we have
\begin{multline}\label{3eee0}
(u-v-(-1)^{\pa{j}_2})\big[[E_{1;i,j}(u),E_{2;j,k}(v)],E_{2;j,g}(v)\big]=\\
-(-1)^{\pa{j}_2}\big[[E_{1;i,j}(v),E_{2;j,k}(v)],E_{2;j,g}(v)\big]
\end{multline}
Note that the right-hand side of (\ref{3eee0}) is independent of the choice of $u$. Specifying $u=v+(-1)^{\pa{j}_2}$ in (\ref{3eee0}), we have
\begin{equation*}
0=-(-1)^{\pa{j}_2}\big[[E_{1;i,j}(v),E_{2;j,k}(v)],E_{2;j,g}(v)\big],
\end{equation*}
and hence
\begin{equation*}
(u-v-(-1)^{\pa{j}_2})\big[[E_{1;i,j}(u),E_{2;j,k}(v)],E_{2;j,g}(v)\big]=0, \,\,\text{for any } u.
\end{equation*}
Choose $u$ such that $u-v-(-1)^{\pa{j}_2}$ is invertible, and then (\ref{63a}) follows.
To show (\ref{63c}), it suffices to show that
\begin{equation}\label{3eeec1}
(u-w)(v-w)(u-v)\big[[E_{1;i,j}(u),E_{2;h,k}(v)],E_{2;f,g}(w)\big]
\end{equation}
is symmetric in $v$ and $w$. We may further assume $j=h$, as in the proof of (\ref{63a}).
By (\ref{61b}), we have
\begin{multline*}
(u-w)(v-w)(u-v)\big[[E_{1;i,j}(u),E_{2;h,k}(v)],E_{2;f,g}(w)\big]
=(v-w)(u-w)\times\\
(-1)^{\pa{j}_2}\big[ \big(E_{1;i,q}(u)-E_{1;i,q}(v) \big)E_{2;q,k}(v)+E_{1,3;i,k}(v)-E_{1,3;i,k}(u)\,,\,E_{2;f,g}(w)\big].
\end{multline*}
Computing the brackets by Lemma \ref{3be}, we have
{\allowdisplaybreaks
\begin{align*}\notag
(u&-w)(v-w)(u-v)\big[[E_{1;i,j}(u),E_{2;h,k}(v)],E_{2;f,g}(w)\big]\\
=&(v-w)(u-w)(-1)^{\pa{j}_2}\big[ [E_{1;i,q}(u) , E_{2;q,k}(v)] , E_{2;f,g}(w) \big]\\
&-(v-w)(u-w)(-1)^{\pa{j}_2}\big[ [E_{1;i,q}(v) , E_{2;q,k}(v)] , E_{2;f,g}(w) \big]\\
&+(v-w)(u-w)(-1)^{\pa{j}_2}\big[ E_{1,3;i,k}(v) , E_{2;f,g}(w) \big]\\
&-(v-w)(u-w)(-1)^{\pa{j}_2}\big[ E_{1,3;i,k}(u) , E_{2;f,g}(w) \big]\\
=&(v-w)(u-w)(-1)^{\pa{j}_2}\big[ [E_{1;i,q}(u) , E_{2;q,k}(v)] , E_{2;f,g}(w) \big]\\
&-(v-w)(u-w)(-1)^{\pa{j}_2}\big[ [E_{1;i,q}(v) , E_{2;q,k}(v)] , E_{2;f,g}(w) \big]\\
&+(v-w)(u-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{i}_1\pa{f}_2+\pa{i}_1\pa{k}_3+\pa{f}_2\pa{k}_3}
E_{2;f,k}(w)\big[ E_{1;i,\ell}(v) , E_{2;\ell,g}(w) \big]\\
&-(v-w)(u-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{i}_1\pa{f}_2+\pa{i}_1\pa{k}_3+\pa{f}_2\pa{k}_3}E_{2;f,k}(w)\big[ E_{1;i,\ell}(u) , E_{2;\ell,g}(w) \big]\\
=&(u-w)(v-w)(-1)^{\pa{j}_2}\big\lbrace E_{1;i,q}(u)E_{2;q,k}(v)E_{2;f,g}(w)\\
&\qquad\qquad\qquad-(-1)^{(\pa{f}_2+\pa{g}_3)(\pa{i}_1+\pa{k}_3)}E_{2;f,g}(w)E_{1;i,q}(u)E_{2;q,k}(v)\big\rbrace\\
&-(u-w)(v-w)(-1)^{\pa{j}_2}\big\lbrace E_{1;i,q}(u)E_{2;q,k}(v)E_{2;f,g}(w) \\
&\qquad\qquad\qquad-(-1)^{(\pa{f}_2+\pa{g}_3)(\pa{i}_1+\pa{k}_3)}E_{2;f,g}(w)E_{1;i,q}(u)E_{2;q,k}(v)\big\rbrace\\
&+(v-w)(u-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{i}_1\pa{f}_2+\pa{i}_1\pa{k}_3+\pa{f}_2\pa{k}_3}
E_{2;f,k}(w)\big[ E_{1;i,\ell}(v) , E_{2;\ell,g}(w) \big]\\
&-(v-w)(u-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{i}_1\pa{f}_2+\pa{i}_1\pa{k}_3+\pa{f}_2\pa{k}_3}E_{2;f,k}(w)\big[ E_{1;i,\ell}(u) , E_{2;\ell,g}(w) \big]\\
=&(u-w)(v-w)(-1)^{\pa{j}_2} E_{1;i,q}(u)\big[E_{2;q,k}(v),E_{2;f,g}(w)\big]\\
&-(u-w)(v-w)(-1)^{\pa{j}_2} E_{1;i,q}(v)\big[E_{2;q,k}(v),E_{2;f,g}(w)\big]\\
&-(u-w)(v-w)(-1)^{\pa{j}_2+(\pa{f}_2+\pa{g}_3)(\pa{i}_1+\pa{k}_3)} \big[E_{2;f,g}(w),E_{1;i,q}(u)\big]E_{2;q,k}(v)\\
&+(u-w)(v-w)(-1)^{\pa{j}_2+(\pa{f}_2+\pa{g}_3)(\pa{i}_1+\pa{k}_3)} \big[E_{2;f,g}(w),E_{1;i,q}(v)\big]E_{2;q,k}(v)\\
&+(u-w)(v-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{g}_3\pa{f}_2+\pa{g}_3\pa{k}_3+\pa{f}_2\pa{k}_3}\big[E_{1;i,\ell}(v),E_{2;\ell,g}(w)\big]E_{2;f,k}(w)\\
&-(u-w)(v-w)(-1)^{\pa{j}_2+\pa{\ell}_2+\pa{g}_3\pa{f}_2+\pa{g}_3\pa{k}_3+\pa{f}_2\pa{k}_3}\big[E_{1;i,\ell}(u),E_{2;\ell,g}(w)\big]E_{2;f,k}(w).
\end{align*}}
We use (\ref{p518}) and Lemma \ref{3be} to compute these brackets, then (\ref{3eeec1}) equals to
\begin{align*}
&\quad \varepsilon(u-w)E_{1;i,q}(u)\big(E_{2;q,g}(v)-E_{2;q,g}(w)\big)\big(E_{2;f,k}(v)-E_{2;f,k}(w)\big)\\
\,&-\varepsilon(u-w) E_{1;i,q}(v)\big(E_{2;q,g}(v)-E_{2;q,g}(w)\big)\big(E_{2;f,k}(v)-E_{2;f,k}(w)\big)\\
\,&+\varepsilon(v-w) \big\lbrace \big(E_{1;i,q}(u)-E_{1;i,q}(w)\big)E_{2;q,g}(w)+E_{1,3;i,g}(w)-E_{1,3;i,g}(u)\big\rbrace E_{2;f,k}(v)\\
\,&-\varepsilon(u-w) \big\lbrace \big(E_{1;i,q}(v)-E_{1;i,q}(w)\big)E_{2;q,g}(w)+E_{1,3;i,g}(w)-E_{1,3;i,g}(v)\big\rbrace E_{2;q,k}(v)\\
\,&+\varepsilon(u-w)\big\lbrace \big(E_{1;i,q}(v)-E_{1;i,q}(w)\big)E_{2;q,g}(w)+E_{1,3;i,g}(w)-E_{1,3;i,g}(v)\big\rbrace E_{2;f,k}(w)\\
\,&-\varepsilon(v-w)\big\lbrace \big(E_{1;i,q}(u)-E_{1;i,q}(w)\big)E_{2;q,g}(w)+E_{1,3;i,g}(w)-E_{1,3;i,g}(u)\big\rbrace E_{2;f,k}(w),
\end{align*}
where the $\varepsilon$ is a sign factor given by $\varepsilon=(-1)^{\pa{j}_2+\pa{g}_3\pa{f}_2+\pa{g}_3\pa{k}_3+\pa{f}_2\pa{k}_3}$, and the index $q$ is summed over $1,\ldots,\mu_2$.
Opening the parentheses of the above identity, one may check that the resulting expression is indeed symmetric in $v$ and $w$. Therefore, (\ref{3eeec1}) is symmetric in $v$ and $w$ and hence (\ref{63c}) is established.
\end{proof}
Suppose now that $\mu=(\mu_1,\ldots,\mu_n)$ with $n\geq 4$. The next lemma is a generalization of \cite[Lemma 5]{Go} and \cite[Lemma 7.2]{Pe1}, in which the results were proved only for one specific index.
Here we show that they in fact hold {\em everywhere} and we require some of them to obtain the desired defining relations; see (\ref{p715}), (\ref{p716}).
\begin{lemma}\label{extra}
Associated to $\mu=(\mu_1,\mu_2,\ldots\mu_n)$ with $n\geq 4$, we have the following identities in $Y_{\mu}$, called the super Serre relations:
\begin{equation}\label{eeee}
\big[\,[E_{a;i,f_1}^{(r)},E_{a+1;f_2,j}^{(1)}],[E_{a+1;h,g_1}^{(1)},E_{a+2;g_2,k}^{(s)}]\,\big]=0,
\end{equation}
\begin{equation}\label{ffff}
\big[\,[F_{a;f_1,i}^{(r)},F_{a+1;j,f_2}^{(1)}],[F_{a+1;g_1,h}^{(1)},F_{a+2;k,g_2}^{(s)}]\,\big]=0,\\[1mm]
\end{equation}
for all $1\leq a\leq n-3$ and all $1\leq i\leq \mu_a$, $1\leq f_1,f_2,h\leq \mu_{a+1}$, $1\leq g_1,g_2,j\leq \mu_{a+2}$, $1\leq k\leq \mu_{a+3}$.
\end{lemma}
\begin{proof}
It suffices to prove the following special case of (\ref{eeee}) when $n=4$, while the general cases and (\ref{ffff}) can be achieved by applying the maps $\psi$ and $\zeta_{M|N}$:
\begin{equation}\label{e4e}
\big[\,[E_{1;i,f_1}^{(r)},E_{2;f_2,j}^{(1)}]\,,\,[E_{2;h,g_1}^{(1)},E_{3;g_2,k}^{(s)}]\,\big]=0.
\end{equation}
We claim that for all $1\leq i\leq \mu_1$, $1\leq h\leq \mu_2$, $1\leq j\leq \mu_3$, $1\leq k\leq \mu_4$, we have\begin{equation}\label{exeee}
[E_{1,3;i,j}(u)\,,\,E_{2;h,q}(v)E_{3;q,k}(v)-E_{2,4;h,k}(v)\,]=0,
\end{equation}
where the index $q$ is summed over $1,\ldots,\mu_3$.
To prove (\ref{exeee}), we multiply the matrix equalities (\ref{T=FDE}) and (\ref{Tp=FDE}) associated to the composition $(\mu_1,\mu_2, \mu_3,\mu_4)$ and derive the following identities.
\begin{eqnarray}
E_{1,3;i,j}(u)&=&D^{\prime}_{1;i,p}(u)t_{p,\mu_1+\mu_2+j}(u),\notag\\
E_{2;h,q}(v)E_{3;q,k}(v)-E_{2,4;h,k}(v)&=&t^{\prime}_{\mu_1+h,\mu_1+\mu_2+\mu_3+r}(v)D_{4;r,k}(v)\notag,
\end{eqnarray}
for all $1\le i\le\mu_1$, $1\le j\le\mu_3$, $1\le h\le\mu_2$, $1\le k\le\mu_4$, and the indices $p$, $q$, $r$ are summed from 1 to $\mu_1$, $\mu_3$, $\mu_4$, respectively. Substituting these identities into the bracket in (\ref{exeee}) and setting a notation $n_a:=\mu_1+\mu_2+\ldots+\mu_a$ for short, we have
\begin{align*}
&[E_{1,3;i,j}(u),E_{2;h,q}(v)E_{3;q,k}(v)-E_{2,4;h,k}(v)]\\
&=[D^{\prime}_{1;i,p}(u)t_{p,n_2+j}(u),t^{\prime}_{\mu_1+h,n_3+r}(v)D_{4;r,k}(v)]\\
&=D^{\prime}_{1;i,p}(u)t_{p,n_2+j}(u)t^{\prime}_{\mu_1+h,n_3+r}(v)D_{4;r,k}(v)\\
&\qquad\qquad\qquad-(-1)^{(\pa{i}_1+\pa{j}_3)(\pa{h}_2+\pa{k}_4)}t^{\prime}_{\mu_1+h,n_3+r}(v)D_{4;r,k}(v)D^{\prime}_{1;i,p}(u)t_{p,n_2+j}(u)\\
&=D^{\prime}_{1;i,p}(u)t_{p,n_2+j}(u)t^{\prime}_{\mu_1+h,n_3+r}(v)D_{4;r,k}(v)\\
&\qquad\qquad\qquad-(-1)^{(\pa{h}_2+\pa{r}_4)(\pa{p}_1+\pa{j}_3)}D^{\prime}_{1;i,p}(u)t^{\prime}_{\mu_1+h,n_3+r}(v)t_{p,n_2+j}(u)D_{4;r,k}(v)\\
&=D^{\prime}_{1;i,p}(u)[t_{p,n_2+j}(u),
t^{\prime}_{\mu_1+h,n_3+r}(v)]D_{4;r,k}(v)=0,
\end{align*}
and (\ref{exeee}) follows.
Note that in the above computation we have used the facts that
\[
D_{1;i,j}(u)=t_{ij}(u) \qquad \text{and} \qquad D^{\prime}_{4;i,j}(u)=t^{\prime}_{n_3+i,n_3+j}(u),
\]
therefore
$[D_{1;i,j}(u),t^{\prime}_{\mu_1+h,n_3+k}(v)]=0$
and $[D^{\prime}_{4;i,j}(u),t_{h,n_2+k}(v)]=0$ \;by (\ref{usefull}).
To show (\ref{e4e}), by (\ref{61b}), we may assume that $f_1=f_2=f$ and $g_1=g_2=g$.
Computing the following bracket by (\ref{61b}), we have
\begin{align*}
(u&-v)(w-z)\big[\,[E_{1;i,f}(u),E_{2;f,j}(v)]\,,\,[E_{2;h,g}(w),E_{3;g,k}(z)]\,\big]\notag\\
&=\big[\,(-1)^{\pa{f}_2}E_{1;i,q}(u)E_{2;q,k}(v)-E_{1;i,q}(v)E_{2;q,k}(v)+E_{1,3;i,k}(v)-E_{1,3;i,k}(u),\notag\\
&\qquad (-1)^{\pa{g}_3}E_{2;h,p}(w)E_{3;p,k}(z)-E_{2;h,p}(z)E_{3;p,k}(z)+E_{2,4;h,k}(z)-E_{2,4;h,k}(w)\,\big].\notag
\end{align*}
Taking its coefficient of $u^{-r}z^{-s}v^0w^0$, we have
\[
(-1)^{\pa{f}_2+\pa{g}_3}\sum_{t=1}^{s-1}[E_{1,3;i,j}^{(r)}\,,\,E_{2;h,p}^{(s-t)}E_{3;p,k}^{(t)}-E_{2,4;h,k}^{(s)}],
\]
which equals to the coefficient of $u^{-r}z^{-s}$ in
\[
(-1)^{\pa{f}_2+\pa{g}_3}[E_{1,3;i,j}(u)\,,\,E_{2;h,p}(z)E_{3;p,k}(z)-E_{2,4;h,k}(z)],
\]
which is zero by (\ref{exeee}).
Finally, the coefficient of $u^{-r}z^{-s}v^0w^0$ in
\begin{equation*}
(u-v)(w-z)\big[\,[E_{1;i,f}(u),E_{2;f,j}(v)]\,,\,[E_{2;h,g}(w),E_{3;g,k}(z)]\,\big]
\end{equation*}
is precisely $-\big[\,[E_{1;i,f}^{(r)},E_{2;f,j}^{(1)}]\,,\,[E_{2;h,g}^{(1)},E_{3;g,k}^{(s)}]\,\big]$ and (\ref{e4e}) follows.
\end{proof}
\section{The general Case}
Recall that our goal is to obtain the defining relations of $Y_\mu(\mathfrak{s})=Y_{M|N}$ in terms of the parabolic generators $\lbrace D_{a;i,j}^{(r)}, D_{a;i,j}^{\prime'(r)}\rbrace$, $\lbrace E_{a;i,j}^{(r)}\rbrace$, and $\lbrace F_{a;i,j}^{(r)}\rbrace$ associated to an arbitrary fixed composition $\mu$ of $M+N$ and an arbitrary fixed $0^M1^N$-sequence $\mathfrak{s}$. The following proposition summarizes the results that we have established earlier.
\begin{proposition}\label{srlns}
The following relations hold in $Y_\mu(\mathfrak{s})$:
\begin{eqnarray}
\label{p701}D_{a;i,j}^{(0)}&=&\delta_{ij}\,,\\
\label{p702}\sum_{p=1}^{\mu_a}\sum_{t=0}^{r}D_{a;i,p}^{(t)}D_{a;p,j}^{\prime (r-t)}&=&\delta_{r0}\delta_{ij}\,,\\
\big[D_{a;i,j}^{(r)},D_{b;h,k}^{(s)}\big]&=&
\delta_{ab}(-1)^{\pa{i}_a\pa{j}_a+\pa{i}_a\pa{h}_a+\pa{j}_a\pa{h}_a}\times \notag\\
&&\sum_{t=0}^{min(r,s)-1}\big(D_{a;h,j}^{(t)}D_{a;i,k}^{(r+s-1-t)}-D_{a;h,j}^{(r+s-1-t)}D_{a;i,k}^{(t)}\big),
\end{eqnarray}
{\allowdisplaybreaks
\begin{multline}\label{p704}
[D_{a;i,j}^{(r)}, E_{b;h,k}^{(s)}]
=\delta_{a,b}\delta_{hj}(-1)^{\pa{h}_a\pa{j}_a}\sum_{p=1}^{\mu_a}\sum_{t=0}^{r-1} D_{a;i,p}^{(t)} E_{b;p,k}^{(r+s-1-t)}\\
-\delta_{a,b+1}(-1)^{\pa{h}_b\pa{k}_a+\pa{h}_b\pa{j}_a+\pa{j}_a\pa{k}_a} \sum_{t=0}^{r-1} D_{a;i,k}^{(t)} E_{b;h,j}^{(r+s-1-t)},
\end{multline}
\begin{multline}\label{p705}
[D_{a;i,j}^{(r)}, F_{b;h,k}^{(s)}]
=\delta_{a,b}(-1)^{\pa{i}_a\pa{j}_a+\pa{h}_{a+1}\pa{i}_a+\pa{h}_{a+1}\pa{j}_a}\sum_{p=1}^{\mu_a}\sum_{t=0}^{r-1} F_{b;h,p}^{(r+s-1-t)}D_{a;p,j}^{(t)}\\
+\delta_{a,b+1}(-1)^{\pa{h}_a\pa{k}_b+\pa{h}_a\pa{j}_a+\pa{j}_a\pa{k}_b} \sum_{t=0}^{r-1} F_{b;i,k}^{(r+s-1-t)}D_{a;h,j}^{(t)},
\end{multline}
\begin{multline}\label{p706}
[E_{a;i,j}^{(r)} , F_{b;h,k}^{(s)}]
=\delta_{a,b}(-1)^{\pa{h}_{a+1}\pa{k}_a+\pa{j}_{a+1}\pa{k}_a+\pa{h}_{a+1}\pa{j}_{a+1}+1}
\sum_{t=0}^{r+s-1} D_{a;i,k}^{\prime (r+s-1-t)} D_{a+1;h,j}^{(t)},
\end{multline}
\begin{multline}\label{p707}
[E_{a;i,j}^{(r)} , E_{a;h,k}^{(s)}]
=(-1)^{\pa{h}_{a}\pa{j}_{a+1}+\pa{j}_{a+1}\pa{k}_{a+1}+\pa{h}_{a}\pa{k}_{a+1}}\times\\
\big( \sum_{t=1}^{s-1} E_{a;i,k}^{(r+s-1-t)} E_{a;h,j}^{(t)}
-\sum_{t=1}^{r-1} E_{a;i,k}^{(r+s-1-t)} E_{a;h,j}^{(t)} \big),
\end{multline}
\begin{multline}\label{p708}
[F_{a;i,j}^{(r)} , F_{a;h,k}^{(s)}]
=(-1)^{\pa{h}_{a+1}\pa{j}_{a}+\pa{j}_{a}\pa{k}_{a}+\pa{h}_{a+1}\pa{k}_{a}}\times\\
\big( \sum_{t=1}^{r-1} F_{a;i,k}^{(r+s-1-t)} F_{a;h,j}^{(t)}
-\sum_{t=1}^{s-1} F_{a;i,k}^{(r+s-1-t)} F_{a;h,j}^{(t)} \big),
\end{multline}
\begin{equation}
\label{p709}[E_{a;i,j}^{(r+1)}, E_{a+1;h,k}^{(s)}]-[E_{a;i,j}^{(r)}, E_{a+1;h,k}^{(s+1)}]
=(-1)^{\pa{j}_{a+1}\pa{h}_{a+1}}\delta_{h,j}\sum_{q=1}^{\mu_{a+1}}E_{a;i,q}^{(r)}E_{a+1;q,k}^{(s)}\,,
\end{equation}
\begin{multline}
\label{p710}[F_{a;i,j}^{(r+1)}, F_{a+1;h,k}^{(s)}]-[F_{a;i,j}^{(r)}, F_{a+1;h,k}^{(s+1)}]=\\
(-1)^{\pa{i}_{a+1}(\pa{j}_{a}+\pa{h}_{a+2})+\pa{j}_a\pa{h}_{a+2}+1}\delta_{i,k}\sum_{q=1}^{\mu_{a+1}}F_{a+1;h,q}^{(s)}F_{a;q,j}^{(r)}\,,
\end{multline}
\begin{align}
\label{p711}&[E_{a;i,j}^{(r)}, E_{b;h,k}^{(s)}] = 0
\qquad\qquad\text{\;\;if\;\; $|b-a|>1$ \;\;or\;\; \;if\;\;$b=a+1$ and $h \neq j$},\\[3mm]
\label{p712}&[F_{a;i,j}^{(r)}, F_{b;h,k}^{(s)}] = 0
\qquad\qquad\text{\;\;if\;\; $|b-a|>1$ \;\;or\;\; \;if\;\;$b=a+1$ and $i \neq k$},\\[3mm]
\label{p713}&\big[E_{a;i,j}^{(r)},[E_{a;h,k}^{(s)},E_{b;f,g}^{(\ell)}]\big]+
\big[E_{a;i,j}^{(s)},[E_{a;h,k}^{(r)},E_{b;f,g}^{(\ell)}]\big]=0 \quad \text{if}\,\,\, |a-b|\geq 1,\\[3mm]
\label{p714}&\big[F_{a;i,j}^{(r)},[F_{a;h,k}^{(s)},F_{b;f,g}^{(\ell)}]\big]+
\big[F_{a;i,j}^{(s)},[F_{a;h,k}^{(r)},F_{b;f,g}^{(\ell)}]\big]=0 \quad \text{if}\,\,\, |a-b|\geq 1,\\[3mm]
\label{p715}&\big[\,[E_{a;i,f_1}^{(r)},E_{a+1;f_2,j}^{(1)}]\,,\,[E_{a+1;h,g_1}^{(1)},E_{a+2;g_2,k}^{(s)}]\,\big]=0 \;\;\text{when\;\;} n\geq 4\, \text{and\;\;} \pa{h}_{a+1}+\pa{j}_{a+2}=1,\\[3mm]
\label{p716}&\big[\,[F_{a;i,f_1}^{(r)},F_{a+1;f_2,j}^{(1)}]\,,\,[F_{a+1;h,g_1}^{(1)},F_{a+2;g_2,k}^{(s)}]\,\big]=0 \;\;\text{when\;\;} n\geq 4\, \text{and\;\;} \pa{j}_{a+1}+\pa{h}_{a+2}=1.
\end{align}}
If $D_{a;i,j}^{(r)}$ appears on the left-hand side of the equation, then it holds for all $1\leq i,j\leq \mu_{a}$ and all $r\geq 0$; if $E_{a;h,k}^{(s)}$ appears on the left-hand side of the equation, then it holds for all $1\leq h\leq \mu_a, 1\leq k\leq \mu_{a+1}$ and all $s\geq 1$; if $F_{a;f,g}^{(\ell)}$ appears on the left-hand side of the equation, then it holds for all $1\leq g\leq \mu_a, 1\leq f\leq \mu_{a+1}$ and all $\ell\geq 1$.
\end{proposition}
\begin{proof}
The first three relations follow from Proposition \ref{dd0}. By Proposition~\ref{n=2} and Lemma \ref{3be}--Lemma \ref{extra}, one can show that the identities hold in smaller Yangians, for example, $Y_{(\mu_2,\mu_3)}$. Then we apply the injective homomorphisms $\psi_{p_1|q_1}=\psi_{\mu_1}$ to these identities so that the corresponding identities also hold in bigger Yangians, for example, $Y_{(\mu_1,\mu_2,\mu_3)}$. Repeating this process and we may eventually deduce that all these relations, in series forms, hold in $Y_{\mu}$.
Finally, the following identity converts the relations from series form into the desired form:
\[
\dfrac{S(v)-S(u)}{u-v}=\sum_{r,s\ge 1}S^{(r+s-1)}u^{-r}v^{-s},
\]
for any formal series $S(u)=\sum_{r\ge 0}S^{(r)}u^{-r}$.
\end{proof}
Our main theorem is that the above relations are enough for a set of defining relations of $Y_\mu(\mathfrak{s})$.
\begin{theorem}\label{Pg}
Let $\mu=(\mu_1,\ldots,\mu_n)$ be a composition of $M+N$ and $\mathfrak{s}$ be a $0^M1^N$-sequence. Associated to this $\mu$ and $\mathfrak{s}$, the super Yangian $Y_\mu(\mathfrak{s})$ is generated by the parabolic generators
\begin{align*}
&\lbrace D_{a;i,j}^{(r)}, D_{a;i,j}^{\prime(r)} \,|\, 1\leq a\leq n, 1\leq i,j\leq \mu_a, r\geq 0\rbrace,\\
&\lbrace E_{a;i,j}^{(r)} \,|\, 1\leq a< n, 1\leq i\leq \mu_a, 1\leq j\leq\mu_{a+1}, r\geq 1\rbrace,\\
&\lbrace F_{a;i,j}^{(r)} \,|\, 1\leq a< n, 1\leq i\leq\mu_{a+1}, 1\leq j\leq \mu_a, r\geq 1\rbrace,
\end{align*}
subject only to the relations (\ref{p701})$-$(\ref{p716}).
\end{theorem}
The remaining part of this article is devoted to the proof of our main theorem, which is built on several technical propositions and lemmas.
Let $\widehat{Y}_{\mu}$ denote the abstract superalgebra generated by the elements and relations as in the statement of Theorem~\ref{Pg}, where the parities of the generator are given explicitly by (\ref{pad})-(\ref{paf}). We may further define all the other $E_{a,b;i,j}^{(r)}$ and $F_{b,a;i,j}^{(r)}$ in $\widehat{Y}_{\mu}$ by the relations (\ref{ter}), and it is straightforward to check that these definitions are independent of the choices of $k$ as in \cite[p.22]{BK1}. Let $\Gamma$ be the map
\[
\Gamma: \widehat{Y}_{\mu}\longrightarrow Y_{\mu}
\]
sending every element in $\widehat{Y}_{\mu}$ into the element in $Y_{\mu}$ with the same notation. By Theorem~\ref{gendef} and Proposition~\ref{srlns}, the map $\Gamma$ is a surjective superalgebra homomorphism. Therefore, it remains to prove that $\Gamma$ is injective.
The injectivity of $\Gamma$ is proved similar to the arguments in \cite{BK1, Go, Pe1}. We first find a spanning set for $\widehat{Y}_{\mu}$, and then show that the image of this spanning set under $\Gamma$ is linearly independent in $Y_{\mu}$.
Let $\widehat{Y}^0_{\mu}$ (respectively, $\widehat{Y}^+_{\mu}$, $\widehat{Y}^-_{\mu}$) denote the subalgebras of
$\widehat{Y}_{\mu}$ generated by the elements $\lbrace D_{a;i,j}^{(r)}\rbrace$ (respectively, $\lbrace E_{a,b;i,j}^{(r)}\rbrace$, $\lbrace F_{b,a;i,j}^{(r)}\rbrace$).
Define a filtration on $\widehat{Y}_{\mu}$ (on $\widehat{Y}^0_{\mu}$, $\widehat{Y}^+_{\mu}$ and $\widehat{Y}_{\mu}^-$ as well) by setting
\begin{equation*}
\text{deg}(D_{a;i,j}^{(r)})=\text{deg}(E_{a,b;i,j}^{(r)})=\text{deg}(F_{b,a;i,j}^{(r)})=r-1,\qquad\text{for all}\;\; r\ge 1,
\end{equation*}
and denote the associated graded superalgebra by $\operatorname{gr}\widehat{Y}_{\mu}$.
Let $\ovl{E}_{a,b;i,j}^{(r)}$ denote the image of $E_{a,b;i,j}^{(r)}$ in the graded superalgebra $\operatorname{gr}_{r-1}\widehat{Y}_{\mu}^+$.
\begin{lemma} The following identities hold in $\operatorname{gr}\widehat{Y}_{\mu}^+$ for all $r,s,t\geq 1$:
\begin{enumerate}
\item[(a)] \begin{equation}\label{L717}
[\ovl{E}_{a,a+1;i,j}^{(r)},\ovl{E}_{b,b+1;h,k}^{(s)}]=0,\;\text{if}\;|a-b|\ne 1,
\end{equation}
\item[(b)] \begin{equation}\label{L718}
[\ovl{E}_{a,a+1;i,j}^{(r+1)},\ovl{E}_{b,b+1;h,k}^{(s)}]=
[\ovl{E}_{a,a+1;i,j}^{(r)},\ovl{E}_{b,b+1;h,k}^{(s+1)}],\; \text{if}\; |a-b|=1,
\end{equation}
\item[(c)] \begin{equation}\label{L719}
\big[\ovl{E}_{a,a+1;i,j}^{(r)},[\ovl{E}_{a,a+1;h,k}^{(s)},\ovl{E}_{b,b+1;f,g}^{(t)}]\big]=
-\big[\ovl{E}_{a,a+1;i,j}^{(s)},[\ovl{E}_{a,a+1;h,k}^{(r)},\ovl{E}_{b,b+1;f,g}^{(t)}]\big],
\end{equation}
\; \text{if}\; $|a-b|=1$,
\item[(d)] \begin{multline}\label{L720}
\ovl{E}_{a,b;i,j}^{(r)}=(-1)^{\pa{h}_{b-1}}[\ovl{E}_{a,b-1;i,h}^{(r)},\ovl{E}_{b-1,b;h,j}^{(1)}]
=(-1)^{\pa{k}_{a+1}}[\ovl{E}_{a,a+1;i,k}^{(1)},\ovl{E}_{a+1,b;k,j}^{(r)}],
\end{multline}
for all $b>a+1$ and any $1\leq h\leq\mu_{b-1}$, $1\leq k\leq\mu_{a+1}$.
\end{enumerate}
Here (\ref{L717}) and (\ref{L718}) hold for all $1\leq i\leq \mu_a$, $1\leq j\leq \mu_{a+1}$, $1\leq h\leq \mu_b$, $1\leq k\leq \mu_{b+1}$; (\ref{L719}) holds for all $1\leq i,h\leq \mu_a$, $1\leq j,k\leq \mu_{a+1}$, $1\leq f\leq \mu_b$, $1\leq g\leq \mu_{b+1}$; (\ref{L720}) holds for all $1\leq i\leq \mu_a$, $1\leq j\leq \mu_b$.
\end{lemma}
\begin{proof}
(\ref{L717}) and (\ref{L718}) follow from (\ref{p711}) and (\ref{p709}), while (\ref{L719}) follows from (\ref{p713}). The first equality of (\ref{L720}) follows from (\ref{ter}), while the second one can be deduced from the first equality by super Jacobi identity, (\ref{L718}) and induction on $b-a$.
\end{proof}
\begin{lemma}
The following identities hold in $\operatorname{gr}\widehat{Y}_{\mu}^+$ for all $r,s\geq 1$:
\begin{enumerate}
\item[(a)]
\begin{equation}\label{LL725}
[\ovl{E}_{a,a+2;i,j}^{(r)},\ovl{E}_{a+1,a+2;h,k}^{(s)}]=0,\;\; \text{for all}\;\; 1\leq a\leq n-2,
\end{equation}
\item[(b)]
\begin{equation}\label{LL726}
[\ovl{E}_{a,a+1;i,j}^{(r)},\ovl{E}_{a,a+2;h,k}^{(s)}]=0,\;\; \text{for all}\;\; 1\leq a\leq n-2,
\end{equation}
\item[(c)]
\begin{equation}\label{L723}
[\ovl{E}_{a,a+2;i,j}^{(r)},\ovl{E}_{a+1,a+3;h,k}^{(s)}]=0,\;\; \text{for all}\;\; 1\leq a\leq n-3,
\end{equation}
\item[(d)]
\begin{equation}\label{L724}
[\ovl{E}_{a,b;i,j}^{(r)},\ovl{E}_{c,c+1;h,k}^{(s)}]=0,\;\; \text{for all}\;\; 1\leq a<c<b\leq n.
\end{equation}
\end{enumerate}
Here (\ref{LL725}) holds for all $1\leq i\leq \mu_a$, $1\leq h\leq \mu_{a+1}$, $1\leq j,k\leq \mu_{a+2}$; (\ref{LL726}) holds for all $1\leq i,h\leq \mu_a$, $1\leq j\leq \mu_{a+1}$, $1\leq j,k\leq \mu_{a+2}$; (\ref{L723}) holds for all $1\leq i\leq \mu_a$, $1\leq h\leq \mu_{a+1}$, $1\leq j\leq \mu_{a+2}$, $1\leq k\leq \mu_{a+3}$; (\ref{L724}) holds for all $1\leq i\leq \mu_a$, $1\leq j\leq \mu_b$, $1\leq h\leq \mu_c$, $1\leq k\leq \mu_{c+1}$.
\end{lemma}
\begin{proof}
Similar to the proof in \cite[Lemma 8.3]{Pe1} so we only show (c) in detail here since it is the place that we actually use the super Serre relations.
Assume first that $\pa{h}_{a+1}+\pa{j}_{a+2}=0$.
Applying (\ref{L720}) on the left-hand side of (\ref{L723}) and using the super Jacobi identity, we have
\begin{align*}
[\ovl{E}_{a,a+2;i,j}^{(r)}&,\ovl{E}_{a+1,a+3;h,k}^{(s)}]=\\
&(-1)^{\pa{h}_{a+1}+\pa{j}_{a+2}}\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}]
\,,\,[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big]\\
&=(-1)^{\pa{h}_{a+1}+\pa{j}_{a+2}}\Big\lbrace\Big[\,\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}],
\ovl{E}_{a+1,a+2;h,j}^{(1)}\big],\ovl{E}_{a+2,a+3;j,k}^{(s)}\Big]\\
&+\varepsilon\Big[\ovl{E}_{a+1,a+2;h,j}^{(1)},
\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}],\ovl{E}_{a+2,a+3;j,k}^{(s)}\big]\,\Big]\Big\rbrace,
\end{align*}
where $\varepsilon=(-1)^{(\pa{i}_{a}+\pa{h}_{a+1})(\pa{h}_{a+1}+\pa{j}_{a+2})}$.
By (\ref{L719}), the first term is zero. Using the super Jacobi identity, (\ref{L717}) and (\ref{L720}) again, we may deduce that the above equals to
\begin{align*}
&\varepsilon\Big[\ovl{E}_{a+1,a+2;h,j}^{(1)},
\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}],\ovl{E}_{a+2,a+3;j,k}^{(s)}\big]\,\Big]\\
=&\varepsilon\Big[\ovl{E}_{a+1,a+2;h,j}^{(1)},
\big[\ovl{E}_{a,a+1;i,h}^{(r)},[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big]\,\Big]+0\\
=&\varepsilon\big[\,[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a,a+1;i,h}^{(r)}],
[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big]+0\\
=&(-1)\varepsilon^2\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}]
,[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big]\\
=&(-1)^{1+\pa{h}_{a+1}+\pa{j}_{a+2}}[\ovl{E}_{a,a+2;i,j}^{(r)},\ovl{E}_{a+1,a+3;h,k}^{(s)}].
\end{align*}
By our assumption, $\pa{h}_{a+1}+\pa{j}_{a+2}=0$ and we have done.
Assume on the other hand that $\pa{h}_{a+1}+\pa{j}_{a+2}=1$. Similarly, we apply (\ref{L720}) on the left-hand side of (\ref{L723}) to obtain
\begin{align*}
[\ovl{E}_{a,a+2;i,j}^{(r)}&,\ovl{E}_{a+1,a+3;h,k}^{(s)}]=\\
&(-1)^{\pa{h}_{a+1}+\pa{j}_{a+2}}\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}]
\,,\,[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big]\\
=&-\big[\,[\ovl{E}_{a,a+1;i,h}^{(r)},\ovl{E}_{a+1,a+2;h,j}^{(1)}]
\,,\,[\ovl{E}_{a+1,a+2;h,j}^{(1)},\ovl{E}_{a+2,a+3;j,k}^{(s)}]\,\big],
\end{align*}
which is zero directly by (\ref{p715}).
\end{proof}
The following lemma, generalizing \cite[Lemma 6.7]{BK1} and \cite[(8.1)]{Pe1}, plays a crucial role in the proof of Theorem~\ref{Pg}.
\begin{lemma}\label{injeq}
For all $1\leq a\leq b\leq n$, $1\leq c\leq d\leq n$, $r,s\geq 0$ and
all $1\leq i\leq \mu_a$, $1\leq j\leq \mu_b$, $1\leq h\leq \mu_c$, $1\leq k\leq\mu_d$, we have
\begin{multline*}
[\ovl{E}_{a,b;i,j}^{(r)},\ovl{E}_{c,d;h,k}^{(s)}]=(-1)^{\pa{j}_b\pa{h}_c}\delta_{b,c}\delta_{h,j}\ovl{E}_{a,d;i,k}^{(r+s-1)}\\
-(-1)^{\pa{i}_a\pa{j}_b+\pa{i}_a\pa{h}_c+\pa{j}_b\pa{h}_c}\delta_{a,d}\delta_{i,k}\ovl{E}_{c,b;h,j}^{(r+s-1)}.
\end{multline*}
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that $a\leq c$. The proof is divided into 7 cases and we discuss them one by one.
\begin{description}
\item[Case 1.] $a<b<c<d$:\\
It follows directly from (\ref{L717}) and (\ref{L720}) that the bracket in Lemma \ref{injeq} is zero.
\item[Case 2.] $a<b=c<d$:\\
By (\ref{L718}) and (\ref{L720}), we have
\begin{equation}\label{L725}
[\ovl{E}^{(r+1)}_{b-1,b;i_1,j}\,,\,\ovl{E}^{(s+1)}_{b,b+1;h,k_1}]
=[\ovl{E}^{(r+s+1)}_{b-1,b;i_1,j}\,,\,\ovl{E}^{(1)}_{b,b+1;h,k_1}]
=\delta_{h,j}(-1)^{\pa{h}_b}\ovl{E}^{(r+s+1)}_{b-1,b+1;i_1,k_1}.
\end{equation}
Note that when $h\neq j$, the bracket is zero by (\ref{61b}) and hence the term $\delta_{h,j}$ shows up.
Taking brackets on both sides of (\ref{L725}) with the elements
\[
\ovl{E}^{(1)}_{b+1,b+2;k_1,k_2}, \ovl{E}^{(1)}_{b+2,b+3;k_2,k_3}, \cdots , \ovl{E}^{(1)}_{d-1,d;k_{d-b+1},k}
\]
from the right then using (\ref{L717}), (\ref{L720}) and the super Jacobi identity, we deduce that
\begin{equation}\label{L726}
[\ovl{E}^{(r+1)}_{b-1,b;i_1,j}\,,\,\ovl{E}^{(s+1)}_{b,d;h,k}]=\delta_{h,j}(-1)^{\pa{h}_b}\ovl{E}^{(r+s+1)}_{b-1,d;i_1,k}\,.
\end{equation}
Taking brackets on both sides of (\ref{L726}) with the elements
\[
\ovl{E}^{(1)}_{b-2,b-1;i_2,i_1}, \ovl{E}^{(1)}_{b-3,b-2;i_3,i_2},\cdots,\ovl{E}^{(1)}_{a,a+1;i,i_{b-a-1}}
\]
from the left and using exactly the same method as above, we have
\[
[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{b,d;h,k}]=\delta_{h,j}(-1)^{\pa{h}_b}\ovl{E}^{(r+s-1)}_{a,d;i,k},
\;\text{as desired}.
\]
\item[Case 3.] $a<c<b=d$:\\
Using the super Jacobi identity together with (\ref{L720}) and (\ref{L724}), we have
\begin{align*}
[\ovl{E}^{(r)}_{a,b;i,j}&,\ovl{E}^{(s)}_{c,b;h,k}]
=\big[\ovl{E}^{(r)}_{a,b;i,j},(-1)^{\pa{f_1}_{c+1}}[\ovl{E}^{(1)}_{c,c+1;h,f_1},\ovl{E}^{(s)}_{c+1,b;f_1,k}]\,\big]\\
&=(-1)^{\pa{f_1}_{c+1}}\big[\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{c,c+1;h,f_1}],\ovl{E}^{(s)}_{c+1,b;f_1,k}\big]\\
&\qquad\qquad\qquad\pm(-1)^{\pa{f_1}_{c+1}}\big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c+1,b;f_1,k}]\,\big]\\
&=0\pm (-1)^{\pa{f_1}_{c+1}}\big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c+1,b;f_1,k}]\,\big]\\
&=\cdots =\pm \Big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,[\ovl{E}^{(1)}_{c+1,c+2;f_1,f_2},\ldots,
[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{b-1,b;f_{b-1-c},k}]\,\big]\cdots\Big].
\end{align*}
The bracket $[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{b-1,b;f_{b-1-c},k}]$ in the middle is zero by (\ref{L724}).
\item[Case 4.] $a<c<d<b$:\\
Using the same technique as in Case 3, we have
\begin{align*}
[\ovl{E}^{(r)}_{a,b;i,j}&,\ovl{E}^{(s)}_{c,d;h,k}]
=\big[\ovl{E}^{(r)}_{a,b;i,j},(-1)^{\pa{f_1}_{c+1}}[\ovl{E}^{(1)}_{c,c+1;h,f_1},\ovl{E}^{(s)}_{c+1,d;f_1,k}]\,\big]\\
&=(-1)^{\pa{f_1}_{c+1}}\big[\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{c,c+1;h,f_1}],\ovl{E}^{(s)}_{c+1,d;f_1,k}]\,\big]\\
&\quad\pm(-1)^{\pa{f_1}_{c+1}}\big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c+1,d;f_1,k}]\,\big]\\
&=0\pm(-1)^{\pa{f_1}_{c+1}}\big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c+1,d;f_1,k}]\,\big]\\
&=\cdots =\pm \Big[\ovl{E}^{(1)}_{c,c+1;h,f_1}\,,\,\big[\ovl{E}^{(1)}_{c+1,c+2;f_1,f_2}\,,\ldots,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{d-1,d;f_{d-1-c},k}]\,\big]\cdots\Big].
\end{align*}
Following from (\ref{L724}) again, the bracket $[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{d-1,d;f_{d-1-c},k}]$ vanishes.
\item[Case 5.] $a<c<b<d$:\\
We prove this case by induction on $d-b\geq 1$. When $d-b=1$, we have
\begin{multline*}
[\ovl{E}^{(r)}_{a,b;i,j},\ovl{E}^{(s)}_{c,b+1;h,k}]
=\big[\ovl{E}^{(r)}_{a,b;i,j},(-1)^{\pa{j}_{b+1}}[\ovl{E}^{(s)}_{c,b;h,j},\ovl{E}^{(1)}_{b,b+1;j,k}]\,\big]\\
=(-1)^{\pa{j}_{b+1}}\big[\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c,b;h,j}]\,,\,\ovl{E}^{(1)}_{b,b+1;j,k}\big]
\pm \big[\ovl{E}^{(s)}_{c,b;h,j}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{b,b+1;j,k}]\,\big].
\end{multline*}
Now the bracket in the first term is zero by Case 3, and we may rewrite the whole second term as
$\pm[\ovl{E}^{(r)}_{a,b+1;i,k},\ovl{E}^{(s)}_{c,b;h,j}]$, which is zero by Case 4.
Assume that $d-b>1$, then $d-1>b$. By (\ref{L720}), the bracket equals to
\begin{align*}
[&\ovl{E}^{(r)}_{a,b;i,j},\ovl{E}^{(s)}_{c,d;h,k}]
=\big[\ovl{E}^{(r)}_{a,b;i,j}\,,\,(-1)^{\pa{f}_{d-1}}[\ovl{E}^{(s)}_{c,d-1;h,f}\,,\,\ovl{E}^{(1)}_{d-1,d;f,k}]\,\big]\\
&=(-1)^{\pa{f}_{d-1}}\big[\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{c,d-1;h,f}]\,,\,\ovl{E}^{(1)}_{d-1,d;f,k}\big]
\pm\big[\ovl{E}^{(s)}_{c,d-1;h,f}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{d-1,d;f,k}]\,\big].
\end{align*}
The first term is zero by the induction hypothesis, while the second term is zero as well by Case 1.
\item[Case 6.] $a=c<b<d$:
\begin{align*}
[\ovl{E}^{(r)}_{a,b;i,j},\ovl{E}^{(s)}_{a,d;h,k}]
&=\big[\ovl{E}^{(r)}_{a,b;i,j}\,,\,(-1)^{\pa{f}_{a+1}}[\ovl{E}^{(1)}_{a,a+1;h,f}\,,\,\ovl{E}^{(s)}_{a+1,d;f,k}]\,\big]\\
&=(-1)^{\pa{f}_{a+1}}\big[\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]\,,\,\ovl{E}^{(s)}_{a+1,d;h,k}\big]\\
&\quad \pm\big[\ovl{E}^{(1)}_{a,a+1;h,f}\,,\,[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{a+1,d;f,k}]\,\big].
\end{align*}
Note that $[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{a+1,d;f,k}]=0$ by Case 5. Hence it suffices to show that
\begin{equation}\label{L727}
[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]=0, \qquad \text{for all}\quad b>a.
\end{equation}
We prove (\ref{L727}) by induction on $b-a\geq 1$. When $b-a=1$, it follows from (\ref{L717}). Now
assume $b-a>1$. By (\ref{L720}), we have
\begin{align*}
[\,\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}\,]
&= \big[\,(-1)^{\pa{g}_{b-1}}\,[\,\ovl{E}^{(r)}_{a,b-1;i,g}\,,\,\ovl{E}^{(1)}_{b-1,b;g,j}\,]\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}\big]\\
&=(-1)^{\pa{g}_{b-1}}\,\big[\ovl{E}^{(r)}_{a,b-1;i,g}\,,\,[\ovl{E}^{(1)}_{b-1,b;g,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]\,\big]\\
&\quad \pm\,\big[\ovl{E}^{(1)}_{b-1,b;g,j}\,,\,[\ovl{E}^{(r)}_{a,b-1;i,g}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]\,\big].
\end{align*}
Note that $[\ovl{E}^{(r)}_{a,b-1;i,g}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]=0$ by the induction hypothesis. Also by (\ref{L717}),
$[\ovl{E}^{(1)}_{b-1,b;g,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]=0$ unless $b-1=a+1$, in which case,
(\ref{L727}) becomes $[\ovl{E}^{(r)}_{a,a+2;i,j}\,,\,\ovl{E}^{(1)}_{a,a+1;h,f}]$, which is zero by (\ref{LL726}).
\item[Case 7.] $a=c<b=d$:\\
We claim that
\begin{equation}\label{L728}
[\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{a,b;h,k}]=0.
\end{equation}
If $b=a+1$, it follows directly from (\ref{L717}). If $b>a+1$, we may expand one term in the bracket of (\ref{L728}) by (\ref{L720}) to deduce that
\begin{align*}
[\,\ovl{E}^{(r)}_{a,b;i,j}\,,\,\ovl{E}^{(s)}_{a,b;h,k}\,]
&=\big[\,(-1)^{\pa{f}_{b-1}}\,[\ovl{E}^{(r)}_{a,b-1;i,f}\,,\,\ovl{E}^{(1)}_{b-1,b;f,j}]\,,\,\ovl{E}^{(s)}_{a,b;h,k}\big]\\
&=(-1)^{\pa{f}_{b-1}}\,\big[\ovl{E}^{(r)}_{a,b-1;i,f}\,,\,[\ovl{E}^{(1)}_{b-1,b;f,j}\,,\,\ovl{E}^{(s)}_{a,b;h,k}]\,\big]\\
&\quad\pm\,\big[\ovl{E}^{(1)}_{b-1,b;f,j}\,,\,[\ovl{E}^{(r)}_{a,b-1;i,f}\,,\,\ovl{E}^{(s)}_{a,b;h,k}]\,\big].
\end{align*}
Note that $[\ovl{E}^{(1)}_{b-1,b;f,j}\,,\,\ovl{E}^{(s)}_{a,b;h,k}]=0$ by Case 3 and $[\ovl{E}^{(r)}_{a,b-1;i,f}\,,\,\ovl{E}^{(s)}_{a,b;h,k}]=0$ by Case~6, which proves (\ref{L728}).
\end{description}
This completes the proof of Lemma \ref{injeq}.
\end{proof}
\begin{proposition}\label{ind1}
$\widehat{Y}_{\mu}$ is spanned as a vector superspace by supermonomials in the elements $\lbrace D_{a;i,j}^{(r)}, E_{a,b;i,j}^{(r)}, F_{b,a;i,j}^{(r)}\rbrace$ taken in a certain fixed order so that $F$'s appear before $D$'s and $D$'s appear before $E$'s.
\end{proposition}
\begin{proof}
Lemma \ref{injeq} implies that the graded algebra $\operatorname{gr}\widehat{Y}_{\mu}^+$ is spanned by supermonomials in $\lbrace\ovl{E}_{a,b;i,j}^{(r)}\rbrace$ in some fixed order and hence $\widehat{Y}_{\mu}^+$ is spanned by supermonomials in $\lbrace E_{a,b;i,j}^{(r)}\rbrace$ in some fixed order as well.
By applying the automorphism $\zeta_{M|N}$, we see that $\widehat{Y}_{\mu}^-$ is spanned by supermonomials in $\lbrace F_{a,b;i,j}^{(r)}\rbrace$ in a certain fixed order as well.
Moreover, $\operatorname{gr}\widehat{Y}^0_{\mu}$ is supercommutative by Proposition \ref{dd0}, and it follows that $\widehat{Y}^0_{\mu}$ is spanned by supermonomials in $\lbrace D_{a;i,j}^{(r)}\rbrace$ in a certain fixed order.
Finally, by the defining relations in Proposition \ref{srlns} and the argument above, we may interchange the order between those $D$'s, $E$'s and $F$'s in a supermonomial such that all the $F$'s appear before all the $D$'s and all the $D$'s appear before all the $E$'s.
As a result, the multiplication map is surjective:
\[
\operatorname{gr} \widehat{Y}^-_{\mu}\otimes \operatorname{gr}\widehat{Y}^0_{\mu}\otimes \operatorname{gr}\widehat{Y}^+_{\mu}\twoheadrightarrow \operatorname{gr}\widehat{Y}_{\mu}
\]
and our proposition is established.
\end{proof}
\begin{proposition}\label{ind2}
The images of the supermonomials in Proposition~\ref{ind1} under $\Gamma$ are linearly independent.
\end{proposition}
\begin{proof}
By Corollary~\ref{Yloop}, we may identify $\operatorname{gr} Y_{M|N}=\operatorname{gr} Y_{\mu}$ with the loop superalgebra $U(\mathfrak{gl}_{M|N}[x])$ via
\[
\operatorname{gr} _{r-1}t_{ij}^{(r)}\longmapsto (-1)^{\pa{i}}e_{ij}x^{r-1}.
\]
We consider the following composition
\[
\operatorname{gr} \widehat{Y}_{\mu}^-\otimes \operatorname{gr} \widehat{Y}_{\mu}^0\otimes \operatorname{gr} \widehat{Y}_{\mu}^+\twoheadrightarrow \operatorname{gr} \widehat{Y}_{\mu} \xrightarrow{\Gamma} \operatorname{gr} Y_{\mu}\cong U(\mathfrak{gl}_{M|N}[x]).
\]
Let $n_a:=\mu_1+\mu_2+\ldots+\mu_a$ for short. By Proposition~\ref{quasi}, the image of $\ovl{E}_{a,b;i,j}^{(r)}$ (respectively, $\ovl{D}_{a;i,j}^{(r)}$, $\ovl{F}_{b,a;i,j}^{(r)}$) under the above composition map is $(-1)^{\pa{i}_{a}}e_{n_a+i, n_b+j}x^{r-1}$ (respectively, $(-1)^{\pa{i}_{a}}e_{n_a+i,n_a+j}x^{r-1}$, $(-1)^{\pa{i}_{b}}e_{n_b+i, n_a+j}x^{r-1}$ ). By the PBW theorem for $U(\mathfrak{gl}_{M|N}[x])$, the image (under the map $\Gamma$) of the set of all supermonomials in the following set
\begin{align*}
&\qquad\big\lbrace \operatorname{gr}_{r-1}\ovl{D}_{a;i,j}^{(r)} \,|\, 1\leq a\leq n, \; 1\leq i,j\leq\mu_a, \, r\geq 1 \big\rbrace \\
&\cup\big\lbrace \operatorname{gr}_{r-1}\ovl{E}_{a,b;i,j}^{(r)} \,|\, 1\leq a<b\leq n, \; 1\leq i\leq\mu_a, 1\leq j\leq\mu_b, \, r\geq 1 \big\rbrace\\
&\cup\big\lbrace \operatorname{gr}_{r-1}\ovl{F}_{b,a;i,j}^{(r)} \,|\, 1\leq a<b\leq n, \; 1\leq i\leq\mu_b, 1\leq j\leq\mu_a, \, r\geq 1 \big\rbrace
\end{align*}
taken in a certain fixed order must be linearly independent in $\operatorname{gr} Y_{\mu}$ and hence Proposition~\ref{ind2} follows.
\end{proof}
\begin{corollary}
The homomorphism $\Gamma:\widehat{Y}_{\mu}\rightarrow Y_{\mu}$ is injective, and Theorem~\ref{Pg} follows.
\end{corollary}
\begin{proof}
We have known that $\Gamma$ is a surjective homomorphism. Now a spanning set for $\widehat{Y}_{\mu}$ is obtained by Proposition~\ref{ind1}, while the image of this spanning set under $\Gamma$ is linearly independent in $Y_{\mu}$ by Proposition~\ref{ind2}. This shows that $\Gamma$ is injective.
\end{proof}
Let $Y_\mu^0$, $Y_\mu^+$ and $Y_\mu^-$ denote the subalgebras of $Y_{\mu}$ generated by all the $D$'s, $E$'s and $F$'s, respectively. The next result follows from the proof of Proposition~\ref{ind1} and the proof of Proposition~\ref{ind2}.
\begin{corollary} We have the PBW bases for the following superalgebras.
\begin{enumerate}
\item[(1)] The set of supermonomials in $\{ D_{a;i,j}^{(r)}\}_{1\leq a\leq n, 1\leq i,j\leq \mu_a, r\geq 1}$ taken in a certain fixed order forms a basis for $Y_\mu^0$.
\item[(2)] The set of supermonomials in $\{ E_{a,b;i,j}^{(r)}\}_{1\leq a<b\leq n, 1\leq i\leq\mu_a,1\leq j\leq\mu_b, r\geq 1}$ taken in a certain fixed order forms a basis for $Y_\mu^+$.
\item[(3)] The set of supermonomials in $\{ F_{b,a;i,j}^{(r)}\}_{1\leq a<b\leq n, 1\leq i\leq \mu_b,1\leq i\leq\mu_a, r\geq 1}$ taken in a certain fixed order forms a basis for $Y_\mu^-$.
\item[(4)] The set of supermonomials in the union of the elements listed in (1), (2) and (3) taken in a certain fixed order forms a basis for $Y_{\mu}$.
\end{enumerate}
\end{corollary}
\subsection*{Acknowledgements}
The author is grateful to Weiqiang Wang and Shun-Jen Cheng for numerous discussions. This work is partially supported by MOST grant 103-2115-M-008-012-MY2 and NCTS Young Theorist Award 2015.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,078 |
Johan Lennart Geijer [], född 14 september 1909 i Ystad, död 16 juni 1999 i Stockholm, var en svensk socialdemokratisk politiker och jurist samt Sveriges justitieminister 1969–1976.
Biografi
Lennart Geijer föddes i Ystad som son till postmästaren Åke Geijer och Anna Sylvan. Han studerade vid Lunds universitet och blev juris kandidat 1933. Efter två års tingstjänst blev han föreståndare vid Hyresgästföreningens juridiska byrå, varefter han hade flera uppdrag som ombudsman och juridisk sakkunnig i hyres- och arbetsrättsliga frågor. Han blev juris licentiat 1957 och juris doktor 1958.
Geijer var statsråd 1966–76, konsultativt statsråd 1966–69 och justitieminister 1969–76. Till den senare posten utsågs han efter att samtliga statsråd begärde entledigande då statsminister Tage Erlander den 9 oktober 1969 meddelade att han skulle avgå. Förutvarande justitieministern Herman Kling utsågs av den tillträdande statsministern Olof Palme till ambassadör, och Geijer, som då var jurist på TCO, sattes istället på dennes post.
Geijer strävade under sin ministertid efter att minska användningen av fängelsestraff och att göra kvarvarande fängelsestraff mer humana och inriktade på vård och rehabilitering; i princip ansåg han att långa frihetsberövande straff skulle vara förbehållna de verkligt grova brotten.
Geijer var svensk justitieminister under ambassadockupationen i Stockholm 1975, då Kommando Holger Meins ur den väpnade vänsterextremistiska organisationen Röda armé-fraktionen (RAF) intog Västtysklands ambassad i Stockholm. Under dramat förhandlade Geijer med terroristerna på plats i ambassadbyggnaden.
Andra större politiska händelser under hans ministertid var spionaffären på Sahlgrenska 1975. Geijer förnekade att Säpo skulle vara inblandat och sände personligen Ebbe Carlsson och Hans Holmér till Göteborg för att utreda vad som hänt. Han hemligstämplade deras slutliga rapport men den läckte trots detta ut.
Geijer avgick tillsammans med den övriga regeringen efter den borgerliga valsegern 1976.
Geijeraffären
Lennart Geijer påstods vara en av de involverade i den så kallade bordellhärvan på 1970-talet, skildrad i bland annat Peter Bratts memoarbok Med rent uppsåt (2007) och Lillemor Östlins självbiografi Hinsehäxan (2005).
I en hemlig promemoria av rikspolischef Carl Persson 1976 uppmanades statsminister Olof Palme att låta undersöka huruvida, och i vilken grad, Geijer kunde vara en säkerhetsrisk på grund av påstådda kontakter med prostituerade från öststaterna. Dessa misstankar krävde en granskning oavsett om de vore befogade eller inte; promemorian påstod alltså inte att Geijer eller andra politiker skulle ha haft några dylika kontakter. Polisen hade nyligen gripit bordellmamman Doris Hopp och vid förhör hade hon uppgett att hon hade flera politiker, bland andra Lennart Geijer, som kunder.
I november 1977 avslöjades promemorians existens av Dagens Nyheter, men oppositionsledaren Olof Palme dementerade skarpt. Tidningen fick be Geijer om ursäkt och betala skadestånd, och reportern Peter Bratt, som skrivit om saken, hamnade ute i kylan. I början av maj 1978 kom historien upp på nytt, nu i tv-programmet Studio S. Statsminister Thorbjörn Fälldin hade efter DN:s artikel lovat att berätta vilka politiker som nämndes i polisens promemoria. Efter Studio S-programmet ändrade han sig dock och avfärdade i riksdagen innehållet som lögner, med motivering att han själv fanns med på listan.
Referenser
Noter
Källor
Vem är det 1961
Födda 1909
Avlidna 1999
Svenska jurister
Sveriges justitieministrar
Sveriges konsultativa statsråd
Ledamöter av Sveriges riksdags första kammare för Socialdemokraterna
Män
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Lennart
Alumner från Lunds universitet
Regeringen Palme I | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,925 |
\section{Introduction}
Pitch is the perceptual correlate of the periodicity in a sound's waveform. It is a fundamental attribute of auditory sensation that forms the basis of both music and speech perception; consequently, understanding the neural foundations of pitch is a major challenge in auditory neuroscience.
A combination of two sounds that simultaneously elicits two different pitches is referred to as a dyad, and the pitch interactions within a dyad give rise to a sensation that can be characterized by its \emph{consonance} or \emph{dissonance}. Loudness, timbre, and the absolute fundamental frequencies of the sounds can have subtle effects on whether a dyad is perceived as consonant or dissonant; however, the dominant factor in determining the degree of consonance is the \emph{relationship} between the fundamental periods of the sounds that make up the dyad \cite{Kameoka1969a}: Simple periodicity ratios result in more consonant sensations; on the other hand, sensation becomes more and more dissonant as the complexity of the periodicity ratio increases \cite{Helmholtz1863, Krueger1913, Plomp1965}. Previously, it was argued that dissonance correlates with the beating, or \emph{roughness} sensation that is elicited by the interfering regularities of the involved sounds \cite{Helmholtz1863, Krueger1913, Plomp1965}. However, listeners that showed impaired pitch perception but were sensitive to beating and roughness have been reported to be unable to differentiate between consonant and dissonant dyads \cite{Cousineau2012, Tramo2006}. This suggests that pitch- rather than roughness-related auditory processing is responsible for the emergence of consonance.
Neurophysiological evidence for a tight link between consonance and pitch has recently been provided by Bidelman and colleagues \cite{Bidelman2014}; they showed, using electroencephalography (EEG), that the amplitude of the cortical pitch onset response (POR) is strongly modulated by a dyad's perceived consonance. The POR is a pitch-selective component of the transient auditory evoked potential/field (AEP/AEF) that occurs within the time range of the well-known N100 wave \cite{Naatanen1987, Alain1997}, around 100 ms after pitch onset. The morphology of the POR is strongly correlated with the perceived pitch in single tones: its latency scales linearly with the period of the sound, and its amplitude increases with the strength of the pitch percept \cite{Krumbholz2003, Ritter2005, Tabas2016}. The neural sources of the POR are located in the anterolateral section of Heschl's gyrus (alHG) in auditory cortex \cite{Krumbholz2003, Ritter2005, Bidelman2014}, consistently with the anatomical location of pitch-selective neurons in non-human primates (e.g.,~\cite{Bendor2006, Bendor2010, Bizley2013, Feng2017}), and with pitch-selective regions that were reported for human listeners \cite{Griffiths2001, Penagos2004, Brugge2009, Norman2013, Moerel2012}. Further experiments in humans have demonstrated that the dyad-evoked frequency-following response in the brainstem is predictive for the perceived consonance of a dyad \cite{Bidelman2009, Bidelman2011, Bidelman2013}; also, functional magnetic resonance imaging (fMRI) studies showing selective activation to consonance/dissonance contrasts in the superior temporal gyrus \cite{Peretz2001} and in frontal cortex \cite{Minati2009} lead the auditory community to link neural representations of consonance and dissonance with higher, cognitive processes \cite{Seger2013}.
In this study, we used a combined experimental and theoretical approach to assess whether consonance and pitch share similar processing mechanisms in human auditory cortex. Towards this goal, we first developed an innovative, realistic model of cortical ensemble responses to pitch, specifically designed to understand the mesoscopic representation of pitch in alHG. The model can account, mechanistically, for the latency effects in the POR that have been robustly reported for decades in multiple experimental settings \cite{Krumbholz2003, Ritter2005, Gutschalk2004} but, up to now, remained poorly understood. Second, we recorded the AEF elicited by consonant and dissonant dyads using magnetoencephalography (MEG); here, our experimental results revealed a strong correlation between POR latency and the degree of consonance, extending previous EEG findings \cite{Bidelman2014}. Finally, we aimed to replicate the results from our MEG experiment using our model. If the hypothesis that consonance and pitch are processed by similar mechanisms in cortex is correct, we would expect the model to explain the dependence of POR latency on the degree of consonance \emph{without} the inclusion of higher processing stages within the auditory hierarchy \cite{Peretz2001, Minati2009}. In line with this hypothesis, the model provided a quantitative, mechanistic explanation for the relationship between POR dynamics and consonance, suggesting that consonance and dissonance perception might be linked to the pitch processing sub-regions of the auditory cortex.
\section{Results}
\subsection{Neural mechanisms underlying pitch processing in auditory cortex}
\paragraph{Model overview}
We introduced a model of cortical pitch processing designed to explain the morphology of the POR as elicited by the onset of consonant and dissonant dyads (see full description in Methods). The model consists of three processing stages located at different levels of the auditory hierarchy. In the first stage, an array of idealized coincidence detector units extracts periodicities from the auditory nerve activity in response to the target stimulus \cite{Meddis2006, Balaguer2008}. Subsequently, the second and third stages, putatively located at adjacent locations within alHG, generate a stable pitch characterization from the first stage, previous representation.
Auditory nerve responses were generated by a recent model of the auditory periphery \cite{Zilany2014, Zilany2009}; periodicity detection was implemented using the principles of the autocorrelation models of pitch \cite{Licklider1951, Meddis1997, Meddis2006, Balaguer2008}. At this stage, the representation of the stimulus shows a harmonic shape along the periodicity axis, with prominent peaks of activation at the neurons which encode the pitch of the stimulus and its lower harmonics (see Figure~\ref{fig:mod:diagram}e).
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{./diagramMute.eps}
\caption{\textbf{Basic schematics of the model.}
a) Average response across populations of the excitatory ensembles of the decoder network, accounting for the evoked fields in the cortical sources of the POR, in response to three stimuli with different pitches.
Stimuli were iterated rippled noises (16 iterations, bandpass filtered between 0.8 and 3.2\,kHz) with pitches: 250\,Hz, 125\,Hz, and 83\,Hz.
b) Excitatory activation, averaged between 150 and 200\,ms after sound onset, in the different ensembles of the decoder network elicited by the same stimuli as above.
c) Model architecture.
d) Connectivity weights between excitatory and inhibitory ensembles in the decoder network.
e) Typical input to the decoder corresponding to the three IRN stimuli used above, averaged between 150 and 200\,ms after sound onset (see also Supplementary Video~S1).}
\label{fig:mod:diagram}
\end{figure}
The array of periodicity detectors provides excitatory input to a first cortical processing stage, termed the \emph{decoder} network in this study. The \emph{decoder} network is putatively located in alHG and effectively extracts the pitch value(s) from the subcortical input. The decoder network connects to a second cortical ensembles network, termed \emph{sustainer}; this stage integrates the decoder network output and modulates it through cortico-cortical top-down efferents. The \emph{sustainer} reinforces and sustains the decoder mechanism, reminiscent to recent models of perceptual decision making \cite{Wimmer2015}.
Both decoder and sustainer comprise a network of cortical microcolumns, each of which is tuned to a specific pitch value along the human perceptual range (see Methods for details). Pitch is encoded in the active pitch-selective populations of the processing network (see Figure~\ref{fig:mod:diagram}b), in agreement with cortical recordings in non-human mammals \cite{Wang2012, Bizley2013, Gao2016}.
Microcolumns in the cortical networks are modelled as blocks comprising an excitatory and inhibitory neural ensemble (see Figure~\ref{fig:mod:diagram}c). Ensembles communicate with each other through realistic synapses. Connectivity weights between populations in the decoder network (see Figure~\ref{fig:mod:diagram}e) are specifically tuned to facilitate the inhibition of the lower harmonics elicited in the periodicity detectors (see Figure~\ref{fig:mod:dec}; a video detailing the integration process is available in Supplementary Video~S1). Similar connectivity patterns have been consistently found in the mammalian auditory cortex (see~\cite{Wang2013} for a review); moreover, neurons mapping harmonic templates to a pitch-selective representation like those introduced in this model have been recently reported in the primate auditory cortex \cite{Feng2017}.
The detailed formulation of the model allows us to perform quantitative predictions regarding the neuromagnetic field that are elicited by the activation of each of the cortical networks within the model (Figure~\ref{fig:mod:diagram}b). More specifically, the equivalent dipole moment elicited by each of the networks is monotonically related to the aggregated excitatory activation of its ensembles \cite{Kiebel2008} (see Methods for details), while the characteristic period of the excitatory population with the largest activity in the network corresponds to the perceived pitch (see Figures~\ref{fig:mod:IRNs}b and~c). Below, we will argue that the characteristic responses of the decoder network during pitch processing can be identified with the responses of the neural sources of the POR.
\paragraph{Dynamics of the decoder network}
Figure~\ref{fig:mod:dec} illustrates an example of the model dynamics in response to a stimulus with a pitch corresponding to $T = 4\,$ms (i.e., $f = 250\,$Hz; details are shown in Supplementary Video~S1). In a first step, periodicity detectors, tuned to $T \simeq 4\,$ms, become active after $t_1 \sim 1.25\,T$ \cite{Wiegrebe2001} (see the top prominent horizontal line at $T = 4$\,ms in Figure~\ref{fig:mod:dec}a). These active neurons provide the bottom-up excitatory input to the excitatory ensemble in the corresponding decoder network column (see Figure~\ref{fig:mod:dec}b). Likewise, the harmonics of the stimulus' period (i.e., $2\,T$, $3\,T$ etc.) are subsequently represented in the subcortical model after $t_2 = 2\,t_1$, $t_3 = 3\,t_1$ etc., and provide the input to the corresponding excitatory populations in the decoder network (see Figure~\ref{fig:mod:dec}b).
\begin{figure}[b!]
\centering
\includegraphics[width=1\textwidth]{./decoding.eps}
\caption{\textbf{Illustration of the decoding process.} The plots show the evolution of key variables of the model during the processing of the first 200\,ms of an iterated rippled noise with a fundamental period of $T = 4\,$ms (parameters were as in Figure~\ref{fig:mod:diagram}). a)--e) Evolution of the model neural ensembles encoding characteristic periods between 0.5\,ms and 15\,ms. a) Activity of periodicity detectors within the first stage of the model. b) and c) Activity of excitatory and inhibitory ensembles in the decoder network. d) and e) Activity of excitatory and inhibitory activities in the sustainer network. f) Aggregated excitatory activity in the decoder. Detailed dynamics of the process are illustrated in an animation in Supplementary Video~S1.}
\label{fig:mod:dec}
\end{figure}
Excitatory ensembles characterized by the periods of the harmonic series $\{T, 2\,T 3\,T\, \dots\}$ are connected to inhibitory neural populations identified by the fundamental period $T$ (see Figure~\ref{fig:mod:diagram}). Synaptic efficacy is tuned such that the inhibitory drive is strong enough to elicit activity only when a sufficient number of excitatory inputs (typically 3) are simultaneously active, thus providing robustness to the upcoming decoding process. Thus, the inhibitory population characterized by $T \simeq 4$\,ms becomes active only when it receives simultaneous synaptic drive from the excitatory ensembles characterizing the periods $T = 4\,$ms, $2T = 8\,$ms, and $3T = 12\,$ms (see Figure~\ref{fig:mod:dec}C).
Correspondingly, the inhibitory ensemble associated with the period $T$ is connected to the excitatory populations encoding the lower harmonics $\{2\,T, 3\,T, 4\,T, \dots\}$ (see Figure~\ref{fig:mod:diagram}). Thus, when active, the inhibitory activity progressively silences populations that are activated by the periodicity detectors but that do not correspond to the fundamental period of the stimulus (see the shunting process in the decoder excitatory network between 60\,ms and 100\,ms in Figure~\ref{fig:mod:dec}b).
The processing dynamics described above explains the neural mechanisms underlying the morphology of the POR. The aggregated excitatory activity in the decoder, monotonically related to the neuromagnetic fields elicited by this network, is shown in Figure~\ref{fig:mod:dec}f. The accumulation of excitatory activity corresponding to the harmonic series of the period of the stimulus $T$ results in the increase of the simulated field magnitude observed between 10\,ms and 65\,ms in Figure~\ref{fig:mod:dec}f. The subsequent decay of the model collective excitatory response between 70\,ms and 120\,ms in the figure, is similarly caused by the action of the most activated inhibitory ensemble on the excitatory populations encoding the lower harmonics. We identify the maximum in the aggregated excitatory activity, corresponding to the time point in which the model performs a perceptual decision about the pitch of the stimulus, with the POR latency (further details regarding this correspondence are shown in Supplementary Video~S1).
The proposed mechanism also explains, quantitatively, the dependence of the POR latency with the period of the stimulus. A periodicity detector tuned to $T$ needs $\gtrsim1.25\,T$ to robustly detect a periodicity $T$ in the auditory nerve activity \cite{Wiegrebe2001}. Thus, the arrival of sufficient excitatory drive to activate an inhibitory ensemble shows a dependence with several periods of the stimulus. This result also provides a mechanistic explanation for the minimum stimulus duration required for robust pitch discrimination, which is around four times the period of the stimulus \cite{Krumbholz2003}.
\paragraph{Dynamics of the sustainer network}
The dynamics of the decoder network are sufficient to explain how the harmonic representations held in the first step of the model are transformed into the final representations shown in Figures~\ref{fig:mod:diagram}b and~\ref{fig:mod:diagram}c. However, after the transformation has taken place, the excitatory ensembles corresponding to the lower harmonics of the stimulus' pitch are no longer active, and hence the inhibitory population silencing them loses its drive. Thus, without top-down control, the decoder network would rapidly reset and need to repeatedly extract the pitch from the harmonic representation, eliciting a series of PORs; this, however, does not reflect the experimental observations (Figure~\ref{fig:mod:diagram}a). The role of the sustainer network is to regulate the dynamics of the decoder network once the pitch value has been extracted from the harmonic representation, in order to effectively \emph{sustain} the perceptual decision until a significant change is produced in the cortical input.
In the absence of external input, the sustainer network rests at equilibrium, with a steady activation in the inhibitory populations and complete deactivation of the excitatory populations (see Figures~\ref{fig:mod:dec}d and~e).
Excitatory/inhibitory ensembles in the sustainer receive direct input from their respective excitatory/inhibitory counterparts in the decoder (see Figure~\ref{fig:mod:diagram}d). Thus, a significantly active inhibitory population in the decoder effectively silences the steady activity of the analogous inhibitory population in the sustainer. If this afferent drive coincides with a strong activation of the corresponding excitatory population in the decoder, the combined bottom-up input results in a strong activation in the equivalent excitatory population within the sustainer (see Figures~\ref{fig:mod:dec}d and~\ref{fig:mod:dec}e).
Top-down efferents connect each excitatory population in the sustainer with its inhibitory counterpart in the decoder network (see~\ref{fig:mod:diagram}b), compensating for the loss of excitatory drive of the silenced populations for as long as the subcortical input remains unchanged (the behaviour of the network under pitch changes is described in Supplementary Figure~S9).
\paragraph{Model predictions}
The POR is defined as the subcomponent of the N100 transient that responds selectively to pitch onset and pitch changes \cite{Krumbholz2003, Ritter2005, Seither2006}. In order to isolate the POR from other subcomponents of the N100 like, e.g., the energy onset response (EOR), experimental setups use iterated rippled noise (IRN) preceded by a noise burst of the same energy and bandwidth \cite{Krumbholz2003, Ritter2005, Bidelman2014}; the POR is then measured as the transient elicited at the transition between noise and IRN (i.e., at the pitch onset). Thus, we tested the predictive power of our model using IRN stimuli with different pitch values (an isolated POR can be elicited using energy-balanced stimuli, but see~Supplementary Figure~S6 for an example of how more general predictions could be drawn for other stimuli.)
Latency predictions of the POR elicited by IRN stimuli are compared with experimental data in Figures~\ref{fig:mod:IRNs}a and~\ref{fig:mod:IRNs}b. Results show that the model replicates the relation between the POR latency and the period of the stimuli as typically reported in the MEG literature \cite{Krumbholz2003, Ritter2005}. Replication of these results for IRN stimuli with different parametrisations are shown in Supplementary Figure~S4.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{./modIRNs.eps}
\caption{\textbf{Model predictions for iterated rippled noise pitch.} a)--b) Latency predictions for iterated rippled noise compared with experimental data obtained by a previous study \cite{Krumbholz2003} for the same stimuli.
c) Comparison of the collective activation of the excitatory ensembles in the decoder (computed as an average across populations) with the equivalent dipole moment elicited at the generator of the POR; stimulus was an IRN of 16 iterations and a delay of 8\,ms.
d)--h) Averaged responses at: (d) periodicity detectors, (e/f) excitatory/inhibitory ensembles in the decoder, (g/h) excitatory/inhibitory ensembles in the sustainer.}
\label{fig:mod:IRNs}
\end{figure}
To ensure that the model is correctly representing the pitch of the IRN stimuli used in the experiment, we also plotted the average activation in the different ensembles in Figures~\ref{fig:mod:IRNs}d--h. Since periodicity detectors and excitatory neurons at the decoder do not show selective activation with pitch (e.g., neurons representing harmonics of the actual pitch value also activate during the decoding process), the perceptual readout of the model can only be robustly measured in the inhibitory populations in the decoder and the populations in the sustainer. Perceptual results in Figures~\ref{fig:mod:IRNs} and Supplementary Figure~S4 indicate that the model is able to robustly represent the pitch of the stimulus when at least two harmonics are present in the cortical input (since we only consider periodicity detectors tuned to periods under $T_{\max} = 30$\,ms, the highest period robustly extracted by the model is $T_{\max} / 2 = 15$\,ms; a larger pitch range could be easily achieved by increasing $T_{\max}$). Robust pitch extraction is shown for IRN stimuli with different parametrisations, pure tones, harmonic complex tones and click trains in Supplementary Figure~S3.
\subsection{Neuromagnetic correlates of consonance and dissonance in auditory cortex}
Next, we recorded neuromagnetic fields evoked by six different dyads from 37 normal hearing subjects. Data were preprocessed using standard MEG procedures, and equivalent current dipoles were fitted, independently, for the POR for each subject and hemisphere, pooled over conditions (see Methods). Dipole locations in Talairach space are plotted in Figure~\ref{fig:aefs}b.
Dyads consisted of two IRN sounds. The lower note pitch was 160\,Hz; the pitch of the upper note was adjusted accordingly to form either a consonant dyad (unison, P1; perfect fifth, P5; major third, M3) or a dissonant dyad (tritone, TT; minor seventh, m7; minor second, m2). To dissociate the EOR in Planum temporale from the POR in alHG, the dyads were preceded by an energy-balanced noise segment, cross-faded with the dyad to avoid discontinuous waveforms (like for the single IRN sounds analyzed in the previous section; see Methods).
Figure~\ref{fig:aefs}A presents the MEG grand-mean source waveforms, for both hemispheres, in response to the six stimulus conditions. The noise onset from silence (depicted in grey below the source waveforms) was followed by a transient P1m-N1m-P2m AEF complex. Since the first stimulus segment did not vary between conditions, we did not expect to find any significant differences in the corresponding neuromagnetic activity at this point.
\begin{figure}[htb!]
\centering
\includegraphics[width=1\textwidth]{./aefs.eps}
\caption{\textbf{Auditory fields evoked at dyad onset.} a) MEG grand-mean source waveforms in response to the pooled stimulus conditions. The course of the stimuli is shown in grey (noise) and black (IRN) below the source waveforms; note the prominent negative POR deflection (N1m) at the transition from the first to the second stimulus segment. BL = baseline. b) Projection of the dipole locations (means and 99,\% Bootstrap confidence intervals) onto the axial view of auditory cortex as suggested by Leonard et al. \cite{Leonard1998}. c) Morphology of the POR in response to the dyad onset in the single experimental conditions (second stimulus segment), pooled over hemispheres. d) 99\,\% Bootstrap confidence intervals for the POR amplitudes and latencies in the single experimental conditions. In subplots, b) and d), confidence intervals are bias-corrected and accelerated, as recommended by Efron and Tibshirani \cite{Efron1993}.}
\label{fig:aefs}
\end{figure}
In contrast, the transition to the second stimulus segment (IRN dyads; black signal below the source waveforms) elicited prominent POR waves, and the morphology of the POR varied considerably between conditions. Figure~\ref{fig:aefs}C show close-up views of the POR. Consonant dyads (pooled conditions [P1+P5+M3]) elicited a much earlier ($p < .0001$) and larger ($p < .0001$) POR than dissonant dyads (pooled conditions [m7+TT+m2]). Figure~\ref{fig:aefs}D depicts 99\,\% bootstrap confidence intervals for the POR amplitudes and latencies, pooled over hemispheres, in response to the experimental conditions; the activity pattern observed here also points to a close relationship between the degree of a dyad's consonance and the morphology of the respective POR.
When pooling across conditions, we found a noticeable difference between the left and the right hemisphere in the POR amplitude ($p = .01$), but not in the POR latency ($p = .36$); also, the difference between the neuromagnetic responses to consonant or dissonant dyads did not significantly vary between hemispheres (latency: $p = .58$; amplitude: $p = .48$).
\subsection{Neural mechanisms underlying the responses of auditory cortex to consonance and dissonance}
The difference in POR latency in response to consonant and dissonant dyads in alGH suggests that consonance and dissonance are computed at relatively early stages of the cortical auditory hierarchy. We used our model of cortical pitch processing, designed to reproduce the neuromagnetic responses elicited by iterated rippled noises, to test this interpretation. If the differential responses to consonance and dissonance in alHG were intrinsic to pitch processing, we would expect our prospective mechanism to be able to reproduce this behaviour.
First, we verified that the model's was able to provide a representation of the individual pitches of the two tones comprised in dyads; results are shown in Figure~\ref{fig:mod:dyads}a--c (see Supplementary Figure~S6 for additional perceptual results obtained with different families of dyads). It should be emphasized that even phenomenological (i.e., non-mechanistic) models of pitch perception are generally unable to perform correct perceptual predictions for sounds with concurrent pitches (e.g.~\cite{Balaguer2009, Patterson1994a}; see~\cite{DeCheveigne2005b} for a review).
\begin{figure}[p!]
\centering
\includegraphics[width=1\textwidth]{./modDyads.eps}
\caption{\textbf{Model responses to the IRN dyads used in the MEG experiment.}
a)--c) Neural representation of the dyads at different levels of the model: (a) periodicity detectors, (b/c) excitatory/inhibitory ensembles in the decoder network; each row shows the activity elicited by each dyad. Excitatory and inhibitory ensembles in the sustainer are precisely correlated to the decoder-inhibitory heatmap. Note that, unlike in the perceptual results for simple IRNs, the neural activity representing the pitch of the second note shifts toward the left because dyads are arranged in ascending order.
d)--i) Examples of the collective excitatory activity at the decoder network (monotonically related to the equivalent dipole moment elicited by the network) in comparison with the elicited dipole moment measured during the experimentation in the neural generator of the POR. The scale of the field derived for the unison dyad was adjusted to account for the comparatively smaller effect on the network of the unison input, which effectively activates half of the populations than the other dyads.
j) Latency predictions for IRN dyads compared with the experimental results reported in the previous section.
k) Latency predictions for the remaining dyads in the chromatic scale.
Strongly consonant dyads are represented with a green triangle, whilst strongly dissonant dyads are represented with a red triangle; dissonance was assessed according to Helmholtz~\cite{Helmholtz1863}, Table on Fig.~61. Dyads were generated using the same parameters as in the experimental procedures: the lower note pitch was set to 160\,Hz and the chromatic scale was generated using the \emph{just intonation} \cite{Helmholtz1863}.}
\label{fig:mod:dyads}
\end{figure}
Figure~\ref{fig:mod:dyads}j shows the latency predictions of the model are compared with the experimental data for the respective dyads.
Although the model predicted a slightly shorter POR latency for the semitone (m2) dyad than observed (see discussion), latency predictions faithfully reproduced the experimental trend; moreover, the differential response to consonant (P1, M3, P5) and dissonant (m2, TT, m7) dyads found in the MEG data was perfectly reproduced by the model (latency of P1 and P5 $<$ latency of dissonant dyads: $p<10^{-7}$, $U > 5150$; latency of M3 $<$ latency of m2: $p = 0.005$, $U = 4120$; latency of M3 $<$ latency of TT: $p = 0.63$, $U = 3571$; latency of M3 $<$ latency of m7: $p = 0.06$, $U = 3927$); according to pair-wise single-tailed Wilcoxon rank-sum tests performed over the results of $n = 60$ runs). The temporal dynamics of the dipole moment predicted by the model is shown in Figures~\ref{fig:mod:dyads}d--i.
Last, we extended the POR latency predictions of the model to include all 13 dyads within the chromatic scale (see Figure~\ref{fig:mod:dyads}), and tested if the differential responses to consonance and dissonance were generalizable to additional dyads. Following Helmholtz~\cite{Helmholtz1863}, we considered an extended set of consonant dyads, including the octave (P8) and the perfect fourth (P4); and an extended set of dissonant dyads, including the major seventh (M7) and the major second (M2). Once again, consonant dyads produced longer latencies than dissonant dyads (latencies of P1, P4, P5 and P8 $<$ latencies of the extended set of dissonant dyads: $p<0.0003$, $U > 4290$; latency of M3 $<$ latency of M2: $p < 10^{-5}$, $U = 4497$; latency of M3 $<$ latency of M7: $p = 0.54$, $U = 3610$); according to pair-wise single-tailed Wilcoxon rank-sum tests performed over the results of 60 runs of the model). These results, fully in line with previous findings, reveal that the differential response of our model to consonance and dissonance in dyads is a general phenomenon caused by the fundamental relationships between the periodicities of the two dyad components. These analyses are extended to further families of dyads in Supplementary Figure~S7, yielding similar results.
The model provides a mechanistic explanation of how the harmonic relationships between the components of the dyads modulate processing time. Consonant dyads consist of tones that share a larger number of lower harmonics than the ones in dissonant dyads. For instance, in the just intonation, the perfect fifth of a given fundamental shares one in every two harmonics with that fundamental, whilst only one in 16 harmonics are shared by a minor second and its fundamental. Our model suggests that cortical pitch processing is triggered by the joint activation of, at least, three periodicity detectors characterizing a specific harmonic series. Consonant dyads elicit a dramatically larger signal-to-noise ratio in the periodicity detectors tuned to their common harmonics, resulting in a collaborative effort towards pitch extraction that effectively speeds up processing dynamics (an animation of this process is depicted in Supplementary Video~S2).
\section{Discussion}
This work combines new theoretical and experimental methods to study how the auditory cortex representations of a sounds' pitch also generate the sensation of consonance and dissonance.
First, we introduced a novel ensemble model of pitch designed to understand the neuromagnetic fields elicited in alHG during pitch processing. The model proposes a mechanistic explanation of the POR morphology and the dependence of its peak latency with the perceived pitch, a phenomenon that has been robustly observed for over two decades \cite{Krumbholz2003, Ritter2005, Seither2006}, yet remained poorly understood. Thereafter, we designed an MEG protocol to investigate whether the POR properties are influenced by the degree of consonance or dissonance, as elicited by different dyads from common Western music. Results revealed a strong correlation between the POR peak latency and the degree of dissonance elicited by each dyad, extending previous EEG results that have also reported a modulation of the POR amplitude by consonance \cite{Bidelman2013}.
Last, we showed that our model, originally designed to explain pitch processing in IRN stimuli with a single pitch, can quantitatively account for the correlation between POR latency and the degree of consonance and dissonance. The model can explain the shorter POR latencies in response to consonant dyads as an effect of harmonic facilitation during pitch extraction. Combined, our results indicate that the neural mechanisms accounting for pitch processing show a differential response to consonant and dissonant dyads, suggesting that the sensation of consonance and dissonance might be elicited as a result of pitch processing in alHG.
\subsection{The POR latency reflects pitch processing time}
The dynamics of the decoder network proposes a new mechanistic interpretation for the POR latency in the sense that it might reflect the amount of time that is necessary for the network to stabilize in an unequivocal pitch (see Figure~\ref{fig:mod:dec}). Although an association between POR latency and processing time has been hypothesized previously in experiments (e.g.,~\cite{Alain1997, Krumbholz2003, Ritter2005, Seither2006}) and a in a model \cite{Balaguer2009}, a biophysical understanding of this phenomenon was still lacking. Our model identifies the magnitude of the POR with the instant in which the net inhibition at the decoder network exceeds the excitatory activity from the periodicity detectors; from a dynamic system perspective, this is equivalent to the instant in which the trajectory in the phase-space is unequivocally directed towards the attractor state dominated by the neural ensemble that is characterized by the perceived pitch (see animation in Supplementary Video~S1).
The model suggests that a robust perceptual decision concerning stimulus pitch is made after the cortical system identifies, typically, three peaks from the harmonic series of the stimulus' period in the representation of the periodicity detectors. This accounts for the relation between the POR latency and the stimulus period \cite{Krumbholz2003, Ritter2005}; moreover, it also may explain why pitch identification is only robust when the stimulus duration exceeds four times the pitch periodicity \cite{Krumbholz2003}. Previous studies have postulated that cortical pitch processing mechanisms must integrate along several period cycles in order to make a perceptual decision \cite{DeCheveigne2010, Gutschalk2004, Krumbholz2003}; however, a specific mechanism for such an integration has not been proposed up to now.
Moreover, since phase-locked activity is not robustly present above 50--200\,Hz in cortex \cite{Brugge2009}, integration along several repetition cycles was only possible in subcortical areas. The decoder network in our model takes advantage of the input harmonic representations provided by an autocorrelation model that does not require phase-locking to transmit information concerning several repetition cycles \cite{Meddis1997}, and thus provides a parsimonious solution to this problem.
\subsection{Effect of consonance and dissonance on cortical processing time}
Combined, our results suggest that cortical processing of dissonant dyads is slower than the processing of consonant dyads, in the sense that it requires a longer processing time. The model provides a physical rationale for this phenomenon: cortical extraction of consonance is based on the accumulation of activity in the columns with preferred periods characterizing the lower harmonics of the target sound; thus, concurrent pitch frequencies sharing common lower harmonics contribute to the build-up of each other's representation, thereby speeding up the stabilization of the network. Since consonant dyads are characterized by simpler frequency ratios, they comprise tones sharing a larger number of lower harmonics than dissonant dyads.
Early phenomenological models based on Helmholtz's \emph{roughness} theory described dissonance as the beating sensation produced by tones with fundamental frequencies that were not harmonically related \cite{Helmholtz1863, Krueger1913, Plomp1965, Kameoka1969a, Sethares1993}. More recent explanations of consonance, based on pitch processing, have linked the regularity of the autocorrelation harmonic patterns elicited by dyads to their evoked consonance and dissonance percepts \cite{Bidelman2009, Bidelman2011, Bidelman2013, Ebeling2008, Tramo2006}. Thus, in one way or another, previous phenomenological models of consonance have consistently related perceived consonance with the amount of harmonics shared between the tones comprising the dyad. The present model confirms that assumption and introduces a potential explanation for its underlying biophysical rationale.
Although our modelling results generally show a good fit with the data from the MEG experiment, the model prediction falls around 5\,ms short when explaining the POR latency evoked by the minor second dyad. This relatively small underestimation might result from the limited number of harmonics considered during the integration step in the decoder network: dissonant dyads, whose components do not share any common harmonic within the first three peaks of their harmonic series, present comparable processing times. An adaptive mechanism adjusting the number of harmonics required to trigger the decoding process according to the degree of overlap of the peaks in the input could potentially yield more accurate results. This adaptive mechanism would be necessary to explain how humans can differentiate dyads that differ in a quarter of a semitone.
Our study did not evaluate, however, if the general (yet not universal \cite{Plantinga2014, Mcdermott2016}) association between consonance and pleasantness might be a consequence of differential processing in alHG. Future work should investigate whether this link could be due to different processing times, as described above, or whether it can be better explained by processes at higher levels of the auditory hierarchy that might be more sensitive to cultural and background modulations.
\subsection{Experimental discussion and comparison with results of the literature}
Our neuromagnetic findings concerning the POR morphology in response to consonant and dissonant dyads resemble and extend recent EEG data reported by Bidelman and Grall~\cite{Bidelman2014}, and by Proverbio et al. \cite{Proverbio2016}. Specifically, Bidelman and Grall~\cite{Bidelman2014} applied EEG in a smaller sample ($N$ = 9) of musically trained listeners and revealed a close relation between their subject's consonance/dissonance ratings and the morphology of the POR that was elicited by the respective dyads in alHG. In their study, the POR latency difference between consonant and dissonant dyads (cf. their figure\,4B) appears to have non-significant ($p = 0.22$) effect size, smaller than the results that were obtained in our study by means of MEG.
One reason for this might be that Bidelman and Grall~\cite{Bidelman2014} applied shorter IRN stimuli with a higher number of iterations, resulting in an increased saliency of the pitch percept; moreover, they employed a dichotic stimulation paradigm in which each ear was presented with only one dyad component, whereas in our experiment, sounds were delivered diotically to the listeners. On the other hand, the POR was found to originate from very similar locations in alHG, in both our MEG experiment and the work of Bidelman and Grall~\cite{Bidelman2014}; this is consistent with earlier fMRI \cite{Patterson2002} and intracranial \cite{Schoenwiesner2008} studies, as well as with other MEG studies that have linked subcomponents of the N100 wave to pitch processing (e.g., \cite{Gutschalk2004, Ritter2005, Andermann2014, Andermann2017}).
\subsection{Relation to previous models of pitch processing}
Previous studies have introduced a wide range of phenomenological models that were originally designed to predict the pitch of complex sounds (e.g.,~\cite{Licklider1951, Patterson1994a, Meddis1997, DeCheveigne1998, Balaguer2007, Balaguer2008, Balaguer2009}, see~\cite{DeCheveigne2005b} for a review). The correlation between pitch and cortical AEFs was previously addressed by the Auditory Image Model's \emph{buffer} \cite{Patterson1994a} and its derivative \cite{Gutschalk2007}; and the derivative of the activation in the population encoding the pitch in \cite{Balaguer2009, Tabas2016}. However, these models do not provide the biophysical mechanisms underlying the generation of the POR, or its latency dependence with pitch.
Other models, designed to explain the precise biophysical mechanisms of pitch perception, were primarily focused on subcortical or early processing. Two of these models describe how neurons, mainly in subcortical nuclei, might process periodicities from the auditory nerve activity: Meddis and O'Mard's model \cite{Meddis2006} proposes a biophysical implementation of the summary autocorrelation function \cite{Licklider1951, Meddis1997}, based on the joint action of chopper neurons in the cochlear nucleus and coincidence detectors in the inferior colliculus. More recently, Huang and Rinzel \cite{Huang2016} describe the neural implementation of a coincidence detector which is able to detect periodicities by comparing neural activity across different cochlear channels. Despite their mechanistic differences, both models present a spectral output comparable to that of the autocorrelation function \cite{Huang2016}. The model presented here is downstream, with respect to Meddis' and Huang's models, with its focus on explaining how pitch-related information is extracted from spectral patterns in alHG. Neurons implementing this transformation have been recently observed in the primary auditory cortex of marmosets \cite{Feng2017}.
\subsection{Relation to previous models of sensory integration}
The decoder network dynamics can be understood as an extension of the decision making model of the well-known winner-take-all ensemble competition \cite{Wang2002, Wong2006}: excitatory populations in the decoding network compete with each other, whilst the inhibitory ensemble that is arbitrating this selective competition is in the column sensitive to the fundamental (see the inhibitory connections in Figure~\ref{fig:mod:diagram}). Since competition is restricted to harmonically related populations, multiple fundamentals can be simultaneously processed by the decoder network in cases where the input presents concurrent pitch values (e.g., Figure~\ref{fig:mod:dyads}).
The hierarchical structure of our model is inspired by a recent model that was designed to explain sensory integration in the medial temporal area of the macaque monkey \cite{Wimmer2015}. In that model, perceptual decisions are first transiently computed in a \emph{sensory circuit} that follows winner-take-all dynamics; after convergence, the decision then propagates to an \emph{integration circuit}, that further modulates the dynamics of the sensory circuit, thus ensuring stability until a significant change occurs in the input to the cortex. Similarly, once a pitch value has been extracted in the decoder network of our model, the activity of the winner column is reinforced by the sustainer network (rather than being repeatedly decoded) until a new change in the subcortical input triggers a new decoding process (see Supplementary Figure~S9).
The sustaining strategy is also reminiscent of \emph{predictive coding} \cite{Auksztulewicz2017, Friston2005, Friston2009} and reversed hierarchical strategies \cite{Hochstein2002, Balaguer2009}, where top-down efferents convey expectations about the input (here: expectations regarding its harmonic structure), whereas bottom-up afferents convey prediction error (peaks not corresponding to the expected harmonic series) \cite{Rauss2013}. Additional top-down expectations could coexist at higher cognitive levels based on, for instance, prior knowledge, experience, or focused attention. Such biases could modulate the sustaining network by increasing the baseline activity of the inhibitory ensembles that characterize the target pitch values, thereby facilitating the extraction of that \emph{privileged} pitch in the decoder.
To summarize, in this study we have introduced a novel model designed to understand the neural mechanisms of cortical pitch processing. The model proposes a mechanistic link between the latency of the POR subcomponent of the N100 wave and the processing time required for the system to achieve convergence, explaining the classical result that tones with a lower pitch elicit PORs with longer latencies. Moreover, our modelling and experimental results indicate that processing time varies with the degree of consonance in dyads, suggesting that the sensation of consonance and dissonance might directly stem from cortical pitch processing.
\section{Methods}
\subsection{Experimentation}
\subsubsection{Participants}
Thirty-seven normal-hearing adults (22 female, 2 left-handed; mean age: 29.1 $\pm$ 8.3 years) participated in the experiment. None of the subjects reported any history of central or peripheral hearing impairments or any neurological or psychiatric disorders. The study and the experimental procedures were approved by the ethics committee of the Heidelberg University's Medical School, and were conducted with written informed consent of each listener.
\subsubsection{Stimuli}
All stimuli were generated on-line using MATLAB 7.1 (The MathWorks, Inc., USA) and a sampling rate of 48000\,Hz. The basic stimulus was a 750\,ms long IRN segment, bandpass-filtered at 125--2000\,Hz, with eight iterations and gain for the delay-and-add filter $g_f = 1$. The delay of the IRN was varied between experimental conditions in an effort to build three consonant and three dissonant musical intervals, as classified by Western music theory. The delay of the lower note was always 6.25\,ms, corresponding to a pitch of 160\,Hz; the delay of the upper note was adjusted accordingly to form either a consonant dyad (unison, P1; perfect fifth, P5; major third, M3) or a dissonant dyad (tritone, TT; minor seventh, m7; minor second, m2). Table~\ref{tab:expCond} presents an overview of the six experimental conditions.
In order to separate the dyad-specific neuromagnetic responses from the cortical activity associated with the onset of sound energy \cite{Biermann2000, Gutschalk2002}, each IRN dyad was preceded by a 750\,ms long, energy-balanced noise segment (bandpass-filtered at 125--2000\,Hz). There were 10\,ms Hanning windows at stimulus onset and offset; moreover, between the first (noise) and the second (IRN) segment of a stimulus, signals were cross-faded for a duration of 10\,ms in an effort to avoid discontinuous waveforms. The overall stimulation level was set to 80\,dB SPL.
\begin{table}
\centering
\input{./tab-stimuli.addtex}
\caption{Overview of the experimental conditions. Dyads are listed in descending consonance order, and are categorized as Perfect consonant (PC), imperfect consonant (IC) or dissonant (D) according to Western music theory and empirical results \cite{Schwartz2003, Itoh2010}.}
\label{tab:expCond}
\end{table}
\subsubsection{Data acquisition and processing}
Gradients of the magnetic field were acquired with a Neuromag-122 whole-head MEG system (Elekta Neuromag Oy, Helsinki, Finland) inside a magnetically shielded room (IMEDCO, H\"{a}gendorf, Switzerland). Raw data were acquired at a sampling rate of 1000\,Hz and low-pass filtered at 330\,Hz. Prior to the recordings, the nasion, two pre-auricular points and 32 surface points were measured as anatomical landmarks, individually for each participant, using a Polhemus 3D-Space Isotrack2 systems. In an effort to keep vigilance stable, participants watched a silent movie of their own choice during data acquisition, and they were asked to direct their attention to the movie and ignore the sounds in the earphones. The IRN dyads were delivered to the subjects via Etymotic Research (ER3) earphones with 90\,cm plastic tubes and foam earpieces. Sounds were presented using a 24-bit sound card (RME ADI 8DS AD/DA interface), an attenuator (Tucker-Davis Technologies PA-5) and a headphone buffer (Tucker-Davis Technologies HB-7). 250 sweeps per stimulus condition were played during the MEG recording, diotically and in pseudo-randomized order. The inter-stimulus interval was 1000\,ms. The total duration of the measurement was 62 minutes.
\subsubsection{Data analysis}
Data were analyzed off-line using the BESA 5.2 software package (BESA GmbH, Germany) with a spherical head model and a homogeneous volume conductor. After visual inspection of the raw data, noisy channels and sweeps with amplitudes greater than 8000\,fT/cm or gradients exceeding 800\,fT/cm/ms were excluded from further analyses. About 235 sweeps per subject and condition remained after artifact rejection; they were averaged, trigger-synchronously, in the epoch from 500\,ms before to 3000\,ms after stimulus onset. The baseline was defined as the average level in the interval of -100\,ms to 0\,ms, relative to stimulus onset.
After pre-processing, we applied spatio-temporal source models \cite{Scherg1989, Scherg1990, Scherg1991} in BESA, in an effort to study the POR component in response to the second stimulus segment, i.e., at the transition from noise to IRN dyads. In this source localization approach, the intracortical sources of the activity observed at the scalp are modeled as equivalent current dipoles, and their spatial position and orientation is varied iteratively until a maximum amount of variance is explained in the scalp data. The source model includes both the spatial information for each dipole, and its physiological activity across time (source waveform). We calculated source models with one dipole per hemisphere for the POR component in the second stimulus segment. Dipole fits were based on pooled conditions [P1+P5+M3+TT+m7+m2]. The fitting interval covered about 30\,ms around its peak, and MEG data were zero-phase filtered 2--20\,Hz.
Individual fits at the AEF components were successful for 36 subjects. In ten participants we included a symmetry constraint in the model to stabilize the individual dipole fits. One participant failed to show stable fits in the dipole model, and was excluded from subsequent analyses. Aside from symmetry, no further constraints were made concerning the orientation and location of the dipoles. The average maximum of explained variance within the fitting window was 64.1\% (SD: 18.9) for the POR dipole model. After fitting, this dipole model was used as spatio-temporal filter, i.e., the source waveforms corresponding to the model were extracted separately for each condition and each subject. Finally, the source waveforms were exported from BESA to MATLAB for statistical analysis.
The statistical evaluation of the MEG source waveforms was conducted using the bootstrap method \cite{Efron1993}. Here, the distribution of a test statistic is approximated by repeated random drawing, with replacement, from the original dataset; based on the resulting bootstrap distribution, confidence intervals can then be derived for that test statistic. Contrary to most standard techniques, the bootstrap method is well-suited for neurophysiological data where peaks cannot be clearly identified for each participant in every condition. Prior to statistical analyses, each source waveform of the POR model was adjusted to the baseline calculated as the average of the last 100\,ms before the transition.
\subsection{Modelling}
\subsubsection{Peripheral model and periodicity detectors}
Neural activity at the auditory nerve was simulated using a recent biophysically realistic model of the auditory periphery \cite{Zilany2009, Zilany2014}. Peripheral parameters were chosen as in~\cite{Meddis2006}, considering 40 cochlear channels with centre frequencies between 125\,Hz and 10\,kHz.
Periodicity detectors were modeled according to the summarized autocorrelation function (SACF) of the auditory nerve activity \cite{Meddis1997, Meddis2006, Balaguer2008}. This highly idealized model yields a harmonic neural representation of pitch-related information (see Figure~\ref{fig:mod:diagram}e), often connected to subcortical processing. The SACF was chosen for its comparably low computational complexity, but more detailed biophysical models yielding similar representations (e.g.~\cite{Meddis2006, Huang2016}) should produce comparable results.
The SACF used here follows the same formulation as the first stage in the cascade autocorrelation model \cite{Balaguer2008}. The $n$th component $A_n(t)$ of the SACF represents a measure of the regularity of the auditory nerve activity with respect to a fixed period $\delta t_n$. The model considers $N = 250$ components $A_n(t)$ with characteristic periods uniformly spaced between $\delta t_1 = 0.5\,$ms, a conservative estimation of the phase-locking limit of the auditory nerve \cite{Bendor2012a}, up to the lower limit of melodic pitch, $\delta t_N = 30$\,ms \cite{Pressnitzer2001}.
The output is further regularized through a procedure $A_n(t) \rightarrow \hat{A}_n(t)$ that reduces the dependence of the SACF with stimulus intensity level and minimizes signal-to-noise variations within sounds with the same pitch but different timbre. The regularization procedure makes use of the principles of neural normalization \cite{Carandini2012} (see details in Supplementary Methods).
\subsubsection{Ensemble dynamics}
Neural ensembles follow mean-field dynamics characterised by their instantaneous firing rate $H^e_n(t)$ (excitatory) and $H^i_n(t)$ (inhibitory), at each cortical column $n$. Evolution dynamics were adapted from~\cite{Wong2006}:
\begin{equation}
\tau^{\text{pop}} \, \dot{H}^{e,i}_n(t) = - H^{e,i}_n(t) + \phi^{e,i}(I^{e,i}_n(t))
\label{eq:Hei}
\end{equation}
\noindent with transfer functions $\phi^{e,i}(I^{e,i}_n(t))$ \cite{Wong2006}:
\begin{equation}
\phi^{e,i}(I) = \frac{a^{e,i} I - b^{e,i}}{1 - e^{-d^{e,i} (a^{e,i} I - b^{e,i})}}
\label{eq:transfer}
\end{equation}
Realistic parameters of the excitatory and inhibitory transfer functions ($a^e$, $b^e$ and $d^e$ for the excitatory; $a^i$, $b^i$ and $d^i$ for the inhibitory) were taken from the literature \cite{Brunel2001, Wong2006}. The total synaptic inputs $I^e_n(t)$ and $I^i_n(t)$ are defined below. Numerically simulations were performed using the Euler's method with a time step $\Delta t = 1$\,ms.
Dynamics of excitatory and inhibitory ensembles at the \emph{decoder} and \emph{sustainer} networks follow the same formulation. In order to differentiate between the two cortical networks, we use $H^{e,i}_n(t)$ and $I^{e,i}_n(t)$ to characterize populations and synaptic inputs of the decoder layer, and $\hat{H}^{e, i}_n(t)$ and $\hat{I}^{e,i}_n(t)$ for the populations and synaptic inputs of the sustainer layer. Population effective time constants $\tau^{\text{pop}}$ are adaptive and depend on the activity of the population \cite{Ostojic2011, Gerstner2014}:
\begin{equation}
\tau^{\text{pop}}(H(t)) = \tau^{\text{pop}}_0 \, \Delta_T \frac{\phi'(I(t))}{H(t)}
\label{eq:taupop}
\end{equation}
\noindent where $\phi'(I(t))$ is the slope of the transfer function (see Equation~\ref{eq:transfer}) and $\Delta_T$ is the sharpness of the action potential initiation. The mean-field dynamics of the populations in our model was based on a LIF neuron \cite{Wong2006} that approximates the action potential initiation as instantaneous \cite{Fourcaud2003}; thus, we used a small $\Delta_T \ll 1 = 0.05$\,mV.
\subsubsection{Synaptic dynamics}
Ensemble connectivity is mediated through realistic AMPA, NMDA and GABA$_\text{A}$ synapses \cite{Brunel2001, Wang2002, Wong2006, Deco2013}. Synaptic dynamics were modelled according to Brunel and Wang formulation \cite{Brunel2001}:
\begin{eqnarray}
\dot{S}_n^{\text{AMPA}}(t) & = &
- \frac{S_n^{\text{AMPA}}(t)}{\tau_{\text{AMPA}}}
+ H_n^e(t) + \sigma \nu_n(t) \label{eq:Sampa} \\
\dot{S}_n^{\text{GABA}}(t) & = &
- \frac{S_n^{\text{GABA}}(t)}{\tau_{\text{GABA}}}
+ H_n^i(t) + \sigma \nu_n(t) \label{eq:Sgaba} \\
\dot{S}_n^{\text{NMDA}}(t) & = &
- \frac{S_n^{\text{NMDA}}(t)}{\tau_{\text{NMDA}}}
+ \gamma \left(1 - S_{\text{NMDA}}(t)\right) H_n^e(t)
+ \sigma \nu_n(t) \label{eq:Snmda}
\end{eqnarray}
NMDA time constant was set to $\tau_{\text{NMDA}} = 30\,$ms; GABA and AMPA time constants $\tau_{\text{GABA}} = 2\,$ms and $\tau_{\text{AMPA}} = 5\,$ms, and the coupling parameter $\gamma = 0.641$, were all taken from the literature \cite{Wong2006, Brunel2001}. Gating variables at the sustainer and decoder layers $\hat{S}_n^{\text{NMDA, AMPA, GABA}}(t)$, $\hat{H}^{e,i}_n(t)$ follow similar dynamics.
\subsubsection{Synaptic inputs}
Total synaptic inputs to populations $I^{i,e}_n(t)$ and $\hat{I}^{i,e}_n(t)$ consist of three different contributions: internal input $I_{\text{int}}$, accounting for inputs from populations within the same network; external input $I_{\text{ext}}$, exerted by sources from other networks; and a constant input drive $I_0$:
\begin{eqnarray}
I^{i,e}_n(t) & = & I^{i,e}_{n, \text{int}}(t) +
I^{i,e}_{n, \text{ext}}(t) +
I^{i,e}_{n, 0}(t)
\label{eq:totalI:dec} \\
\hat{I}^{i,e}_n(t) & = & \hat{I}^{i,e}_{n, \text{int}}(t) +
\hat{I}^{i,e}_{n, \text{ext}}(t) +
\hat{I}^{i,e}_{n, 0}(t)
\label{eq:totalI:sus}
\end{eqnarray}
\paragraph{Internal input} Connectivity weights between any two ensembles in the decoder network are provided by the matrices $C^{ee}, C^{ei}, C^{ie}, C^{ii}$. $C^{ei}$ and $C^{ie}$. $C^{ei}$ and $C^{ie}$ present a harmonic structure inspired in connectivity patterns reported in the mammal auditory cortex (see \cite{Wang2013} for a review); this matrices are plotted in Figure~\ref{fig:mod:diagram}d. $C^{ee}$ is the identity matrix, and $C^{ii}$ has a similar diagonal structure: $C^{ii}_{\alpha\beta} = (1 - c^{ie}_0) \delta_{\alpha\beta} + c^{ie}_0$, where $c^{ie}_0$ is the baseline inhibitory weight $c^{ie}_0 = 0.1$ and $\delta_{\alpha\beta}$ is the Kronecker delta.
Altogether, the internal inputs at the decoder $I_{\text{int}}(t)$ are defined as follows:
\begin{eqnarray}
I^e_{n, \text{int}}(t) & = & \sum_k C^{ee}_{nk} \left(
J^{ee}_{\text{NMDA}} \, S_k^{\text{NMDA}}(t) +
J^{ee}_{\text{AMPA}} \, S_k^{\text{AMPA}}(t)
\right) - \sum_k C^{ie}_{nk}
J^{ie}_{\text{GABA}} \, S_k^{\text{GABA}}(t)
\label{eq:Iint:edec} \\
I^i_{n, \text{int}}(t) & = & \sum_k C^{ei}_{nk} \left(
J^{ie}_{\text{NMDA}} \, S_k^{\text{NMDA}}(t) +
J^{ei}_{\text{AMPA}} \, S_k^{\text{AMPA}}(t)
\right) - \sum_k C^{ii}_{nk}
J^{ii}_{\text{GABA}} \, S_k^{\text{GABA}}(t)
\label{eq:Iint:idec}
\end{eqnarray}
Ensembles ni the sustainer network only communicate internally with ensembles within the same block:
\begin{eqnarray}
\hat{I}^e_{n, \text{int}}(t) & = &
\hat{J}^{ee}_{\text{NMDA}} \, \hat{S}_n^{\text{NMDA}}(t) +
\hat{J}^{ee}_{\text{AMPA}} \, \hat{S}_n^{\text{AMPA}}(t) -
\hat{J}^{ie}_{\text{GABA}} \, \hat{S}_n^{\text{GABA}}(t)
\label{eq:Iint:esus} \\
\hat{I}^i_{n, \text{int}}(t) & = &
\hat{J}^{ei}_{\text{NMDA}} \, \hat{S}_n^{\text{NMDA}}(t) +
\hat{J}^{ei}_{\text{AMPA}} \, \hat{S}_n^{\text{AMPA}}(t) -
\hat{J}^{ii}_{\text{GABA}} \, \hat{S}_n^{\text{GABA}}(t)
\label{eq:Iint:isus}
\end{eqnarray}
\noindent Conductivities $J_{\text{NMDA}, \text{AMPA}, \text{GABA}}$ and $\hat{J}_{\text{NMDA}, \text{AMPA}, \text{GABA}}$ (see Table~\ref{tab:pars}) were initialized to typical values in the literature $J \simeq 0.15\,$nA \cite{Wong2006}, and fine-tuned within a range of realistic values to ensure the convergence of the ensembles activity to match perceptual results in iterated rippled noises \cite{Krumbholz2003}. Model's final parameters are listed in Table~\ref{tab:pars}.
\begin{table}[p]
\centering
\input{./tab-pars.addtex}
\caption{\textbf{Values for the parameters used in the cortical model} source in the literature.}{The Last column specifies the source of the parameter value; entries with the label \emph{fitted} were tuned according to the indications described in the main text; entries with the label \emph{fixed} were selected to a fixed value according to theoretical considerations (see main text).}
\label{tab:pars}
\end{table}
\paragraph{External input} Excitatory ensembles in the decoder network receive bottom-up input $\hat{A}_n(t)$ via AMPA-driven synapses, according to previous studies in perceptual integration \cite{Wong2006}:
\begin{equation}
I^e_{n, \text{ext}}(t) = J^{th}_{\text{AMPA}} \,
S_n^{th, \text{AMPA}}(t) \label{eq:Iext:esus}
\end{equation}
\noindent The conductivity $J^{th}_{\text{AMPA}}$ was adjusted to ensure a smooth and robust propagation of the activity in the periodicity detectors to the decoder's excitatory populations. The corresponding gating variables $S_n^{th, \text{AMPA}}(t)$ follow AMPA-like dynamics:
\begin{equation}
\dot{S}_n^{th, \text{AMPA}}(t) =- \frac{S_n^{th, \text{AMPA}}(t)}
{\tau_{\text{AMPA}}} + A_n(t) \label{eq:SampaTh}
\end{equation}
Inhibitory ensembles in the decoder receive efferent external input from the sustainer network. Top-down excitatory processes in cortex are typically dominated by NMDA dynamics \cite{Friston2005}; thus, efferent AMPA synapses were not considered:
\begin{equation}
I^i_{n, \text{ext}}(t) = J^{e}_{\text{NMDA}} \,
\hat{S}_n^{th, \text{NMDA}}(t) \label{eq:isus}
\end{equation}
The efferent conductivity $J^{e}_{\text{NMDA}}$ (Table~\ref{tab:pars}) was tuned to enable the top-down enhancement of the inhibitory ensembles at the decoder once the lower harmonics that have been inhibited after the decoding process (see \emph{Dynamics of the decoder network} in Results).
Sustainer's external inputs are sourced in the decoder network, driven by inhibitory GABAergic and excitatory AMPAergic synapses \cite{Wong2006, Friston2005}:
\begin{eqnarray}
\hat{I}^e_{n, \text{ext}}(t) & = &
\hat{J}^{a}_{\text{AMPA}} \, S_n^{\text{AMPA}}(t)
\label{eq:hIext:esus} \\
\hat{I}^i_{n, \text{ext}}(t) & = &
\hat{J}^{a}_{\text{GABA}} \, S_n^{\text{GABA}}(t)
\label{eq:hIext:isus}
\end{eqnarray}
Afferent conductivities $\hat{J}^{a}_{\text{AMPA, GABA}}$ (Table~\ref{tab:pars}) were set to make the sustainer both sensitive to decoded decisions, yet robust to spurious activations.
\paragraph{Constant input drive} Constant inputs to the decoder $I^e_{n, 0}(t) = I^e_0$ and $I^i_{n, 0}(t) = I^i_0$ (Table~\ref{tab:pars}) were selected to enable the system to be reactive to external input, yet silent in absence of a significant input. An additional constant drive $I^{\text{sus}}_0 = 0.24\,$nA was applied to the populations at the sustainer (see \emph{Dynamics of the sustainer network} in Results).
\subsubsection{Derivation of the evoked fields}
Assuming that all microcolumns within each of the two cortical networks present similar orientations, the total dipolar moment representing the neuromagnetic field elicit by each network is monotonicallt related to the collective excitatory activity in the network \cite{Kiebel2008, Kerr2008, Bruyns2016}:
\begin{equation}
m(t) = \sum_n H^e_n(t + \Delta t_{\text{subcort}})
\end{equation}
The subcortical delay $\Delta t$ accounts for the time elapsed from tone onset until the signal first arrives in the decoder network. This delay reflects not only propagation time, but also processing time of secondary processes like the regularization of the output of the periodicity detectors.
The delay was fixed to $\Delta t = 75\,$ms, according to the experimentally observed latency of the POR elicited by an IRN with a delay of 8\,ms. We used a longer $\Delta t^{\text{dyads}} = 95\,$ms in dyads to compensate for a systematic 20\,ms delay of the experimental observations, most likely due to the different rescaling factors used for the regularized SACF of simple tones and dyads (see Supplementary Methods for details).
| {
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} | 2,133 |
Behdad Shambayati
学術・一般書 > 技術 > Medicine & health > Human anatomy, cytology, histology
Fundamentals of Biomedical Science
Cytopathology provides a broad-ranging overview of the microscopic study of normal and abnormal cells, which embraces the latest imaging and visualization methods to study the structure of cells. The full colour presentation features over 400 figures. Written with the needs of the biomedical scientist centre-stage, it provides a firm grounding in normal cell structure, and the abnormal features that are indicative of different clinical conditions. It also explains how screening programmes can be used to detect changes early, giving an invaluable opportunity for treatment regimes to be implemented in a timely way. Crucially, it demonstrates throughout how an understanding of cellular physiology underpins the key investigations carried out by a biomedical scientist to forge a clear link between science and practice. The series is developed in partnership with and endorsed by the Institute of Biomedical Science. See ibms.org for more information. Online Resource Centre The Online Resource Centre to accompany Cytopathology features: For registered adopters of the book: * Figures from the book, available to download For students: * An interactive Digital Microscope, encouraging the exploration of tissue samples * Self-assessment modules to help you to check and reinforce understanding of the basic science introduced in each volume * Video podcasts including interviews with practicing biomedical scientists, and 'in the lab' footage showing biomedical science in practice The Fundamentals of Biomedical Science series is written to reflect the challenges of practicing biomedical science today. It draws together essential basic science with insights into laboratory practice to show how an understanding of the biology of disease is coupled to the analytical approaches that lead to diagnosis. Assuming only a minimum of prior knowledge, the series reviews the full range of disciplines to which a Biomedical Scientist may be exposed - from microbiology to cytopathology to transfusion science. Alongside volumes exploring specific biomedical themes and related laboratory diagnosis, an overarching Biomedical Science Practice volume gives a grounding in the professional and experimental skills with which any Biomedical Scientist must be equipped.
1. Introduction: A glance at the past, a glimpse of the future
2. Preparation techniques
3. The Cervical Screening Process
4. Normal cervical cytology
5. Abnormal cervical cytology
6. Quality Issues in cervical screening and cytology
7. Diagnostic cytopathology
8. Cytology of Urine
9. Serous effusions
10. Lower respiratory tract cytology
11. Fine needle aspiration cytology
12. Basic Semen Analysis
13. Advances in Cytopathology
Behdad Shambayati is a Consultant Clinical Cytologist at Ashford and St. Peter's Cytology Department. Behdad was part time lecturer on the IBMS fellowship course at Bromley College of Technology from 1992 to 1996 (when the course ended), and is periodically invited to lecture for the BSc and MSc courses at University of Westminster, NESCOT and University of West of England. He regularly teaches Cytology on courses at Northwick Park Cytology Training School, Welsh Cytology Training School, Sheffield Cytology Training School and South West Cytology Training School. Behdad has been a member of the IBMS Scientific Advisory Panel in cytology since 1999, and is currently their Chief Examiner for cytology.
The Guide to Interpersonal Psychotherapy (Updated and Expanded Edition)
Cunningham's Manual of Practical Anatomy VOL 3 Head, Neck and Brain (16th edition)
Neuroanatomical Terminology: A Lexicon of Classical Origins and Historical Foundations
Anatomy for Dental Students (4th edition)
Oxford Handbook of Genitourinary Medicine, HIV, and Sexual Health (3rd edition)
The Human Eye: Structure and Function
The Deeper Genome: Why There is More to the Human Genome Than Meets the Eye
Atlas of Descriptive Histology
Malleable Anatomies: Models, Makers, and Material Culture in Eighteenth-Century Italy | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 311 |
{"url":"https:\/\/iacr.org\/cryptodb\/data\/paper.php?pubkey=11322","text":"## CryptoDB\n\n### Paper: Comparing Entropies in Statistical Zero-Knowledge with Applications to the Structure of SZK\n\nAuthors: Oded Goldreich Salil P. Vadhan URL: http:\/\/eprint.iacr.org\/1998\/026 Search ePrint Search Google We consider the following (promise) problem, denoted ED (for Entropy Difference): The input is a pairs of circuits, and YES instances (resp., NO instances) are such pairs in which the first (resp., second) circuit generates a distribution with noticeably higher entropy. On one hand we show that any language having a (honest-verifier) statistical zero-knowledge proof is Karp-reducible to ED. On the other hand, we present a public-coin (honest-verifier) statistical zero-knowledge proof for ED. Thus, we obtain an alternative proof of Okamoto's result by which HVSZK (i.e., Honest-Verifier Statistical Zero-Knowledge) equals public-coin HVSZK. The new proof is much simpler than the original one. The above also yields a trivial proof that HVSZK is closed under complementation (since ED easily reduces to its complement). Among the new results obtained is an equivalence of a weak notion of statistical zero-knowledge to the standard one.\n##### BibTeX\n@misc{eprint-1998-11322,\ntitle={Comparing Entropies in Statistical Zero-Knowledge with Applications to the Structure of SZK},\nbooktitle={IACR Eprint archive},\nkeywords={Zero-Knowledge, Universal Hashing},\nurl={http:\/\/eprint.iacr.org\/1998\/026},\nnote={Appeared in the THEORY OF CRYPTOGRAPHY LIBRARY and has been included in the ePrint Archive. salil@theory.lcs.mit.edu 10500 received Dec 24th, 1998.},\nauthor={Oded Goldreich and Salil P. Vadhan},\nyear=1998\n}","date":"2021-10-19 06:30:03","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\": 0.7741116285324097, \"perplexity\": 3049.7756670605263}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585242.44\/warc\/CC-MAIN-20211019043325-20211019073325-00120.warc.gz\"}"} | null | null |
\section{Introduction}
\subsection{Background}
We are interested in the continued fractions and Stieltjes continued fractions
defined by automatic sequences in finite characteristic, and more precisely their algebraicity or transcendence. We give here the background and motivation
for studying such problems. The definitions of related notions will given in subsection
\ref{prel}.
The link of automaticity and algebraicity goes back to the well-known Theorem
of Christol, Kamae, Mend\`es France and Rauzy \cite{Christol1980KMFR}
which states that
a formal power series in $\mathbb{F}_q[[x]]$ is algebraic over $\mathbb{F}_q(x)$
if and only if the sequence of its coefficients is $q$-automatic.
The situation is completely different for real numbers.
In 2007, Adamczewski and Bugeaud \cite{Adamczewski2007B} proved that for an integer
$b\geq 2$, if
the $b$-ary expansion of an irrational real number $u$ form an automatic
sequence, then $u$ must be transcendental.
In 2013, Bugeaud \cite{Bugeaud2013} proved that the continued fraction
expansion of an algebraic real number of degree at least $3$ is not automatic.
As with real numbers, a formal Laurent series can also be represented by
a continued fraction whose partial quotients are polynomials.
Unlike for real numbers, the continued fraction expansion of an algebraic
Laurent series of degree at least $3$ may or may not have automatic partial
quotients \cite{Baum1976S, Baum1977S, Mills1986R, Allouche1988, Mkaouar1995,
Lasjaunias2015Y, Lasjaunias2016Y, Lasjaunias2017Y}; see also the introduction
of \cite{hh2}.
We could also ask the converse question: what can we say about
the algebraicity of a continued fraction whose partial quotients form an
automatic sequence? To our knowledge, little has been done in this direction.
The authors \cite{hh2} proved that the Stieltjes continued
fractions defined by the Thue-Morse sequence and the period-doubling sequence
in $\mathbb{Z}[[x]]$ are congruent, modulo $4$, to algebraic series
in $\mathbb{Z}[[x]]$.
In 2020, Wu \cite{Wu2020} obtained similar results concerning the Stieltjes
continued fractions defined by the paperfolding sequence and the Golay-Shaprio-Rudin sequence.
In this article we propose to approach this problem with the most classical
example of automatic sequences, the Thue-Morse sequence.
\subsection{Preliminaries}\label{prel}
We introduce the necessary notions for stating the conjectures and our main results.
\subsubsection{Automatic sequnces}
A sequence is said to be {\it $k$-automatic} if it can be generated by a $k$-DFAO ({\it deterministic finite automaton with output}).
For an integer $k\geq 2$, a $k$-DFAO
is defined to be a $6$-tuple
$$M=(Q,\Sigma, \delta, q_0, \Delta, \tau)$$
where $Q$ is the set of states with $q_0\in Q$ being the initial state, $\Sigma=\{0,1,\ldots,k-1\}$ the input alphabet, $\delta:Q\times \Sigma\rightarrow Q$ the transition function, $\Delta$ the output alphabet, and $\tau:Q\rightarrow \Delta$ the output function.
The $k$-DFAO $M$ generates a sequence $(c_n)_{n\geq 0}$ in the following way: for each non-negative integer $n$, the base-$k$ expansion of $n$ is read by $M$ from right to left starting from the initial state $q_0$, and the automaton moves from state to state according to its transition funciton $\delta$. When the end of the string is reached, the automaton halts in a state $q$, and the automaton outputs the symbol $c_n=\tau (q)$.
A necessary and sufficient condition \cite{Eilenberg1974} for a sequence to be
$k$-automatic is that its {\it $k$-kernel}, defined as
$$\{ (u_{k^d n+j})_{n\geq 0} \mid d\in \mathbb{N},\, 0\leq j\leq k^d-1 \},$$
is finite.
If we let $\Lambda_{i}^{(k)}$ denote the operator that sends a
sequence $(u(n))_{n\geq 0}$ to its subsequence $(u(k n+i))_{n\geq 0}$,
then the $k$-kernel can be defined alternatively
as the smallest set containing $\bf{u}$ that is stable under $\Lambda_i^{(k)}$ for $0\leq i<k-1$. We write $\Lambda_i$ instead of $\Lambda_i^{(k)}$ when the value of $k$ is clear from the context.
We will use the fact that for an integer $m\geq 1$, a sequence is $k$-automatic
if and only if it is $k^m$-automatic \cite{Eilenberg1974}.
For a $k$-automatic sequence $\mathbf{u}$, we can construct a $k$-DFAO
that generates it from its $k$-kernel. The set of states $Q$ will be in
bijection with the $k$-kernel, so we choose to identify them.
For $q\in Q$ and $0\leq j< k$, the
value of the transition function $\delta(q,j)$ is defined as $\Lambda_j q$;
the output function $\tau$ maps $q$ to the $0$-th term of $q$.
This automaton has the property that leading $0$'s in the input
does not change the output. It is minimal
among $k$-automata with this property that generates $\mathbf{u}$.
We refer the readers to \cite{Allouche2003Sh} for a comprehensive exposition
of automatic sequences.
In this article we will consider the Thue-Morse sequence and the period-doubling
sequence. For two distinct element $a$ and $b$ from an alphabet, the
$(a,b)$-Thue-Morse sequence is the sequence $\mathbf{t}$ defined as
the fixed point $s^\infty(a)$
of the substitution $s: a\mapsto ab,\; b\mapsto ba$;
the $(a,b)$-period-doubling sequence $\mathbf{p}$ is defined as the fixed point
$\sigma(a)$ of the substitution $\sigma: a\mapsto ab,\; b\mapsto aa$.
\subsubsection{Continued fractions}
Let $K$ be a field.
Given a sequence of polynomials $a_j(z)\in K[z]\backslash K$,
we may define the infinite continued fraction
\begin{equation}\label{eq:ncf}
\CF(\mathbf{a}(z)):=\cfrac{1}{a_0(z)+\cfrac{1}{a_1(z)+\cfrac{1}{a_2(z)+\cfrac{1}{\ddots}}}}
\end{equation}
as the limit of the finite continued fractions
\begin{equation}\label{eq:fncf}
\CF_n(\mathbf{a}(z))=\cfrac{1}{a_0(z)+\cfrac{1}{a_1(z)+\cfrac{1}{\ddots+\cfrac{1}{a_n(z)}}}}
\in K((1/z)).
\end{equation}
The existence of the limit is guaranteed by the convergence theorem,
whose proof is completely analogous to that for the classical continued fracions
with positive integer partial quotients.
Define the sequences $(P_n(z))$ and $(Q_n(z))$ by
\begin{equation}\label{eq:pq}
\begin{pmatrix} P_n(z)&Q_{n}(z)\\ P_{n-1}(z)&Q_{n-1}(z) \end{pmatrix}
:=
\begin{pmatrix} a_n(z)&1\\ 1&0 \end{pmatrix}
\begin{pmatrix} a_{n-1}(z)&1\\ 1&0 \end{pmatrix}
\cdots
\begin{pmatrix} a_0(z)&1\\ 1&0 \end{pmatrix}
\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}
\end{equation}
for $n\geq 0$,
then
$$\CF_n(z;a,b)=P_n(z)/Q_n(z)\in K((1/z)),$$
for $n\geq 0$.
Note that here rational fractions are expanded in $1/z$.
The unsimplified fraction $P_n(z)/Q_n(z)$ is called the
$n$-th {\it convergent} of $\CF(x;\mathbf{a})$.
\medskip
Conversely, let
\begin{equation}
f(z)=c_nz^n+c_{n-1}z^{n-1}+\cdots+c_0+c_{-1}z^{-1}+\cdots
\end{equation}
be an arbitrary element of $K((1/z))$.
Define the integer part of $f(z)$ as
\begin{equation}
[f(z)]=c_nz^n+c_{n-1}z^{n-1}+\cdots+c_0.
\end{equation}
Set $f_0=f$, $a_0=[f_0]$, $f_0=a_0+1/f_1$, $a_1=[f_1]$, $f_1=a_1+1/f_2$,
$a_2=[f_1]$...
Then $a_0\in K[z]$ and $a_j\in K[z]\backslash K$ for $j\geq 1$, and $f(z)$
admits the following continued fraction expansion
\begin{equation}\label{eq:ncfe}
f(z)=a_0(z)+\cfrac{1}{a_1(z)+\cfrac{1}{a_2(z)+\cfrac{1}{a_3(z)+\cfrac{1}{\ddots}}}}.
\end{equation}
\subsubsection{Stieltjes continued fractions}
Let $(u_j)_{j\geq 0}$ be a sequence taking value in $K^\times$, the Stieltjes continued
fraction
\begin{equation}\label{eq:sfrac}
\Stiel(x;\mathbf{u}):
=\cfrac {u_0}{ 1+ \cfrac{u_1x}{ 1+ \cfrac{u_2x}{ 1+\cfrac {u_3x}{\ddots}}} }
\end{equation}
is defined to be the limit of the finite Stieltjes continued fractions
\begin{equation}\label{eq:tms}
\Stiel_n(x;\mathbf{u}):
=\cfrac {u_0}{ 1+ \cfrac{u_1x}{ 1+ \cfrac{u_2x}{ 1+\cfrac{\ddots} { 1+u_nx}}} }
\in K[[x]].
\end{equation}
It can be easily shown that the sequence $\Stiel_n(x;\mathbf{u})$ is convergent.
Define the sequence $(P_n(x))$ and $(Q_n(x))$ by
\begin{equation}
\begin{pmatrix}P_n(x)&Q_n(x)\\P_{n-1}(x)&Q_{n-1}(x)\end{pmatrix}:=
\begin{pmatrix} 1&u_{n}x\\ 1&0\end{pmatrix}
\begin{pmatrix} 1&u_{n-1}x\\ 1&0\end{pmatrix}
\cdots \begin{pmatrix} 1&u_0x \\1&0\end{pmatrix}
\begin{pmatrix}0&1\\1/x&0\end{pmatrix}
\end{equation}
for $n\geq 0$. Then
$$\Stiel_n(x;\mathbf{u})=\frac{P_n(x)}{Q_n(x)},$$
for $n\geq 0$.
The unsimplified fraction $P_n(x)/Q_n(x)$ is called the
$n$-th {\it convergent} of $\Stiel(x;\mathbf{u})$.
Unlike continued fractions, every formal power series in $K[[x]]$ can not be
expanded as a Stieltjes continued fraction.
\subsection{Conjectures}
We put forward the following conjectures concerning the Thue-Morse
and period-doubling continued
fractions and Stieltjes continued fractions.
\begin{Conjecture}\label{conj:tmncf}
Let $a,b$ be two distinct elements from
$\mathbb{F}_2[z]\backslash\mathbb{F}_2$.
Let $\mathbf{u}(z)$ be the $(a,b)$-Thue-Morse sequence.
The continued fraction $\CF(\mathbf{u}(z))$
is algebraic of degree $4$ over $\mathbb{F}_2(z)$.
\end{Conjecture}
\begin{Conjecture}\label{conj:tms}
Let $k\geq 2$ be an integer. Let $a,b$ be two distinct elements from
$\mathbb{F}_{2^k}^\times$. Let $\mathbf{u}$ be the $(a,b)$-Thue-Morse sequence.
The Stieltjes continued fraction $\Stiel(x;\mathbf{u})$ is algebraic
over $\mathbb{F}_2^k(x)$. Its minimal polynomial is
$$p_0(x)+p_1(x)y+p_2(x)y^2+p_4(x)y^4,$$
where
\begin{align*}
p_0(x)&=(a^{2} b^{4} + b^{6})/{a^{4}} x^{2} + {b^{5}}/(a^{5} + a^{4} b),\\
p_1(x)&=((ab^4 + b^5)/a^5)x + b^4/a^5,\\
p_2(x)&=b^4/a^5x + b^4/(a^6 + a^5b),\\
p_4(x)&=(b^4/(a^6 + a^5b))x^2.
\end{align*}
\end{Conjecture}
\begin{comment}
always minimal?
Answer: Yes.
If not minimal then of degree 3 because quadratic is periodic ( if
f(x)=\Stiel(a;x) is quadratic, then so is f(x^2)=\Stiel(a;x^2), which can be
converted easily to NCF.)
Suppose poly factors into (y+b)(p_4 y^3 +c y^2+d y+ e)
then b p_4+c=0, bc+d=p_2, bd+e=p_1, de=p_0. Therefore b=1,
p_0+p_1+p_2+p_4=0. Not possible because coeff in x^1 not 0.
\end{comment}
Let $\mathbf{v}$ be the $(a/b,1)$-Thue-Morse sequence, then
$$\Stiel(x;\mathbf{u})=b\cdot \Stiel(bx;\mathbf{v}).$$
Therefore conjecture \ref{conj:tms} admits the following equivalent form:
\begin{manualtheorem}{\ref{conj:tms}a}\label{conj:tmsa}
Let $k\geq 2$ be an integer. Let $a$ an elements from $\mathbb{F}_{2^k}^\times$
distinct from $1$. Let $\mathbf{u}$ be the $(a,1)$-Thue-Morse sequence.
The Stieltjes continued fraction $\Stiel(x;\mathbf{u})$ is algebraic
over $\mathbb{F}_2^k(x)$. Its minimal polynomial is
$$p_0(x)+p_1(x)y+p_2(x)y^2+p_4(x)y^4,$$
where
\begin{align*}
p_0(x)&=(a^2 + 1)/a^4)x^2 + 1/(a^5 + a^4),\\
p_1(x)&=((a + 1)/a^5)x + 1/a^5,\\
p_2(x)&=1/a^5x + 1/(a^6 + a^5),\\
p_4(x)&=(1/(a^6 + a^5))x^2.
\end{align*}
\end{manualtheorem}
Or still
\begin{manualtheorem}{\ref{conj:tms}b}\label{conj:tmsb}
We regard $a$ as a formal variable. Let $\mathbf{u}$ be the $(a,1)$-Thue-Morse
sequence. Then
the Stieljtes continued fraction
$\Stiel(x;\mathbf{u})\in \mathbb{F}_2(a)[[x]]$
is algebraic over $\mathbb{F}_2(a)(x)$.
Its minimal polynomial is
$$p_0(x)+p_1(x)y+p_2(x)y^2+p_4(x)y^4,$$
where
\begin{align*}
p_0(x)&=(a^2 + 1)/a^4)x^2 + 1/(a^5 + a^4),\\
p_1(x)&=((a + 1)/a^5)x + 1/a^5,\\
p_2(x)&=1/a^5x + 1/(a^6 + a^5),\\
p_4(x)&=(1/(a^6 + a^5))x^2.
\end{align*}
\end{manualtheorem}
It is clear that conjecture \ref{conj:tmsb} implies conjecture \ref{conj:tmsa},
noticing that the only roots of the denominators of the coefficients of
$p_j(x)$, $j=0,1,2,4$, are $0$ and $1$.
On the other hand, if conjecture \ref{conj:tmsb} does not hold, then
$$0\neq p_0(x)+p_1(x)\Stiel(x;\mathbf{u})+p_2(x)\Stiel(x;\mathbf{u})^2+p_4(x)\Stiel(x;\mathbf{u})^4=:\sum_{n=0}^\infty
c_n(a) x^n,$$
and there exists an $n\in\mathbb{N}$ for which $c_n(a)\in \mathbb{F}_2(a)$
is not the zero. Necessarily there exists a $k\geq 2$ and an element
$u\in \mathbb{F}_{2^k}\backslash\{0,1\}$ that is not a root of the
numerator of $c_n(a)$, and consequently conjecture \ref{conj:tmsa} does not hold for $a=u$.
\medskip
Based on our calculation, we believe that the period-doubling continued
fractions are also algebraic. However, the period-doubling Stieltjes continued
fractions seem to be transcendental.
\begin{Conjecture}\label{conj:pdncf}
Let $a,b$ be two distinct elements from $\mathbb{F}_2[z]\backslash\mathbb{F}_2$.
Let $\mathbf{u}(z)$ be the $(a,b)$-period-doubling sequence.
The continued fraction $\CF(\mathbf{u}(z))$
is algebraic over $\mathbb{F}_2(z)$.
\end{Conjecture}
\begin{Conjecture}\label{conj:pds}
Let $J\in\mathbf{F}_4\backslash \{0,1\}$.
Let $\mathbf{u}$ be the $(1,J)$-period-doubling sequence.
The Stieltjes continued fraction $\Stiel(x;\mathbf{u})$
is transcendental over $\mathbb{F}_2(z)$.
\end{Conjecture}
\subsection{Main results}
We developped a method for checking conjecture \ref{conj:tmncf} and
\ref{conj:tmsa} and implemented it.
Using this method, we checked that conjecture \ref{conj:tmncf} holds for all pairs $(a,b)$ of
elements from $\mathbb{F}_2[x]\backslash \mathbb{F}_2$
such that $\deg a+ \deg b \leq 7$,
and that conjecture \ref{conj:tmsa} holds for all
$a\in \mathbb{F}_{2^k}\backslash\{0,1\}$ for $k=2,3,4$.
For the verification of conjectures \ref{conj:tmncf} and \ref{conj:tmsa},
we use the Guess'n'Prove method.
For conjecture \ref{conj:tmncf}, our program takes
the pair $(a,b)$ as input, and, for the $(a,b)$-Thue-Morse sequence
$\mathbf{u}$, uses the Derksen algorithm for Pad\'e-Hermite approximants
\cite{Bostan2017S} to guess the minimal polynomial of $\CF(\mathbf{u})(z)$.
To prove that the guess is correct, it then guesses and proves several lemmas,
whose forms depend on the choice of $(a,b)$, that would lead to the final
result.
In Section \ref{section:tmncf} we illustrate our method with an example
of computer generated proof.
The proofs for the other pairs that we have tested can be found
on the personal web page of the authors \footnote{\;
\texttt{http://irma.math.unistra.fr/\~{}guoniu/frconj/}}.
For conjecture \ref{conj:tmsa}, the situation is similar, except that
we choose to regard $a$ as a formal variable whenever we can. In this way
we prepare a common part for all $a$, and to prove that conjecture
\ref{conj:tmsa} holds for a certain $a$, we only need to fill in the rest
of the proof for this specific $a$.
In the proofs, we exploit the structure of the automata that generate
the algebraic series in question. For this, we need to first obtain
a $k$-automaton of an algebraic series from an annihilating polynomial of it.
In Section \ref{sec:eq2kern} we explain this part of our program. The algorithm
is based on the proof of theorem 1 of \cite{Christol1980KMFR}.
\medskip
Our method for checking conjecture \ref{conj:tmncf} can be adpated for the
verification of conjecture \ref{conj:pdncf}. We give two examples in
Section \ref{section:pd}.
\section{Thue-Morse Continued Fraction}\label{section:tmncf}
Our program tests conjecture \ref{conj:tmncf} for a given
pair of distinct elements $(a,b)$ from $\F2[x]\backslash \F2$.
We have checked that the conjecture holds in the case
where $\deg a+ \deg b \leq 7$.
The following is an example of proof that conjecture \ref{conj:tmncf}
holds for $(a,b)=(z, z^2 + z + 1)$. Both the statement of the theorem
and its proof are generated automatically by our program.
The exact statement of theorem \ref{th:ab}, lemma \ref{lem:tmncf2} and
\ref{lem:tmncf1} depends on the choice of $(a,b)$.
\subsection{Statement of the theorem for $(a,b)=(z, z^2 + z + 1)$}
\begin{Theorem}\label{th:ab}
Let $(a,b)=(z, z^2 + z + 1)\in (\F2[z]\backslash \F2)^2$. Let $\mathbf{t}$
be the $(a,b)$-Thue-Morse sequence, and $\bar{\mathbf{t}}$, the
$(b,a)$-Thue-Morse sequence.
The two power series $\CF(\mathbf{t}(z))$ and $\CF(\bar{\mathbf{t}}(z))$ are
algebraic over ${\mathbb F}_2(z)$, with minimal polynomials of the form
\begin{align*}
p_4(z)y^4+p_3(z)y^3+p_2(z)y^2+p_1(z)y+p_0(z)=0.
\end{align*}
For $\CF(\mathbf{t}(z))$
\begin{align*}
p_{0}(z)&=z^9 +z^7 +z^6 +z^5 +z^4 +z +1 ,\\
p_{1}(z)&=z^{11} +z^{10} +z^8 +z^6 +z^5 +z^3 +z^2 +z ,\\
p_{2}(z)&=z^{12} +z^{10} +z^2 ,\\
p_{3}(z)&=z^{11} +z^{10} +z^8 +z^6 +z^5 +z^3 +z^2 +z ,\\
p_{4}(z)&=z^{10} +z^9 +z^7 +z^6 +z^5 +z^2 +z ,
\end{align*}
and for $\CF(\bar{\mathbf{t}}(z))$
\begin{align*}
p_{0}(z)&=z^9 +z^8 +z^7 +z^6 +z^5 +z^4 +z ,\\
p_{1}(z)&=z^{11} +z^{10} +z^8 +z^6 +z^5 +z^3 +z^2 +z ,\\
p_{2}(z)&=z^{12} +z^{10} +z^2 ,\\
p_{3}(z)&=z^{11} +z^{10} +z^8 +z^6 +z^5 +z^3 +z^2 +z ,\\
p_{4}(z)&=z^{10} +z^9 +z^8 +z^7 +z^6 +z^5 +z^2 +z +1 .
\end{align*}
\end{Theorem}
\subsection{Proof}
Define
\begin{align*}
M_n(x)&=
x^{\deg\left(t_{2^n-1}\right)}
\begin{pmatrix} t_{2^n-1}(1/x)&1\\1&0\end{pmatrix}
x^{\deg\left(t_{2^n-2}\right)}
\begin{pmatrix} t_{2^n-2}(1/x)&1\\1&0\end{pmatrix}\cdots\\
&\quad x^{\deg\left(t_{0}\right)}
\begin{pmatrix} t_0(1/x)&1\\1&0\end{pmatrix}
\end{align*}
and
\begin{align*}
W_n(x)&=
x^{\deg\left(\bar{t}_{2^n-1}\right)}
\begin{pmatrix} \bar{t}_{2^n-1}(1/x)&1\\1&0\end{pmatrix}
x^{\deg\left(\bar{t}_{2^n-2}\right)}
\begin{pmatrix} \bar{t}_{2^n-2}(1/x)&1\\1&0\end{pmatrix}\cdots\\
&\quad x^{\deg\left(\bar{t}_{0}\right)}
\begin{pmatrix} \bar{t}_0(1/x)&1\\1&0\end{pmatrix}
\end{align*}
where $\mathbf{\bar{t}}$ is the $(b,a)$-Thue-Morse sequence. By the property
of the Thue-Morse sequence, we have for all $n\geq 0$
\begin{align*}
M_{n+1}(x)&=W_n(x)\cdot M_n(x), \\W_{n+1}(x)&=M_n(x)\cdot W_n(x).
\end{align*}
Define $x:=1/z$. For an non-zero polynomial $P(z)$, we define $\tilde{P}(x)$
to be $P(1/x)$. Then
\begin{equation*}
\CF_n(\mathbf{t}(z))=
\frac{P_n(z)}{Q_n(z)}=\frac{\tilde{P}_n(x)}{\tilde{Q}_n(x)}
\in\mathbb{F}_2((x))=\mathbb{F}_2((1/z)).
\end{equation*}
Comparing the definition of $M_n(x)$ with definition \eqref{eq:pq}, we see that
\begin{align*}
M_n(x)_{0,1}&= x^{d_n} \tilde{P}_{2^n-1}(x),\\
M_n(x)_{0,0}&= x^{d_n} \tilde{Q}_{2^n-1}(x),
\end{align*}
for some positive integer $d_n$,
and
\begin{align}\label{eq:cm}
\CF_{{2^{2n}-1}}(\mathbf{t}(z))
=\frac{\tilde{P}_{2^{2n}-1}(x)}{\tilde{Q}_{2^{2n}-1}(x)}
= \frac{M_{2n}(x)_{0,1}}{M_{2n}(x)_{0,0}}.
\end{align}
Our strategy is to first prove that
both $M_{2n}(x)_{0,1}$ and $M_{2n}(x)_{0,0}$ converge to algebraic series in
$\mathbb{F}_2[[x]]$, and then use their minimal polynomials to obtain that
of $ \CF_n(\mathbf{t}(z))$.
Actually, we will prove that for all $0\leq i,j\leq 1$ the four sequences
$(M_{2n}(x)_{i,j})_n$, $(M_{2n+1}(x)_{i,j})_n$, $(W_{2n}(x)_{i,j})_n$, and
$(W_{2n+1}(x)_{i,j})_n$ converge to algebraic series in $\mathbb{F}_2[[x]]$.
For this purpose, we define four $2\times 2$ matrices $M^e, M^o, W^e, W^o$ as
follows: For each $T\in\{M^e, M^o, W^e, W^o\}$ and $0\leq i,j\leq 1$,
$T_{i,j}$ is defined to be the unique solution in $\mathbb{F}_2[[x]]$ of
the polynomial $\phi(T,i,j)$ under certain initial conditions; the polynomials
$\phi(T,i,j)$ and initial conditions are given in Subsection \ref{data:tmncf}.
We will prove that these four matrices, whose components are algebraic
by definition, are the limits of
$(M_{2n}(x))_n$, $(M_{2n+1}(x))_n$, $(W_{2n}(x))_n$, and $(W_{2n+1}(x))_n$.
Let us explain how the polynomials $\phi(T,i,j)$ and initial conditions
are found, and why the solutions exist and are unique.
For $0\leq i,j\leq 1$, the the coefficients of the polynomial
$\phi(M^e,i,j)$
(resp. $\phi(M^o,i,j)$, $\phi(W^e,i,j)$, and $\phi(W^o,i,j)$)
are the Pad\'e-Hermite approximants
of type
$$(75,75,75,75,75)$$
of
the vector
$$(1, f^3, f^6, f^9, f^{12}),$$
where
$f=M_{12,i,j}$ (resp. $M_{11,i,j}$, $W_{12,i,j}$,
and $W_{11,i,j}$). See Chapter 7 of \cite{Bostan2017S} for a description
of the Derksen algorithm that is used here to find the Pad\'e-Hermite
approximants.
We take the first eight terms of $M_{12,i,j}$ (resp. $M_{11,i,j}$,
$W_{12,i,j}$, and $W_{11,i,j}$) as the initial conditions for $\phi(M^e,i,j)$
(resp. $\phi(M^o,i,j)$, $\phi(W^e,i,j)$, and $\phi(W^o,i,j)$).
The following fact will be used to ensure that the solution exists and is
unique:
let $P(x,y)\in\mathbb{F}_2[x,y]$ and for each series
$f(x)=\sum_{n=0}^\infty a_n x^n\in \F2[[x]]$ denote the partial sum
$\sum_{j=0}^{n-1}a_j x^j$ by $f_n(x)$ for $n\geq 0$. If
for some $n\geq 0$ and $a_0, a_1, \ldots, a_{n-1}\in\F2$
$P(x,\sum_{j=0}^{n-1}a_j x^j)=O(x^n)$
and $Q(x,y):= P(x, \sum_{j=0}^{n-1}a_j x^j+x^ny)$ can be written as
$x^m\sum_{j=0}^\infty q_j(x)y^j$ where $q_j(x)$ are polynomials for
$j\geq 0$, $q_1(0)=1$,
and $q_j(0)=0$ for $j>1$, then there exists a unique solution
$f(x)\in \F2[[x]]$ of $P(x,f(x))=0$ that satisfies the initial condition
$f_n(x)=\sum_{j=0}^{n-1}a_j x^j$.
\medskip
We state two lemmas concerning the four matrices $M^e, M^o, W^e, W^o$.
The first one is about relations between them;
the second, about the structure of the each matrix.
\begin{Lemma}\label{lem:tmncf0}
We have
\begin{align}
M^e&=W^o\cdot M^o,\label{eq:me}\\
M^o&=W^e\cdot M^e,\label{eq:mo}\\
W^e&=M^o\cdot W^o,\label{eq:we}\\
W^o&=M^e\cdot W^e.\label{eq:wo}
\end{align}
\end{Lemma}
\begin{proof}
We give the proof of the identity
$$M^e_{0,0}=W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0},$$
the proofs of the others being similar.
First, we compute the minimal polynomials of $W^o_{0,0}M^o_{0,0}$ and
$W^o_{0,1}M^o_{1,0}$.
We know that
$$P(x,y)=\Res_z\left(\phi(W^o,0,0)(x,z),\ z^{12}\cdot \phi(M^o, 0,0)(x,y/z)\right)$$
is an annihilating polynomial of $W^o_{0,0}M^o_{0,0}$; here
$Res_z$ means the resultant with respect to the variable $z$ (
see Chapter 6 of \cite{Bostan2017S}).
We use Pad\'e-Hermite approximation
to find a candidate for the minimal polynomial of $W^o_{0,0}M^o_{0,0}$, that
will be called $\phi_0(x,y)$. To prove that $\phi_0(x,y)$ is
indeed the minimal polynomial, it suffices to prove that it is an
irreducible factor of $P(x,y)$ of multiplicity $m$ and that
$Q(x,y):=P(x,y)/\phi_0(x,y)^m$ is not an annihilating polynomial of
$W^o_{0,0}M^o_{0,0}$. We verify the first point directly. For the second
point, we truncate $W^o_{0,0}M^o_{0,0}$ to order $270$
and substitute it
for $y$ in $Q(x,y)$. We get a series of valuation less than $270$, which
proves that $Q(x,y)$ is not an annihilating polynomial of $W^o_{0,0}M^o_{0,0}$.
We find the minimal polynomial $\phi_1(x,y)$ of $W^o_{0,1}M^o_{1,0}$
in a similar way.
Now we prove that $\phi(M^e,0,0)$ is the minimal polynomial of
$W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$.
We know that
$$S(x,y)=\Res_z\left(\phi_0(x,z),\; \phi_1(x,y+z)\right)$$
is an annihilating of $W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$.
We verify that $\phi(M^e,0,0)$ is an irreducible factor of $S(x,y)$ of
multiplicity $\mu$, and that $Q(x,y):=S(x,y)/\phi(M^e,0,0)^\mu$ is not
an annihilating polynomial of $W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$.
To see the last point, we truncate $W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$
to order $330$ and substitute it for $y$ in $Q(x,y)$. We get a series of
valuation less than $330$, and therefore $Q(x,y)$ is not an annihilating
polynomial of $W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$.
Finally, the first $8$ terms of $M^e_{0,0}$ and
$W^o_{0,0}M^o_{0,0}+W^o_{0,1}M^o_{1,0}$ coincide. As these first terms
determine a unique solution of $\phi(M^e,0,0)$, we know that the two
series are one and the same.
\end{proof}
Define
$$R^e=\begin{pmatrix}1&0\\
0&1\end{pmatrix} \qquad\text{and}\qquad R^o=\begin{pmatrix}1&0\\
0&1\end{pmatrix}.$$
\begin{Lemma}\label{lem:tmncf2}
For all integers $k\geq 2$ and $u=2^{2k-1}$, the following identities hold.
\begin{align*}
M^e[:\!6u]&=M^e[:\!3u] +x^uM^e[:\!2u] +x^uM^e[:\!3u] +x^{2u}M^e[:\!2u] +x^{2u}M^e[:\!3u]\\
& \quad +x^{3u}M^e[:\!2u] +(x^{3u}+x^{4u}+x^{5u})R^e,\\
W^e[:\!6u]&=W^e[:\!3u] +x^uW^e[:\!2u] +x^uW^e[:\!3u] +x^{2u}W^e[:\!2u] +x^{2u}W^e[:\!3u]\\
& \quad +x^{3u}W^e[:\!2u] +(x^{3u}+x^{4u}+x^{5u})R^e.
\end{align*}
For all integers $k\geq 2$ and $u=2^{2k}$,
\begin{align*}
M^o[:\!6u]&=M^o[:\!3u] +x^uM^o[:\!2u] +x^uM^o[:\!3u] +x^{2u}M^o[:\!2u] +x^{2u}M^o[:\!3u]\\
& \quad +x^{3u}M^o[:\!2u] +(x^{3u}+x^{4u}+x^{5u})R^o,\\
W^o[:\!6u]&=W^o[:\!3u] +x^uW^o[:\!2u] +x^uW^o[:\!3u] +x^{2u}W^o[:\!2u] +x^{2u}W^o[:\!3u]\\
& \quad +x^{3u}W^o[:\!2u] +(x^{3u}+x^{4u}+x^{5u})R^o.
\end{align*}
\end{Lemma}
\begin{proof}
To prove Lemma \ref{lem:tmncf2} we first construct an automaton for each
sequence concerned, and then transform the conditions on infinitely many
$k$'s into finitely many conditions on the states of the automaton.
In the following, we will prove that for $T=M^o_{1,0}$, for all integer
$k\geq 2$ and $u=2^{2k}$,
\begin{align*}
T[:\!6u]&=T[:\!3u] +x^uT[:\!2u] +x^uT[:\!3u] +x^{2u}T[:\!2u] +x^{2u}T[:\!3u] +x^{3u}T[:\!2u] .
\end{align*}
The proofs of the other $15$ cases are similar.
We break down the above identity into $3$ parts:
\begin{align*}
0&=x^{3u}T[:\!u] +x^uT[2u\!:\!3u] +T[3u\!:\!4u],\\
0&=x^{3u}T[u\!:\!2u] +x^{2u}T[2u\!:\!3u] +T[4u\!:\!5u],\\
0&=T[5u\!:\!6u],
\end{align*}
which can be rewritten as
\begin{align}
0&=T[[w]_2] +T[[10w]_2] +T[[11w]_2],\label{labelp2l3}\\
0&=T[[1w]_2] +T[[10w]_2] +T[[100w]_2],\label{labelp2l4}\\
0&=T[[101w]_2],\label{labelp2l5}
\end{align}
for all binary word $w$ of length $2k$ and $w\neq 0^{2k}$, and
\begin{align}
0&=T[[w]_2] +T[[10w]_2] +T[[11w]_2],\label{labelp3l3}\\
0&=T[[1w]_2] +T[[10w]_2] +T[[100w]_2],\label{labelp3l4}\\
0&=T[[101w]_2],\label{labelp3l5}
\end{align}
for $w=0^{2k}$.
First we calculate an $2$-automaton that generates $T$ from its minimal
polynomial and
its first terms. This automaton
has $124$ states; its transition function and output function can be found in
the annex.
Let $A(s,w)$ denote the state reached after reading $w$ from right to
left starting from the state $s$, and $\tau$ the output function.
Define $$E_{2k}=\{A(i,w) : |w|=2k, w\neq 0^{2k}\}.$$
Identities \eqref{labelp2l3} through \eqref{labelp3l5} can be written as
\begin{align}
0&=\tau(A(s,\epsilon)) +\tau(A(s,10)) +\tau(A(s,11)),\label{labelp4l3}\\
0&=\tau(A(s,1)) +\tau(A(s,10)) +\tau(A(s,100)),\label{labelp4l4}\\
0&=\tau(A(s,101)),\label{labelp4l5}
\end{align}
for all $s\in E_{2k}$, and
\begin{align}
0&=\tau(A(s,\epsilon)) +\tau(A(s,10)) +\tau(A(s,11)),\label{labelp5l3}\\
0&=\tau(A(s,1)) +\tau(A(s,10)) +\tau(A(s,100)),\label{labelp5l4}\\
0&=\tau(A(s,101)),\label{labelp5l5}
\end{align}
for $s=A(i,0^{2k})$.
We find that $(A(i,0^{24}),E_{24})=(A(i,0^{16}),E_{16})$, so that we only
have to verify that identities \eqref{labelp4l3} through \eqref{labelp5l5} hold for
$2\leq k\leq 12$, which turns out to be true.
\end{proof}
In the following lemma, we express $M_{2k}$, $M_{2k+1}$, $W_{2k}$, and
$W_{2k+1}$ in terms of $M^e$, $M^o$, $W^e$, and $W^o$.
\begin{Lemma}\label{lem:tmncf1}
For all integer $k\geq 2$, and $u=2^{2k-1}$,
\begin{align*}
M_{2k}&=M^e[:\!3u] +x^uM^e[:\!2u] +x^{3u}R^e,\\
W_{2k}&=W^e[:\!3u] +x^uW^e[:\!2u] +x^{3u}R^e.
\end{align*}
For all integer $k\geq 2$, and $u=2^{2k}$,
\begin{align*}
M_{2k+1}&=M^o[:\!3u] +x^uM^o[:\!2u] +x^{3u}R^o,\\
W_{2k+1}&=W^o[:\!3u] +x^uW^o[:\!2u] +x^{3u}R^o.
\end{align*}
\end{Lemma}
\begin{proof}
Call the four identities in Lemma \ref{lem:tmncf1} also by the name
$M_{2k}$, $W_{2k}$, $M_{2k+1}$, and $W_{2k+1}$.
For $n=2$, the identities can be verified directly. For $n\geq 2$,
we claim that
\begin{align*}
M_{2k} \wedge W_{2k} &\Rightarrow M_{2k+1} \wedge W_{2k+1},\\
M_{2k+1} \wedge W_{2k+1} &\Rightarrow M_{2k+2} \wedge W_{2k+2}.
\end{align*}
We give the proof of
$$M_{2k} \wedge W_{2k} \Rightarrow M_{2k+1},$$
the proofs of the other ones being similar. Set $u=2^{2k}$ and $v=2^{2k-1}$.
By definition and induction hypothesis, the left side of identity
$M_{2k+1}$ is equal to
\begin{align}
W_{2k}M_{2k}&=\bigl( W^e[:\!3v] +x^vW^e[:\!2v] +x^{3u}R^e \bigr)\times\bigl( M^e[:\!3v] +x^vM^e[:\!2v]\nonumber\\
& \quad +x^{3u}R^e \bigr).\label{label:lhs}
\end{align}
Call this expression $lhs$.
Note that both sides of identity $M_{2k+1}$ have the same term of highest
degree $x^{6v}R^o$. Therefore we only need to prove that their difference
is $O(x^{6v})$. Using Lemma \ref{lem:tmncf0} it can be seen that the right
side of identity $M_{2k+1}$ is congruent, modulo $x^{6v}$, to
\begin{equation*}
W^e[:\!6v]M^e[:\!6v] +x^{2v}W^e[:\!4v]M^e[:\!4v].\label{}
\end{equation*}
For all $n\leq 6$, replace the occurrences of $W^e[:\!n\cdot v]$ and
$M^e[:\!n\cdot v]$ in the above expression
by the reduction modulo $x^{n\cdot v}$ of the right side of the
corresponding identity in Lemma \ref{lem:tmncf2} and get a new
expression,
which we call $rhs$.
Define
\begin{align*}
X&:=x^v,\\
a_n&:=W^e[n\cdot v:(n+1)\cdot v]/X^n,\\
b_n&:=M^e[n\cdot v:(n+1)\cdot v]/X^n,\\
c&:=R^e.
\end{align*}
Using the notation introduced above, we can represent the expressions $lhs$ \eqref{label:lhs}
and $rhs$ as polynomials in $\mathbb{F}_2[a_1,...,a_{6},
b_1,...,b_{6},c][X]$. Note that it is not a problem that
$a_j$ commutes with $b_k$ while $W^e$ does not commute with $M^e$, because
in the expressions concerned, the products of $W^e$-terms and $M^e$-terms
are always in the same order.
We let the computer do the simplification and
check that the difference between these
two polynomials is indeed $O(X^{6})$, which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:ab}]
We prove the theorem for $\CF(\mathbf{t}(z))$; for $\CF(\bar{\mathbf{t}}(z))$,
the proof is similar.
By Lemma \ref{lem:tmncf1}, we have
For all $0\leq j,k\leq 1$,
\begin{align*}
\lim_{n\rightarrow\infty} M_{2n,j,k}&=M^e_{j,k},\\
\lim_{n\rightarrow\infty} M_{2n+1,j,k}&=M^o_{j,k},\\
\lim_{n\rightarrow\infty} W_{2n,j,k}&=W^e_{j,k},\\
\lim_{n\rightarrow\infty} W_{2n+1,j,k}&=W^o_{j,k}.
\end{align*}
Let $z=1/x$. By the convergence theorem and identity \eqref{eq:cm},
$$\CF(\mathbf{t}(z))=\frac{M^e_{0,1}(x)}{M^e_{0,0}(x)}.$$
By definition, that $\phi(M^e,0,1)$ and
$\phi(M^e,0,0)$ are minimal polynomials of $M^e_{0,1}$ and $M^e_{0,0}$.
Therefore
$$P(x,y)=\Res_t\left(\phi(M^e,0,1)(x,t),y^{12}\phi(M^e,0,1)(x,t/y)\right)$$
is an annihilating polynomial of $f(x)=M^e_{0,1}/M^e_{0,0}$.
Define
$$Q(x,y)=q_4(x)y^4+q_3(x)y^3+q_2(x)y^2+q_1(x)y+q_0(x),$$
where
\begin{align*}
q_{0}(x)&=x^{12} +x^{11} +x^8 +x^7 +x^6 +x^5 +x^3 ,\\
q_{1}(x)&=x^{11} +x^{10} +x^9 +x^7 +x^6 +x^4 +x^2 +x ,\\
q_{2}(x)&=x^{10} +x^2 +1 ,\\
q_{3}(x)&=x^{11} +x^{10} +x^9 +x^7 +x^6 +x^4 +x^2 +x ,\\
q_{4}(x)&=x^{11} +x^{10} +x^7 +x^6 +x^5 +x^3 +x^2 .
\end{align*}
The polynomial $Q(x,y)$ is the candidate for the minimal polynomial of $f(x)$ found by
Pad\'e-Hermite approximation.
To prove that it is indeed the minimal polynomial of $f(x)$,
we only need to prove that it is an irreducible factor of $P(x,y)$ of
multiplicity $m$ and $R(x,y):=P(x,y)/Q(x,y)^m$
is not an annihilating polynomial of $f(x)$.
We verify the first point directly. For the second point, we find that
when we truncate $f(x)$ to order $96$, and substitute it for $y$ in
$R(z,y)$, we get a series with valuation smaller than $96$, which proves that
$R(z,y)$ is not an annihilating polynomial of $f(x)$.
Finally, $z^{12}Q(1/z,y)$ is the minimal polynomial of $\CF(\mathbf{t}(z))=f(1/z)$.
\end{proof}
\section{Thue-Morse Stieltjes Continued Fraction}\label{sec:s}
Using our program, we checked that conjecture \ref{conj:tmsa} holds for all
$a\in \mathbb{F}_{2^k}\backslash\{0,1\}$ for $k=2,3,4$.
In this section we present our method.
\subsection{Testing of the conjecture}\label{sub:tms1}
For $k\geq 2$, instead of all $a$ in $\mathbb{F}_{2^k}\backslash\{0,1\}$, we
only need to test one $a$ in each of the orbits of the Frobenius morphism
$\phi: a\mapsto a^2$, because if we let $\mathbf{t}$ denote the $(a,1)$-Thue-Morse
seuqence and $\phi(\mathbf{t})$ the $(\phi(a),1)$-Thue-Morse sequence, then
$$\Stiel(x;\phi(\mathbf{t}))=\phi(\Stiel(x;\mathbf{t})),$$
and they are either both algebraic or both transcendental.
For example, $\mathbb{F}_{8}\cong \mathbb{F}_2[u]/<u^3+u+1>$
is partitioned into orbits
$$\{0\},\{1\},\;\{\bar{u},\bar{u}^2,\;\bar{u}^4\},\mbox{ and } \{\bar{u}^3,\bar{u}^6,\bar{u}^5\}.$$
Therefore for $\mathbb{F}_8$, we only have to test the conjecture for
$a=\bar{u}$ and $a=\bar{u}^3$.
Furthermore, we only have to test those $a$ in $\mathbb{F}_{2^k}\backslash\{0,1\}$
whose orbit contains $k$ elements, because elements whose orbit has size
$l<k$ are already tested in $\mathbb{F}_{2^l}$. For example, for
$\mathbb{F}_{16}\cong \mathbb{F}_2[u]/<u^4+u+1 >$,
the orbit of $a=\bar{u}^5$ contains only itself and $a^2$. This means
that $a^4=a$, and therefore it is already treated in $\mathbb{F}_4$.
\subsection{Our method}
The same method for testing conjecture \ref{conj:tmncf} can be used here to
test conjecture \ref{conj:tmsa}
for $a\in \mathbb{F}_{2^k}\backslash\{0,1\}$ ($k\geq 2$), with only slight
modifications.
As most of the following have a uniform expression for all $a$, we first
regard $a$ as a formal variable.
As in Section \ref{section:tmncf}, we define
\begin{equation}
M_{n}=
\begin{pmatrix} 1&t_{2^n-1}x\\ 1&0\end{pmatrix}
\begin{pmatrix} 1&t_{2^n-2}x\\ 1&0\end{pmatrix}
\cdots \begin{pmatrix} 1&t_0x \\1&0\end{pmatrix},
\end{equation}
and
\begin{equation}
W_{n}=
\begin{pmatrix} 1&\bar{t}_{2^n-1}x\\ 1&0\end{pmatrix}
\begin{pmatrix} 1&\bar{t}_{2^n-2}x\\ 1&0\end{pmatrix}
\cdots \begin{pmatrix} 1&\bar{t}_0x \\1&0\end{pmatrix},
\end{equation}
where $\mathbf{t}$ is the $(a,1)$-Thue-Morse sequence, and
$\mathbf{\bar{t}}$, $(1,a)$-Thue-Morse sequence.
We have $M_{n+1}=W_n\cdot M_n$ and $W_{n+1}=M_n\cdot W_n$ for all $n$.
We define four $2\times 2$ matrices $M^e$, $M^o$, $W^e$ and $W^o$ as follows:
For all $T\in \{M^e, M^o,W^e,W^o\}$, and all $i,j \in \{0,1\}$,
$T_{i,j}$ is defined to be the unique solution in $\mathbb{F}_2(a)[[x]]$
of the polynomial
$\phi(T,i,j)$ under certain initial conditions. The polynomials
$\phi(T,i,j)$ and initial conditions can be found in the annex.
The reason for defining these matrices and how the polynomials $\phi(T,i,j)$
and initial conditions are found are similar to those given in Section
\ref{section:tmncf}.
As expected, the following Lemma holds:
\begin{Lemma}\label{lem:4relations}
\begin{align}
M^e&=W^o\cdot M^o,\label{eq:mes}\\
M^o&=W^e\cdot M^e,\label{eq:mos}\\
W^e&=M^o\cdot W^o,\label{eq:wes}\\
W^o&=M^e\cdot W^e.\label{eq:wos}
\end{align}
\end{Lemma}
\begin{proof}
Similar to the proof of Lemma \ref{lem:tmncf0}.
\end{proof}
We have the following observation concerning the structure of the four matrices.
\begin{Observation}\label{th:obs1}
For $k\geq 2$ and $u=2^{2k-1}$ the following identities hold:
\begin{align}
M^e[u\!:\!2u]&=x^u\cdot (a^{u}+1)\label{eq:obs}
\cdot M^e[:\!u] + a^{u/2}x^{u}\cdot I_2,\\
W^e[u\!:\!2u]&=x^u\cdot (a^{u}+1)\label{eq:obs1}
\cdot W^e[:\!u] + a^{u/2}x^{u}\cdot I_2;
\end{align}
for $k\geq 2$ and $u=2^{2k}$,
\begin{align}
M^o[u\!:\!2u]&=x^u\cdot (a^{u}+1)\label{eq:obs2}
\cdot M^o[:\!u] + a^{u/2}x^{u}\cdot I_2,\\
W^o[u\!:\!2u]&=x^u\cdot (a^{u}+1)\label{eq:obs3}
\cdot W^o[:\!u] + a^{u/2}x^{u}\cdot I_2.
\end{align}
\end{Observation}
\begin{Observation}\label{th:obs2}
For $k\geq 2$ and $u=2^{2k-1}$,
\begin{align*}
M_{2k}&=M^e[:\!u]+a^{u/2}x^{u}\cdot I_2,\\
W_{2k}&=W^e[:\!u]+a^{u/2}x^{u}\cdot I_2;
\end{align*}
for $k\geq 2$ and $u=2^{2k}$,
\begin{align*}
M_{2k+1}&=M^o[:\!u]+a^{u/2}x^{u}\cdot I_2,\\
W_{2k+1}&=W^o[:\!u]+a^{u/2}x^{u}\cdot I_2.
\end{align*}
\end{Observation}
\begin{Lemma}\label{lem:obs}
Observation \ref{th:obs1} implies observation \ref{th:obs2}.
\end{Lemma}
\begin{proof}
Let us call the four identities in observation \ref{th:obs2} also by
the name $M_{2k}$, $W_{2k}$, $M_{2k+1}$ and $W_{2k+1}$.
Suppose observation \ref{th:obs2} is true.
We want to prove observation \ref{th:obs1} by induction. For $n=2$, the
identities are verified directly.
The inductive step is
\begin{align*}
M_{2k} \wedge W_{2k} &\Rightarrow M_{2k+1} \wedge W_{2k+1},\\
M_{2k+1} \wedge W_{2k+1} &\Rightarrow M_{2k+2} \wedge W_{2k+2}.
\end{align*}
Let us show for example how to prove
\begin{equation}
M_{2k} \wedge W_{2k} \Rightarrow M_{2k+1}.
\end{equation}
By definition, the left side of the identity $M_{2k+1}$ is equal to
$$W_{2k}\cdot M_{2k},$$
which, by induction hypothesis, is equal to
$$(W^e[:\!u]+a^{u/2}x^{u}I_{2})\cdot
(M^e[:\!u]+a^{u/2}x^{u}I_{2}),$$
where $u=2^{2k-1}$.
Therefore only need to prove that
\begin{equation}\label{eq:1s}
W^e[:\!u]\cdot M^e[:\!u]+a^{u/2}x^{u}(W^e[:\!u]+
M^e[:\!u])-M^o[:\!2u]
\end{equation}
is equal to zero.
As the degree of the above polynomial is at most $2u-1$,
we only need to prove that it is
$O(x^{2^{2n}})$. By \eqref{eq:mos},
\begin{align*}
&\quad M^o[:\!2u]\\&\equiv W^e[:\!2u]\cdot M^e[:\!2u] \mod x^{2u}\\
&\equiv W^e[:\!u]\cdot M^e[:\!u]+W^e[u\!:\!2u]\cdot
M^e[:\!u] +W^e[:\!u]\cdot M^e[u\!:\!2u] \mod x^{2u}
\end{align*}
Therefore \eqref{eq:1s} is congruent modulo $ x^{2u}$
to
\begin{align*}
a^{u/2}x^{u}(W^e[:\!u]+ M^e[:\!u])+
W^e[u:2u]\cdot M^e[:\!u] +W^e[:\!u]\cdot M^e[u\!:\!2u].
\end{align*}
Substitute $W^e[u\!:\!2u]$ and $M^e[u\!:\!2u]$ by the
expressions in observation \ref{th:obs1} and we obtain that the quantity
above is $O(x^{2u})$. That is, expression \eqref{eq:1s} is congruent
to $0$ modulo $x^{2u}$; since it has no term of order higher than $2u-1$,
it is equal to~$0$.
\end{proof}
\begin{Proposition}
Observation \ref{th:obs2} implies conjecture \ref{conj:tmsb}.
\end{Proposition}
\begin{proof}
First, taking the limit of the identities in observation \ref{th:obs2},
we have for all $j,k\in \{0,1\}$,
\begin{align*}
\lim_{n\rightarrow \infty} M_{2n,j,k}&=M^e_{j,k},\\
\lim_{n\rightarrow \infty} M_{2n+1,j,k}&=M^o_{j,k},\\
\lim_{n\rightarrow \infty} W_{2n,j,k}&=W^e_{j,k},\\
\lim_{n\rightarrow \infty} W_{2n+1,j,k}&=W^o_{j,k}.
\end{align*}
Therefore
\begin{equation*}
\Stiel(x;\mathbf{t})=\lim_{n\rightarrow \infty} \frac{P_n}{Q_n}\\
=\lim_{n\rightarrow \infty} \frac{P_{2^{2n}-1}}{Q_{2^{2n}-1}}\\
=\lim_{n\rightarrow \infty} \frac{M_{2n,0,1}/x}{M_{2n,0,0}}\\
=\frac{M^e_{0,1}/x}{M^e_{0,0}}.
\end{equation*}
We obtain the minimal polynomial of $\Stiel(x;\mathbf{t})$
from those of $M^e_{0,1}$ and
$M^e_{0,0}$, using the same method described in the proof of Theorem
\ref{th:ab}.
\end{proof}
\begin{Remark}
The above proposition says that observation \ref{th:obs2} implies
conjecture \ref{conj:tms} when $a$ is regarded as a formal variable.
The implication also holds when $a$ specializes as an element in
$\mathbb{F}_{2^k}\backslash\{0,1\}$ ($k\geq 2$).
\end{Remark}
Therefore, to prove that conjecture \ref{conj:tmsa} holds for a certain
$a\in\mathbb{F}_{2^k}\backslash\{0,1\}$ ($k\geq 2$), we only need to prove
that observation \ref{th:obs1} holds for $a$.
Because of the following
argument, we only have to check \eqref{eq:obs} through \eqref{eq:obs3} for finitely
many $k$'s instead of for all $k\geq 2$:
For $k\geq 2$ and $u=2^{2k-1}$, identity \eqref{eq:obs} and \eqref{eq:obs1} can be written as
\begin{equation}\label{eq:w}
T[[1w]_2]=(a^{u}+1)\cdot T[[w]_2]
\end{equation}
for every component $T$ of $M^e$ and $W^e$
and all binary words $w$ of length
$2k-1$ and $w\neq 0^{2k-1}$; and
\begin{equation}\label{eq:0}
T[[1w]_2]=(a^{u}+1)\cdot T[[w]_2]+a^{u/2}
\end{equation}
for $w=0^{2k-1}$.
We calculate the an automaton of $T$ from the algebraic equation that
defines it, following the method in \cite{Christol1980KMFR} (see Section
\ref{sec:eq2kern}).
Let $A(s,w)$ denote the state reached after reading $w$ from right to
left starting from the state $s$, and $\tau$ the output function.
Define
$$E_{2k-1}=\{A(i,w) \mid |w|=2k-1, w\neq 0^{2k-1}\}.$$
Identity \eqref{eq:w} and \eqref{eq:0} can be written as
\begin{equation}\label{eq:aw}
\tau(A(s,1))=(a^{u}+1)\cdot \tau(A(s,\epsilon))
\end{equation}
for all $s\in E_{2k-1}$, and
\begin{equation}\label{eq:a0}
\tau(A(s,1))=(a^{u}+1)\cdot \tau(A(s,\epsilon))+a^{u/2}
\end{equation}
for $s=A(i,0^{2k-1})$.
As $E_{2k+1}$ is completely determined by $E_{2k-1}$, the sequence
$(E_{2k+1})_k$ is ultimately periodic. The sequences $(A(i,0^{2k-1}))_k$
and $a^{2^{2k-1}}$ are also periodic.
Therefore we only need to check \eqref{eq:aw} and \eqref{eq:a0}
for finitely many $k$'s.
\subsection{An example}
For $a\in \mathbb{F}_{4}\backslash \{0,1\}$ and
$T=Me_{0,0}$, we find that the minimal $2$-DFAO of $T$ has as transition
function
$(n,j)\mapsto \delta(n,j)$ ($\Lambda(n):=[\delta(n,0),\delta(n,1)]$):
{
\renewcommand{\arraystretch}{1.2}
\begin{longtable}{| c c | c c | c c | c c |}
\hline
$n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ \\
\hline
0 & [1, 2]& 5 & [2, 8]& 10 & [7, 8]& 15 & [13, 17]\\
1 & [3, 4]& 6 & [9, 4]& 11 & [8, 13]& 16 & [18, 4]\\
2 & [5, 6]& 7 & [10, 4]& 12 & [14, 4]& 17 & [19, 12]\\
3 & [1, 7]& 8 & [11, 6]& 13 & [15, 16]& 18 & [16, 6]\\
4 & [4, 4]& 9 & [6, 12]& 14 & [12, 16]& 19 & [17, 8]\\
\hline
\end{longtable}
}
and output function $n\mapsto \tau(n)$:
{
\renewcommand{\arraystretch}{1.2}
$$
\begin{array}{| c c | c c | c c | c c | c c | c c | c c |}
\hline
n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) \\
\hline
0 & 1& 3 & 1& 6 & a + 1& 9 & a + 1& 12 & 1& 15 & 1& 18 & a\\
1 & 1& 4 & 0& 7 & 0& 10 & 0& 13 & 1& 16 & a& 19 & a + 1\\
2 & a& 5 & a& 8 & a& 11 & a& 14 & 1& 17 & a + 1& &\\
\hline
\end{array}
$$
}
The tuple $(A(i,0^{2k-1})$, $E_{2k-1})$ has the following values:
\begin{align*}
k&=3: ({1}, \{2, 4, 7, 8, 9\}),\\
k&=5: ({1}, \{2, 4, 7, 8, 9, 13, 14\}),\\
k&=7: ({1}, \{2, 4, 7, 8, 9, 13, 14, 17, 18\}),\\
k&=9: ({1}, \{2, 4, 7, 8, 9, 13, 14, 17, 18\}).
\end{align*}
For all $k\geq 1$, $a^{2^{2k-1}}=a+1$. Therefore we only have to check that
identity \eqref{eq:obs} holds for $k=3,5,7$, which turns out to be true.
\section{Period-doubling Continued Fractions}\label{section:pd}
The method for checking conjecture \ref{conj:tmncf} can be adapted for the
verification of conjecture \ref{conj:pdncf}. In this section, we give
two examples. First we introduce the notation.
For $(a,b)\in (\F2[z]\backslash \F2)^2$. Let $\mathbf{p}$ be the
$(a,b)$-period-doubling sequence.
Define two sequence $A_n(x)$ and $B_n(x)$ by
\begin{align*}
A_0(x)&=x^{\deg(a)}\begin{pmatrix}
a(1/x)&1\\
1&0
\end{pmatrix} \\
B_0(x)&=x^{\deg(b)}\begin{pmatrix}
b(1/x)&1\\
1&0
\end{pmatrix} \\
A_{n+1}(x)&=B_n(x)A_n(x)\forall n\geq 0\\
B_{n+1}(x)&=A_n(x)A_n(x)\forall n\geq 0\\
\end{align*}
Define $x:=1/z$. For an non-zero polynomial $P(z)$, we define $\tilde{P}(x)$
to be $P(1/x)$. Then
\begin{equation*}
\CF_n(\mathbf{p}(z))=
\frac{P_n(z)}{Q_n(z)}=\frac{\tilde{P}_n(x)}{\tilde{Q}_n(x)}
\in\mathbb{F}_2((x))=\mathbb{F}_2((1/z)).
\end{equation*}
Comparing the definition of $A_n(x)$ with definition \eqref{eq:pq}, we see that
\begin{align*}
A_n(x)_{0,1}&= x^{d_n} \tilde{P}_{2^n-1}(x),\\
A_n(x)_{0,0}&= x^{d_n} \tilde{Q}_{2^n-1}(x),
\end{align*}
for some positive integer $d_n$,
and
\begin{align}\label{eq:cm}
\CF_{{2^{2n}-1}}(\mathbf{p}(z))
=\frac{\tilde{P}_{2^{2n}-1}(x)}{\tilde{Q}_{2^{2n}-1}(x)}
= \frac{A_{2n}(x)_{0,1}}{A_{2n}(x)_{0,0}}.
\end{align}
\subsection{ The $(z^2,z)$-period-doubling sequence}\label{subsection:pd1}
In this subsection, we prove the following theorem.
\begin{Theorem}\label{thm:pd1}
Let $(a,b)=(z^2,z)\in (\F2[z]\backslash \F2)^2$. Let $\mathbf{p}$ be the
$(a,b)$-period-doubling sequence. The power series $\CF(\mathbf{p}(z))$ is algebraic
over $\F2(z)$; its minimal polynomial is
$$
z^4+x^3z^2+ (x^5+x^4)z+x^3+x^2+1=0.
$$
\end{Theorem}
We define four $2\times 2$ matrices $A^e$, $A^o$, $B^e$ and $B^o$ as follows:
For all $T\in \{A^e, A^o,B^e,B^o\}$, and all $i,j \in \{0,1\}$,
$T_{i,j}$ is defined to be the unique solution in $\mathbb{F}_2(a)[[x]]$
of the polynomial
$\phi(T,i,j)$ under certain initial conditions. The polynomials
$\phi(T,i,j)$ and initial conditions can be found in the annex.
The reason for defining these matrices and how the polynomials $\phi(T,i,j)$
and initial conditions are found are similar to those given in Section
\ref{section:tmncf}.
\begin{Lemma}\label{lem:0}
The following identities hold:
\begin{align*}
A^e&=B^o\cdot A^o,\\
A^o&=B^e\cdot A^e,\\
B^e&=A^o\cdot A^o,\\
B^o&=A^e\cdot A^e.
\end{align*}
\end{Lemma}
\begin{proof}
Similar to the proof of Lemma \ref{lem:tmncf0}.
\end{proof}
\begin{Lemma}\label{lem:1}
For $n\geq 2$ even and $u=(5\cdot 2^n+1)/3$,
\begin{align*}
A^e[:2u]&=A^e[:u]+x^u\cdot I_2\\
B^e[:2u]&=B^e[:u]+x^u\begin{pmatrix}x^{u-1}&x^{u-2}+x^{u-3}\\0&x^{u-1}
\end{pmatrix}
\end{align*}
For $n\geq 1$ odd, for $u=(5\cdot 2^n+2)/3$,
\begin{align*}
A^o[:2u-1]&=A^o[:u]+x^{2u-2}\cdot I_2\\
B^o[:2u-1]&=B^o[:u]+x^u\begin{pmatrix}1&x^{u-2}+x^{u-3}\\0&1
\end{pmatrix}
\end{align*}
\end{Lemma}
\begin{proof}
We give proof of
\begin{equation}\label{eq:pdlem1}
A^e_{0,0}[:2u]=A^e_{0,0}[:u]+x^u
\end{equation}
for $n\geq 2$ even and $u=(5\cdot 2^n+1)/3$; the proofs of the other $15$
cases are similar. Let $T=A^e_{0,0}$. Identity \eqref{eq:pdlem1} can be
written as
\begin{equation}\label{eq:pdlem1p}
T[u:2u]=x^u.
\end{equation}
From the minimal polynomial and the first terms of $T$, we find its minimal
automaton. Its transition function $\delta$
($\Lambda(n):=[\delta(n,0),\delta(n,1)]$)
and output function $\tau$
are as follows:
\begin{longtable}{| c c | c c | c c | c c |}
\hline
$n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ \\
\hline
0 & [1, 2]& 2 & [4, 5]& 4 & [4, 4]& 6 & [6, 4]\\
1 & [3, 2]& 3 & [6, 6]& 5 & [2, 6]& & \\
\hline
\end{longtable}
\begin{longtable}{| c c | c c | c c | c c |}
\hline
$n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ \\
\hline
0 & 1& 2 & 0& 4 & 0& 6 & 1\\
1 & 1& 3 & 1& 5 & 0& & \\
\hline
\end{longtable}
Let $A(s,w)$ denote the state reached after reading $w$ from right
to left starting from the state $s$.
For $n\geq 2$ even, $k=n/2-1$, and $u=(5\cdot 2^n+1)/3$
the binary expansions of integers $j$ in $[u, 2u[$ have the following forms:
\begin{longtable}{| c | c |}
\hline
$j$ & $[j]_2$ \\
\hline
$j=u$ & $1(10)^k11$ \\
$u<j<2^{n+1}$ & $ 1(10)^{k-l}11\{0,1\}^{2l}$, $1\leq l \leq k$ \\
$2^{n+1}\leq j<2u-2$ & $ 1(10)^{l}0\{0,1\}^{n-2l}$, $0\leq l \leq k$ \\
$j=u-1, u-2$ & $1(10)^k10\{0,1\}$\\
\hline
\end{longtable}
Consider the sets
\begin{align*}
B_0&=\{1(10)^k11 \mid k\geq 0 \}\\
B_1&=\{ 1(10)^{m}11\{0,1\}^{2l} \mid l\geq 1,\;m\geq 0\}\\
B_2&=\{ 1(10)^{m}0\{0,1\}^{2l} \mid l\geq 1,\;m\geq 0\}\\
B_3&=\{ 1(10)^m10\{0,1\}\mid m\geq 0\}.
\end{align*}
For $i=0,1,2,3$, define
$$E_i=\{ A(0,w) \mid w\in B_i \}.$$
We find that
\begin{align*}
E_0&=\{6\} \\
E_1&= \{4\}\\
E_2&= \{4, 5\}\\
E_3&= \{4\}\\
\end{align*}
We verify that for all $s\in E_0$, $\tau(s)=1$, and for all $s\in E_i$, $i=1,2,3$, $\tau(s)=0$. This proves identity \eqref{eq:pdlem1p} for all $n\geq 2$ even,
and $u=(5\cdot 2^n+1)/3$.
\end{proof}
\begin{Lemma}\label{lem:2}
For $n\geq 2$ even and $u=(5\cdot 2^n+1)/3$
\begin{align*}
A_{n}&=A^e[:u]+x^u\cdot I_2,\\
B_{n}&=B^e[:u].
\end{align*}
For $n\geq 1$ odd, for $u=(5\cdot 2^n+2)/3$
\begin{align*}
A_{n}&=A^o[:u],\\
B_{n}&=B^o[:u]+x^u\cdot I_2.\\
\end{align*}
\end{Lemma}
\begin{proof}
Let us call the identities involving $A_n$ and $B_n$ also by the name
$A_n$ and $B_n$. We will prove the lemma by induction.
It can be verified directly that $A_n$ and $B_n$ holds for $n=1,\; 2$.
For the inductive step, we want to prove that for $n\geq 2$,
$$
A_n \wedge B_n \Rightarrow A_{n+1} \wedge B_{n+1}.
$$
We give the proof of
$$
A_{n} \wedge B_n \Rightarrow A_{n+1}
$$
when $n$ is even. The proofs of the other cases are similar.
Now suppose that for some $n\geq 2$ even and $u=(5\cdot 2^n+1)/3$
it holds that
\begin{align*}
A_{n}&=A^e[:u]+x^u\cdot I_2, \\
B_{n}&=B^e[:u]
\end{align*}
We want to prove that
\begin{equation}\label{eq:ao}
A_{n+1}=A^o[:2u].
\end{equation}
By definition and induction hypothesis,
\begin{align*}
A_{n+1}&= B_{n}A_{n},\\
&=B^e[:u]\cdot (A^e[:u]+x^u\cdot I_2).
\end{align*}
As the degrees of both $A_{n+1}$ and $A^o[:2u]$ are at most $2u-1$, to prove
that they are equal, we only need to prove that they are congruent modulo
$x^{2u}$.
By lemma \ref{lem:0} and lemma \ref{lem:1},
\begin{align*}
A^o[:2u]&\equiv B^e[:2u]\cdot A^e[:2u]\\
&\equiv
\left (B^e[:u]+x^u\begin{pmatrix}x^{u-1}&x^{u-2}+x^{u-3}\\0&x^{u-1}
\end{pmatrix}\right)\cdot
(A^e[:u]+x^u\cdot I_2) \mod x^{2u}.
\end{align*}
Therefore
\begin{align*}
A^o[:2u]-A_{n+1}&\equiv x^u\begin{pmatrix}x^{u-1}&x^{u-2}+x^{u-3}\\0&x^{u-1}
\end{pmatrix} \cdot A^e[:u]\\
&\equiv x^u\begin{pmatrix}x^{u-1}&x^{u-2}+x^{u-3}\\0&x^{u-1}
\end{pmatrix} \cdot (A^e[:u]\mod x^3)\\
&\equiv x^u\begin{pmatrix}x^{u-1}&x^{u-2}+x^{u-3}\\0&x^{u-1}
\end{pmatrix} \cdot
\begin{pmatrix}
1&x^2\\x^2&0
\end{pmatrix}\\
&\equiv 0 \mod x^{2u}.\qedhere
\end{align*}
\end{proof}
Theorem \ref{thm:pd1} can be derived from lemma \ref{lem:2} and the definition
of $A^e$, using the same method as in the proof of theorem \ref{th:ab}.
\subsection{ The $(z^3, z^2+z+1)$-period-doubling sequence}\label{subsection:pd2}
In this subsection, we prove the following theorem.
\begin{Theorem}\label{thm:pd2}
Let $(a,b)=(z^3,z^2+z+1)\in (\F2[z]\backslash \F2)^2$. Let $\mathbf{p}$ be the
$(a,b)$-period-doubling sequence. The power series $\CF(\mathbf{p}(z))$ is algebraic
over $\F2(z)$; its minimal polynomial is
$$
z^4+ (x^5 + x^4 + x^3)z^2+(x^8 + x^6 + x^5 + x^3)z+x^5 + x^3 + x^2=0
$$
\end{Theorem}
We define four $2\times 2$ matrices $A^e$, $A^o$, $B^e$ and $B^o$ as follows:
For all $T\in \{A^e, A^o,B^e,B^o\}$, and all $i,j \in \{0,1\}$,
$T_{i,j}$ is defined to be the unique solution in $\mathbb{F}_2(a)[[x]]$
of the polynomial
$\phi(T,i,j)$ under certain initial conditions. The polynomials
$\phi(T,i,j)$ and initial conditions can be found in the annex.
The reason for defining these matrices and how the polynomials $\phi(T,i,j)$
and initial conditions are found are similar to those given in Section
\ref{section:tmncf}.
\begin{Lemma}\label{lem:00}
The following identities hold:
\begin{align*}
A^e&=B^o\cdot A^o,\\
A^o&=B^e\cdot A^e,\\
B^e&=A^o\cdot A^o,\\
B^o&=A^e\cdot A^e.
\end{align*}
\end{Lemma}
\begin{proof}
Similar to the proof of Lemma \ref{lem:tmncf0}.
\end{proof}
\begin{Lemma}\label{lem:4}
For $n\geq 2$ even, $u=(2^{n+3}+1)/3$, $v=2^n$,
\begin{align*}
A^e[:2u]&=(1+x^v+x^{2v})\cdot (A^e[:u]+x^v A^e[:u-v])+x^{4v}A^e[:2u-4v]+\\
&\quad (x^u+x^{u+v}+x^{u+2v})\begin{pmatrix}1&1\\0&1\end{pmatrix},\\
B^e[:2u]&=(1+x^v+x^{2v})\cdot (B^e[:u]+x^v B^e[:u-v])+x^{4v}B^e[:2u-4v]+\\
&\quad \begin{pmatrix}x^{2u-1}&x^{2u-1}+x^{2u-4}\\0&x^{2u-1}\end{pmatrix}.
\end{align*}
For $n\geq 1$ odd, $u=(2^{n+3}+2)/3$, $v=2^n$,
\begin{align*}
A^o[:2u-1]&=(1+x^v+x^{2v})\cdot (A^o[:u]+x^v A^o[:u-v])+x^{4v}A^o[:2u-4v-1]+\\
&\quad (x^{2u-2})\begin{pmatrix}1&0\\0&1\end{pmatrix},\\
B^o[:2u-1]&=(1+x^v+x^{2v})\cdot (B^o[:u]+x^v B^o[:u-v])+x^{4v}B^o[:2u-4v-1]+\\
&\quad \begin{pmatrix}x^{u}+x^{u+v}+x^{u+2v}& x^{2u-2}+x^{2u-4}
\\0& x^{u}+x^{u+v}+x^{u+2v}\end{pmatrix}.
\end{align*}
\end{Lemma}
\begin{proof}
We give the proof of the $(0,0)$-th component of the first identity. The
proofs of the other $15$ identities are similar.
Let $T=A^e_{0,0}$. We want to prove that
\begin{equation}\label{eq:lem4}
T[:\!2u]=(1+x^v+x^{2v})\cdot (T[:\!u]+x^v T[:\!u-v])+x^{4v}T[:\!2u-4v]+
x^u+x^{u+v}+x^{u+2v}.
\end{equation}
for all $n\geq 2$ even, $u=(2^{n+3}+1)/3$, and $v=2^n$.
Equation \eqref{eq:lem4} can be rewritten as
\begin{align*}
&\quad T[u\!:\!2u]\\
&=x^vT[u-v\!:\!u]+x^{2v}T[u-v\!:\!u]+x^{3v}T[:\!u-v]+\\
&\quad x^{4v}T[:\!2u-4v]+ x^u+x^{u+v}+x^{u+2v}\\
&=(x^u+x^vT[u-v\!:\!2v])+ (x^vT[2v\!:\!u]+x^{3v}T[:\!u-2v])+\\
&\quad (x^{u+v}+x^{3v}T[u-2v\!:v]+ x^{2v}T[u-v:2v])+\\
&\quad (x^{2v}T[2v\!:\!u]+x^{3v}T[v\!:\!u-v]+x^{4v}T[\!:\!u-2v])+\\
&\quad (x^{u+2v}+x^{4v}T[u-2v\!:\!2u-4v])
\end{align*}
noting that $u<3v<u+v<4v<u+2v<2u$. The above identity can be decomposed into
five parts:
\begin{align*}
T[u:3v]&=x^u+x^vT[u-v\!:\!2v] \\
T[3v:u+v]&=x^vT[2v\!:\!u]+x^{3v}T[:\!u-2v]\\
T[u+v:4v]&=x^{u+v}+x^{3v}T[u-2v\!:v]+ x^{2v}T[u-v:2v]\\
T[4v:u+2v]&=x^{2v}T[2v\!:\!u]+x^{3v}T[v\!:\!u-v]+x^{4v}T[\!:\!u-2v]\\
T[u+2v:2u]&=x^{u+2v}+x^{4v}T[u-2v\!:\!2u-4v]
\end{align*}
That the above five identities hold for all $n\geq 2$, $u=(2^{n+3}+1)/3$,
and $v=2^n$ is equivalent to the following identities:
\begin{align}
\label{eq:tf}T[[10w]_2]&=1+T[[1w]_2] &\forall w\in L_0\\
T[[10w]_2]&=T[[1w]_2] &\forall w\in L_1\\
T[[11w]_2]&=T[[10w]_2]+T[[w]_2] &\forall w\in L_2\\
T[[11w]_2]&=1+T[[w]_2]+T[[1w]_2] &\forall w\in L_0\\
T[[11w]_2]&=T[[w]_2]+T[[1w]_2] &\forall w\in L_1\\
T[[100w]_2]&=T[[10w]_2]+T[[1w]_2]+T[[w]_2] &\forall w\in L_2\\
T[[100w]_2]&=1+T[[w]_2]&\forall w \in L_0\\
T[[100w]_2]&=T[[w]_2]&\forall w\in L_1\\
T[[10w]_2]&=T[[w]_2]&\forall w\in L_3\\
\label{eq:tl} T[[10w]_2]&=T[[w]_2]&\forall w\in L_4
\end{align}
where
\begin{align*}
L_0&=L((10)^*11),\\
L_1&=L((10)^*11\{00,01,10,11\}^+),\\
L_2&=L((10)^*0\{0,1\} \{00,01,10,11\}^*+ (10)^+),\\
L_3&=L((10)^+0\{00,01,10,11\}^+),\\
L_4&=L((10)^+\{0,1\}).
\end{align*}
From the minimal polynomial and the first terms of $T$, we find its minimal
automaton. Its transition function $\delta$
($\Lambda(n):=[\delta(n,0),\delta(n,1)]$)
and output function $\tau$
are as follows:
{
\renewcommand{\arraystretch}{1.2}
\begin{longtable}{| c c | c c | c c | c c | c c |}
\hline
$n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ \\
\hline
0 & [1, 2]& 6 & [11, 12]& 12 & [15, 20]& 18 & [23, 12]& 24 & [27, 15]\\
1 & [3, 4]& 7 & [13, 14]& 13 & [9, 21]& 19 & [24, 23]& 25 & [15, 18]\\
2 & [5, 6]& 8 & [5, 15]& 14 & [14, 14]& 20 & [14, 26]& 26 & [17, 20]\\
3 & [7, 8]& 9 & [16, 14]& 15 & [22, 10]& 21 & [14, 16]& 27 & [28, 14]\\
4 & [9, 10]& 10 & [17, 18]& 16 & [23, 9]& 22 & [5, 17]& 28 & [27, 17]\\
5 & [8, 9]& 11 & [19, 6]& 17 & [24, 25]& 23 & [16, 16]& & \\
\hline
\end{longtable}
}
{
\renewcommand{\arraystretch}{1.2}
$$
\begin{array}{| c c | c c | c c | c c | c c | c c |}
\hline
n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) & n & \tau(n) \\
\hline
0 & 1& 5 & 1& 10 & 0& 15 & 1& 20 & 0& 25 & 1\\
1 & 1& 6 & 0& 11 & 0& 16 & 1& 21 & 0& 26 & 0\\
2 & 1& 7 & 1& 12 & 1& 17 & 0& 22 & 1& 27 & 0\\
3 & 1& 8 & 1& 13 & 1& 18 & 1& 23 & 1& 28 & 0\\
4 & 1& 9 & 1& 14 & 0& 19 & 0& 24 & 0& & \\
\hline
\end{array}
$$
}
Let $A(s,w)$ denote the state reached after reading $w$ from right
to left starting from the state $s$.
For $j=0,1,\ldots, 4$, define
$$E_j=\{ A(0,w)\mid w\in L_j\}.$$
We can compute $E_j$ explicitly and find
\begin{align*}
E_0&=\{6\} \\
E_1&=\{14,15,16,17,18,20\} \\
E_2&=\{3, 4, 5, 13, 14, 15, 16, 17, 19, 27\} \\
E_3&= \{9, 14, 21, 23\}\\
E_4&= \{8,9\}.
\end{align*}
Equations \eqref{eq:tf} through \eqref{eq:tl} can be written as
\begin{align}
\label{eq:tauf} \tau(A(s,10))&=1+\tau(A(s,1)) &\forall s\in E_0\\
\tau(A(s,10))&=\tau(A(s,1)) &\forall s\in E_1\\
\tau(A(s,11))&=\tau(A(s,10))+\tau(s) &\forall s\in E_2\\
\tau(A(s,11))&=1+\tau(s)+\tau(A(s,1)) &\forall s\in E_0\\
\tau(A(s,11))&=\tau(s)+\tau(A(s,1)) &\forall s\in E_1\\
\tau(A(s,100))&=\tau(A(s,10))+\tau(A(s,1))+\tau(s) &\forall s\in E_2\\
\tau(A(s,100))&=1+\tau(s)&\forall s \in E_0\\
\tau(A(s,100))&=\tau(s)&\forall s\in E_1\\
\tau(A(s,10))&=\tau(s)&\forall s\in E_3\\
\label{eq:taul} \tau(A(s,10))&=\tau(s)&\forall s\in E_4
\end{align}
We verify equations \eqref{eq:tauf} through \eqref{eq:taul} directly.
\end{proof}
\begin{Lemma}\label{lem:3}
For $n\geq 2$ even, $u=(2^{n+3} + 1)/3$, $ v=2^n$,
\begin{align*}
A_{n}&=A^e[:u]+x^v\cdot A^e[:u-v]+x^u\cdot \begin{pmatrix}1&1\\0&1\end{pmatrix}\\
B_{n}&=B^e[:u]+x^v\cdot B^e[:u-v]
\end{align*}
For $n\geq 1$ odd, $u=(2^{n+3} + 2)/3$, $ v=2^n$,
\begin{align*}
A_{n}&=A^o[:u]+x^v\cdot A^o[:u-v]\\
B_{n}&=B^o[:u]+x^v\cdot B^o[:u-v]+x^u\cdot I_2
\end{align*}
\end{Lemma}
\begin{proof}
For $n\geq 2$ even, we prove that
$$A_n \wedge B_n \Rightarrow A_{n+1}.$$
Set $u=(2^{n+3} + 1)/3$, $ v=2^n$. Identity $A_{n+1}$ can be written as
\begin{equation}\label{eq:anp1}
A_{n+1}=A^o[:2u]+x^{2v}\cdot A^o[2u-2v]
\end{equation}
The left side of Eq. \eqref{eq:anp1} is
\begin{align*}
&\quad B_n A_n\\
&=(B^e[:u]+x^v\cdot B^e[:u-v])\left(A^e[:u]+x^v\cdot A^e[:u-v]+x^u\cdot \begin{pmatrix}1&1\\0&1\end{pmatrix}\right)\\
&=(B^e[:u]+x^v\cdot B^e[:u-v])(A^e[:u]+x^v\cdot A^e[:u-v])\\
&\quad x^uB^e[:u]\begin{pmatrix}1&1\\0&1\end{pmatrix}+x^{u+v}B^e[:u-v]\begin{pmatrix}1&1\\0&1\end{pmatrix}
\end{align*}
By lemma \ref{lem:00}, the right side of Eq. \eqref{eq:anp1} is congruent, modulo $x^{2u}$ to
$$(1+x^{2v})A^o[:2u]= (1+x^{2v})B^e[:2u]A^e[:2u].$$
We recall that
\begin{align*}
A^e[:2u]&=(1+x^v+x^{2v})\cdot (A^e[:u]+x^v A^e[:u-v])+x^{4v}A^e[:2u-4v]+\\
&\quad (x^u+x^{u+v}+x^{u+2v})\begin{pmatrix}1&1\\0&1\end{pmatrix},\\
B^e[:2u]&=(1+x^v+x^{2v})\cdot (B^e[:u]+x^v B^e[:u-v])+x^{4v}B^e[:2u-4v]+\\
&\quad \begin{pmatrix}x^{2u-1}&x^{2u-1}+x^{2u-4}\\0&x^{2u-1}\end{pmatrix}.
\end{align*}
Noticing that
$$ \begin{pmatrix}x^{2u-1}&x^{2u-1}+x^{2u-4}\\0&x^{2u-1}\end{pmatrix} A^e[:2u]\equiv 0 \mod x^{2u},$$
and
$$(1+x^{2v})(1+x^v+x^{2v})^2=1+x^{6v}\equiv 0\mod x^{2u},$$
we have
\begin{align*}
&\quad (1+x^{2v})\cdot B^e[:2u]A^e[:2u]\\
&\equiv(1+x^{2v})\cdot\left((1+x^v+x^{2v})\cdot (B^e[:u]+x^v B^e[:u-v])+x^{4v}B^e[:2u-4v]\right)\\
&\quad ((1+x^v+x^{2v})\cdot (A^e[:u]+x^v A^e[:u-v])+x^{4v}A^e[:2u-4v])\\
&\quad(1+x^{2v})\cdot\left((1+x^v+x^{2v})\cdot (B^e[:u]+x^v B^e[:u-v])+x^{4v}B^e[:2u-4v]\right)\\
&\quad (x^u+x^{u+v}+x^{u+2v})\begin{pmatrix}1&1\\0&1\end{pmatrix}\\
&\equiv (B^e[:u]+x^v B^e[:u-v])\cdot(A^e[:u]+x^v A^e[:u-v])\\
&\quad x^u\cdot (B^e[:u]+x^v B^e[:u-v])\cdot
\begin{pmatrix}1&1\\0&1\end{pmatrix} \mod x^{2u}.
\end{align*}
Thus we have proved that both sides of Eq. \eqref{eq:anp1} are congruent
modulo $x^{2u}$, so that they must be equal as both have degree at
most $2u$.
\end{proof}
Theorem \ref{thm:pd1} can be derived from lemma \ref{lem:3} and the definition
of $A^e$, using the same method as in the proof of theorem \ref{th:ab}.
\section{From equation to automaton}\label{sec:eq2kern}
In this section we give an description of the algorithm that we use to calculate
a $p$-automaton of an algebraic series $T(x)$
in $\mathbb{F}_{q}[[x]]$
from an annihilating polynomial of it, where $\mathbb{F}_q$ is a finite field
of characteristic $p$. The algorithm is based on the
proof of theorem 1 in \cite{Christol1980KMFR}.
\medskip
\noindent
{\bf Step one: Normalization}\\
Input: an annihilating polynomial $P(x,y)\in \mathbb{F}_{q}(x)[y]$ of $T(x)$.\\
Output: an annihilating polynomial $Q(x,y)\in \mathbb{F}_{q}(x)[y]$ of $T(x)$
the form
$$y+\frac{a_1(x)}{b_1(x)}y^{p^1}+\frac{a_2(x)}{b_2(x)}y^{p^2}+\cdots+
\frac{a_n(x)}{b_n(x)}y^{p^n}.$$
Method:
use the relation $P(x,T(x))=0$ to express $T(x)^{p^j}$ as
$\mathbb{F}_{p}(x)$-linear combination of $T(x)^k$, $k=0,1,\ldots, d-1$,
where $d$ is the degree of $P(x,y)$ as a polynomial in $y$. In practice,
to find the expression of $T(x)^{p^j}$, we first calculate that of
$T(x)^{p^{j-1}}$, then raise it to the $p$-th power, and finally reduce
again using the relation $P(x,T(x))=0$.
We know that the family $T(x)^{p^j}$,
$j=0,1,\ldots, d$ is necessarily linearly dependent.
However, as it can be
costly to compute $T(x)^{p^j}$ when $j$ is large, in reality we stop once
the rank of the family $T(x)^{p^j}$, $k=0,1,\ldots, j_0$ is less than $j_0+1$.
\medskip
\noindent
{\bf Step two: From normalized equation to kernel}\\
Input: the relation
\begin{equation}\label{eq:rel}
T(x)=\frac{a_1(x)}{b_1(x)}T(x)^{p^1}+\frac{a_2(x)}{b_2(x)}T(x)^{p^2}+\cdots+
\frac{a_n(x)}{b_n(x)}T(x)^{p^n}.
\end{equation}
Output: the $p$-kernel of $T(x)$.\\
Method:
We let $\phi$ denote the Frobenius morphism and $\Lambda_j$ the Cartier operator
that maps $\sum a_l x^l$ to $\sum a_{pl+j} x^l$ for $j=0,1,\ldots, p-1$.
We recall that for a series $f(x)=\sum_{l\geq l_0} c_lx^l\in \mathbb{F}_q((x))$
and polynomials $a(x)$ and $b(x)$,
$$\Lambda_j(a(x)f(x)^p)=\Lambda_j(a(x))\Lambda_0 (f(x)^p)$$
for $j=0,1,\ldots,p$ and
$$\Lambda_0(f(x)^p)=\Lambda_0 \sum_{l\geq l_0} c_l^p x^{p\cdot l}=\sum_{l\geq l_0} c_l^p x^l=\phi(f)(x).$$
Combining the above two identities and we get
\begin{equation}\label{eq:rule}
\Lambda_j\left(\frac{a(x)}{b(x)}f(x)^p\right)
=\Lambda_j\left(a(x)b(x)^{p-1} \frac{f(x)^p}{b(x)^p}\right)
=\Lambda_j(a(x)b(x)^{p-1})\frac{ \phi(f)(x)}{\phi(b)(x)}.
\end{equation}
When we apply repeatedly $\Lambda_j$, $j=0,1, \ldots, p-1$
to both sides of \eqref{eq:rel} using the above computation rule and rewrite
$\phi^k(T)(x)$ using relation \eqref{eq:rel},
we always get an expression of the form (this will be illustrated by
example \ref{eg:ker} below)
\begin{equation}\label{eq:ker2}
\frac{c_1(x)}{d_1(x)}\phi^k(T)(x)^{p^1}+\frac{c_2(x)}{d_2(x)}\phi^k(T)(x)^{p^2}+\cdots+
\frac{c_n(x)}{d_n(x)}\phi^k(T)(x)^{p^n},
\end{equation}
where $k=0,1,\ldots, \log q/\log p-1$, and $c_j(x)$ and $d_j(x)$ are polynomial
of bounded degree for $j=1,2,\ldots, n$.
To see the last point, note that for $d_j(x)$ is always a factor of
$$\prod_{l=1}^{n} \phi^k(b_l(x))$$
for some $0\leq k < \log q/\log p$,
and
$$
\deg c_j\leq \deg d_j +
\max \{\deg a_l -\deg b_l\mid l=1,2,\ldots, n\}.
$$
The set of expression of the form \eqref{eq:ker2} is therefore finite and
the process must terminate. In the end we get a finite set that is the
$p$-kernel of $T(x)$.
In our program, the expression \eqref{eq:ker2} is encoded by the tuple
$$
\left(\left \{1: {c_1(x)}/{d_1(x)}, 2:{c_2(x)}/{d_2(x)}, \ldots,
n: {c_n(x)}/{d_n(x)}\right\}, k\right).
$$
\begin{Remark}
Note that the reason we use powers of $p$ instead of powers of $q$ in
\eqref{eq:rel} is that the latter usually needs much larger coefficients.
\end{Remark}
\medskip
\medskip
\begin{Example}\label{eg:ker}
Set $a=\bar{u}\in \mathbb{F}_4=\mathbb{F}_2[u]/<u^2+u+1>$.
Let $T(x)$ be the unique solution in $\mathbb{F}_4[[x]]$ of
$$(x^2+ax)y^3+y+a+x=0.$$
We write the equation in the normalized form, which is really easy for this
example:
\begin{equation}\label{eq:eg}
T(x)=\frac{1}{x+a}T(x)^2+x T(x)^4.
\end{equation}
We use computation rule \eqref{eq:rule} to calculate $\Lambda_0 T(x)$ and $\Lambda_0\Lambda_0 T(x)$ to illustrate
of this process:
\begin{align*}
\Lambda_0 T(x)&=\Lambda_0(\frac{1}{x+a}T(x)^2)+\Lambda_0(x T(x)^4)\\
&=\Lambda_0\left((x+a)\frac{T(x)^2}{(x+a)^2}\right)
+\Lambda_0(x)\cdot \phi(T)(x)^2\\
&=a \frac{\phi(T)(x)}{x+a+1}.
\end{align*}
To calculate $\Lambda_0\Lambda_0 T(x)$, we need to first put the above
expression into form \eqref{eq:ker2}. Applying $\phi$ to both sides of
\eqref{eq:eg} we get
\begin{equation*}
\phi( T)(x)=\frac{1}{x+a+1}\phi(T)(x)^2+x \phi(T)(x)^4.
\end{equation*}
Therefore
\begin{align*}
\Lambda_0 T(x)&=\frac{a}{(x+a+1)^2}\phi(T)(x)^2+\frac{ax}{(x+a+1)}
\phi(T)(x)^4,
\end{align*}
and
\begin{align*}
\Lambda_0 \Lambda_0 T(x)&=\Lambda_0\left(\frac{a}{(x+a+1)^2}\phi(T)(x)^2\right)
+\Lambda_0\left(\frac{ax(x+a+1)}{(x+a+1)^2} \phi(T)(x)^4, \right)\\
&=\frac{a}{x+a} \phi^2(T)(x)+ \frac{a}{x+a} \phi^2(T)(x)^2\\
&=\frac{a}{x+a} T(x)+ \frac{ax}{x+a}T(x)^2.\\
\end{align*}
In our program, the series $T(x)$, $\Lambda_0 T(x)$ and $\Lambda_0\Lambda_0
T(x)$ are encoded by
$$(\{0:1\},\; 0),$$
$$(\{0:a/(x + a + 1)\},\;1),$$
and
$$(\{0: a/(x + a),\; 1: ax/(x + a)\},\; 0).$$
\end{Example}
\begin{comment}
For example, set $a=\bar{u}\in \mathbb{F}_4=\mathbb{F}_2[u]/<u^2+u+1>$.
Let $T(x)$ be the unique solution in $\mathbb{F}_4[[x]]$ of
$$(ax^2+1)y^3+x+a+1=0.$$
We write the equation in the normalized form, which is really easy for this
example:
\begin{equation}\label{eq:eg}
T(x)=\frac{ax^2+1}{x+a+1}T(x)^4.
\end{equation}
Let us calculate $\Lambda_0 T(x)$ and $\Lambda_0\Lambda_0 T(x)$ to illustrate
this step.
\begin{align*}
\Lambda_0 T(x)
&= \Lambda_0 \left(\frac{(ax^2+1)(x+a+1)}{(x+a+1)^2}T(x)^4\right)\\
&=\Lambda_0((ax^2+1)(x+a+1))\frac{\phi(T)(x)^2}{\phi(x+a+1)}\\
&= (x+a+1)\frac{\phi(T)(x)^2}{x+a}.
\end{align*}
So that in our program, $\Lambda_0 T(x)$ represented by
$$ (\{1: (x + a + 1)/(x + a)\}, 1).$$
\begin{align*}
\Lambda_0\Lambda_0 T(x)
&= \Lambda_0 \left(\frac{x+a+1}{x+a}{\phi(T)(x)^2}\right)\\
&= \Lambda_0 ((x+a+1)(x+a)) \frac{\phi^2(T)(x)}{\phi(x+a)}\\
&=(x+1) \frac{T(x)}{(x+a+1)}
\end{align*}
If we want to continue to calculate $\Lambda_0\Lambda_0\Lambda_0 T(x)$,
we need to replace the $T(x)$ that appear in the expression above
by the right side of equation \eqref{eq:eg}.
\end{comment}
\medskip
\noindent
{\bf Step three: Output function}
In step two, each element from the $p$-kernel of $T(x)$ is expressed as an
$\mathbb{F}_{q}(x)$-linear combination of powers of $\phi^k(T)(x)$, for
some $0\leq k< \log q/\log p$. The output
function maps the corresponding state to the constant term of the series.
To calculate it, we simply plug in $T(x) \mod x^{D+1}$, where $D$ is the maximum
of $0$ and the minus of the orders of the coefficients of the
linear combination.
\newpage
\section{Annex}
\subsection{Data for Section \ref{section:tmncf}}\label{data:tmncf}
All $16$ polynomials are of the form
$$p_0(x)+p_3(x)y^3+p_6(x)y^6+p_9(x)y^9+p_{12}(x)y^{12}.$$
The coefficients $p_j(x)$ and the $8$ initial terms to determine the solutions uniquely
are given below.
For $\phi(M^e,0,0)$:
\begin{align*}
p_{0}(x)&=x^{66} +x^{64} +x^{62} +x^{60} +x^{58} +x^{56} +x^{52} +x^{50} +x^{36} +x^{32} +x^{30}\\
& \quad +x^{20} +x^{16} +x^{14} +x^{12} ,\\
p_{3}(x)&=x^{62} +x^{60} +x^{58} +x^{56} +x^{52} +x^{50} +x^{48} +x^{44} +x^{42} +x^{38} +x^{36}\\
& \quad +x^{32} +x^{28} +x^{22} +x^{20} +x^{18} +x^{14} +x^{12} ,\\
p_{6}(x)&=x^{56} +x^{44} +x^{40} +x^{38} +x^{36} +x^{32} +x^{30} +x^{26} +x^{20} +x^{18} +x^{16}\\
& \quad +x^{14} +x^2 +1 ,\\
p_{9}(x)&=x^{64} +x^{62} +x^{58} +x^{56} +x^{54} +x^{52} +x^{48} +x^{42} +x^{32} +x^{30} +x^{26}\\
& \quad +x^{20} +x^{18} +x^{14} +x^{12} +x^{10} +x^8 +x^2 ,\\
p_{12}(x)&=x^{64} +x^{56} +x^{40} +x^{32} +x^{16} +x^8 +1 ,
\end{align*}
and the initial terms are [1, 0, 0, 0, 0, 0, 0, 0].
For $\phi(M^e,0,1)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{62} +x^{60} +x^{59} +x^{56} +x^{55} +x^{53} +x^{51} +x^{50} +x^{49} +x^{48}\\
& \quad +x^{47} +x^{45} +x^{44} +x^{42} +x^{41} +x^{39} +x^{37} +x^{36} +x^{35} +x^{34} +x^{32}\\
& \quad +x^{28} +x^{27} +x^{25} +x^{24} +x^{23} +x^{21} +x^{20} +x^{18} +x^{15} ,\\
p_{3}(x)&=x^{61} +x^{59} +x^{51} +x^{49} +x^{43} +x^{41} +x^{37} +x^{33} +x^{21} +x^{19} +x^{13} +x^9 ,\\
p_{6}(x)&=x^{60} +x^{59} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{32} +x^{31} +x^{29} +x^{28} +x^{26} +x^{25} +x^{24} +x^{22} +x^{21}\\
& \quad +x^{19} +x^{18} +x^{16} +x^{15} +x^{13} +x^{12} +x^{10} +x^9 +x^7 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{53} +x^{51} +x^{49} +x^{47} +x^{45} +x^{43} +x^{41} +x^{37} +x^{29} +x^{27}\\
& \quad +x^{25} +x^{21} +x^{15} +x^{13} +x^7 +x^5 +x^3 ,\\
p_{12}(x)&=x^{58} +x^{57} +x^{56} +x^{54} +x^{53} +x^{52} +x^{50} +x^{49} +x^{48} +x^{42} +x^{41}\\
& \quad +x^{40} +x^{38} +x^{37} +x^{36} +x^{34} +x^{33} +x^{32} +x^{10} +x^9 +x^8 +x^6 +x^5\\
& \quad +x^4 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 1, 0, 0, 1, 1, 0, 0].
For $\phi(M^e,1,0)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{62} +x^{60} +x^{59} +x^{56} +x^{55} +x^{53} +x^{51} +x^{50} +x^{49} +x^{48}\\
& \quad +x^{47} +x^{45} +x^{44} +x^{42} +x^{41} +x^{39} +x^{37} +x^{36} +x^{35} +x^{34} +x^{32}\\
& \quad +x^{28} +x^{27} +x^{25} +x^{24} +x^{23} +x^{21} +x^{20} +x^{18} +x^{15} ,\\
p_{3}(x)&=x^{61} +x^{59} +x^{51} +x^{49} +x^{43} +x^{41} +x^{37} +x^{33} +x^{21} +x^{19} +x^{13} +x^9 ,\\
p_{6}(x)&=x^{60} +x^{59} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{32} +x^{31} +x^{29} +x^{28} +x^{26} +x^{25} +x^{24} +x^{22} +x^{21}\\
& \quad +x^{19} +x^{18} +x^{16} +x^{15} +x^{13} +x^{12} +x^{10} +x^9 +x^7 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{53} +x^{51} +x^{49} +x^{47} +x^{45} +x^{43} +x^{41} +x^{37} +x^{29} +x^{27}\\
& \quad +x^{25} +x^{21} +x^{15} +x^{13} +x^7 +x^5 +x^3 ,\\
p_{12}(x)&=x^{58} +x^{57} +x^{56} +x^{54} +x^{53} +x^{52} +x^{50} +x^{49} +x^{48} +x^{42} +x^{41}\\
& \quad +x^{40} +x^{38} +x^{37} +x^{36} +x^{34} +x^{33} +x^{32} +x^{10} +x^9 +x^8 +x^6 +x^5\\
& \quad +x^4 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 1, 0, 0, 1, 1, 0, 0].
For $\phi(M^e,1,1)$:
\begin{align*}
p_{0}(x)&=x^{60} +x^{56} +x^{54} +x^{50} +x^{48} +x^{44} +x^{38} +x^{34} +x^{30} +x^{28} +x^{26}\\
& \quad +x^{20} +x^{18} ,\\
p_{3}(x)&=x^{54} +x^{50} +x^{44} +x^{38} +x^{28} +x^{20} +x^{18} +x^{12} +x^{10} +x^6 ,\\
p_{6}(x)&=x^{54} +x^{52} +x^{50} +x^{38} +x^{36} +x^{24} +x^{18} +x^{16} +x^{10} +x^6 +x^4 +1 ,\\
p_{9}(x)&=x^{52} +x^{44} +x^{42} +x^{40} +x^{30} +x^{24} +x^{22} +x^{20} +x^{16} +x^{10} +x^8 +x^6\\
& \quad +x^4 +x^2 ,\\
p_{12}(x)&=x^{52} +x^{50} +x^{48} +x^{36} +x^{34} +x^{32} +x^4 +x^2 +1 ,
\end{align*}
and the initial terms are [0, 0, 1, 0, 0, 0, 0, 0].
For $\phi(W^e,0,0)$:
\begin{align*}
p_{0}(x)&=x^{60} +x^{50} +x^{44} +x^{40} +x^{30} +x^{28} +x^{24} +x^{18} +x^{12} ,\\
p_{3}(x)&=x^{54} +x^{44} +x^{34} +x^{28} +x^{26} +x^{22} +x^{20} +x^{14} +x^{12} +x^{10} ,\\
p_{6}(x)&=x^{54} +x^{52} +x^{48} +x^{46} +x^{42} +x^{38} +x^{34} +x^{32} +x^{30} +x^{22} +x^{20}\\
& \quad +x^{18} +x^{14} +x^{12} +x^6 +1 ,\\
p_{9}(x)&=x^{52} +x^{44} +x^{42} +x^{40} +x^{30} +x^{24} +x^{22} +x^{20} +x^{16} +x^{10} +x^8 +x^6\\
& \quad +x^4 +x^2 ,\\
p_{12}(x)&=x^{52} +x^{50} +x^{48} +x^{36} +x^{34} +x^{32} +x^4 +x^2 +1 ,
\end{align*}
and the initial terms are [1, 0, 1, 0, 1, 0, 0, 0].
For $\phi(W^e,0,1)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{60} +x^{59} +x^{56} +x^{52} +x^{51} +x^{50} +x^{47} +x^{43} +x^{39} +x^{36}\\
& \quad +x^{35} +x^{28} +x^{25} +x^{22} +x^{20} +x^{18} +x^{16} +x^{15} ,\\
p_{3}(x)&=x^{61} +x^{59} +x^{51} +x^{49} +x^{43} +x^{41} +x^{37} +x^{33} +x^{21} +x^{19} +x^{13} +x^9 ,\\
p_{6}(x)&=x^{60} +x^{59} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{32} +x^{31} +x^{29} +x^{28} +x^{26} +x^{25} +x^{24} +x^{22} +x^{21}\\
& \quad +x^{19} +x^{18} +x^{16} +x^{15} +x^{13} +x^{12} +x^{10} +x^9 +x^7 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{53} +x^{51} +x^{49} +x^{47} +x^{45} +x^{43} +x^{41} +x^{37} +x^{29} +x^{27}\\
& \quad +x^{25} +x^{21} +x^{15} +x^{13} +x^7 +x^5 +x^3 ,\\
p_{12}(x)&=x^{58} +x^{57} +x^{56} +x^{54} +x^{53} +x^{52} +x^{50} +x^{49} +x^{48} +x^{42} +x^{41}\\
& \quad +x^{40} +x^{38} +x^{37} +x^{36} +x^{34} +x^{33} +x^{32} +x^{10} +x^9 +x^8 +x^6 +x^5\\
& \quad +x^4 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 0, 1, 1, 1, 1, 0, 1].
For $\phi(W^e,1,0)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{60} +x^{59} +x^{56} +x^{52} +x^{51} +x^{50} +x^{47} +x^{43} +x^{39} +x^{36}\\
& \quad +x^{35} +x^{28} +x^{25} +x^{22} +x^{20} +x^{18} +x^{16} +x^{15} ,\\
p_{3}(x)&=x^{61} +x^{59} +x^{51} +x^{49} +x^{43} +x^{41} +x^{37} +x^{33} +x^{21} +x^{19} +x^{13} +x^9 ,\\
p_{6}(x)&=x^{60} +x^{59} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{32} +x^{31} +x^{29} +x^{28} +x^{26} +x^{25} +x^{24} +x^{22} +x^{21}\\
& \quad +x^{19} +x^{18} +x^{16} +x^{15} +x^{13} +x^{12} +x^{10} +x^9 +x^7 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{53} +x^{51} +x^{49} +x^{47} +x^{45} +x^{43} +x^{41} +x^{37} +x^{29} +x^{27}\\
& \quad +x^{25} +x^{21} +x^{15} +x^{13} +x^7 +x^5 +x^3 ,\\
p_{12}(x)&=x^{58} +x^{57} +x^{56} +x^{54} +x^{53} +x^{52} +x^{50} +x^{49} +x^{48} +x^{42} +x^{41}\\
& \quad +x^{40} +x^{38} +x^{37} +x^{36} +x^{34} +x^{33} +x^{32} +x^{10} +x^9 +x^8 +x^6 +x^5\\
& \quad +x^4 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 0, 1, 1, 1, 1, 0, 1].
For $\phi(W^e,1,1)$:
\begin{align*}
p_{0}(x)&=x^{66} +x^{60} +x^{58} +x^{56} +x^{52} +x^{48} +x^{42} +x^{40} +x^{36} +x^{32} +x^{30}\\
& \quad +x^{28} +x^{26} +x^{20} +x^{18} ,\\
p_{3}(x)&=x^{58} +x^{50} +x^{46} +x^{44} +x^{42} +x^{38} +x^{36} +x^{32} +x^{30} +x^{24} +x^{20}\\
& \quad +x^{18} +x^8 +x^6 ,\\
p_{6}(x)&=x^{62} +x^{56} +x^{52} +x^{46} +x^{40} +x^{38} +x^{36} +x^{32} +x^{26} +x^{22} +x^{18}\\
& \quad +x^{16} +x^6 +x^4 +x^2 +1 ,\\
p_{9}(x)&=x^{64} +x^{62} +x^{58} +x^{56} +x^{54} +x^{52} +x^{48} +x^{42} +x^{32} +x^{30} +x^{26}\\
& \quad +x^{20} +x^{18} +x^{14} +x^{12} +x^{10} +x^8 +x^2 ,\\
p_{12}(x)&=x^{64} +x^{56} +x^{40} +x^{32} +x^{16} +x^8 +1 ,
\end{align*}
and the initial terms are [0, 0, 0, 0, 1, 0, 0, 0].
For $\phi(M^o,0,0)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{62} +x^{61} +x^{60} +x^{59} +x^{57} +x^{55} +x^{53} +x^{51} +x^{50} +x^{49}\\
& \quad +x^{47} +x^{45} +x^{44} +x^{42} +x^{40} +x^{39} +x^{37} +x^{36} +x^{31} +x^{30} +x^{28}\\
& \quad +x^{27} +x^{26} +x^{24} +x^{23} +x^{21} +x^{18} +x^{17} +x^{15} +x^{14} +x^{13} +x^{12} ,\\
p_{3}(x)&=x^{62} +x^{59} +x^{58} +x^{56} +x^{55} +x^{53} +x^{52} +x^{51} +x^{46} +x^{45} +x^{41}\\
& \quad +x^{39} +x^{38} +x^{37} +x^{36} +x^{34} +x^{32} +x^{31} +x^{30} +x^{28} +x^{27} +x^{25}\\
& \quad +x^{23} +x^{20} +x^{18} +x^{17} +x^{16} +x^{13} +x^{11} +x^{10} +x^9 +x^8 ,\\
p_{6}(x)&=x^{62} +x^{59} +x^{58} +x^{57} +x^{54} +x^{52} +x^{51} +x^{48} +x^{44} +x^{42} +x^{41}\\
& \quad +x^{37} +x^{36} +x^{35} +x^{33} +x^{31} +x^{30} +x^{29} +x^{26} +x^{24} +x^{23} +x^{18}\\
& \quad +x^{17} +x^{14} +x^{13} +x^{12} +x^{10} +x^8 +x^7 +x^2 +x +1 ,\\
p_{9}(x)&=x^{62} +x^{61} +x^{60} +x^{54} +x^{53} +x^{52} +x^{46} +x^{45} +x^{44} +x^{38} +x^{37}\\
& \quad +x^{36} +x^{14} +x^{13} +x^{12} +x^6 +x^5 +x^4 ,\\
p_{12}(x)&=x^{62} +x^{61} +x^{60} +x^{58} +x^{57} +x^{56} +x^{50} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{44} +x^{42} +x^{41} +x^{40} +x^{34} +x^{33} +x^{32} +x^{14} +x^{13} +x^{12} +x^{10}\\
& \quad +x^9 +x^8 +x^2 +x +1 ,
\end{align*}
and the initial terms are [1, 0, 1, 1, 0, 1, 0, 0].
For $\phi(M^o,0,1)$:
\begin{align*}
p_{0}(x)&=x^{64} +x^{61} +x^{59} +x^{55} +x^{54} +x^{52} +x^{51} +x^{50} +x^{48} +x^{47} +x^{46}\\
& \quad +x^{45} +x^{42} +x^{41} +x^{40} +x^{38} +x^{36} +x^{34} +x^{33} +x^{32} +x^{31} +x^{30}\\
& \quad +x^{29} +x^{28} +x^{26} +x^{23} +x^{20} +x^{19} +x^{17} +x^{16} +x^{15} ,\\
p_{3}(x)&=x^{57} +x^{56} +x^{55} +x^{51} +x^{50} +x^{48} +x^{47} +x^{43} +x^{42} +x^{40} +x^{39}\\
& \quad +x^{35} +x^{34} +x^{33} +x^{17} +x^{16} +x^{15} +x^{11} +x^{10} +x^9 ,\\
p_{6}(x)&=x^{58} +x^{56} +x^{54} +x^{50} +x^{48} +x^{44} +x^{42} +x^{30} +x^{28} +x^{24} +x^{20}\\
& \quad +x^{18} +x^{14} +x^{12} +x^8 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{58} +x^{56} +x^{54} +x^{52} +x^{51} +x^{43} +x^{42} +x^{40} +x^{38} +x^{36}\\
& \quad +x^{35} +x^{11} +x^{10} +x^8 +x^6 +x^4 +x^3 ,\\
p_{12}(x)&=x^{60} +x^{56} +x^{48} +x^{44} +x^{40} +x^{32} +x^{12} +x^8 +1 ,
\end{align*}
and the initial terms are [0, 1, 0, 1, 0, 1, 0, 0].
For $\phi(M^o,1,0)$:
\begin{align*}
p_{0}(x)&=x^{66} +x^{65} +x^{64} +x^{62} +x^{60} +x^{59} +x^{58} +x^{54} +x^{51} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{41} +x^{40} +x^{35} +x^{33} +x^{31} +x^{29} +x^{28} +x^{27} +x^{23}\\
& \quad +x^{22} +x^{21} +x^{20} +x^{18} +x^{15} ,\\
p_{3}(x)&=x^{65} +x^{64} +x^{63} +x^{61} +x^{60} +x^{58} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51}\\
& \quad +x^{47} +x^{46} +x^{44} +x^{43} +x^{41} +x^{40} +x^{38} +x^{37} +x^{35} +x^{34} +x^{33}\\
& \quad +x^{25} +x^{24} +x^{23} +x^{21} +x^{20} +x^{18} +x^{16} +x^{14} +x^{13} +x^{11} +x^{10}\\
& \quad +x^9 ,\\
p_{6}(x)&=x^{66} +x^{64} +x^{60} +x^{58} +x^{46} +x^{44} +x^{42} +x^{38} +x^{36} +x^{34} +x^{30}\\
& \quad +x^{28} +x^{26} +x^{10} +x^8 +x^6 ,\\
p_{9}(x)&=x^{67} +x^{66} +x^{64} +x^{63} +x^{55} +x^{54} +x^{52} +x^{50} +x^{48} +x^{47} +x^{39}\\
& \quad +x^{38} +x^{36} +x^{35} +x^{19} +x^{18} +x^{16} +x^{15} +x^7 +x^6 +x^4 +x^3 ,\\
p_{12}(x)&=x^{68} +x^{48} +x^{36} +x^{32} +x^{20} +x^4 +1 ,
\end{align*}
and the initial terms are [0, 0, 1, 1, 1, 0, 0, 1].
For $\phi(M^o,1,1)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{61} +x^{59} +x^{58} +x^{57} +x^{56} +x^{55} +x^{53} +x^{49} +x^{48} +x^{45}\\
& \quad +x^{44} +x^{43} +x^{42} +x^{35} +x^{31} +x^{30} +x^{29} +x^{28} +x^{26} +x^{25} +x^{20}\\
& \quad +x^{18} ,\\
p_{3}(x)&=x^{62} +x^{60} +x^{56} +x^{55} +x^{53} +x^{52} +x^{51} +x^{50} +x^{49} +x^{48} +x^{46}\\
& \quad +x^{45} +x^{42} +x^{40} +x^{39} +x^{38} +x^{37} +x^{35} +x^{34} +x^{33} +x^{31} +x^{30}\\
& \quad +x^{28} +x^{27} +x^{25} +x^{23} +x^{19} +x^{17} +x^{16} +x^{13} +x^{12} +x^{10} ,\\
p_{6}(x)&=x^{62} +x^{60} +x^{57} +x^{56} +x^{55} +x^{49} +x^{46} +x^{45} +x^{44} +x^{43} +x^{41}\\
& \quad +x^{38} +x^{35} +x^{34} +x^{30} +x^{28} +x^{25} +x^{23} +x^{20} +x^{19} +x^{18} +x^{15}\\
& \quad +x^{13} +x^{12} +x^{10} +x^5 +x^4 +x^2 +x +1 ,\\
p_{9}(x)&=x^{62} +x^{61} +x^{60} +x^{54} +x^{53} +x^{52} +x^{46} +x^{45} +x^{44} +x^{38} +x^{37}\\
& \quad +x^{36} +x^{14} +x^{13} +x^{12} +x^6 +x^5 +x^4 ,\\
p_{12}(x)&=x^{62} +x^{61} +x^{60} +x^{58} +x^{57} +x^{56} +x^{50} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{44} +x^{42} +x^{41} +x^{40} +x^{34} +x^{33} +x^{32} +x^{14} +x^{13} +x^{12} +x^{10}\\
& \quad +x^9 +x^8 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 0, 0, 1, 1, 1, 1, 0].
For $\phi(W^o,0,0)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{62} +x^{61} +x^{60} +x^{59} +x^{57} +x^{55} +x^{53} +x^{51} +x^{50} +x^{49}\\
& \quad +x^{47} +x^{45} +x^{44} +x^{42} +x^{40} +x^{39} +x^{37} +x^{36} +x^{31} +x^{30} +x^{28}\\
& \quad +x^{27} +x^{26} +x^{24} +x^{23} +x^{21} +x^{18} +x^{17} +x^{15} +x^{14} +x^{13} +x^{12} ,\\
p_{3}(x)&=x^{62} +x^{59} +x^{58} +x^{56} +x^{55} +x^{53} +x^{52} +x^{51} +x^{46} +x^{45} +x^{41}\\
& \quad +x^{39} +x^{38} +x^{37} +x^{36} +x^{34} +x^{32} +x^{31} +x^{30} +x^{28} +x^{27} +x^{25}\\
& \quad +x^{23} +x^{20} +x^{18} +x^{17} +x^{16} +x^{13} +x^{11} +x^{10} +x^9 +x^8 ,\\
p_{6}(x)&=x^{62} +x^{59} +x^{58} +x^{57} +x^{54} +x^{52} +x^{51} +x^{48} +x^{44} +x^{42} +x^{41}\\
& \quad +x^{37} +x^{36} +x^{35} +x^{33} +x^{31} +x^{30} +x^{29} +x^{26} +x^{24} +x^{23} +x^{18}\\
& \quad +x^{17} +x^{14} +x^{13} +x^{12} +x^{10} +x^8 +x^7 +x^2 +x +1 ,\\
p_{9}(x)&=x^{62} +x^{61} +x^{60} +x^{54} +x^{53} +x^{52} +x^{46} +x^{45} +x^{44} +x^{38} +x^{37}\\
& \quad +x^{36} +x^{14} +x^{13} +x^{12} +x^6 +x^5 +x^4 ,\\
p_{12}(x)&=x^{62} +x^{61} +x^{60} +x^{58} +x^{57} +x^{56} +x^{50} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{44} +x^{42} +x^{41} +x^{40} +x^{34} +x^{33} +x^{32} +x^{14} +x^{13} +x^{12} +x^{10}\\
& \quad +x^9 +x^8 +x^2 +x +1 ,
\end{align*}
and the initial terms are [1, 0, 1, 1, 0, 1, 0, 0].
For $\phi(W^o,0,1)$:
\begin{align*}
p_{0}(x)&=x^{66} +x^{65} +x^{64} +x^{62} +x^{60} +x^{59} +x^{58} +x^{54} +x^{51} +x^{46} +x^{45}\\
& \quad +x^{43} +x^{42} +x^{41} +x^{40} +x^{35} +x^{33} +x^{31} +x^{29} +x^{28} +x^{27} +x^{23}\\
& \quad +x^{22} +x^{21} +x^{20} +x^{18} +x^{15} ,\\
p_{3}(x)&=x^{65} +x^{64} +x^{63} +x^{61} +x^{60} +x^{58} +x^{57} +x^{55} +x^{54} +x^{52} +x^{51}\\
& \quad +x^{47} +x^{46} +x^{44} +x^{43} +x^{41} +x^{40} +x^{38} +x^{37} +x^{35} +x^{34} +x^{33}\\
& \quad +x^{25} +x^{24} +x^{23} +x^{21} +x^{20} +x^{18} +x^{16} +x^{14} +x^{13} +x^{11} +x^{10}\\
& \quad +x^9 ,\\
p_{6}(x)&=x^{66} +x^{64} +x^{60} +x^{58} +x^{46} +x^{44} +x^{42} +x^{38} +x^{36} +x^{34} +x^{30}\\
& \quad +x^{28} +x^{26} +x^{10} +x^8 +x^6 ,\\
p_{9}(x)&=x^{67} +x^{66} +x^{64} +x^{63} +x^{55} +x^{54} +x^{52} +x^{50} +x^{48} +x^{47} +x^{39}\\
& \quad +x^{38} +x^{36} +x^{35} +x^{19} +x^{18} +x^{16} +x^{15} +x^7 +x^6 +x^4 +x^3 ,\\
p_{12}(x)&=x^{68} +x^{48} +x^{36} +x^{32} +x^{20} +x^4 +1 ,
\end{align*}
and the initial terms are [0, 0, 1, 1, 1, 0, 0, 1].
For $\phi(W^o,1,0)$:
\begin{align*}
p_{0}(x)&=x^{64} +x^{61} +x^{59} +x^{55} +x^{54} +x^{52} +x^{51} +x^{50} +x^{48} +x^{47} +x^{46}\\
& \quad +x^{45} +x^{42} +x^{41} +x^{40} +x^{38} +x^{36} +x^{34} +x^{33} +x^{32} +x^{31} +x^{30}\\
& \quad +x^{29} +x^{28} +x^{26} +x^{23} +x^{20} +x^{19} +x^{17} +x^{16} +x^{15} ,\\
p_{3}(x)&=x^{57} +x^{56} +x^{55} +x^{51} +x^{50} +x^{48} +x^{47} +x^{43} +x^{42} +x^{40} +x^{39}\\
& \quad +x^{35} +x^{34} +x^{33} +x^{17} +x^{16} +x^{15} +x^{11} +x^{10} +x^9 ,\\
p_{6}(x)&=x^{58} +x^{56} +x^{54} +x^{50} +x^{48} +x^{44} +x^{42} +x^{30} +x^{28} +x^{24} +x^{20}\\
& \quad +x^{18} +x^{14} +x^{12} +x^8 +x^6 ,\\
p_{9}(x)&=x^{59} +x^{58} +x^{56} +x^{54} +x^{52} +x^{51} +x^{43} +x^{42} +x^{40} +x^{38} +x^{36}\\
& \quad +x^{35} +x^{11} +x^{10} +x^8 +x^6 +x^4 +x^3 ,\\
p_{12}(x)&=x^{60} +x^{56} +x^{48} +x^{44} +x^{40} +x^{32} +x^{12} +x^8 +1 ,
\end{align*}
and the initial terms are [0, 1, 0, 1, 0, 1, 0, 0].
For $\phi(W^o,1,1)$:
\begin{align*}
p_{0}(x)&=x^{63} +x^{61} +x^{59} +x^{58} +x^{57} +x^{56} +x^{55} +x^{53} +x^{49} +x^{48} +x^{45}\\
& \quad +x^{44} +x^{43} +x^{42} +x^{35} +x^{31} +x^{30} +x^{29} +x^{28} +x^{26} +x^{25} +x^{20}\\
& \quad +x^{18} ,\\
p_{3}(x)&=x^{62} +x^{60} +x^{56} +x^{55} +x^{53} +x^{52} +x^{51} +x^{50} +x^{49} +x^{48} +x^{46}\\
& \quad +x^{45} +x^{42} +x^{40} +x^{39} +x^{38} +x^{37} +x^{35} +x^{34} +x^{33} +x^{31} +x^{30}\\
& \quad +x^{28} +x^{27} +x^{25} +x^{23} +x^{19} +x^{17} +x^{16} +x^{13} +x^{12} +x^{10} ,\\
p_{6}(x)&=x^{62} +x^{60} +x^{57} +x^{56} +x^{55} +x^{49} +x^{46} +x^{45} +x^{44} +x^{43} +x^{41}\\
& \quad +x^{38} +x^{35} +x^{34} +x^{30} +x^{28} +x^{25} +x^{23} +x^{20} +x^{19} +x^{18} +x^{15}\\
& \quad +x^{13} +x^{12} +x^{10} +x^5 +x^4 +x^2 +x +1 ,\\
p_{9}(x)&=x^{62} +x^{61} +x^{60} +x^{54} +x^{53} +x^{52} +x^{46} +x^{45} +x^{44} +x^{38} +x^{37}\\
& \quad +x^{36} +x^{14} +x^{13} +x^{12} +x^6 +x^5 +x^4 ,\\
p_{12}(x)&=x^{62} +x^{61} +x^{60} +x^{58} +x^{57} +x^{56} +x^{50} +x^{49} +x^{48} +x^{46} +x^{45}\\
& \quad +x^{44} +x^{42} +x^{41} +x^{40} +x^{34} +x^{33} +x^{32} +x^{14} +x^{13} +x^{12} +x^{10}\\
& \quad +x^9 +x^8 +x^2 +x +1 ,
\end{align*}
and the initial terms are [0, 0, 0, 1, 1, 1, 1, 0].
Below is the transition function and output function of an $2$-automaton that
generates $T=M^o_{0,0}$:
Transition function $(n,j)\mapsto \delta(n,j)$ ($\Lambda(n):=[\delta(n,0),\delta(n,1)]$):
{\small
{
\renewcommand{\arraystretch}{1.2}
\begin{longtable}{| c c | c c | c c | c c | c c |}
\hline
$n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ & $n$ & $\Lambda(n)$ \\
\hline
0 & [1, 2]& 25 & [14, 42]& 50 & [71, 44]& 75 & [99, 51]& 100 & [10, 39]\\
1 & [3, 4]& 26 & [26, 26]& 51 & [78, 29]& 76 & [100, 97]& 101 & [28, 9]\\
2 & [5, 6]& 27 & [43, 44]& 52 & [32, 14]& 77 & [16, 101]& 102 & [84, 36]\\
3 & [7, 8]& 28 & [45, 46]& 53 & [44, 79]& 78 & [102, 101]& 103 & [79, 41]\\
4 & [9, 10]& 29 & [47, 48]& 54 & [80, 81]& 79 & [103, 104]& 104 & [115, 40]\\
5 & [11, 12]& 30 & [49, 50]& 55 & [82, 36]& 80 & [54, 95]& 105 & [81, 93]\\
6 & [13, 14]& 31 & [51, 52]& 56 & [83, 84]& 81 & [64, 105]& 106 & [87, 81]\\
7 & [15, 16]& 32 & [53, 54]& 57 & [30, 71]& 82 & [48, 60]& 107 & [116, 42]\\
8 & [17, 18]& 33 & [55, 20]& 58 & [85, 79]& 83 & [61, 49]& 108 & [117, 35]\\
9 & [19, 20]& 34 & [56, 51]& 59 & [86, 87]& 84 & [97, 23]& 109 & [118, 41]\\
10 & [20, 21]& 35 & [57, 49]& 60 & [88, 37]& 85 & [23, 19]& 110 & [52, 56]\\
11 & [22, 23]& 36 & [58, 59]& 61 & [73, 31]& 86 & [106, 64]& 111 & [95, 59]\\
12 & [24, 19]& 37 & [60, 61]& 62 & [89, 75]& 87 & [59, 68]& 112 & [101, 104]\\
13 & [25, 26]& 38 & [62, 56]& 63 & [40, 2]& 88 & [107, 105]& 113 & [69, 18]\\
14 & [27, 4]& 39 & [63, 64]& 64 & [90, 26]& 89 & [108, 98]& 114 & [18, 27]\\
15 & [12, 28]& 40 & [21, 16]& 65 & [1, 75]& 90 & [105, 54]& 115 & [119, 106]\\
16 & [29, 30]& 41 & [65, 47]& 66 & [42, 12]& 91 & [6, 47]& 116 & [104, 98]\\
17 & [31, 32]& 42 & [66, 60]& 67 & [91, 92]& 92 & [109, 27]& 117 & [120, 61]\\
18 & [33, 9]& 43 & [67, 68]& 68 & [93, 87]& 93 & [68, 106]& 118 & [121, 21]\\
19 & [34, 30]& 44 & [50, 69]& 69 & [77, 84]& 94 & [110, 2]& 119 & [37, 92]\\
20 & [35, 29]& 45 & [70, 71]& 70 & [46, 6]& 95 & [111, 93]& 120 & [122, 28]\\
21 & [36, 37]& 46 & [72, 73]& 71 & [94, 95]& 96 & [112, 39]& 121 & [92, 89]\\
22 & [2, 38]& 47 & [74, 10]& 72 & [96, 48]& 97 & [8, 77]& 122 & [123, 69]\\
23 & [39, 40]& 48 & [75, 73]& 73 & [4, 97]& 98 & [113, 31]& 123 & [38, 8]\\
24 & [41, 35]& 49 & [76, 77]& 74 & [98, 46]& 99 & [114, 32]& & \\
\hline
\end{longtable}
}
}
Output function $n\mapsto \tau(n)$:
{\small
{
\renewcommand{\arraystretch}{1.2}
\begin{longtable}{| c c | c c | c c | c c | c c | c c | c c |}
\hline
$n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ & $n$ & $\tau(n)$ \\
\hline
0 & 0& 18 & 0& 36 & 1& 54 & 1& 72 & 1& 90 & 1& 108 & 1\\
1 & 0& 19 & 1& 37 & 0& 55 & 0& 73 & 1& 91 & 1& 109 & 1\\
2 & 0& 20 & 0& 38 & 1& 56 & 1& 74 & 0& 92 & 1& 110 & 0\\
3 & 0& 21 & 1& 39 & 1& 57 & 0& 75 & 0& 93 & 1& 111 & 0\\
4 & 1& 22 & 0& 40 & 1& 58 & 1& 76 & 0& 94 & 0& 112 & 1\\
5 & 0& 23 & 1& 41 & 0& 59 & 0& 77 & 0& 95 & 0& 113 & 0\\
6 & 1& 24 & 0& 42 & 1& 60 & 0& 78 & 1& 96 & 1& 114 & 0\\
7 & 0& 25 & 1& 43 & 1& 61 & 1& 79 & 0& 97 & 1& 115 & 0\\
8 & 1& 26 & 0& 44 & 0& 62 & 1& 80 & 1& 98 & 0& 116 & 0\\
9 & 1& 27 & 1& 45 & 1& 63 & 1& 81 & 1& 99 & 0& 117 & 1\\
10 & 0& 28 & 1& 46 & 1& 64 & 1& 82 & 0& 100 & 0& 118 & 1\\
11 & 0& 29 & 0& 47 & 0& 65 & 0& 83 & 1& 101 & 1& 119 & 0\\
12 & 0& 30 & 0& 48 & 0& 66 & 1& 84 & 1& 102 & 1& 120 & 1\\
13 & 1& 31 & 1& 49 & 0& 67 & 1& 85 & 1& 103 & 0& 121 & 1\\
14 & 1& 32 & 0& 50 & 0& 68 & 1& 86 & 0& 104 & 0& 122 & 1\\
15 & 0& 33 & 0& 51 & 1& 69 & 0& 87 & 0& 105 & 1& 123 & 1\\
16 & 0& 34 & 1& 52 & 0& 70 & 1& 88 & 0& 106 & 0& & \\
17 & 1& 35 & 0& 53 & 0& 71 & 0& 89 & 1& 107 & 0& & \\
\hline
\end{longtable}
}
}
\subsection{Data for Section \ref{sec:s}}
All $16$ polynomials are of the form
$$p_0(x)+p_3(x)y^3+p_6(x)y^6+p_9(x)y^9+p_{12}(x)y^{12}.$$
For $(i,j)=(1,0)$ and all $T\in\{M^e, M^o, W^e, W^o\}$,
$$p_6(x)=p_9(x)=p_{12}(x)=0.$$
The coefficients $p_j(x)$ and the $2$ initial terms to determine the solutions uniquely
are given below.
For $\phi(M^e,0,0)$:
\begin{align*}
p_0(x)&=\left(a^{12} + a^{11} + a^{4} + a^{3}\right) x^{12} + \left(a^{10} + a^{9} + a^{7} + a^{5} + a^{3} + a^{2}\right) x^{10} \\ &\quad +\left(a^{8} + a^{7} + a^{2} + a\right) x^{8} + \left(a^{6} + a^{5} + a^{3} + a + 1\right) x^{6},\\
p_3(x)&=\left(a^{9} + a\right) x^{9} + \left(a^{8} + 1\right) x^{8} + \left(a^{6} + a^{5} + a^{2} + a\right) x^{7} + \left(a^{5} + a\right) x^{5}, \\
&\quad + \left(a^{4} + 1\right) x^{4} + \left(a^{2} + a\right) x^{3},\\
p_6(x)&=\left(a^{9} + a\right) x^{9} + \left(a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1\right) x^{8} \\&\quad + \left(a^{6} + a^{5} + a^{2} + a\right) x^{7} + \left(a^{6} + a^{5} + a^{4} + a^{2} + a + 1\right) x^{6} + \left(a^{4} + 1\right) x^{5} \\&\quad + \left(a^{4} + a^{3} + a^{2} + a\right) x^{4}+ \left(a^{2} + a\right) x^{3} + \left(a^{2} + a + 1\right) x^{2} + \left(a + 1\right) x + 1,\\
p_9(x)&=\left(a^{8} + 1\right) x^{8} + \left(a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1\right) x^{7}\\&\quad + \left(a^{6} + a^{4} + a^{2} + 1\right) x^{6} + \left(a^{5} + a^{4} + a + 1\right) x^{5} + \left(a^{4} + 1\right) x^{4}\\&\quad + \left(a^{3} + a^{2} + a + 1\right) x^{3} + \left(a^{2} + 1\right) x^{2}+ \left(a + 1\right) x,\\
p_{12}(x)&=\left(a^{8} + 1\right) x^{8} + 1,
\end{align*}
and the initial terms are $[1,a]$.
For $\phi(M^e,0,1)$:
\begin{align*}
p_0(x)&=(a^{24} + a^{22} + a^{8} + a^{6}) x^{24} + (a^{22} + a^{21} + a^{6} + a^{5}) x^{22} + (a^{20} + a^{19} + a^{18} +\\ &\quad
a^{17} + a^{15} + a^{14} + a^{13} + a^{11} + a^{10} + a^{9} + a^{7} + a^{6} + a^{5} + a^{4}) x^{20} + (a^{18}
\\&\quad + a^{17} + a^{16} + a^{15} + a^{14} + a^{13} + a^{12} + a^{11} + a^{10} + a^{9} + a^{8} + a^{7} + a^{6} + a^{5}\\
&\quad + a^{4} + a^{3}) x^{18} + (a^{15} + a^{12} + a^{11} + a^{10} + a^{8} + a^{7} + a^{6} + a^{3}) x^{16}
+ (a^{12} \\&\quad + a^{11} + a^{10} + a^{9} + a^{6} + a^{5} + a^{4} + a^{3}) x^{14} + (a^{9} + a^{8} + a^{6} + a^{4} + a^{3}) x^{12}, \\
p_3(x)&=(a^{14} + a^{12} + a^{10} + a^{8} + a^{6} + a^{4} + a^{2} + 1) x^{14} + (a^{13} + a^{12} + a^{9} + a^{8}\\&\quad + a^{5} + a^{4} + a + 1) x^{13} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} ,\\
p_6(x)&=(a^{12} + a^{8} + a^{4} + 1) x^{12} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{8} + 1) x^{8}\\&\quad + (a^{6} + a^{4} + a^{2} + 1) x^{6} ,\\
p_9(x)&=(a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} + (a^{8} + 1) x^{8} + (a^{7} + a^{6}\\&\quad + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{6} + a^{4} + a^{2} + 1) x^{6} + (a^{5} + a^{4} + a\\&\quad + 1) x^{5} + (a^{4} + 1) x^{4} + (a^{3} + a^{2} + a + 1) x^{3} ,\\
p_{12}(x)&=(a^{8} + 1) x^{8} + 1,
\end{align*}
and the initial terms are $[0,a]$.
For $\phi(M^e,1,0)$:
\begin{align*}
p_0(x)&=1,\\
p_3(x)&=(a^{2} + 1) x^{2} + 1 ,
\end{align*}
and the initial terms are $[1,0]$.
For $\phi(M^e,1,1)$:
\begin{align*}
p_0(x)&=(a^{12} + a^{11} + a^{4} + a^{3}) x^{12} + (a^{9} + a^{7} + a^{5} + a^{3}) x^{10} + (a^{6} + a^{5} + a^{4} + a^{3})\\&\quad x^{8} + a^{3} x^{6} ,\\
p_3(x)&=(a^{9} + a) x^{9} + (a^{8} + 1) x^{8} + (a^{7} + a^{4} + a^{3} + 1) x^{7} + (a^{5} + a) x^{5} + (a^{4} + 1) x^{4} \\&\quad+ (a^{3} + 1) x^{3} ,\\
p_6(x)&=(a^{9} + a) x^{9} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{8} + (a^{7} + a^{4} + a^{3}\\&\quad + 1) x^{7} + (a^{5} + a) x^{6} + (a^{4} + 1) x^{5} + (a^{4} + a^{3} + a^{2} + a) x^{4} + (a^{3} + 1) x^{3}\\&\quad + a x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{8} + 1) x^{8} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{6} + a^{4} + a^{2}\\&\quad + 1) x^{6} + (a^{5} + a^{4} + a + 1) x^{5} + (a^{4} + 1) x^{4} + (a^{3} + a^{2} + a + 1) x^{3}\\&\quad + (a^{2} + 1) x^{2} + (a + 1) x ,\\
p_{12}(x)&=(a^{8} + 1) x^{8} + 1,
\end{align*}
and the initial terms are $[0,a]$.
For $\phi(M^o,0,0)$:
\begin{align*}
p_0(x)&=(a^{9} + a^{8} + a + 1) x^{12} + (a^{8} + a^{7} + a^{5} + a^{3} + a + 1) x^{10} + (a^{7} + a^{6} + a + 1)\\&\quad x^{8} + (a^{6} + a^{5} + a^{3} + a + 1) x^{6} ,\\
p_3(x)&=(a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{8} + (a^{7} + a^{5} + a^{3} + a) x^{7} + (a^{6}\\&\quad + a^{2}) x^{6} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{5} + (a^{4} + a^{3} + a + 1) x^{4}+ (a^{2} + a) x^{3} ,\\
p_6(x)&=(a^{6} + a^{4} + a^{2} + 1) x^{7} + (a^{6} + a^{5} + a^{2} + a) x^{6} + (a^{3} + a^{2} + a + 1) x^{5}\\&\quad + (a^{2} + 1) x^{4} + (a^{3} + 1) x^{3} + a x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{5} + a^{4} + a + 1) x^{5} + (a + 1) x ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[1,a]$.
For $\phi(M^o,0,1)$:
\begin{align*}
p_0(x)&=(a^{21} + a^{19} + a^{5} + a^{3}) x^{24} + (a^{20} + a^{4}) x^{22} + (a^{19} + a^{16} + a^{12} + a^{8} + a^{4}\\&\quad
+ a^{3}) x^{20} + (a^{18} + a^{14} + a^{10} + a^{6}) x^{18} + (a^{15} + a^{14} + a^{9} + a^{6} + a^{5} + a^{3})\\&\quad x^{16} + (a^{12} + a^{10} + a^{6} + a^{4}) x^{14} + (a^{9} + a^{8} + a^{6} + a^{4} + a^{3}) x^{12} ,\\
p_3(x)&=(a^{16} + 1) x^{16} + (a^{15} + a^{14} + a^{13} + a^{12} + a^{11} + a^{10} + a^{9} + a^{8} + a^{7} + a^{6} \\&\quad+ a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{15} + (a^{14} + a^{12} + a^{10} + a^{8} + a^{6} + a^{4}\\&\quad + a^{2} + 1) x^{14} + (a^{13} + a^{12} + a^{9} + a^{8} + a^{5} + a^{4} + a + 1) x^{13} + (a^{12} + a^{8}\\&\quad + a^{4} + 1) x^{12} + (a^{11} + a^{10} + a^{9} + a^{8} + a^{3} + a^{2} + a + 1) x^{11} + (a^{10} + a^{8}\\&\quad + a^{2} + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} ,\\
p_6(x)&=(a^{12} + a^{8} + a^{4} + 1) x^{12} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{8} + 1) x^{8} + (a^{6} + a^{4} +\\&\quad a^{2} + 1) x^{6} ,\\
p_9(x)&=(a^{8} + 1) x^{8} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{4} + 1) x^{4} + (a^{3}\\&\quad + a^{2} + a + 1) x^{3} ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[0,a]$.
For $\phi(M^o,1,0)$:
\begin{align*}
p_0(x)&=1\\
p_3(x)&=(a + 1) x + 1 ,
\end{align*}
and the initial terms are $[1,a+1]$.
For $\phi(M^o,1,1)$:
\begin{align*}
p_0(x)&=(a^{12} + a^{11} + a^{4} + a^{3}) x^{12} + (a^{9} + a^{7} + a^{5} + a^{3}) x^{10} + (a^{6} + a^{5} + a^{4} + a^{3})\\&\quad x^{8} + a^{3} x^{6} ,\\
p_3(x)&=(a^{8} + a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a) x^{8} + (a^{6} + a^{4} + a^{2} + 1) x^{7} + (a^{6} +\\&\quad a^{2}) x^{6} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{5} + (a^{3} + a) x^{4} + (a^{3} + 1) x^{3} ,\\
p_6(x)&=(a^{7} + a^{5} + a^{3} + a) x^{7} + (a^{5} + a^{4} + a + 1) x^{6} + (a^{3} + a^{2} + a + 1) x^{5}\\&\quad + (a^{2} + 1) x^{4} + (a^{2} + a) x^{3} + (a^{2} + a + 1) x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{5} + a^{4} + a + 1) x^{5} + (a + 1) x ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[0,a]$.
For $\phi(W^e,0,0)$:
\begin{align*}
p_0(x)&=(a^{9} + a^{8} + a + 1) x^{12} + (a^{8} + a^{7} + a^{5} + a^{3} + a + 1) x^{10} + (a^{7} + a^{6} + a + 1)\\&\quad x^{8}+ (a^{6} + a^{5} + a^{3} + a + 1) x^{6} ,\\
p_3(x)&=(a^{8} + 1) x^{9} + (a^{8} + 1) x^{8} + (a^{6} + a^{5} + a^{2} + a) x^{7} + (a^{4} + 1) x^{5} + (a^{4} + 1) x^{4}\\&\quad + (a^{2} + a) x^{3} ,\\
p_6(x)&=(a^{8} + 1) x^{9} + (a^{8} + a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a) x^{8} + (a^{6} + a^{5} + a^{2}\\&\quad + a) x^{7} + (a^{6} + a^{5} + a^{4} + a^{2} + a + 1) x^{6} + (a^{5} + a) x^{5} + (a^{3} + a^{2} + a + 1)\\&\quad x^{4} + (a^{2} + a) x^{3} + (a^{2} + a + 1) x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{8} + 1) x^{8} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{6} + a^{4} + a^{2}\\&\quad + 1) x^{6} + (a^{5} + a^{4} + a + 1) x^{5} + (a^{4} + 1) x^{4} + (a^{3} + a^{2} + a + 1) x^{3}\\&\quad + (a^{2} + 1) x^{2} + (a + 1) x ,\\
p_{12}(x)&=(a^{8} + 1) x^{8} + 1 ,
\end{align*}
and the initial terms are $[1,1]$.
For $\phi(W^e,0,1)$:
\begin{align*}
p_0(x)&=(a^{18} + a^{16} + a^{2} + 1) x^{24} + (a^{17} + a^{16} + a + 1) x^{22} + (a^{16} + a^{15} + a^{14} + a^{13}\\&\quad + a^{11} + a^{10} + a^{9} + a^{7} + a^{6} + a^{5} + a^{3} + a^{2} + a + 1) x^{20} + (a^{15} + a^{14} + a^{13}\\&\quad + a^{12} + a^{11} + a^{10} + a^{9} + a^{8} + a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{18}\\&\quad + (a^{13} + a^{10} + a^{9} + a^{8} + a^{6} + a^{5} + a^{4} + a) x^{16} + (a^{11} + a^{10} + a^{9} + a^{8}\\&\quad + a^{5} + a^{4} + a^{3} + a^{2}) x^{14} + (a^{9} + a^{8} + a^{6} + a^{4} + a^{3}) x^{12} ,\\
p_3(x)&=(a^{14} + a^{12} + a^{10} + a^{8} + a^{6} + a^{4} + a^{2} + 1) x^{14} + (a^{13} + a^{12} + a^{9} + a^{8} + a^{5}\\&\quad + a^{4} + a + 1) x^{13} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} ,\\
p_6(x)&=(a^{12} + a^{8} + a^{4} + 1) x^{12} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{8} + 1) x^{8} + (a^{6} + a^{4}\\&\quad + a^{2} + 1) x^{6} ,\\
p_9(x)&=(a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} + (a^{8} + 1) x^{8} + (a^{7} + a^{6}\\&\quad + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{6} + a^{4} + a^{2} + 1) x^{6} + (a^{5} + a^{4} + a\\&\quad + 1) x^{5} + (a^{4} + 1) x^{4} + (a^{3} + a^{2} + a + 1) x^{3} ,\\
p_{12}(x)&=(a^{8} + 1) x^{8} + 1 ,
\end{align*}
and the initial terms are $[0,1]$.
For $\phi(W^e,1,0)$:
\begin{align*}
p_0(x)&=1\\
p_3(x)&=(a^{2} + 1) x^{2} + 1 ,\\
\end{align*}
and the initial terms are $[1,0]$.
For $\phi(W^e,1,1)$:
\begin{align*}
p_0(x)&=(a^{9} + a^{8} + a + 1) x^{12} + (a^{7} + a^{5} + a^{3} + a) x^{10} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{8}\\&\quad + a^{3} x^{6} ,\\
p_3(x)&=(a^{8} + 1) x^{9} + (a^{8} + 1) x^{8} + (a^{7} + a^{4} + a^{3} + 1) x^{7} + (a^{4} + 1) x^{5} + (a^{4} + 1) x^{4}\\&\quad + (a^{3} + 1) x^{3} ,\\
p_6(x)&=(a^{8} + 1) x^{9} + (a^{8} + a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a) x^{8} + (a^{7} + a^{4} + a^{3}\\&\quad + 1) x^{7} + (a^{5} + a) x^{6} + (a^{5} + a) x^{5} + (a^{3} + a^{2} + a + 1) x^{4} + (a^{3} + 1) x^{3}\\&\quad + a x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{8} + 1) x^{8} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{6} + a^{4} + a^{2}\\&\quad + 1) x^{6} + (a^{5} + a^{4} + a + 1) x^{5} + (a^{4} + 1) x^{4} + (a^{3} + a^{2} + a + 1) x^{3}\\&\quad + (a^{2} + 1) x^{2} + (a + 1) x ,\\
p_{12}(x)&=(a^{8} + 1) x^{8} + 1 ,
\end{align*}
and the initial terms are $[0,1]$.
For $\phi(W^o,0,0)$:
\begin{align*}
p_0(x)&=(a^{12} + a^{11} + a^{4} + a^{3}) x^{12} + (a^{10} + a^{9} + a^{7} + a^{5} + a^{3} + a^{2}) x^{10} + (a^{8}\\&\quad + a^{7} + a^{2} + a) x^{8} + (a^{6} + a^{5} + a^{3} + a + 1) x^{6} ,\\
p_3(x)&=(a^{8} + a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a) x^{8} + (a^{6} + a^{4} + a^{2} + 1) x^{7}\\&\quad + (a^{4} + 1) x^{6} + (a^{3} + a^{2} + a + 1) x^{5} + (a^{4} + a^{3} + a + 1) x^{4} + (a^{2} + a) x^{3} ,\\
p_6(x)&=(a^{7} + a^{5} + a^{3} + a) x^{7} + (a^{5} + a^{4} + a + 1) x^{6} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{5}\\&\quad + (a^{4} + a^{2}) x^{4} + (a^{3} + 1) x^{3} + a x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{5} + a^{4} + a + 1) x^{5} + (a + 1) x ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[1,1]$.
For $\phi(W^o,0,1)$:
\begin{align*}
p_0(x)&=(a^{21} + a^{19} + a^{5} + a^{3}) x^{24} + (a^{18} + a^{2}) x^{22} + (a^{17} + a^{16} + a^{12} + a^{8} + a^{4}\\&\quad + a) x^{20} + (a^{12} + a^{8} + a^{4} + 1) x^{18} + (a^{13} + a^{11} + a^{10} + a^{7} + a^{2} + a) x^{16}\\&\quad + (a^{10} + a^{8} + a^{4} + a^{2}) x^{14} + (a^{9} + a^{8} + a^{6} + a^{4} + a^{3}) x^{12} ,\\
p_3(x)&=(a^{16} + 1) x^{16} + (a^{15} + a^{14} + a^{13} + a^{12} + a^{11} + a^{10} + a^{9} + a^{8} + a^{7} + a^{6}\\&\quad + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{15} + (a^{14} + a^{12} + a^{10} + a^{8} + a^{6} + a^{4} + a^{2}\\&\quad + 1) x^{14} + (a^{13} + a^{12} + a^{9} + a^{8} + a^{5} + a^{4} + a + 1) x^{13} + (a^{12} + a^{8} + a^{4} \\&\quad+ 1) x^{12} + (a^{11} + a^{10} + a^{9} + a^{8} + a^{3} + a^{2} + a + 1) x^{11} + (a^{10} + a^{8} + a^{2}\\&\quad + 1) x^{10} + (a^{9} + a^{8} + a + 1) x^{9} ,\\
p_6(x)&=(a^{12} + a^{8} + a^{4} + 1) x^{12} + (a^{10} + a^{8} + a^{2} + 1) x^{10} + (a^{8} + 1) x^{8} + (a^{6} + a^{4}\\&\quad + a^{2} + 1) x^{6} ,\\
p_9(x)&=(a^{8} + 1) x^{8} + (a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{7} + (a^{4} + 1) x^{4} + (a^{3}\\&\quad + a^{2} + a + 1) x^{3} ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[0,1]$.
For $\phi(W^o,1,0)$:
\begin{align*}
p_0(x)&=1,\\
p_3(x)&=(a + 1) x + 1 ,
\end{align*}
and the initial terms are $[1,a+1]$.
For $\phi(W^o,1,1)$:
\begin{align*}
p_0(x)&=(a^{9} + a^{8} + a + 1) x^{12} + (a^{7} + a^{5} + a^{3} + a) x^{10} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{8} \\&\quad+ a^{3} x^{6} ,\\
p_3(x)&=(a^{7} + a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) x^{8} + (a^{7} + a^{5} + a^{3} + a) x^{7} + (a^{4}\\&\quad + 1) x^{6} + (a^{3} + a^{2} + a + 1) x^{5} + (a^{3} + a) x^{4} + (a^{3} + 1) x^{3} ,\\
p_6(x)&=(a^{6} + a^{4} + a^{2} + 1) x^{7} + (a^{6} + a^{5} + a^{2} + a) x^{6} + (a^{5} + a^{4} + a^{3} + a^{2}) x^{5}\\&\quad + (a^{4} + a^{2}) x^{4} + (a^{2} + a) x^{3} + (a^{2} + a + 1) x^{2} + (a + 1) x + 1 ,\\
p_9(x)&=(a^{5} + a^{4} + a + 1) x^{5} + (a + 1) x ,\\
p_{12}(x)&=(a^{4} + 1) x^{4} + 1 ,
\end{align*}
and the initial terms are $[0,1]$.
\subsection{Data for subsection \ref{subsection:pd1}}
All $16$ polynomials are of the form
$$p_0(x)+p_1(x)y+p_2(x)y^2+p_{4}(x)y^{4}.$$
The coefficients $p_j(x)$ and the two initial terms to determine
the solutions uniquely are given below.
For $\phi(A^e,0,0)$:
\begin{align*}
p_{0}(x)&=x^8 + x^6 + x^5 + x^2 + 1\\
p_{1}(x)&=x^2 + x,\\
p_{2}(x)&=x,\\
p_{4}(x)&=1.
\end{align*}
and the initial terms are [1, 0].
For $\phi(A^e,0,1)$:
\begin{align*}
p_{0}(x)&=x^{13} + x^9 + x^4 + x^3 + x^2,\\
p_{1}(x)&=x^3 + x^2 + x + 1,\\
p_{2}(x)&=0,\\
p_{4}(x)&=x^5.
\end{align*}
and the initial terms are [0, 0].
For $\phi(A^e,1,0)$:
\begin{align*}
p_{0}(x)&=x^2,\\
p_{1}(x)&=1.
\end{align*}
and the initial terms are [0, 0].
For $\phi(A^e,1,1)$:
\begin{align*}
p_{0}(x)&=x^8 + x^6 + x^5,\\
p_{1}(x)&=x^2 + x,\\
p_{2}(x)&=x,\\
p_{4}(x)&=1.
\end{align*}
and the initial terms are [0, 0].
For $\phi(A^o,0,0)$:
\begin{align*}
p_{0}(x)&=x^6 + x^4 + x^3 + x^2 + 1,\\
p_{1}(x)&=x+1,\\
p_{2}(x)&=x,\\
p_{4}(x)&=x^2.
\end{align*}
and the initial terms are [1, 0].
For $\phi(A^o,0,1)$:
\begin{align*}
p_{0}(x)&=x^{11} + x^5 + x^4 + x^3 + x^2,\\
p_{1}(x)&=x^3 + x^2 + x + 1,\\
p_{2}(x)&=0,\\
p_{4}(x)&=x^7.
\end{align*}
and the initial terms are [0, 0].
For $\phi(A^o,1,0)$:
\begin{align*}
p_{0}(x)&=x,\\
p_{1}(x)&=1.
\end{align*}
and the initial terms are [0, 1].
For $\phi(A^o,1,1)$:
\begin{align*}
p_{0}(x)&=x^6 + x^4 + x^3,\\
p_{1}(x)&=x+1,\\
p_{2}(x)&=x,\\
p_{4}(x)&=x^2.
\end{align*}
and the initial terms are [0, 0].
For all $0\leq i,j\leq 1$, $\phi(B^e,i,j)=\phi(A^o,i,j)$, $\phi(B^o,i,j)=
\phi(Ae,i,j)$. The corresponding initial conditions are also the same.
We happen to have $Ae=Bo$ and $Ao=Be$ here.
\subsection{Data for subsection \ref{subsection:pd2}}
All $16$ polynomials are of the form
$$p_0(x)+p_3(x)y^3+p_6(x)y^6+p_9(x)y^9+p_{12}(x)y^{12}.$$
For $(i,j)=(1,0)$ and all $T\in\{A^e, A^o, B^e, B^o\}$,
$$p_6(x)=p_9(x)=p_{12}(x)=0.$$
The coefficients $p_j(x)$ and the initial terms to determine the solutions uniquely
are given below.
For $\phi(A^e,0,0)$:
\begin{align*}
p_0(x)&=x^{24} + x^{22} + x^{20} + x^{17} + x^{16} + x^{14} + x^{13} + x^{5} + x^{3} + x + 1,\\
p_3(x)&=x^{15} + x^{14} + x^{12} + x^{9} + x^{6} + x^{2},\\
p_6(x)&= x^{11} + x^{9} + x^{8} + x^{7} + x^{5} + x^{3} + x^{2} + x,\\
p_9(x)&= x^6 + x^4 + x^3 + x,\\
p_{12}(x)&=x^2 + x + 1.
\end{align*}
and the initial terms are $[1,1]$.
For $\phi(A^e,0,1)$:
\begin{align*}
p_0(x)&=x^{42} + x^{41} + x^{39} + x^{35} + x^{34} + x^{33} + x^{32} + x^{30} + x^{26} + x^{25} + x^{24}\\&\quad + x^{20} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{12} + x^{11} + x^{10} + x^{9},\\
p_3(x)&=x^{29} + x^{27} + x^{26} + x^{24} + x^{21} + x^{19} + x^{18} + x^{16} + x^{13} + x^{11} + x^{10}\\&\quad + x^{8} + x^{5} + x^{3} + x^{2} + 1,\\
p_6(x)&=x^{28} + x^{27} + x^{25} + x^{24} + x^{12} + x^{11} + x^{9} + x^{8},\\
p_9(x)&= x^{27} + x^{24} + x^{19} + x^{16}\\
p_{12}(x)&=x^{26} + x^{25} + x^{24}.
\end{align*}
and the initial terms are $[0,0,0 ,1]$.
For $\phi(A^e,1,0)$:
\begin{align*}
p_0(x)&=x^9,\\
p_3(x)&=x^2+x+1.
\end{align*}
and the initial terms are $[0,0,0 ,1]$.
For $\phi(A^e,1,1)$:
\begin{align*}
p_0(x)&=x^{30} + x^{29} + x^{28} + x^{25} + x^{24} + x^{23} + x^{21},\\
p_3(x)&=x^{21} + x^{19} + x^{17} + x^{16} + x^{15} + x^{14} + x^{11} + x^{9} + x^{8} + x^{7} + x^{4} + x^{3},\\
p_6(x)&=x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{10} + x^{7} + x^{5} + x^{3} + x^{2} ,\\
p_9(x)&=x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x,\\
p_{12}(x)&=x^{8} + x^{4} + 1.
\end{align*}
and the initial terms are $[0, 0, 0, 0, 0, 0, 1, 1]$.
For $\phi(A^o,0,0)$:
\begin{align*}
p_0(x)&=x^{30} + x^{28} + x^{27} + x^{25} + x^{24} + x^{23} + x^{22} + x^{20} + x^{14} + x^{13} + x^{7}\\ &\quad + x^{6} + x^{5} + x^{4} + 1,\\
p_3(x)&=x^{22} + x^{21} + x^{17} + x^{16} + x^{15} + x^{11} + x^{8} + x^{7} + x^{6} + x^{2} + x + 1,\\
p_6(x)&=x^{23} + x^{22} + x^{20} + x^{16} + x^{15} + x^{13} + x^{12} + x^{9} + x^{7} + x^{6} + x^{5} + x^{2},\\
p_9(x)&=x^{21} + x^{18} + x^{17} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{8} + x^{7} + x^{4},\\
p_{12}(x)&=x^{22} + x^{14} + x^{6}.
\end{align*}
and the initial terms are $[1,1]$.
For $\phi(A^o,0,1)$:
\begin{align*}
p_0(x)&=x^{48} + x^{47} + x^{43} + x^{42} + x^{41} + x^{40} + x^{39} + x^{36} + x^{35} + x^{34} + x^{33}\\ &\quad + x^{32} + x^{27} + x^{25} + x^{24} + x^{14} + x^{12} + x^{11} + x^{9},\\
p_3(x)&=x^{43} + x^{42} + x^{41} + x^{40} + x^{27} + x^{26} + x^{25} + x^{24} + x^{19} + x^{18} + x^{17}\\ &\quad + x^{16} + x^{3} + x^{2} + x + 1,\\
p_6(x)&=x^{44} + x^{42} + x^{36} + x^{34} + x^{20} + x^{18} + x^{12} + x^{10},\\
p_9(x)&=x^{45} + x^{44} + x^{21} + x^{20},\\
p_{12}(x)&=x^{46} + x^{38} + x^{30}.
\end{align*}
and the initial terms are $[0,0,0 ,1]$.
For $\phi(A^o,1,0)$:
\begin{align*}
p_0(x)&=x^6,\\
p_3(x)&=x^4+x^2+1.
\end{align*}
and the initial terms are $[0,0,1]$.
For $\phi(A^o,1,1)$:
\begin{align*}
p_0(x)&=x^{24} + x^{23} + x^{22} + x^{19} + x^{18} + x^{17} + x^{15},\\
p_3(x)&=x^{20} + x^{19} + x^{17} + x^{14} + x^{13} + x^{11} + x^{10} + x^{9} + x^{6} + x^{4} + x^{3} + 1,\\
p_6(x)&=x^{21} + x^{20} + x^{17} + x^{13} + x^{12} + x^{11} + x^{10} + x^{7} + x^{6} + x^{4} + x^{3} + x^{2},\\
p_9(x)&=x^{21} + x^{18} + x^{17} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{8} + x^{7} + x^{4},\\
p_{12}(x)&=x^{22} + x^{14} + x^{6}.
\end{align*}
and the initial terms are $[0,0,0,0,0,1 ]$.
For $\phi(B^e,0,0)$:
\begin{align*}
p_0(x)&=x^{30} + x^{28} + x^{27} + x^{25} + x^{24} + x^{23} + x^{22} + x^{20} + x^{14} + x^{13} + x^{7}\\ &\quad + x^{6} + x^{5} + x^{4} + 1,\\
p_3(x)&=x^{20} + x^{18} + x^{17} + x^{14} + x^{12} + x^{11} + x^{6} + 1,\\
p_6(x)&=x^{19} + x^{18} + x^{17} + x^{14} + x^{13} + x^{10} + x^{6} + x^{5} + x^{4} + x^{2},\\
p_9(x)&=x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4},\\
p_{12}(x)&=x^{14} + x^{10} + x^{6}.
\end{align*}
and the initial terms are $[1,0]$.
For $\phi(B^e,0,1)$:
\begin{align*}
p_0(x)&=x^{48} + x^{47} + x^{43} + x^{42} + x^{41} + x^{40} + x^{39} + x^{36} + x^{35} + x^{34} + x^{33}\\ &\quad + x^{32} + x^{27} + x^{25} + x^{24} + x^{14} + x^{12} + x^{11} + x^{9},\\
p_3(x)&=x^{41} + x^{38} + x^{37} + x^{35} + x^{34} + x^{32} + x^{31} + x^{29} + x^{28} + x^{26} + x^{23}\\ &\quad + x^{21} + x^{20} + x^{18} + x^{15} + x^{13} + x^{12} + x^{10} + x^{9} + x^{7} + x^{6} + x^{4} + x^{3} + 1,\\
p_6(x)&=x^{40} + x^{36} + x^{34} + x^{32} + x^{30} + x^{26} + x^{24} + x^{20} + x^{18} + x^{16} + x^{14} + x^{10},\\
p_9(x)&=x^{39} + x^{36} + x^{35} + x^{32} + x^{27} + x^{24} + x^{23} + x^{20},\\
p_{12}(x)&=x^{38} + x^{34} + x^{30}.
\end{align*}
and the initial terms are $[0,0,0,1 ]$.
For $\phi(B^e,1,0)$:
\begin{align*}
p_0(x)&=x^6,\\
p_3(x)&=x^2+x+1.
\end{align*}
and the initial terms are $[0,0,1,1 ]$.
For $\phi(B^e,1,1)$:
\begin{align*}
p_0(x)&=x^{24} + x^{23} + x^{22} + x^{19} + x^{18} + x^{17} + x^{15},\\
p_3(x)&=x^{18} + x^{16} + x^{14} + x^{13} + x^{12} + x^{11} + x^{8} + x^{6} + x^{5} + x^{4} + x + 1,\\
p_6(x)&=x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{10} + x^{7} + x^{5} + x^{3} + x^{2},\\
p_9(x)&=x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4},\\
p_{12}(x)&=x^{14} + x^{10} + x^{6}.
\end{align*}
and the initial terms are $[ 0,0,0,0,0,1 ]$.
For $\phi(B^o,0,0)$:
\begin{align*}
p_0(x)&=x^{24} + x^{22} + x^{20} + x^{17} + x^{16} + x^{14} + x^{13} + x^{5} + x^{3} + x + 1 ,\\
p_3(x)&=x^{17} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{4} + x^{3} + x^{2},\\
p_6(x)&=x^{15} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{2} + x,\\
p_9(x)&=x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x,\\
p_{12}(x)&=x^{10} + x^{9} + x^{8} + x^{6} + x^{5} + x^{4} + x^{2} + x + 1.
\end{align*}
and the initial terms are $[1,0]$.
For $\phi(B^o,0,1)$:
\begin{align*}
p_0(x)&=x^{42} + x^{41} + x^{39} + x^{35} + x^{34} + x^{33} + x^{32} + x^{30} + x^{26} + x^{25} + x^{24}\\ &\quad + x^{20} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{12} + x^{11} + x^{10} + x^{9},\\
p_3(x)&=x^{31} + x^{30} + x^{25} + x^{24} + x^{23} + x^{22} + x^{17} + x^{16} + x^{15} + x^{14} + x^{9}\\ &\quad + x^{8} + x^{7} + x^{6} + x + 1,\\
p_6(x)&=x^{32} + x^{31} + x^{30} + x^{26} + x^{25} + x^{24} + x^{16} + x^{15} + x^{14} + x^{10} + x^{9} + x^{8},\\
p_9(x)&=x^{33} + x^{32} + x^{29} + x^{28} + x^{21} + x^{20} + x^{17} + x^{16},\\
p_{12}(x)&=x^{34} + x^{33} + x^{32} + x^{30} + x^{29} + x^{28} + x^{26} + x^{25} + x^{24}.
\end{align*}
and the initial terms are $[0,0,0,1 ]$.
For $\phi(B^o,1,0)$:
\begin{align*}
p_0(x)&=x^9,\\
p_3(x)&=x^4+x^2+1.
\end{align*}
and the initial terms are $[0,0,0,1 ]$.
For $\phi(B^o,1,1)$:
\begin{align*}
p_0(x)&=x^{30} + x^{29} + x^{28} + x^{25} + x^{24} + x^{23} + x^{21},\\
p_3(x)&=x^{23} + x^{22} + x^{20} + x^{17} + x^{16} + x^{14} + x^{13} + x^{12} + x^{9} + x^{7} + x^{6} + x^{3}
,\\
p_6(x)&=x^{21} + x^{20} + x^{17} + x^{13} + x^{12} + x^{11} + x^{10} + x^{7} + x^{6} + x^{4} + x^{3} + x^{2},\\
p_9(x)&=x^{18} + x^{15} + x^{14} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{5} + x^{4} + x,\\
p_{12}(x)&=x^{16} + x^{8} + 1.
\end{align*}
and the initial terms are $[0,0,0,0,0,0,1 ]$.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,671 |
Delightful Cooling Mats #8 Are Pet Cooling Mats Worth Your Money?
Cooling Mats was published on September 6, 2018 at 9:19 am. It is posted in the Mat category. Cooling Mats is labelled with Cooling Mats, Cooling, Mats..
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"redpajama_set_name": "RedPajamaC4"
} | 568 |
\section{Other point sources of interest}\label{apx:sourceX}
Here we list the 3 non-candidate point sources that meet the selection criteria (color, shape, and $z_{ph}$), but have large $I_{814}$\ uncertainties and land outside of the selection box. A deeper $I_{814}$\ image will determine their $I_{814}$$-$$J_{125}$\ color.
\input{targX2}
\section{Composite quasar spectrum}\label{apx:qsotemp}
We generate a composite quasar spectrum for the SED fits in Sec.~\ref{sec:photz}. We combine the low redshift templates \citep{vandenberk2001,Glikman2006ApJ} and $z\sim6$ spectra from SDSS \citep{sdssQ14}. Since the neutral hydrogen absorption due to the intergalactic medium is unknown at $z\sim7$-8, we assume full absorption at $\lambda<1215$ \AA. We do not adjust the emission lines such as: ${\rm Ly\alpha}$, which will be affected by intergalactic absorption, [\textrm{C}~\textsc{iv}], which is expected to show a blueshifted profile \citep[e.g.,][]{yang2021ApJ}, or ${\rm H\beta}$\ $+$ [\textrm{O}~\textsc{iii}]. As we can seen in Figure \ref{fig:eazybagsSED}, our composite quasar spectrum lacks the strong nebular emission lines, which are predicted by the \bags\ star-forming galaxy SEDs. Detailed spectroscopic analysis is beyond the scope of this study, and will be left for future study. The final composite quasar spectrum is shown in Figure \ref{fig:qsotemp}.
\begin{figure*}[!th]
\begin{center}
\includegraphics[width=0.7\textwidth]{qso_template.png}
\end{center}
\caption{Low-redshift template (blue), $z\sim6$ SDSS composite (gray), and full composite quasar spectrum (red).}
\label{fig:qsotemp}
\end{figure*}
\section{Dwarf star selections}\label{apx:BDlist}
\input{BDlist}
\begin{figure*}[!th]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{goodss-01-G141_45889_1D_v4b_BD.png} &
\includegraphics[width=0.45\textwidth]{cosmos-09-G141_16730_1D_v4b_BD.png} \\
\includegraphics[width=0.45\textwidth]{cosmos-15-G141_25286_1D_v4b_BD.png} &
\includegraphics[width=0.45\textwidth]{aegis-03-G141_17053_1D_v4b_BD.png} \\
\end{tabular}
\end{center}
\caption{The observed \HSTsurvey\ grism spectra in black with the best-fit SpeX templates in red.}
\label{fig:grismBD}
\end{figure*}
Due to the nature of point source selection, it is inevitable that we find dwarf star contaminants as discussed in Section \ref{subsec:lowz}. From the short list of color dropout point sources, we identify 4 dwarf stars (GDS 45889, COS 16730, COS 25286, and EGS 17053) based on \HSTsurvey\ grism spectroscopic data. We perform a least-squares fit, minimizing the residual, $(F_d-F_m)/\sigma$. $F_d$ is the observed grism flux; $F_m$ is the scaled template flux; and $\sigma_d$ is the uncertainty of the grism flux. The sources are best fit with T7, T7.5, T8, and T5.5 templates respectively to within $3\sigma$. We note that the best fitting templates are approximate and are only used to identify dwarf star contaminants. Accurate characterization will require careful stellar modeling, as demonstrated by \cite{Aganze2021}. With the exception of COS 25286, all of these sources were identified by \cite{Aganze2021}; it is possible that COS 25286 did not make their selection due to its low $S/N$ ratio. These targets have F125W magnitudes ranging between $25\gtrsim I_{125}\gtrsim 22$ mags. While three out of four dwarf stars are much brighter than our final targets, some like COS 25286 can appear as faint as $z\sim8$ candidates. If we calculate the $\beta_{UV}$\ of these targets using Eq.\ref{eq:Buv}, we find very blue slopes of $\bar{\beta_{UV}}=-4.1\pm 0.5$. We show the target fluxes in Table \ref{tab:BDflux}; the grism spectra and their best fitting SpeX templates \citep{Rayner2003PASP,spexprism2014} are shown in Figure \ref{fig:grismBD}.
\section{Fitting the luminosity function}\label{apx:chi2min}
We calculate the absolute rest-frame UV magnitude, $M_{UV}$, at $z_{ph}$\ as follows:
\begin{equation}
M_{UV} = m_{UV/(1+z_{ph})} - 25 - 5\log_{10}\bigg[(3 \textrm{ Mpc}^{-1})\times(1+z_{ph}) \times DA(z_{ph}) - 5\log_{10}h \bigg]+2.5\log_{10}(1+z_{ph}),
\label{eq:muv}
\end{equation}
where $h$ is the Hubble parameter and $DA(z_{ph})$ is the angular diameter distance in Mpc, calculated with \astropy.
To fully reflect the estimated number densities, including upper limits,
we fit the luminosity function by minimizing Eq.~\ref{eq:chi2} as derived by \cite{Sawicki2012PASP}, which is also re-written in a more convenient form for computation:
\begin{equation}
\begin{split}
\chi^2_{mod} & = \sum_{i} \bigg(\frac{\phi_{d,i}-\phi_{m,i}}{\sigma_i}\bigg)^2
- 2\sum_{j}\ln\int_{-\infty}^{\phi_{\ulim,j}}\exp \bigg[-\frac{1}{2}\bigg(\frac{\phi'-\phi_{m,i}}{\sigma_j} \bigg) \bigg]d\phi' \\
& = \sum_{i} \bigg(\frac{\phi_{d,i}-\phi_{m,i}}{\sigma_i}\bigg)^2
- 2\sum_{j}\ln \bigg\{ \sqrt{\frac{\pi}{2}}\sigma_j \bigg[1+\erf\bigg(\frac{\phi_{ulim,j}-\phi_{m,j}}{\sigma_j} \bigg) \bigg]\bigg\},
\end{split}
\label{eq:chi2}
\end{equation}
where $\erf(x)=(s/\sqrt{\pi})\int_0^x e^{-t^2}dt$. Here $\phi_{d,i}$ is the observed number density at a given $\Delta$$M_{UV}$\ bin; $\phi_{m,i}$ is the model luminosity function value at the same $M_{UV}$\ bin; $\phi_{\ulim,j}$ is the upper limit number density; and $\sigma$ is the uncertainty in observed number density. When detections are made at all bands, the second summation ($j$-index) goes to zero, revealing the standard $\chi^2$ form
\section{Introduction} \label{sec:intro}
Statistical studies of the first galaxies and quasars are crucial to understanding their formation and evolution processes. To-date, a tremendous amount of effort has been made to probe the early universe with high redshift surveys like CANDELS \citep{Koekemoer2011ApJS197, Grogin2011ApJS197, Bouwens2019ApJ880}, BoRG \citep{Trenti2011ApJ, Bradley2012ApJ, Morishita2018ApJ, morishita2021sb}, HUDF12 \citep{HUDF12}, XDF \citep{XDF}, CLASH \citep{CLASH}, HFF \citep{HFF}, RELICS \citep{RELICS}, ULTRAVISTA \citep{McCracken2012AA, Stefanon2017ApJ,Stefanon2019ApJ,Bowler2020MNRAS}, among others. These surveys combined with follow-up spectroscopy have successfully identified some of the earliest galaxies up to $z\sim9$\,-\,10, yet characterizing the number densities and physical properties of these early sources remain incomplete.
It is necessary to accurately quantify the early populations with observational constraints.
Characterizing the luminosity function is a fundamental step in estimating the contribution from various luminous sources; it describes the number density of sources as a function of luminosity, or absolute magnitude. Since ultraviolet (UV) emission is primarily dominated by ionizing sources, the rest-frame UV luminosity function is a useful tool in investigating the early galaxy populations. In particular, the shape of the luminosity function can provide insights into the different physical processes such as star-formation and quasar activity that drive galaxy formation.
The faint end of the luminosity function is believed to be the key driver for cosmic reionization \citep[e.g.,][]{Ishigaki2018ApJ, atek2018MNRAS}, in which the early universe transitioned from completely neutral to almost ionized \citep{Ouchi2010ApJ723, konno14, pentericci14, Robertson2015ApJ802L, Mason2018ApJ857L,Mason2019MNRAS485,Hoag2019ApJ}. It is believed that reionization paved the way for the formation of the first galaxies \citep{LoebBarkana2001ARAA}, yet the question of what astrophysical objects are primarily responsible for reionization remain debated.
The bright end of the luminosity function is composed of the brightest sources that may be signposts of in-situ star formation or even quasar activity. The discovery of luminous quasars \citep{Mortlock2011nat,Banados2018Nature, Yang2020ApJ, Wang2021z76} and luminous star forming galaxies at $z\simgt7$ \citep{zitrin15,Oesch16,Hashimoto2018Natur557,jiang21} may indicative of populations of luminous sources that are unaccounted for. There is no consensus on the shape of the bright end luminosity function; some even suggest that the early galaxy luminosity function may depart from the standard Schechter form \citep{Harikane2022ApJS}. This departure manifest as a bright end excess \citep[e.g.,][]{Morishita2018ApJ}, and its origins remain unclear. Theoretical studies suggest that this bright excess may be caused by intense and compact star-forming clumps \citep[e.g.,][]{Ma2018MNRAS} or even stochastic quasar activity \citep[e.g.,][]{Ren2020ApJ}. Luminous source are also believed to contribute to cosmic reionization to some degree, yet the consensus on their contribution remain controversial \citep[e.g.,][]{willott2010,Finkelstein2015ApJ,Jiang2016ApJ, Matsuoka2019ApJ883, naidu2020ApJ}. The characterization of the brightest sources at high redshifts remain elusive.
High-redshift sources are typically identified with the Lyman dropout photometric selection \citep{Steidel1996AJ} combined with follow-up spectroscopic confirmation. However, the complication is that the redshifted spectral energy distribution (SED) of these sources at the end of cosmic reionization, $z\sim7$-8, overlaps with those of low-mass foreground stars. As a result, previous studies have often excluded compact, unresolved sources with star-like morphology in preference for more galaxy-like sources with extended morphology. Nevertheless, there is evidence from lensing surveys that early galaxies are very compact \citep[e.g.,][]{Bouwens2017ApJ, salmon2020ApJ}. Some even predict compact star-forming clumps \citep{Ma2018MNRAS}. So there is a possibility that certain population of quasars and compact galaxies at at $z\sim7$-8 that are rejected with the standard selection.
There is a renewed interest in examining these overlooked point sources. A recent medium-depth, wide $\sim0.4 \textrm{ deg}^2$ \textit{HST}\ survey ({SuperBoRG}; \citealt{Morishita2020z8, morishita2021sb}) has identified several $z\gtrsim8$ point sources as potential quasar candidates. Key features of these $z\sim8$ point sources are their blue rest-frame UV slopes and the \textit{Spitzer}/IRAC flux excess in the SED, which may be indicative of significant ${\rm H\beta}$\ and [\textrm{O}~\textsc{iii}]\ emission often seen in quasars. Observations suggest that these point sources are unlikely to be foreground stars and may contribute to the bright end of the luminosity function.
In this study, as part of our \textit{HST}\ archival program (AR 15804; PI. Morishita), we reexamine the selection of high-redshift compact unresolved (point) sources that have been overlooked in previous studies. We take advantage of the successful {SuperBoRG}\ study to revisit $z\sim7$-8 point sources in the \candels\ legacy \textit{HST}\ fields. Since the selection criteria of our study and of {SuperBoRG}\ are complementary, we combine the results of both studies to characterize the $z\sim7$-8 dropout point sources and to quantify their contribution to the total galaxy luminosity function. For simplicity, we will refer to these sources as ``point sources'' throughout the paper.
This paper is organized as follows. In Section \ref{sec:data} we describe the data reduction and target selection from \HSTsurvey. In Section \ref{sec:res} we explore the properties of the targets selected. And in Section \ref{sec:disc} we discuss the physical implications of these objects. We use the AB-magnitude system \citep{OkeGunn1983ApJ,fukugita96} and adopt the $h=0.7$, $\Omega_M=0.3$, and $\Omega_{\Lambda}=0.7$ cosmology.
\section{Target Selection} \label{sec:data}
\subsection{Source Catalog}\label{sec:select}
Our primary focus is to identify sources that satisfy the dropout color selection and have point source morphology in the \candels\ fields. We begin our analyses with the publicly available photometric catalogs provided by the \HSTsurvey\ team \citep{Brammer2012ApJS200,vandokkum13}. The \HSTsurvey\ is a \textit{HST}\ near-infrared spectroscopic survey designed to study galaxies across the universe. It surveyed nearly 700 arcmin$^2$\ of the well-studied \textit{HST}/\candels\ Treasury fields to obtain direct images and spectroscopic data with the ACS/G800L and WFC3/G141 grisms. \HSTsurvey\ covers about 75\% of the original \candels\ area. When all the fields are combined, their photometric observations of $H_{160}$\ reach median $5\,\sigma$ depths at 26 mags at $1\,\arcsec$ aperture. Further details of the survey and the published catalog can be found in \cite{Skelton2014ApJS214, Momcheva2016ApJS225}.
Our choice of using the \HSTsurvey\ catalog over the catalogs published by the \candels\ team \citep{guo13,Galametz13,Stefanon17,Nayyeri17,Barro2019ApJS243} is that uniform analysis is performed on all 5 \candels\ fields by the \HSTsurvey\ team to create the source catalog. This vastly simplifies the source detection procedure (Sec.~\ref{subsec:color}) and the completeness simulation analysis (Sec.~\ref{sec:LF}), to calculate the number density of the target population. However, due to inconsistencies in the filter coverage, we only analyze 4 of the 5 \candels\ fields (AEGIS, COSMOS, GOODS South, and UKIDSS-UDS), where F814W, F125W, and F160W filters are available (Sec.~\ref{subsec:color}).
We also exploit the published G141 grism data to identify low redshift interlopers (Sec.~\ref{subsec:lowz}).
We obtain deep \textit{HST}\ data from the publicly available \HSTsurvey\ database. The \textit{HST}\ image mosaics used have already been corrected for distortions and drizzled to the plate scale of $0.\!\arcsec06 \textrm{ pixel}^{-1}$. The photometric source catalogs were produced using point-spread function (PSF)-matched aperture photometry, reduced using \sex\ \citep{Bertin1996}, and flux calibrated to an aperture radius of $0.\!\arcsec7$. The \textit{HST}\ ACS and WFC3 images were convolved to match the \textit{HST}/F160W PSF ($\sim0.\!\arcsec14$). Ground-based optical, NIR, and \textit{Spitzer}/IRAC fluxes are similarly PSF-matched to a combination of F125W, F140W, and F160W priors and aperture corrected to F160W (or F140W, otherwise).
\subsection{Color-dropout and shape selection}\label{subsec:color}
Our strategy in identifying $z\sim7$-8 point source candidates from the \HSTsurvey\ is twofold. First, we identify sources with the Lyman-break dropout technique \citep{Steidel1996AJ} from the photometric catalog. Then, we select point sources from the list of Lyman-dropout sources. The color selection is only based on deep \textit{HST}\ photometry.
A caveat is that unlike the \cite{Morishita2020z8, morishita2021sb} selection, which uses the F105W/F125W/F160W ($Y_{105}$/$J_{125}$/$H_{160}$) filters, the \HSTsurvey\ catalog does not include $Y_{105}$\ fluxes. Instead, we use the \cite{Bouwens2015ApJ} color-dropout criteria, which is based on the F814W/F125W/F160W ($I_{814}$/$J_{125}$/$H_{160}$) selection:
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.97\columnwidth]{hst3d_colorcolor6.png}
\end{tabular}
\end{center}
\caption{Color-color diagram for the target selections from the \HSTsurvey. The targets fall into the marked by the dashed lines that define the \cite{Bouwens2015ApJ} color selection. Our final point sources from Table \ref{tab:targlist} are shown as red stars. Point sources that do not meet the photometric redshift selection are shown as dark red stars. Spectroscopically confirmed dwarf stars are shown as gray diamonds. We also compare with dwarf star template colors \citep{spexprism2014} in gray dots, predicted quasar colors from composite SDSS quasar spectrum (\citealt{sdssQ14}) in blue, and simple powerlaw spectra at $\beta=-1,-2,-3$ in green. Strong overlap of known dwarf star colors and our Lyman-dropout sources indicate the need for further follow-up to disentangle the degeneracy.}
\label{fig:colorcolor}
\end{figure}
\begin{equation}
\begin{split}
S/N_{125,160} > 5.0 \\
S/N_{\textrm{blue}} < 2.0 \\
I_{814} - J_{125} > 2.2 \\
J_{125} - H_{160} < 0.4 \\
I_{814} - J_{125} > 2\times(J_{125} - H_{160})+2.2
\end{split}
\end{equation}
\input{targlist}
\begin{figure*}[!hbt]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.98\textwidth]{hst3d_images5.png}
\end{tabular}
\end{center}
\caption
\textit{HST}\ ACS and WFC3 3 arcsec$^2$ postage stamp images of the point source targets shown at 95 percentile fluxes. \textit{Spitzer}/IRAC images are shown in 6~arcsec$^2$ boxes with 3 arcsec$^2$ box inserts in black. Filter images with no observations are indicated as ``No Data.'' Since the IRAC Ch3 and CH4 fluxes are dominated by noise, their fluxes are not included in the analysis.}
\label{fig:imgTOT}
\end{figure*}
Compared to the \citet{Morishita2020z8} selection, this color selection results in a broader $z\sim7$-8 selection. Also, the GOODS-North catalog does not include $I_{814}$\ data, so we only examine 4 of the 5 \candels\ fields. Where available, we use the \textit{HST}/ACS blue filters ($I_{814}$\ and bluer) to determine the Lyman-dropout with strict blue signal-to-noise constraints, combined with the non-detection flag from the \textit{HST}\ pipeline catalog. Since we require $2\,\sigma$ non-detections for $I_{814}$ fluxes, we calculate the resulting $I_{814} - J_{125}$ lower limit color as:
\begin{equation}
(I_{814} - J_{125})_{\textrm{lim}} = -2.5\log_{10}(2\sigma_{814} / J_{125}).
\end{equation}
We do not include sources with $I_{814} - J_{125}$ that fall below the non-detection limit (i.e. falls outside the selection box in Figure \ref{fig:colorcolor}). While additional ground-based fluxes are available, higher $S/N$ \textit{HST}\ fluxes are prioritized for color selection here (but see Sec.~\ref{subsec:lowz}).
From the color dropouts, we identify point sources based on two morphological parameters, elongation and flux concentration, measured in the $H_{160}$\ filter. The selection criteria are defined by \cite{Morishita2020z8}:
\begin{equation}
\begin{split}
e<1.2 \\
f_{5}/f_{10}>0.5.
\end{split}
\end{equation}
\input{targphotom}
Elongation, $e$ (ratio of semi-major/semi-minor axes), describes the circularity of the source, and the flux concentration (flux ratio between inner and outer radii) describes the compactness. \cite{morishita2021sb} find light concentration is an appropriate metric for point source selection. We obtain $e$ from the \HSTsurvey\ catalog.
To calculate flux concentration, we run \sex\ on the image mosaics, matching the \HSTsurvey\ detection parameters, and obtain detailed aperture photometry of the targets. We extract the aperture fluxes to calculate the flux ratios.
After careful comparison of the different $H_{160}$\ flux ratios at different radii, we determine the $f_{5}/f_{10}$ flux concentration, the flux ratios taken within the 5 pixel ($0.\!\arcsec3$) and 10 pixel ($0.\!\arcsec6$) radii, to be the appropriate criteria. This decision is based on the ability to concurrently recover known dwarf star contaminants due to the point source selection (see Sec.~\ref{subsec:lowz}). Although the \HSTsurvey\ source catalogs include the \texttt{star\_class} flag that classify whether a source is star-like, \citet{Finkelstein2015ApJ} and \citet{morishita2021sb} have demonstrated that these flags are not complete down to fainter magnitudes; it fails to distinguish between fuzzy circular objects and compact point sources in the faint magnitude ranges up to $\sim24$ mag. We note that other studies \citep[e.g.,][]{rborsani2016ApJ, Bouwens2015ApJ} have successfully identified sources by combining color dropout, stellarity parameters, and SED properties. Our aim is to identify additional sources that may be missed with the standard method.
Of the 169,614 objects listed in the \HSTsurvey\ catalog, we identify 22 $I_{814}$/$J_{125}$/$H_{160}$\ dropout point sources. Of these point sources, 7 meet the photometric redshift selection of $z_{ph}>7$, discussed in Section \ref{sec:photz}. We show these $J_{125} - H_{160}$ vs. $I_{814} - J_{125}$ color-color diagram of all color and $z_{ph}>7$ selected point sources in Figure \ref{fig:colorcolor}. Then, we check the grism spectra to further eliminate any low redshift interlopers, discussed in Section \ref{subsec:lowz}. Our final $z\sim7$-8 point source candidates list consists of 3 point sources, which are listed in Table \ref{tab:targlist}. We also visually inspect the \textit{HST}\ images of the $z_{ph}$\ selected targets to eliminate image artifacts and/or other spurious detections; postage stamp images of the final sample are shown in Figure \ref{fig:imgTOT} and discussed in Section \ref{sec:disc}. The observed fluxes of the point sources are shown in Table \ref{tab:photom}. We also list low confidence, $I_{814} - J_{125}$ limited, non-candidate point sources in Appendix \ref{apx:sourceX}.
\subsection{Photometric redshifts and SED fits}\label{sec:photz}
The Lyman-break color selection is comprehensive but also allows low-redshift sources with similar colors to migrate into the selection window. To filter out these contaminants, we apply further selection based on the photometric redshift measurement discussed here.
\input{BAGpriors}
Following the similar approach by \cite{rborsani2021superB}, we estimate the photometric redshift, $z_{ph}$, using the photometric redshift code, \eazy\ \citep{Brammer2008ApJ686}.
While photometric redshifts are also included in the public \HSTsurvey\ catalog, they were calculated with a maximum limiting redshift of $z=6$. Hence, we re-reduce the redshift estimates for all of our color-selected samples.
We use \eazy\ in the default setup (v1.3 templates) to derive the best-fit SED and redshift posterior probability, $p(z_{ph})$. To ensure a more accurate redshift derivation, we use all available photometric data points, including ground-based fluxes that were excluded in our initial color selection in Sec.~\ref{subsec:color}. We turn off magnitude priors in the fit to avoid any biased redshift selections, and fit between $0\leq z_{ph}\leq9$. The $z_{ph}$\ fit range is based on the redshift probability from the survey completeness (Sec.~\ref{sec:LF}). From this analysis, we eliminate low redshift interlopers by making a cut in which the probability of $z_{ph}>6$ is greater than 70\%. The $z_{ph}$\ and corresponding probability of our targets are shown in Table \ref{tab:targlist}.
\input{BAGbestfit}
\begin{figure*
\begin{center}
\includegraphics[width=\textwidth]{hst3d_sedfit_v4.png}
\end{center}
\caption{Best-fit \bags\ galaxy SEDs (black) and the $1\sigma$ distribution (light gray) based on the broadband photometric data (red diamonds) with $1\sigma$ uncertainties. The x-axis errorbars indicate the filter bandwidth. We note that the we enforce $5\sigma$ cuts on \textit{HST}\ $J_{125}$\ and $H_{160}$\ fluxes. The large J-band fluxes, relative to $J_{125}$\ fluxes, in EGS 515 and EGS 29337 correspond to shallower ground-based CFHT data \citep{Bielby2012AA}. The $z_{ph}$\ are fixed based on the best \eazy\ estimates. The non-detections are indicated with $2\sigma$ upper limits. We plot the best-fit quasar template SEDs (light blue). Quasars and starbursts are nearly degenerate at the predicted $z_{ph}$. We can see that a galaxy model is most appropriate for EGS 29337, as expected from \cite{rborsani2016ApJ,Stark2017MNRAS}. We also plot the best-fit dwarf star templates in dark gray, which also show nearly degenerate fits. The insert plot shows the \eazy\ $z_{ph}$\ probability distribution in blue. The resulting $p(z_{ph})$ distribution across $6.5\lesssim z\lesssim8.5$ is broad since $Y_{105}$\ data was not included.}
\label{fig:eazybagsSED}
\end{figure*}
Upon determining $z_{ph}$, we refine the SED fits using \bags\ \citep{Carnall2018} to determine their physical properties. \cite{Morishita2020z8} notes that distinguishing between luminous galaxies and quasars at $z\sim7$-8 is ambiguous and challenging without spectroscopy. In Figure \ref{fig:eazybagsSED} we plot both the best-fit quasar (described in Appendix \ref{apx:qsotemp}) and star-forming galaxy SEDs (from \bags\ fitting described next), which clearly show degenerate profiles, with the exception of EGS 29337 (to be discussed later). Since precise modeling is beyond the scope of this study, in this paper we instead assume that the sources are well represented with a young stellar spectrum with nebular emission with \bags\ modeling and explore the inferred properties.
We describe our \bags\ fit methodology here. The redshift is fixed to $z_{ph}$\ from \eazy, and we freely fit for other properties. The model fit priors are listed in Table \ref{tab:BAGpriors}, following the treatment in \cite{rborsani2021superB} as a guide. The best-fit \bags\ SEDs and \eazy\ $z_{ph}$\ probability distributions are shown in Figure \ref{fig:eazybagsSED}. For each source, we use the best fitting SED model to extract the source's rest-frame UV luminosity, $M_{UV}$, and other stellar parameters of interest for subsequent analysis, which is discussed in Section \ref{sec:res}. The best-fit \bags\ model parameters are shown in Table \ref{tab:bagsSED}.
\subsection{Excluding low-redshift contaminants}\label{subsec:lowz}
To further exclude low redshift interlopers among the selected point sources, we utilize G141 grism spectra made available by the \HSTsurvey\ team. As alluded to earlier, dwarf star SEDs have a sharp $1\,\micron$ drop-off that mocks the Lyman break of $z\sim7$-8 objects, making them likely interloper contaminants. There are notable spectral features at $1\,\micron$, $1.25\,\micron$, and $1.6\,\micron$ that are captured by the G141 grism.
We find that some of our point sources are not listed in the grism catalog, due either to the extraction limit ($JH_{140}\approx26$\,mag) or incomplete spectral coverage (landing on/outside of detector edge). It is noted that the G141 grism does not cover the redshifted Ly$\alpha$ break at $z\sim7$-8, making the confirmation as high redshift sources difficult. Therefore, the objective of our inspection here is to exclude interlopers through the detection of continuum spectral features instead of characterizing their spectra.
When extracted grism spectra are available for the sources selected in Sec.~\ref{subsec:color}, we perform spectral fits to low-mass L and T dwarf template spectra, which were observed with the SpeX spectrograph on NASA InfraRed Telescope Facility \citep{Rayner2003PASP}. Template spectra are obtained from the SpeX Prism Library \citep{spexprism2014}. Based the spectral fits, we identified 4 T dwarfs with clear spectral features. Their 1D spectra are shown in Appendix \ref{apx:BDlist}. In fact, 3 of these dwarf stars were also identified in a recent \HSTsurvey\ dwarf star study \citep{Aganze2021}. We exclude these targets from the final point source list. We note that none of our final redshift selected, point source candidates were identified by \cite{Aganze2021}. However, the \cite{Aganze2021} selection was limited to spectra with $S/N>10$, and thus their selected sources are all brighter than our final targets ($H_{160}\simlt24$\,mag). This may simply reflect the limitations of the \HSTsurvey\ grism data instead of differences in selection. We also fit the photometric SEDs with SpeX dwarf star templates using \eazy. In Table \ref{tab:targlist} we list the $\chi_{\nu,BD}$, and in Figure \ref{fig:eazybagsSED} we show the best-fit models. When compared to the $\chi_{\nu}$ of $z_{ph}$\ fits, we find that our final point sources are better constrained as $z\sim7$-8 sources.
\subsection{Visual inspection of point sources}
As the final step of our sample selection, here we examine the images to identify and to eliminate any spurious fluxes in \textit{HST}\ and \textit{Spitzer}. Once we eliminate false detections, we repeat and refine the \eazy\ $z_{ph}$\ estimates and \bags\ SED fits.
The postage stamp images of our final point sources in Table \ref{fig:imgTOT}. Images are extracted for available deep \textit{HST}\ \citep{Grogin2011ApJS197,Koekemoer2011ApJS197,Skelton2014ApJS214} and \textit{Spitzer}\ \citep{Dickinson2003,Ashby2013ApJ} that make up the \HSTsurvey\ catalogs. From the images, we clearly see the blue color dropouts, which are also reflected in the SED fits.
Some sources show suspicious blue detections. For example, the GDS 29369 images show suspicious $I_{814}$\ fluxes, despite meeting the non-detection criteria. After comparing the different filter images, we concluded that this is likely noise artifacts because their flux centroids do not match and the size is on the same order as the surrounding noise structure. The catalog also suggested spurious ground-based blue fluxes observed with \textit{Subaru} \citep{taniguchi2007ApJS} for GDS 45797. However, careful examination of the \textit{Subaru} images suggested that they are artifacts due to diffraction spikes from a nearby star. So, we treat these blue fluxes as non-detections in our analysis. Fortunately, the inclusion of these blue fluxes did not have a major affect on the $z_{ph}$\ or the SED fits results. Another uncertainty comes from \textit{Spitzer}/IRAC fluxes, which suffer from lower spatial resolution. The \HSTsurvey\ catalog includes IRAC contamination flags for each channel; however, the flags are based on contamination within 3\!\arcsec apertures, whereas our fluxes are taken at 0.\!\arcsec7 apertures. As a result, we individually inspected the IRAC images to decide whether or not to include them in our analysis. If we visually confirmed obvious contamination within a channel, we omitted its flux.
\begin{figure*}[!th]
\begin{tabular}{cc}
\includegraphics[width=0.97\columnwidth]{hst3d_sizeLum7.png} &
\includegraphics[width=0.97\columnwidth]{hst3d_sfrd7.png} \\
\includegraphics[width=0.97\columnwidth]{hst3d_sfrdwZ_7.png} &
\includegraphics[width=0.97\columnwidth]{hst3d_EW7.png} \\
\end{tabular}
\caption
$R_{\textrm{eff}}$ (top left) and $\Sigma_{\textrm{SFR}}$ (top right) vs. $M_{UV}$\ relations. We also plot the inferred $R_{\textrm{eff}}$-$M_{UV}$\ relations from \cite{Kennicutt1998,Ono2013ApJ}. We compare with results from {SuperBoRG}\ point sources \citep{Morishita2020z8} and other galaxy studies \citep{rborsani2021superB,Oesch2010ApJ}. We also show the $\Sigma_{\textrm{SFR}}$ (bottom left) and $EW_{H\beta+[\textrm{O}~\textsc{iii}]}$ (bottom right) vs. redshift, and compare with \cite{Oesch2010ApJ,Ono2013ApJ, labbe2013ApJ,Schenker2013ApJ,Smit2014ApJ,Smit2015ApJ,Morishita2020z8}. We plot the inferred $\Sigma_{\textrm{SFR}}$ redshift evolution based on the empirically derived $R_{\textrm{eff}}\propto(1+z)^{-1.3}$ relations for $L^*_{z=3}$ (dotted line) and $(0.3-1)L^*_{z=3}$ (dashed line) from \cite{Ono2013ApJ}. The corresponding values for the \HSTsurvey\ point sources are shown in Table \ref{tab:bagsSED}.}
\label{fig:LFz8_sizeLum}
\end{figure*}
\subsection{Cross-matching with \textit{Chandra} X-ray catalog}
\input{xraytable}
Finally, we also cross-matched our \textit{HST}\ selected source with the deep X-ray \textit{Chandra} catalogs for GOODS-South \citep{Luo2017ApJS} and AEGIS \citep{Nandra2015ApJS}. Significant X-ray emission would strengthen the case for quasar candidacy.
Previous targeted observations, at shallower depths, have
detected $z\sim7$ low-luminosity quasars \citep[e.g.,][]{banados2018ApJxray, Wang2021ApJ}; however, no clear X-ray emission was detected at $0.5-7 \textrm{ keV}$. We place upper limits on the fluxes and luminosities of our targets in Table \ref{tab:xray}. Moreover, there are also known Compton-thick $z>7$ quasar candidates with faint X-rays \citep{Fujimoto2022Nat}, so this is not entirely unusual. Detailed X-ray analyses will be left for future study.
\section{Results: Nature of point sources}\label{sec:res}
\subsection{On the point source selection}\label{subsec:res1}
We cross-match our point source selection with findings from \cite{Bouwens2015ApJ}, which examined the CANDELS, HUDF09, HUDF12, ERS, and BoRG/HIPPIES fields to estimate the galaxy UV luminosity function. Both studies appply similar Lyman-dropout color selections. The main difference is that we explicitly search for point-sources with the morphology selection defined in Sec.~\ref{subsec:color} using the $f_{5}/f_{10}$ flux ratios, while \cite{Bouwens2015ApJ} implements the stellarity parameter (e.g., \texttt{star\_class} flag), combined with SED fit photometry, to eliminate point-sources.
We find that only EGS 29337 is detected in both catalogs (EGSY-0120800269) with the separation of $\delta r=0.\!\arcsec11$, which is well within the PSF uncertainty. In fact, this object has also been spectroscopically confirmed as as a $z_{ph}=7.4$ galaxy \citep{rborsani2016ApJ,Stark2017MNRAS}. This means that some near point-like sources are in fact bona-fide galaxies. We discuss the implications in Section \ref{sec:disc}. Our SED modeling in Figure \ref{fig:eazybagsSED} also supports these observations.
We note that the other sources were not identified by other $z\sim$7-8 studies \citep[e.g.]{Bouwens2015ApJ, rborsani2016ApJ, Stark2017MNRAS}. In fact, these sources have larger $f_{5}/f_{10}$ values compared EGS 29337, which suggest that they appear more point-like, and thus more likely to have been rejected in the galaxy selection. Thus, a unique aspect of this study is that we explore the $z\sim7$-8 dropout point-sources that are often excluded in earlier studies of high-redshift galaxies.
\subsection{Stellar SED fit properties of point sources}
We list the best-fit stellar population properties of the point sources, which were estimated with \bags\ SED fits, in Table \ref{tab:bagsSED}. Due to the limitation of data, we assume that our detected point sources are
well represented by a young stellar spectrum. The fit results predict sub-solar metallicities for nearly all of our point sources. Considering the young age of the universe
, the metallicity estimates are not surprising. However, since uncertainties in broadband SED fits are strongly influenced by assumptions in the star formation history and age-metallicity-dust degeneracies \citep[e.g., Fig.~12 in][]{Morishita2019ApJ}, it is difficult to place any confident constraints on the metallicity evolution in the early universe. Instead we examine the size and star formation rate (SFR) density properties of the point sources.
We derive the projected physical size, $R_{\textrm{eff}}$, from the half-light radius, which is calculated with \sex. When compared to the \HSTsurvey\ PSF limit, $0.\!\arcsec14$,
we find that 2 of the calculated $R_{\textrm{eff}}$ are upper limits. This is similar to the \cite{Morishita2020z8} results, as shown in Figure \ref{fig:LFz8_sizeLum}. This means that these sources are likely unresolved by \textit{HST}, despite meeting our point source selection criteria.
Using the SFR estimated from the \bags\ modeling, we also calculate the SFR density, $\Sigma_{\textrm{SFR}}$, which is defined as the average SFR within a circle with radius $R_{\textrm{eff}}$:
\begin{equation}
\Sigma_{\textrm{SFR}} = \frac{\textrm{SFR} }{ 2\pi R^2_{\text{eff}} }.
\label{eq:SFRdensity}
\end{equation}
The calculated $\Sigma_{\textrm{SFR}}$ serves as lower limits since its uncertainty is dependent on the upper limit uncertainty in $R_{\textrm{eff}}$. We compare the inferred $\Sigma_{\textrm{SFR}}$, which is calculated from the $M_{UV}$-SFR relation \citep{Kennicutt1998,Ono2013ApJ} defined as follows:
\begin{equation}
M_{UV} = -2.5 \log_{10} \bigg[ \frac{ \Sigma_{\textrm{SFR}} \cdot \pi R^2_{\text{eff}} }{2.8\times10^{-28} (M_{\astrosun}yr^{-1})} \bigg]+51.59.
\label{eq:sigSFR}
\end{equation}
It appears that our point source candidates are highly compact star-forming objects. In Figure \ref{fig:LFz8_sizeLum}, we plot $R_{\textrm{eff}}$ and $\Sigma_{\textrm{SFR}}$ against their corresponding $M_{UV}$. We also compare the $\Sigma_{\textrm{SFR}}$ redshift evolution. Our results appear to be consistent with trends of high redshift galaxies discussed in \cite{Ono2013ApJ,Holwerda2018AA}, which predicts a greater number of compact galaxies with high star-formation at earlier epochs.
Using the best-fit \bags\ SEDs, we calculate the UV continuum slope $\beta_{UV}$. We adopt the formula defined \citep[e.g.,][]{Dunlop2013MNRAS}:
\begin{equation}
\beta_{UV} = -2.0+4.39\times(J_{125}-H_{160}),
\label{eq:Buv}
\end{equation}
where $J_{125}$\ and $H_{160}$\ here are the best-fit magnitudes from the best-fit \bags\ SEDs. The calculated $\beta_{UV}$\ are listed in Table \ref{tab:bagsSED} with the mean slope of $\bar{\beta_{UV}}=-1.90\pm 0.35$. The resulting $\bar{\beta_{UV}}$ is consistent with $\beta_{UV}$\ of known bright galaxies at $z\sim$7-8 \citep[e.g.,][]{Dunlop2013MNRAS,Bouwens2014ApJ793}.
Lastly, we estimate the rest-frame equivalent width due to the H$\beta+[\textrm{O}~\textsc{iii}]$ emission lines from the best-fit SED for objects with sufficient IRAC fluxes. This is possible because $H\beta+[\textrm{O}~\textsc{iii}]$ emission from $z\sim7$-9 sources is well sampled by IRAC CH1 and CH2, $3.6\,\micron$ and $4.5\,\micron$, \citep{rborsani2016ApJ}. We calculate the equivalent widths as,
\begin{equation}
EW_{H\beta+[\textrm{O}~\textsc{iii}]} = \frac{(f_{\rm ch2}-f_{\rm cont})}{f_{\rm cont}} \frac{\Delta\lambda_{\rm ch2}}{(1+z_{ph})} ,
\label{eq:EW_hbo3}
\end{equation}
where $f_{\rm cont}$ is the underlying continuum flux obtained from the best-fit \bags\ spectrum, $f_{\rm ch2}$ is the observed \textit{Spitzer}/IRAC CH2 flux, $\Delta\lambda_{\rm ch2}\sim1$\,\micron\ is the full-width half maximum of the CH2 filter, and $z_{ph}$ is from \eazy. Comparing the values in Table \ref{tab:bagsSED} and the SED plots in Figure \ref{fig:eazybagsSED}, we see that the $EW_{H\beta+[\textrm{O}~\textsc{iii}]}$ estimates can be applied to three targets. These objects show a moderate $EW_{H\beta+[\textrm{O}~\textsc{iii}]}$ of 500-1000\,\AA, similar to estimates from other high redshift surveys \citep{labbe2013ApJ,Schenker2013ApJ,Smit2014ApJ,Smit2015ApJ,Morishita2020z8}. However, this is based on the assumption that $z_{ph}$\ is correct.
We compare the redshift evolution of the measured $EW_{H\beta+[\textrm{O}~\textsc{iii}]}$ in Figure \ref{fig:LFz8_sizeLum}, and our results appear to be consistent with other survey results.
Future infrared observations with higher resolution and sensitivity are needed to better characterize the predicted $H\beta+[\textrm{O}~\textsc{iii}]$ emission.
\subsection{Number density of point sources}\label{sec:LF}
From the survey data, we constrain the point source luminosity function at $z\sim7$-8. We produce our own completeness simulation to calculate the effective volume, $V_{\rm eff}$, probed by the \HSTsurvey. We follow the completeness simulation treatment in \cite{Leethochawalit2021MNRAS} to calculate $V_{\rm eff}$\ \citep{Oesch2012ApJ745, GLACiAR2018,Calvi2016ApJ817, Morishita2018ApJ}.
We inject 500 sources into each \HSTsurvey\ image at each $(M_{UV},z)$ bin: 100 $\Delta M_{UV}$ bins across $-26\leq M_{UV} \leq-16$ and 13 $\Delta z$ bins across $7\leq z \leq9.4$. All simulated sources in a given $(M_{UV},z)$ grid have the same UV slope, which are randomly drawn from a Gaussian distribution with a mean slope of $\bar{\beta_{UV}}=-2.2\pm 0.4$. Source fluxes are calculated in the same way as in \cite{Skelton2014ApJS214}. We extract the simulated point sources according to our selection criteria described in Section \ref{sec:select}. We repeat this process for every field. We show the redshift and magnitude probability distribution function of the extraction completeness in Figure \ref{fig:compsimul}.
After we calculate $V_{\rm eff}$, we estimate the number density of the point sources shown in Table \ref{tab:LFNdensity} within each $\Delta M_{UV}=0.5$ mag bin. We quote Poisson uncertainties for the number density \citep{Gehrels1986ApJ}. In Figure \ref{fig:numdensity}, we plot the estimated number density of our point sources and compare with results from previous surveys of point sources and galaxies at $z\sim7$-8.
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.96\columnwidth]{hst3d_compSimul3.png}
\end{tabular}
\end{center}
\caption{The $M_{UV}$\ and redshift dependent selection probability distribution as determined from our completeness simulation. The plot shows the ratio of color-selected point sources recovered to all input sources as a function of redshift and $M_{UV}$. The colorbar to the right indicates this recovery fraction. At brighter $M_{UV}\lesssim-21$ magnitudes, at least 50\% of simulated color-selected point sources are recovered (blue contour line). At fainter magnitudes, the recovery fraction decreases due to the combined effect of color-selection and source detection (gray contour lines at 40 \% and 75 \% detection). We also indicate the observed point source candidates at their respective $z_{ph}$\ and $M_{UV}$\ (red stars).}
\label{fig:compsimul}
\end{figure}
\input{LFNdensity2}
\begin{figure
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.96\columnwidth]{hst3d_LF20_mcmc.png} \\
\end{tabular}
\end{center}
\caption{The derived number density of the \HSTsurvey\ point sources in red, {SuperBoRG}\ point sources \citep{Morishita2020z8} in blue, and galaxies at $z\sim7$-8 \citep{Bouwens2021AJ} in black. The $z\sim7$ number density from \cite{Harikane2022ApJS} are in green. Open symbols indicate observed data; and filled symbols indicate upper limits, which are estimated from the completeness simulation. Our number density values are listed in Table \ref{tab:LFNdensity}. }
\label{fig:numdensity}
\end{figure}
We fit the point source number density with both the Schechter function (Eq.\ref{eq:schechter}; \citealt{Schechter1976ApJ}) and with the Double powerlaw (Eq.\ref{eq:DPL}; \citealt{Hopkins2007ApJ}), where $\phi^*$ is the characteristic normalization, $M^*_{UV}$ is the characteristic UV luminosity defined at $M_{UV}$\ (1450\,\AA), $\alpha$ defines the faint end slope, and $\beta$ defines the bright end slope.
\input{LFbestfit2}
\begin{equation}
\begin{split}
\phi_{\textrm{Sch}} = \frac{\ln{10}}{2.5}\phi^* & \times 10^{-0.4(M'-M_*)(\alpha+1)}\\
& \times\exp{ \bigg[-10^{-0.4(M_{UV}-M^*_{UV})} \bigg]}
\end{split}
\label{eq:schechter}
\end{equation}
\begin{equation}
\begin{split}
\phi_{\textrm{DPL}} = \frac{\ln{10}}{2.5}\phi^* \times & \bigg[10^{0.4(\alpha+1)(M_{UV}-M^*_{UV})} \\
& +10^{0.4(\beta+1)(M_{UV}-M^*_{UV})} \bigg]
\end{split}
\label{eq:DPL}
\end{equation}
\begin{figure
\begin{center}
\includegraphics[width=\columnwidth]{hst3d_LF20c_mcmc_gfract.png}
\end{center}
\caption{(Top) Comparison of \cite{Bouwens2021AJ} galaxy vs. point source fraction. Point sources dominate a larger fraction of the bright end starting at $\sim-22\textrm{ mags}$. We compare with the \cite{Harikane2022ApJS} galaxy fraction in gray. (Bottom) We compare the best-fit $z\sim7$-8 luminosity functions for the combined, freely-fit point sources and galaxies in purple curves against other galaxy-only models by \cite{Bowler2020MNRAS, Bouwens2021AJ} in black and gray: Schechter in solid, double powerlaw in dashed lines. The purple squares indicate the combined effective number density. EGS 29337 has been removed from the point source number density to avoid duplicate counting. Open symbols indicate observed data; and filled symbols indicate upper limits. The curves are shown with $1\sigma$ uncertainties.}
\label{fig:LFz8}
\end{figure}
\begin{figure*}[!hbt]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.9\columnwidth]{hst3d_LF20e_mcmc.png} &
\includegraphics[width=1.1\columnwidth]{hst3d_LF20f_mcmc.png}
\end{tabular}
\end{center}
\caption{Contour plots of the best-fit luminosity function parameters. (Left) Combined point source + galaxy Schechter fits with free parameters. (Right) Combined point source + galaxy double powerlaw fits with free parameters. The best-fit parameters are indicated in Table \ref{tab:LFbestfit}. The 1D histogram of the fit parameters are shown with the 16\%, 50\%, and 84\% percentile values indicated.}
\label{fig:LFz8_contour}
\end{figure*}
To properly include the bins of non-detection of the point source population at the faintest and brightest ends in fitting evaluation, we incorporate the upper limits from non-detections following the derivations from \cite{Sawicki2012PASP} for $\chi^2$ minimization. The derivations for $M_{UV}$\ and the $\chi^2$ minimization are shown in Appendix \ref{apx:chi2min}.
We perform Markov Chain Monte Carlo (MCMC) sampling, using \emcee\ package \citep{emcee2013PASP}, to constrain the luminosity function. First we fit the luminosity function for point sources from this study (\HSTsurvey\ selections). Then we apply the fits on all point sources selected from both this study and {SuperBoRG}.
If we freely fit for all of the parameters, the fit parameters do not converge to physically meaningful values. This is likely due to the lack of data points at both the faint and bright magnitude ranges.
Instead, we opt for a more conservative approach and follow the known luminosity function shapes of $z\sim7$-8 galaxies and $z\sim6$ quasars. We fix both the faint end and bright end slopes and freely fit for $\phi^*$ and $M^*_{UV}$. For the Schechter model fits, we fix the faint end slope to $\alpha=-2.2$, which is the observed galaxy luminosity function at $z\sim7$-8 \citep{Bouwens2021AJ}. For the double powerlaw mode fits, we fix both the faint end slope to $\alpha=-1.2$ and the bright end slope to $\beta=-2.7$ based on the extrapolations of the quasar luminosity at $z\sim6$ \citep{Matsuoka2018ApJ869, Harikane2022ApJS}.
With the deeper exposures of the \HSTsurvey\ survey, we improve the point source luminosity function fits estimated by \cite{Morishita2020z8}. The best-fit luminosity function parameters of the point sources are shown in Table \ref{tab:LFbestfit}. If we freely fit for $M_{UV}$, we find that only the double powerlaw fit produces more reasonable parameters ($M^*_{UV}\approx-24$) than the best-fit Schechter function, which instead suggests an unrealistically bright UV cut-off ($M^*_{UV}\approx-38$, not shown). This may suggest that the point source luminosity function is more consistent with the high redshift quasar luminosity function (at $z\sim6$; \citealt{Matsuoka2018ApJ869}). On the other hand, if we force a lower-bound on the $M_{UV}$\ fit, then it is difficult to confidently favor either functions over the other. For both cases, the best-fit normalization $\phi^*$ deviates from the \cite{Matsuoka2018ApJ869} extrapolation by nearly $\times100$. This may be because \HSTsurvey\ is volume-limited, similar to the results in \cite{Morishita2020z8}.
\subsection{Number density of point source+galaxy populations}
Finally, we fit the combined point source and galaxy luminosity functions. Here, we remove EGS 29337 from the point source luminosity function since it is already included in the \cite{Bouwens2015ApJ} determination (discussed in Section \ref{subsec:res1}). First, we fit with the slopes fixed, and then we freely fit over all parameters. With the current survey volume by \textit{HST}, it is difficult to confidently favor either functions over the other for combined luminosity function. With the Schechter model, both the fixed-slope and freely-fit runs produce brighter UV cut-offs at $M^*_{UV}\approx-21.9$, compared to $M^*_{UV}=-22.8$ by \citet{Bouwens2021AJ}, suggesting an excess of bright sources. The freely-fit double powerlaw model produces a superposition of point source Schechter and galaxy Schechter functions also with a slight excess of $M^*_{UV}=-20.0$ and a steep $\beta=-3.6$.
The exact shape of the galaxy luminosity function is under debate. For example, \cite{Harikane2022ApJS} performed a two-component luminosity function fit (i.e. DPL+DPL or DPL+Schechter) to the combined quasar+galaxy populations. In contrast, we model a single component function across all magnitudes for both the point source and galaxy number densities (i.e. Schechter or DPL). The main difference between the two studies is that \cite{Harikane2022ApJS} extrapolates the $z\sim6$ \cite{Matsuoka2018ApJ869} quasar luminosity function to estimate the $z\sim7$ relation, while we use observed sources to anchor the point source luminosity function. Despite the differences in modeling, both results demonstrate that the inclusion of bright $M_{UV}\lesssim-24$ {SuperBoRG}\ sources at $z\sim7$-8 results in a departure from the previously measured galaxy luminosity function, i.e. a bright excess. We plot the $z\sim7$-8 point source number density and the best-fit luminosity functions in Figure \ref{fig:LFz8}. The best-fit parameters of all fits are shown in Table \ref{tab:LFbestfit}. The contours of the combined luminosity function fits are shown in Figure \ref{fig:LFz8_contour}.
We compare the point source number densities and luminosity function against galaxy luminosity function measured at $z\sim7$-8 \citep{Bouwens2021AJ}. We quantify the fraction of extended galaxies to all sources (galaxies and point sources) as a function of $M_{UV}$\ as the following:
\begin{equation}
\begin{split}
f_{galaxy}(M_{UV}) = \frac{\phi_{galaxy}}{\phi_{galaxy}+\phi_{points}}
\end{split}
\label{eq:gfract}
\end{equation}
where $\phi_{galaxy}$ is the galaxy number density from the luminosity function \citep{Bouwens2021AJ} and $\phi_{points}$ is the observed point source number density listed in Table \ref{tab:LFNdensity}. The uncertainty in $f_{galaxy}$ simply reflects the uncertainty in the number count of point sources (i.e. poisson). We plot $f_{galaxy}$ alongside the luminosity function fits in Figure \ref{fig:LFz8} and compare with results from \cite{Harikane2022ApJS}. We find that these point sources dominate at the bright $M_{UV}$\ magnitudes. This suggests that the bright end excess implicated by the new point sources is not likely dominated by a typical population that has been identified in previous studies of high-redshift galaxies. We discuss the physical interpretations of this measured excess in the following section.
\section{Discussion}\label{sec:disc}
Although our sources are selected with slightly different colors and from different surveys, the inferred number density of our point sources at $M_{UV}<-21.5$ mags agree with the {SuperBoRG}\ point source study. This indicates that these $z\sim7$-8 dropout point sources are abundant enough to be detected in both surveys and are representative of similar populations. The inferred $M_{UV}$\ suggests that these objects are driven by intense phenomena that occurs in a small physical scale, such as central starburst or quasar activity, which also shape their observed morphology point-like.
In this section we explore the physical properties and implications of these sources.
\subsection{Point sources as compact starburst galaxies}
With the exception of EGS 29337, it is currently difficult to distinguish our final candidates between non-active galaxies or quasar-hosting galaxies, the inferred sizes of the point sources is consistent with the observed trends of smaller galaxy sizes in the early universe \citep{Oesch2010ApJ,Ono2013ApJ,Holwerda2015ApJ}. If these point sources are compact non-active galaxies, we predict high $\Sigma_{SFR}$ based on our SED fitting analysis. Previously, \cite{Oesch2010ApJ,Ono2013ApJ} observed constant $\Sigma_{SFR}$ from $z\sim4$ to $z\sim7$ with a weak increase towards higher redshifts. Our SED analysis of the point sources appears to support this increasing $\Sigma_{SFR}$ trend, albeit we predict even larger values, as shown in Figure \ref{fig:LFz8_sizeLum}. Finally, given their predicted SFR and stellar masses, these sources may be progenitors of massive quiescent galaxies that are already present in the early universe \citep[e.g.,][]{vandokkum2008ApJ, Damjanov2009ApJ}. This may suggest that our sources are UV enhanced starbursts and/or that additional physics may be at play. In fact, EGS 29337 has been shown to be one of the brightest $z>7$ known with large SFR \citep{Stark2017MNRAS,rborsani2016ApJ}.
\subsection{Point sources as low-luminosity quasars}
While much of our SED analyses assumed the star-forming SED properties, Figure \ref{fig:eazybagsSED} clearly shows that the SEDs of quasars and star-forming galaxies are degenerate. Theoretical predictions of early quasar properties suggest that variations in the quasar duty cycle may lead to an enhancement of UV bright quasars \citep{RenTrenti2021ApJ}. With the detection of potentially UV bright quasars, there are also implications on the obscured quasar fraction, which is unknown at these redshifts \citep{vito2018MNRAS473, vito2019AA, Inayoshi2020araa}. Either there is an enhancement in the population of unobscured quasars or some physical mechanisms, such as powerful outflows, may drive obscured quasars to appear more luminous like unobscured quasars. Also, while non-detections from deep \textit{Chandra} images suggest that no luminous quasar is present, the possibility of heavy Compton-thick obscuration may complicate this result \citep{Ni2020MNRAS}. Another possibility is that these sources are quasars embedded in star-forming galaxies, similar to $z\sim7$ sources identified by \cite{Laporte2017ApJ}. In fact, the possibility of either a compact starburst or quasar is supported by the recent discovery of an UV compact, red bright, X-ray faint object at $z\sim7$, which is hypothesized to be either a compact dusty, star-forming region or a Compton-thick super-Eddington quasar \citep{Fujimoto2022Nat}.
Although we cannot distinguish between these possibilities with the current \textit{HST}\ data, the \textit{James Webb Space Telescope}'s spatial resolution and sensitivity is expected reach below the predicted sizes and magnitudes of our sources at the range of $ -22 \lesssim M_{UV}\lesssim -18$. \cite{Marshall2021MNRAS} predicts that deep NIRCam observations may allow us to study the quasar and its host galaxy at $z\sim7$. Its imaging and spectroscopic capability may enable us to confidently distinguish them as quasars or as compact star-forming galaxies. If they are revealed as quasars, they will become one of the most distant quasars ever discovered.
\subsection{Investigating the impact on the bright-end excess}
In this section, we focus on understanding the point sources' contribution to the bright end of the galaxy luminosity function. In Figure \ref{fig:LFz8}, we show the combined point source and galaxy luminosity function. Once the point sources from \HSTsurvey\ and {SuperBoRG}\ are incorporated, we find that the best-fit luminosity functions suggest the existence of a bright point source population, that may be missed by the galaxy surveys. This may support the existence of a bright end excess in the early universe. While ground-based observations \citep[e.g.]{Bowler2020MNRAS} may detect unresolved sources, these studies are limited to select fields. Thus, we stress the importance of large-volume studies to accurately quantify the luminosity function.
Indeed, \citealt{Harikane2022ApJS} present a comprehensive analysis of the luminosity function by combining quasar and galaxy populations identified in the HSC program. They proposed several explanations for the apparent bright end excess seen in galaxy luminosity functions. Physical mechanisms such as inefficient mass quenching due to high star-formation and/or poor quasar feedback, low dust obscuration in the host galaxy \citep{MarquesChaves2020, MarquesChaves2021}, or even additional hidden quasar activity may increase the observed rest-frame UV luminosity \citep{Mirocha2020MNRAS}. Variations in the quasar duty cycle can also enhance the UV luminosity and contribute to the bright end \citep{RenTrenti2021ApJ}. Our predictions as compact star-forming galaxies or quasars are consistent with these possibilities.
Other possibilities include superposition of lensed galaxies or even merging galaxies. However, these may be unlikely since the point source criteria requires small elongation values. Our sources also do not appear to be close to potential lensing sources (see Figure \ref{fig:imgTOT}). Moreover, \cite{mason2015ApJ} showed that the effect of magnification bias on the luminosity function determination is small. \cite{Shibuya2022PASJ} calculated the merger fraction of 10\% to 70\% for bright $ -24 \lesssim M_{UV}\lesssim -22$ galaxies \citep{Harikane2022ApJS}. Considering the inferred $M_{UV}$\ of our sources, there is a non-zero possibility of merger contaminants, especially since galaxy formation in the early universe may involve major mergers.
If the point sources are revealed as quasars by future spectroscopic follow-ups, then it may imply substantial population of low-luminous quasars in the early universe. Since quasar activity is associated with rapid accretion, this may allow a pathway for the rapid formation of massive black hole seeds. Distinguishing between compact sources and contaminants may require higher spatial resolution than the capabilities of \textit{HST}.
What is clear is that point sources selected from deep \HSTsurvey\ and medium-deep {SuperBoRG}\ both consistently suggest a bright end excess. With the derived upper limit in their number density, we also find that the number of the point-source population dominates only $\simlt5$\,\% over the galaxy population at $M_{\rm UV}\simgt-21$\,mag (Figure \ref{fig:LFz8}). This suggests that their contribution to the faint end luminosity function, and thus cosmic reionization, is likely limited.
\subsection{Caveat: low-redshift interlopers}\label{sec:contam}
As demonstrated in this paper, the color and morphology selection using current \textit{HST}\ capabilities is still challenging in distinguishing between $z\sim7$-8 sources and Galactic interloper stars. The difficulty is further compounded by the fact that these objects are detected at low signal to noise ratio. We also note that the astrometry analysis did not show significant differences in the apparent proper motion following the analysis used in \cite{Morishita2020z8}, so high-quality spectroscopic analysis was the only reliable, albeit incomplete, metric in eliminating low redshift contaminants. We also calculated $\beta_{UV}$, defined in Eq.\ref{eq:Buv}, of all SpeX dwarf star template spectra, in the event they are misidentified as galaxies. We find a mean slope of $\bar{\beta_{UV}}=-1.2\pm 1.2$, which is nearly indistinguishable from those of galaxies \citep{Dunlop2013MNRAS}. However, the range of predicted $\beta_{UV}$\ is also large, spanning between $-5.32 \lesssim \beta_{UV} \lesssim +6.76$, which suggests that $\beta_{UV}$\ is not a useful metric to distinguish galaxies (including quasars) from dwarf stars. In fact, the $\beta_{UV}$\ of spectroscopically confirmed dwarf stars (see Appendix \ref{apx:BDlist}) have a mean slope of $\bar{\beta_{UV}}=-4.1$. Thus, we require spectroscopic follow-up to confirm the redshifts and spectroscopic properties of the targets identified in this study. Therefore, for now, our luminosity function estimates serve as an upper limit to the quasar number density at $z\sim7$-8.
Despite this fact, our study shows that low-resolution spectroscopy around $1\micron$ is an effective method in identifying foreground stars.
It is noted that when we opt to use the $Y_{105}$/$J_{125}$/$H_{160}$\ color selection \citep{Morishita2020z8, morishita2021sb, rborsani2021superB} for the field available (i.e. the GOODS-South field), no dwarf star contaminants are confirmed by the grism data (Ishikawa, in prep.).
This may be indicative that $Y_{105}$\ is an effective $z>7$ selector in the absence of other filter observations as proposed by \citet{morishita2021sb}. Despite this challenge, we eliminate a few dwarf stars contaminants with grism spectroscopy. For very faint point sources that do not have sensitive spectroscopic data, we demonstrate that the SED fits favor $z_{ph}\sim7$-8 sources over dwarf stars.
\section{Conclusion}\label{sec:concl}
We searched for $z\sim8$ Lyman dropout point sources with the archival \HSTsurvey\ data. \HSTsurvey\ surveys nearly 700 square arcminutes of the CANDELS fields, reaching the depth of $J_{125}$\ and $H_{160}$\ $\sim26$ mags.
We combined Lyman dropout color and point source selections, with additional photometric redshift estimates and grism spectroscopy to eliminate low-redshift contaminants, and identified three $z\sim7$-8 point source candidates.
We then investigated the physical properties of the point sources by using the available multi-band photometric data. SED analyses suggest that these sources are potentially quasars or compact star-forming galaxies. Assuming these sources are galaxies, the fitting results revealed high star formation surface density. This is consistent with the redshift trend of previously identified luminous galaxies of comparable luminosity, $M_{UV}\sim-22$; however, we find even larger $\Sigma_{\textrm{SFR}}$ values than those predicted by the relation. We measured the $EW_{H\beta+[\textrm{O}~\textsc{iii}]}$ emission for the 3 targets with \textit{Spitzer}/IRAC photometry available and found moderately high equivalent widths of $\sim500-1000$\,\AA.
We calculated the number density distribution and derived the luminosity function of the point sources.
Similar to previous \textit{HST}\ $z\sim7$-8 point source surveys, we found that the inclusion of point sources revealed an excess in the bright end at $M_{UV}\lesssim-22$, consistent with the {SuperBoRG}\ point source survey.
We combined the $z\sim7$-8 point source point source candidates with published galaxy number densities to estimate the total galaxy luminosity function. We found that the best-fitting models all point to a bright $M^*_{UV}$ cut-off, which departs from the known galaxy luminosity function.
The deeper observations of \HSTsurvey\ allowed us to extend the dynamic range covered by our previous work in {SuperBoRG}. We did not identify point sources in the faint end.
Moreoever, we found that they make up less than 5\% of the galaxy fraction at the faint end. Thus it is unlikely that these point sources have major contributions to cosmic reionization.
If these $z\sim7$-8 point sources are confirmed to be quasars, our results suggest that quasars in this luminosity range may be more abundant in the early universe.
If they turn out to be a galaxy population, it would indicate the presence of compact and intense star-formation in the early universe. Further follow-up observations are required to confirm the inferred properties of our point sources.
Future surveys using the infrared optimized \textit{James Webb Space Telescope} and large field-of-view capable \textit{Roman Space Telescope} may resolve the current limitations of \textit{HST}.
\acknowledgments
Support for this work was provided by NASA through grant numbers HST-GO-15212.002 and HST-AR-15804.002-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.
This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
This research has benefitted from the SpeX Prism Library (and/or SpeX Prism Library Analysis Toolkit), maintained by Adam Burgasser at \url{http://www.browndwarfs.org/spexprism}.
The authors would like to thank the anonymous referee for the helpful suggestions in improving the paper. NL acknowledges that parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. CAM acknowledges support by the VILLUM FONDEN under grant 37459, the Danish National Research Foundation through grant DNRF140. YI also thanks Nadia Zakamska for helpful discussions.
\software{\astropy\ \citep{Astropy2013},
\eazy\ \citep{Brammer2008ApJ686},
\bags\ \citep{Carnall2018},
\sex\ \citep{Bertin1996},
\lmfit\ \citep{lmfit2014zndo},
\emcee\ \citep{emcee2013PASP},
\texttt{xspec} \citep{xspec1996},
\texttt{CIAO} \citep{ciao2006SPIE}}
| {
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Chrysanthemums are a genus (Chrysanthemum) of about 30 species of perennial flowering plants in the family Asteraceae, native to Asia and northeastern Europe. The genus once included many more species, but was split several decades ago into several genera; the naming of the genera has been contentious, but a ruling of the International Code of Botanical Nomenclature in 1999 has resulted in the defining species of the genus being changed to Chrysanthemum indicum, thereby restoring the economically important florist's chrysanthemum to the genus Chrysanthemum. These species were, after the splitting of the genus but before the ICBN ruling, commonly treated under the genus name Dendranthema.
The other species previously treated in the narrow view of the genus Chrysanthemum are now transferred to the genus Glebionis. The other genera split off from Chrysanthemum include Argyranthemum, Leucanthemopsis, Leucanthemum, Rhodanthemum, and Tanacetum. Naming in the horticultural trade is quite inconsistent, with Chrysanthemum and Dendranthema still in common usage.
The species of Chrysanthemum are herbaceous perennial plants growing to 50–150 cm tall, with palmately 3-7 lobed, thick leaves and large flowerheads, white, yellow or pink in the wild species.
Modern chrysanthemums are much more showy than their wild relatives. The flowers occur in various forms, and can be daisy-like, decorative, pompons or buttons. This genus contains many hybrids and thousands of cultivars developed for horticultural purposes. In addition to the traditional yellow, other colours are available, such as white, purple, and red. The most important hybrid is Chrysanthemum × morifolium (syn. C. × grandiflorum), derived primarily from C. indicum but also involving other species.
Chrysanthemum leaves resemble its cousin, the mugwort (Aremisia vulgaris), which is sometimes called wild chrysanthemum.
Fertile, well-drained, consistently moist soils in full sun to light shade.
Pompons are miniature and ball-shaped.
Chrysanthemums make a welcome addition to the garden, in particular because their late flowering offers advantages when the other garden plants are winding down for winter. Outdoor flowering chrysanthemums will need protection from hard frosts.
Chrysanthemums were cultivated in China as a flowering herb as far back as the 15th century BCE. An ancient Chinese city was named Ju-Xian, meaning "chrysanthemum city". The flower was introduced into Japan probably in the 8th century CE, and the Emperor adopted the flower as his official seal. There is a "Festival of Happiness" in Japan that celebrates the flower.
The flower was brought to Europe in the 17th century. Linnaeus named it from the Greek prefix chrys-, which means golden (the colour of the original flowers), and -anthemon, meaning flower.
Yellow or white chrysanthemum flowers are boiled to make a sweet drink in some parts of Asia. The resulting beverage is known simply as "chrysanthemum tea" (菊花茶, pinyin: jú huā chá, in Chinese).
Chrysanthemum tea has many medicinal uses, including an aid in recovery from influenza. In Korea, a rice wine flavored with chrysanthemum flowers is called gukhwaju (국화주).photo 1photo 2 The leaves of several species such as Chrysanthemum coronarium, the Garland chrysanthemum, which is grown commercially in East Asia as a leaf vegetable, known as tung ho (Template:Zh-cp), shungiku (Template:Lang-ja[シュンギク]) or ssukkat (Template:Lang-ko). In China, the greens are often stir-fried simply with garlic and dried chili peppers. The colour of the cooked greens is dark, their texture dense and mucilaginous, and their flavour fragrant and complex.
Fall-blooming species benefit from being cut back about 1/2 way down in midsummer in order to encourage a bushy habit.
Christopher Brickell and Judith D. Zuk (1997). The American Horticultural Society A-Z Encyclopedia of Garden Plants. DK Publishing. pp. 353.
Pirone, Pascal P. (1978). Diseases & Pests of Ornamental Plants (Fifth Edition ed.). John Wiley & Sons, New York. pp. 194–201.
Cranshaw, Whitney (2004). Garden Insects of North America: The Ultimate Guide to Backyard Bugs. Princeton University Press. pp. 588.
Pippa Greenwood, Andrew Halstead, A.R. Chase, Daniel Gilrein (2000). American Horticultural Society Pests & Diseases: The Complete Guide to Preventing, Identifying, and Treating Plant Problems (First Edition ed.). Dorling Kindersley (DK) Publishing, inc.. pp. 198. | {
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The Chicago Botanic Garden, and its Windy City Harvest urban agriculture program, has partnered with the Lawndale Christian Health Center to open a new 20,000 square foot center called the Farm on Ogden, a multiuse facility that supports and sustains a healthy urban community by bring food, health and jobs together in one location.
The Farm on Ogden will open on June 22, 2018.
The Chicago Botanic Garden ranks fourth in this Associated Press story by Donna Bryson.
(October 19, 2015 – Washington, DC) Building on more than a decade of research into successful STEM collaborations, the STEM Funders Network and 350 committed local funders and organizations are announcing a five-year effort to create 100 tight-knit local STEM networks—STEM ecosystems—that bring together broad-scale, cross-sector collaborations to nurture and scale effective science, technology, engineering and math (STEM) learning opportunities for all young people, especially girls and underserved populations.
Enjoy Illinois touts Wonderland Express. Click below to learn more.
Your museum guide to the Chicago Botanic Garden in summer.
The Windy City Harvest farm at McCormick Place was profiled in this article in Urban Farm Magazine. Learn about what they are harvesting in this second year of production on the roof of the convention facility.
Read more about "Thin Farming" | {
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} | 2,903 |
Q: Recursion to compare two Arrays without using loop [pseudo code]
Consider the given two arrays A and B without repetitions (that
is, no double occurrences of the same element). The task is to check whether
each element of B is also an element of A without regard to the order.
For instance if A = [1, 2, 3, 4] and B = [2, 3, 1] then the answer is YES. If
however B = [1, 2, 5] then the answer is NO because 5 is not in A.
Design a recursive algorithm (no use of loops) for the above problem.
I am trying to solve the above problem, and there is no way I can find to solve it without using the loop. Would anyone know a way to solve this with recursion without using a loop?
I can not use any builtin functions, this is a recursion exercise for algorithms and data structures.
A: Pseudocode for a function that recursively tests whether the array A contains a given element, x might look something like this:
function isMember(x, A):
if A = [] then return false
if x = A[0] then return true
return isMember(x, A[1..-1])
end
This function is built on the premise that to test if x is a member of A, we can test to see if the first element of A, denoted A[0], is the same as x. If so, we can terminate the function and return true. If not, then call the function again, passing it all the elements of array A except the first one that we already tested. I've denoted the remaining elements of A with A[1..-1], i.e. element numbers 1 through to -1, which in some languages is another way to refer to the last element.
Now during the second call to this function, x is being compared to the first element of A[1..-1], which is, of course, the second element of A. This recurses over and over, each time shrinking the size of array A the element at the top of the list is tested and discarded.
Then we eventually reach the final case, where there are no more elements left in A to test, which results in the final recursive call to the function being passed an empty array. In this situation, we can infer that every element in A failed to match with x, and so we can safely return false, stating the x is not a member of A.
Now, in order to determine whether a given array B is contained by array A, each element in B needs to undergo the test described above. If isMember() returns true for every element in B, then B must be contained by A. If any one element causes isMember() to return false, then we can stop further testing because B contains elements that are not in A, so it cannot be a subarray.
Here's some pseudocode that illustrates this recursive process:
function containedBy(B, A):
if B = [] then return true
let x := B[0]
if not isMember(x, A) then return false
return containedBy(B[1..-1], A)
end
It's very similar in many ways to the first function, which isn't surprising. This time, however, array B is reduced in size with each recursion, as its lead element is passed through to the isMember() function, then discarded upon isMember() returning true. As before, the final case after the last recursive call to containedBy() passes through an empty list in place of B. This can only mean that every element of B successfully passed the membership test with A, so we return true to confirm that B is, indeed, contained by A.
A: One way is to turn a simple iterative algorithm -- with a nested loop -- into a recursive one.
An iterative solution, which is not optimised to use a map, but has a O(n²) time complexity, would look like this:
function includesValue(A, v):
for i = 0 to length(A) - 1:
if A[i] == v:
return true
return false
function includesArray(A, B):
for j = 0 to length(B) - 1:
if not includesValue(A, B[j]):
return false
return true
If you translate this into a recursive pattern, you could pass the current index as an extra argument:
function recIncludesValueAfter(A, v, i):
if i >= length(A):
return false
if A[i] == v:
return true
return recIncludesValuesAfter(A, v, i + 1)
function recIncludesSubArray(A, B, j):
if j >= length(B):
return true
if not recIncludesValueAfter(A, B[j], 0):
return false
return recIncludesSubArray(A, B, j + 1)
You would call it with the third argument as 0:
recIncludesSubArray(A, B, 0)
Each of the two recursive functions uses this pattern:
*
*The first if block corresponds to the end of the loop in the iterative version
*The second if block corresponds to the body of the for loop in the iterative version (including its potential break-out)
*The final recursive call corresponds to the launch of a next iteration in the iterative version.
Optimised by using a Map/Set
If you would need an optimised version, using a set (a map where only the key is important, not the value associated with it), then the iterative version would look like this:
function includesArray(A, B):
setA = new Set
for i = 0 to length(A):
setA.add(A[i])
for j = 0 to length(B) - 1:
if setA.has(B[j]):
return false
return true
Again, we can convert this to a recursive version:
function recAddSubArrayToSet(setA, B, j):
if j >= length(B):
return setA
setA.add(B[j])
return recAddSubArrayToSet(setA, B, j + 1)
function recSetIncludesSubArray(setA, B, j):
if j >= length(B):
return true
if not setA.has(B[j]):
return false
return recSetIncludesSubArray(A, B, j + 1)
function recIncludesSubArray(A, B):
setA = new empty Set
recAddSubArrayToSet(setA, B, 0)
return recSetIncludesSubArray(setA, B, 0)
About built-in functions
You wrote that built-in functions are not allowed. This is a constraint that only makes sense when you have a specific target programming language in mind. In pseudo code there is no concept of built-in functions.
Some languages will provide maps/sets/dictionaries in a way where you can only do something useful with them by calling methods on them (built-in), while other languages will allow you to apply operators (like +) on them, and use an in operator to test membership.
But even getting the size of an array may need a function call in some languages. So this constraint really only makes sense in the context of a specific programming language.
A: You can convert a loop to recursion by designing a recursive function/method that operates on a part of your original array (technically on a sub-array) (initially the sub-array will be your complete array) and reducing the size of the array (each time you are passing to your recursive method) by 1.
The following method (I've the implementation in Java) simply checks if the given number is present in the array/list. But notice that it also takes startIndex and endIndex which specifically denotes our boundaries of sub-array/sub-list.
In simple words the following method checks whether the given number is present in list or not but the check is done only between startIndex and endIndex both inclusive. Consider that you pass each element of your array B (listB in my case) to this method, and list argument is actually a reference to your array A (listA in my case).
/**
* This method recursively checks whether given
* number is contained in the given list or not.
*
* For this method startIndex and endIndex
* correspond to the indices of listA
*/
private static boolean contains(List<Integer> list, int startIndex, int endIndex, int number) {
if (startIndex == endIndex) {
return list.get(startIndex) == number;
} else if (startIndex < endIndex) {
return list.get(startIndex) == number || contains(list, startIndex + 1, endIndex, number);
}
// should never be the case
return true;
}
Now, once you have the above method, you can now device a recursive method that pics up all the elements of listB one at a time, and "plugs it in" inside the above method. This can be preciously done as follows:
/**
* This method recurse over each element of listB and checks
* whether the current element is contained in listA or not
*
* for this method startIndex and endIndex correspond to the
* indices of listB
*/
private static boolean contains(List<Integer> listA, List<Integer> listB, int startIndex, int endIndex) {
if (startIndex > endIndex) {
return true;
}
boolean c = contains(listA, 0, listA.size() - 1, listB.get(startIndex));
if (!c) {
return false;
}
return contains(listA, listB, startIndex + 1, endIndex);
}
And a call to above method will look like contains(listA, listB, 0, listB.size() - 1)
Bingo!! You are done.
I'd like you to think of recursive functions in a specific manner. Think of them as like what arguments it take, and what it does. So when you have a recursive call inside the recursive method, you don't need to think how that recursive call will work, rather have an abstraction there and believe that the recursive call will give you the result. Now focus on how you can use this returned result to make this recursive method correctly work.
| {
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I realize I only have a 5.1 setup (I don't plan on hooking up a rear center yet) but the 2 power outputs are listing 6 and 8 ohm outputs whereas the speaker is 4 ohm. I don't have the receiver yet (special ordered, apparently they're hard to come by as they are so new) but I'm wondering if there will be any problems/damage to the speakers or amplifier due to the lower impedance. Can anyone help me out or direct me somewhere that I can figure out what might happen? Thanks.
The Polk web site lists the RTi28 speakers as "Nominal 8 ohm impedance", and the CSi30 as "Nominal 8 ohm impedance"... when you download the PDF manual is says it is "Compatible with 8 ohm outputs"... However, the back of the actual printed manual for each of these products lists themselves as a "Nominal 4 ohm impedance." I tried calling Polk, and they weren't open today (Sunday) so I'll try calling again tomorrow, but does anyone else here have any more information than I about this?
i think the rt-28i & cs-30i are 8 ohms, the only polk you should worry about is the polk lsi line they are 4ohms.
so you should be alright, go ahead and hook them up and injoy. | {
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# The Secrets of My Success
**Table of Contents**
Acknowledgements
Introduction
Part I: My Boost journey
1 Blitzing in the blender of life
> Just a simple girl from a simple world
>
> First job, bad hair and many lessons
>
> The adventure that was supposed to last three months
>
> Sex, drugs and rock 'n' roll
2 Smoothly settling down (not!)
> No sex, drugs or rock 'n' roll
>
> Working with movie stars and finding my soulmate
>
> Dormant DNA
>
> Spinning my wheels
3 Holding all the apples
> Boost or bust
>
> Vision to fruition
>
> Making the decision to franchise
>
> Surrounding yourself with greatness
>
> Running at full tilt
>
> Risking it all
>
> Picking the right partner
>
> Adding a Boost to Viva
>
> Business Woman of the Year
4 Keeping the juices flowing
> Cracks in the foundation
>
> Proactive boosters
>
> Hitting the wall
>
> Building a Zoo
>
> Pausing to reflect
>
> The secret
Part II: Thirty recipes for success
5 Executive juices
> Integrity
>
> Motivation
>
> Passion
>
> Confidence
>
> Reputation
>
> Discipline
>
> Listener
>
> Solutions
>
> Honesty
>
> Partners
>
> Positive thoughts
>
> Negative thoughts
6 Staff smoothies
> Teamwork
>
> People
>
> Relationships
>
> Protégés
>
> Leaders
>
> Delegation
>
> Meetings
7 Power blends
> Assumptions
>
> Mentors
>
> Principles
>
> Respect
>
> Confrontation
>
> Money
>
> Negotiation
>
> Success
>
> Credibility
>
> Communicating
>
> Customers
Part III: Expanding skills and overcoming obstacles
8 Blending a great team
> Hiring the right ones
>
> Firing the wrong 'uns
9 Mixing up your marketing
> Why our marketing strategy worked
10 Picking the franchising fruit
> The Boost franchising method
11 Loving life
> There's more to life
>
>> Prioritise
>>
>> It takes two
>>
>> Get help
>>
>> Work out how much you need
>>
>> Enjoy extracurricular activities
>
> Taking care of yourself and your health
12 Throwing away the pulp
> Handling conflict with your colleagues
>
> Setting realistic expectations
>
> Learning to say no
>
> Respecting others
>
> Acting the part
>
> Eliminating the fear
>
> Avoiding burnout
Conclusion
The Boost Juice timeline
First published in 2013 by John Wiley & Sons Australia, Ltd 42 McDougall St, Milton Qld 4064
Office also in Melbourne
Typeset in 11/13.5 pt ITC Berkeley Oldstyle Std
© Allis Investments Pty Ltd 2013
The moral rights of the author have been asserted.
National Library of Australia Cataloguing-in-Publication data:
Author: Allis, Janine, 1965-
Title: The secrets of my success and the story of Boost Juice — juicy bits and all/Janine Allis.
ISBN: 9781118648179 (pbk.)
Notes: Includes index.
Subjects: Allis, Janine, 1965-
Success in business.
Businesswomen — Biography.
Women-owned business enterprises — Management.
Dewey Number: 658.4092082
All rights reserved. Except as permitted under the Australian Copyright Act 1968 (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above.
Cover design by saso content & design
Internal design by Peter Reardon, pipelinedesign.com.au
Internal images:
© iStock.com/Viktar;
© iStock.com/CGissemann;
© iStock.com/Natikka;
© iStock.com/photomaru;
© iStock.com/AlasdairJames;
© iStock.com/kosziv;
© iStock.com / Antagain
Every effort has been made to trace copyright holders and to obtain their permission for the use of any copyright material. Please contact the publisher if you are the copyright holder of any material used in this book which may have been incorrectly credited so we can correct any future reprints or editions
Printed in Australia by Ligare Book Printer
10 9 8 7 6 5 4 3 2 1
Disclaimer
The material in this publication is of the nature of general comment only, and does not represent professional advice. It is not intended to provide specific guidance for particular circumstances and it should not be relied on as the basis for any decision to take action or not take action on any matter which it covers. Readers should obtain professional advice where appropriate, before making any such decision. To the maximum extent permitted by law, the author and publisher disclaim all responsibility and liability to any person, arising directly or indirectly from any person taking or not taking action based on the information in this publication.
To my patient husband, Jeff, who does not always get the credit he deserves. To my mum, Joan, for her selfless commitment to me and my four beautiful children. And to those four very different, yet individually amazing children — thank you for your patience. Thank you for your forced flexibility, and sitting through more than a few dinner conversations with your Dad and me talking Boost! Without these loves of my life, nothing would be worthwhile. Thank you Jeff, Mum, Samuel, Oliver, Riley and Tahlia for making my life complete.
Acknowledgements
I have mentioned throughout this book that if you surround yourself with great people, great things happen; I am who I am because of the people I choose to be surrounded by. One of these people is my best friend, Amy Hart-Doering, who is an accomplished businesswoman and mother, and also had the challenging job of helping with the writing of this book. I thank her for her blunt honesty, humour, and her endless patience with my grammar and all the words that I made up along the way — remember I did go to Knox Secondary College.
Amy has the amazing ability to get the best out of me, she has a kind and compassionate heart, and I love having her in my life. This book would never have been completed without her bossiness, her timelines and her insistence that after every yoga session we did together (which is a lot because we do it five times a week!) we work to get this book just right. I love you and thank you.
Introduction
I walked into a book store last week and realised there really is a book on anything you could possibly be interested in. If you want to know how to speak to the dead, a book is available; if you want to make a cupcake, over 300 books are available on this subject alone (I checked online). There are over 10 000 business books printed each year — what you can learn from many very successful (and some not so successful) people is mind-boggling.
There are books on dealing with generation Y and on how to work less for more money, and hundreds of books with steps or particular units — like Covey's The 7 Habits of Highly Effective People or de Bono's Six Thinking Hats. There are many more 'how-to' books, such as How to Win Friends & Influence People and, one of my favourites, Good to Great. I also like the animal business books, like Eat That Frog, Who Says Elephants Can't Dance?, Purple Cow or (my favourite title) Swim with the Sharks without Being Eaten Alive.
So why on earth am I writing another book to add to the enormous number that is already out there on shelves and in e-stores? I can tell you one reason I'm not doing it for, and that's the money! As you will read in part I, I tried publishing with my husband, Jeff, and now know is not for the faint-hearted — unless you are a JK Rowling or a James Patterson, writing books is not a business that you get into for the money. (And I'm not holding my breath waiting for Universal Studios to pick up the phone and tell me they want to do my life story and that Julia Roberts has agreed to play me. Mind you, I think Andy MacDowell would do a better job.)
There are, however, three main reasons I'm writing this book.
The first is that I often get stopped by young women or men who have started or are about to start their business, and they tell me that I have inspired them to do so; that they have followed my journey, believe that they have a similar spark and now want to start on their own business journey.
The second reason is for me — when I started to write my story, it was the first time that I had actually stopped to look back and revisit the Boost Juice journey and how it all unfolded. Two of the most common questions I'm asked is whether I could have imagined that the business would grow to where it is today, and whether I knew when I started that it would become a real Australian icon. The honest answer to these questions is I never really thought about it, but I always believed that there was no other option but to succeed in creating a profitable business that I was proud of. The reality was that I was too busy growing the business and putting out the fires along the way to think about the business not being where it should be. I was (and still am) a terrible networker, and the isolation caused by this meant I didn't know whether what I was doing was better or worse than another businesses — I just assumed that it was worse, so I kept pushing forward. I also had people along the way who kept raising the bar, so I never really felt that I had arrived at the final destination. (When you have a friend like Geoff Harris, whose business turns over billions of dollars, your business always appears small.) Writing the book enabled me to relive my thoughts on each part of the journey and see the young scared woman who was winging it along the way. Through the writing, I was also able to watch her as she grew up. Writing the book has also enabled me to think back in other ways, and focus on the people who were with me along the journey. I can now look at these people with a more mature head on my shoulders, and understand some of what they did through their eyes — not the eyes of a scared, stressed young mum — and appreciate everything that they had to offer.
The third reason for writing this book is you — the person who is reading these words that I am typing on my laptop while in the background my young daughter is playing dress ups, my oldest is interrupting me by sending me texts saying he arrived in NZ safely (and no, he has no plans yet), my second son is also interrupting me by asking to go for a surf and my youngest boy has 'Call of Duty' turned up far too loud — all of which I would not have any other way.
The story on how Boost started may inspire you to make a change in your life, or you may simply find it an interesting read. The tips and insights through the second and third parts of the book are things I wish I had known before I started in business — all the mistakes I have made have cost money, often at a time when we could least afford it. Most of the tips and lessons learned are what I use in everyday life and all are what I do business by. Throughout the book, I've also included insights and extra detail from three people who were very important in my Boost journey: Jeff, my husband, Geoff Harris, co-founder of the hugely successful Flight Centre, and Kristie Piniuta, our in-house lawyer for many years and now with her own legal practice (Kubed Legal). These add another perspective to my journey.
And (as the title promises) there truly are some of my secrets in this book — there are many things in the book that I have not written about before or that are not in any interview you may have read about me.
Or the book may be one of those that you pick up, read a bit and put down to be finished at another time. I know I have at least five books on the go at any one stage, and usually an audio book as well.
Whatever your reasons for reading this book, I hope you enjoy my journey, and I will feel I have done my job if you do take something away from the tips and lessons learned. I hope through Boost Juice bars and our other businesses that we can continue to grow and produce products that you love. And I truly wish you as much luck as I had in creating my business that I love.
And remember — love life!
Part I
My Boost journey
When you ask many successful businesspeople how they got started, they may tell you about the little businesses they started in primary school — the ingenious corner lemonade stand, the school candy-bar sales or the lawn-mowing service employing other 12 year olds. The types of businesses that led these overachievers to climb that first rung on the ladder to success. Sales charts, forecasts and ROI calculations lined these kids' walls like Andy Gibb lined mine. The entrepreneurial spirit seems to be part of their DNA.
My story is drastically different. You could say that my entrepreneurial spirit was ... umm — dormant. Okay, it was non-existent. To be honest, if you'd asked me what an entrepreneur was in primary school, I may have thought it had something to do with food and would have had no idea how to spell it. It was 20 years after primary school that foreign entrepreneurial DNA somehow began to morph my behaviour.
During the 20-year, pre-DNA takeover, I travelled around the United States, Europe, parts of Asia and Australia. I had 30 jobs, got fired from some, moved on to others, made money, lost more, met the wrong man, had a beautiful baby and met my soulmate. What I didn't realise at the time was all the lessons and tools I was picking up with each triumph and pitfall. Each piece of my journey was enabling me to have the strength to take a tiny idea and turn it into a passion.
Of course, I don't really think having the skills to become a successful entrepreneur literally needs to part of your DNA. I also don't believe there is a cookie-cutter process for success, or that success has to be hard or come easy. What I have attempted to do in this and the following chapters is to share with you my journey; it has many ups and equally as many downs. (You can also get a summary of the main events in the Boost journey from the timeline.) If someone had done the same for me, perhaps the learning curve would have been less bumpy. This is a short, honest glimpse into my archives so that you can see I'm human, just like you. I too trip over the kids' toys, go to work with my children's fears and problems running through my head, laugh, cry, make mistakes, learn from them and try to grow.
I hope that you take something from the following and follow your dreams.
Chapter 1
Blitzing in the blender of life
Growing up, I was a typical suburban kid. My passion was netball and I spent as much time as possible outside throwing a ball at the brick wall in our garden. After leaving technical college, my first job was in advertising — during the 1980s (think shoulder pads, big hair and liquid lunches) — and even gave modelling a go. Sensing there was more to life, I worked three jobs to save for a travelling adventure. Telling my mother I would be away for three months, I set off — returning six years later with a two year old.
Just a simple girl from a simple world
I once read a book that suggested we actually 'pick' our parents. If that's the case, I picked the quintessential Aussie Mum and Dad. Mum stayed home and Dad made the bacon. Dad worked for Fibremakers, a carpet-making company, in a management position. His aim was to move up the corporate ladder during the week and enjoy his time off on the weekends.
I'm the youngest of their four kids, born in Knoxfield, about 30 kilometres east of the Melbourne CBD. Back in the 1970s, the suburb was semirural. Our home was a tiny green weatherboard house — only 10 squares — but it was set on a quarter-acre block of land that had previously been an orchard. It was full of fruit trees. (Uhmm... perhaps the love of fruit started here?) We were outside children by necessity. Weekends were spent at the football oval for my brother, Greg, or the netball courts for my sisters and me. Our family was obsessed with sport. Netball was the one thing I was truly interested in during those years. I played and trained six days a week (even as an adult, I played netball until I stumbled into yoga at 41). Okay — healthy living and a bit of obsessiveness started to shine through during my childhood, but the availability of fruit and overachieving netball skills do not a businesswoman make.
Jeff says...
Janine always had spunk and threw herself into any challenge with great zeal. The best sign of what was to come was watching her on the netball court; there was no lack of vigour, for a fierce contest was within this woman.
My childhood was relatively uneventful; I and my siblings were much loved, and it was a stable upbringing. Life was simple, with not too much money being left over after the expenses were paid, so everything we did have was appreciated. I remember as a child the joy of seeing black and white television for the first time. I also remember going to the movies and watching that huge man on a horse, telling everyone how good for you it was to smoke Alpine Cigarettes — as opposed to the other horrible, unhealthy cigarettes. I wasn't sold on the habit of smoking but, on the big movie screen, I did notice the vibrant green of the grass, so when I returned to the black and white TV, I made a point of telling my whole family what colours we were missing.
Holidays were 8-hour road trips to Robe, in a car without air conditioning or seatbelts. For Christmas one year, I got a bike that was second-hand with a damaged seat. (Mum told me Santa had damaged it on the way down the chimney and, of course, I believed every word because I knew Santa existed.) Looking back at my childhood, my memories are happy ones; my parents ensured we never felt like we missed out.
Even though my parents were encouraging of anything and everything we did, their aspirations for my siblings and me were minimal. Neither thought that someday we would own our own business, become a lawyer or even a doctor. This had nothing to do with not believing in us, and everything to do with expectations and our environment. My parents sent me to Knox Secondary College for two reasons: it was close to home and it had a business course. In our neighbourhood, you completed your Leaving Certificate and then you got a really good job as a secretary, preferably in a bank. My school only went to year 11; my parents had no expectations that I would finish or go to university. In fact, it was never discussed. Being the youngest, I could slip through the cracks. I was never the class clown or class dunce; I was smack in the middle — Miss Average. I never pushed myself too hard and rarely did my homework. How is that for dormant entrepreneurial DNA? I seemed to be always thinking, 'What is the point to all of this?' In contrast, my older sisters, Rae and Lisa, were diligent, smart students. Not seeming to match them in potential or politeness, I was a bitter disappointment to the teachers who had taught my sisters prior to me.
My school was a technical college, focusing on practical skills like woodwork, typing, basic bookkeeping, graphics and metal work. As a result, I can type, build a solid birdhouse and do basic drafting, and I'm very handy with a soldering iron. But don't ask me the capital of Azerbaijan or where the country is located on a map!
At the age of sixteen years and ten months, I left tech school and could type 100 words per minute. At the time, I didn't realise that this was probably the most useful skill I had learned; everyone on Earth was about to switch to computers. I could also handle very basic bookkeeping, which would serve me well later when Boost was without a CFO. The technical drawing class came in handy when building the birdhouse, but also when designing the first Boost Juice stores. You never know what subjects are going to be helpful in the future.
When I left, my mother made me sit for the Commonwealth Bank test. She thought working in a bank would be the perfect job for me; I could think of nothing worse. My parents' plan for me was to finish school, get a good stable job, marry well and live happily ever after. God forbid you not having a child by the time you were 21 (this was Mum's expiration date for starting a family). All I wanted was an adventure. But, to please Mum, I attended the Commonwealth Bank test to see if I could get a job. I doodled my way through the test and I didn't get the job (surprise, surprise).
I would like to be able to say that it was during this time that a wise teacher saw the flicker of an entrepreneurial spirit in me and encouraged me to think higher, but I would be making it up. My childhood was loving, yet simple. I was happy, but somewhere buried deep within, I knew there was bigger a point to this, that there was more to life. I just needed to figure out where more was.
First job, bad hair and many lessons
After turning my back on a safe bank job, I managed to get a job in advertising. My sister Rae was working for a huge ad agency at the time and she recommended I go to the employment agency she used to get her job. So, in I went, even though I had absolutely no experience. The woman I met with told me she thought she had the perfect job, and with a quick phone call she'd arranged an interview (telling my future boss I was a 'freebie' for him and that she thought I would be perfect, even though I was a bit green). After a ten-minute interview, and answering the question on whether I made good coffee ('Absolutely!'), I got my first job.
I was a very junior, junior (did I mention I was junior?) media assistant at an advertising agency. Advertising in the '80s was all about short skirts, bad hair and long boozy lunches. Each Friday, lunch started at noon and ended at 5 pm. For a while, the fun in advertising significantly outweighed the boredom of my first job. (And it was a very dull job, mostly just typing little numbers into little squares, which, to be honest, after many liquid lunches, was a challenge.)
The ad agency was very advanced and had some nifty devices to help me out. They had these boxlike things called 'Apple computers' that allowed me to do a spell check (after coming from Knox Secondary College, I thought all my dreams had come true). Three months after I started, they also purchased a brand-new machine where you could insert a photo (or whatever) in one end, and it would print out on a similar machine somewhere else. (If it was a photo, it would print out a bit grainy, but if you looked really hard you could see what it was.) They called this machine a 'fax'. Still, the spirit within wanted more.
One of the many terrific things my mother did was to continually tell her daughters how beautiful they were. Personally, I think a degree of rose-coloured glasses was involved when she looked at us, but it was always nice to hear. While I was at school, I completed a Suzan Johnston modelling course, like my sisters had before me. Twelve months into my new job at the agency, the people who ran the course called and asked if I wanted to audition for a job promoting Australian-made products. The promotion was to be government-funded and they wanted one girl from every state. Never one to die wondering, I went to the audition — and, to my surprise, was given the role of the Victorian model. So I handed in my resignation and off I went to Brisbane to start my very short-lived stab at modelling. After settling in to Brisbane and meeting all the girls from each state we started our 'training'. Unfortunately, however, after about three weeks we heard the government had decided not to go ahead with the promotion — and I found myself out of a job.
Still, with the confidence I gained after getting the role, I thought Why not try modelling more seriously? I had some photos taken and did the advertising rounds with my new photo book. It became fairly clear fairly quickly that my mother's view and reality did not quite match. Tall and thin I was; Elle McPherson I was not. However, I did land the in-house modelling job at Adidas and made a few front covers — admittedly not the cover of Vogue; more like Greyhound News and CB Action magazine. In the end, modelling was not for me — a fact cemented after an appearance on The Bert Newton Show. I was modelling the new Olympic Uniforms and went in the complete opposite direction to everyone else, tried to turn, tripped and fell. Not my finest moment and the end of a very short modelling career.
Next, it was back to the wheel of advertising for me with a job as an account coordinator. Multiple lessons were learned in this place. One senior male had octopus arms, which he used for big hugs and touches. When I complained to one of the bosses, I was told that I just had to put up with it (got to love the 1980s — see lesson 1 at the end of this chapter). The same male spent absolutely no time teaching me anything and kept everything regarding his work to himself. When he was sacked, I was given his accounts to run (Johnson Tiles and the SEC) and found myself way out of my depth. I tried my best to swim, but I simply did not have the experience or knowledge to do an effective job. In the end, the agency lost the accounts and I lost my job (see lesson 2).
So there I was — 20 years old and jobless — when my friend Deborah asked me if I wanted to go travelling around the world with her. That was it. That was exactly what I wanted to do! I said I would join her, but the pull of a good party and buying new clothes meant I had very little money saved. When she packed her bags and took off without me, I knew that I had missed out. It was time to get serious so I could investigate my deep-seated knowledge that there was more. I started to work at night for two clubs. One was called The Chevron. If you're from Melbourne and over the age of 40, you probably remember that this was the hippest place to be — and I most likely checked your ID. I was hired as 'The Door Bitch' (a term that was not always affectionate). The nightclub life was an eye-opener for a girl from the 'burbs. I saw all sorts of things: girls being taken out the back for a quickie, drugs and gangsters. I worked six nights a week at these clubs and got a job at a little advertising agency during the day. I was too busy working to spend any money, so rather quickly I had saved enough to start my travelling. During this period I was so determined, most nights I worked until 2 am. I remember driving home thinking that if I drove in the centre lane, I might wake up before I hit anything. Young people can be dumb and, once again, I was no exception. (I can only hope my own children are wiser than I was.)
The adventure that was supposed to last three months
With a blue backpack, $6000, a plane ticket and a determined look, I set out on my own. I can still see Mum's bewildered face as I kissed her goodbye at the airport. To this day, she still complains that I didn't turn around to wave goodbye like all the rest of the travellers; my sights were firmly set on the future. I was off to Marine County, San Francisco, to work as a camp counsellor during the American summer.
The camp was for children of different backgrounds, some with health challenges. Many were deaf and in one of the sessions all the children were blind. At the camp I taught the kids about trees and nature, and how to swim, make candles and light a camp fire. It was a wonderful camp. Not only did I learn a bit of American Sign Language, but I also learned patience and appreciation for what I had as I watched these children with extreme physical challenges overcome daily obstacles.
When the camp ended, I travelled with some of the camp counsellors I had befriended. We travelled up and down the California coast, hiked the Grand Canyon, sat by Lake Tahoe and eventually ended up in New York. From there we flew to London. I found the city a bit too depressing — grey skies, little houses and lots of rain. I contacted an agency and quickly scored a job as a nanny in Le Château-Cambrésis, a little village in France, about an hour from Paris. It was the birthplace of Matisse and the site of much fighting during World War I.
I arrived in the village and couldn't find anyone who spoke English, except the woman I worked for. She was, not to put too fine a point on it, a cow. I was hired to look after her three children and ended up in the basement doing all the ironing and most of the cleaning. She wouldn't talk to me for days on end; at other times, she would shout at me for mispronouncing the little French I knew. The kids were lovely — or at least I think they were. They spoke no English and I no French; perhaps they actually said awful things to me. I will never know. Overall, it was a horrible situation, but at the time I couldn't see many alternatives. All I could think was that I should give it my best shot. And sticking with the job was good grounding in finding solutions to problems; when you travel you have to rely on your own resources.
I had been playing Cinderella for the evil French woman for a few months when a friend from Oz called me. She was visiting her father in Munich for two weeks and invited me to meet up with her there. I was so miserable in France, I jumped at the chance. Days later I was in Munich with the one line of German I knew, 'eins Weißbier, bitte' (which literally translates as 'one white beer, please'). A much-needed and well-used phrase when travelling through Germany. With the same friend, I travelled on to Denmark and that's where I spent my first Christmas away from home. In Australia, my family celebrates Christmas on Christmas Day; Mum makes a big Christmas lunch and we all sit around eating and opening presents. Denmark celebrates on Christmas Eve, so my Christmas lunch there was in a local hotel eating a sandwich and drinking a beer. Even the white Christmas didn't lighten my mood — I'd been travelling for nearly a year and I was starting to miss home.
After Denmark money was running low. My friend and I heard there was work in the Canary Islands selling time share, so we made our way to an island called Tenerife. Tenerife was a major tourist attraction for the English; its beaches had velvet-soft, black sand attributed to the local volcano, which not everybody thought was a good thing. (Two years before the time I was there, the council thought having white sand would help tourism and dumped 200 tonnes on the beaches. Within 48 hours, the white beaches turned back into the black sandy beaches they were meant to be.) My job on the island was to get tourists to visit the timeshare resorts that were popping up everywhere. One of the many downfalls to the job was 'promoting' was considered illegal. 'Illegal' in the Canary Islands was a grey area as far as I could tell. As long as the police were making money off the promoters, they turned a blind eye to the dozen or so on each corner. This is how the system worked: a police officer would issue an 'on the spot fine' to the promoter, the promoter would give the police cash, the police would then give the promoter a receipt, the promoter would then take the receipt to their boss to be reimbursed for the 'fine'.
This all appeared to be a viable way to earn money, until my friend, who was now also my flatmate, revealed her dodgy side. She and an equally dodgy policeman decided she would purchase a receipt book off him. My flatmate then used the receipts to fraudulently claim reimbursement from the timeshare company. In addition to the scam, my flatmate used the money to buy drugs, thus leaving herself with no money to pay off the policeman for his part in the arrangement. My problem wasn't that I did anything wrong, it was that I was associated with her. I knew over a dozen English friends who were arrested and held for months without charge for simply being in the vicinity of a pub fight. In a country where corruption exists, association often means guilt; I was in as much hot water as my flatmate. I was facing the same fate as her if she didn't come up with the money 'owed' to the police.
The straw that broke the camel's back was when my flatmate asked me to leave the front door open because she had lost her key. I awoke at 2 am to find a six-foot-five security guard standing next to my bed complete with baton, handcuff and gun. One of his hands was heading under my sheet and the other was undoing his pants. You never know how you're going to react in these situations. Strangely what went through my mind at that moment was not fear — it was pure fury! Who does he think he is?, How dare he touch me!, Oh my God, is that a gun? — these were the outraged thoughts that were running through my head. Making a split-second decision, I yelled, 'GET OUT!' To my surprise, he did. I kept yelling and he backed away saying something in Spanish as I stood at the front door. I slammed the door shut, returned to bed, and slept. Thinking back, I can't believe that was my reaction. If asked, I would have assumed that I would be a dribbling mess at such a frightening close call. However, the next morning the full extent of what might have happened sunk in.
After that night, with only a few days until my flatmate's debts were to be paid back to the police, I decided it was time to leave. My flatmate decided to come with me. After living in Tenerife for four months, we hitched a ride with friends on a catamaran heading for Portugal. Ten days later, we found ourselves in The Algarve in southern Portugal. I was funding both myself and my friend, who kept promising she would pay me back, but never did. In The Algarve we came across some fellow backpackers who had just returned from the south of France. They had been working on yachts for the rich and famous and their stories convinced me this seemed like the direction to head, so I packed my bags and headed back to France, alone.
So, I'm in Antibes, France, with $40 to my name, no ticket home and $2000 in credit card debt. (I had cashed in my return airline ticket months ago.) Yet, interestingly, I wasn't the slightest bit concerned. Was it the arrogance of youth or perhaps I knew I would figure it out? I'm not sure — but I do know, if it was today, I would be having heart palpitations. But in 1985, I just knew all would be fine.
The South of France was magical, complete with cobblestone lanes and old men playing boules in the parks. Restaurants and cafes spilt out onto the streets and dogs sat at tables like people, eating off china plates. The quays were full of large white palace-like boats. I was off on another adventure.
At the local pub, an Englishman informed me there was a job on a boat called the Deneb Star, based in Villefranche-sur-Mer, near the border with Italy and a 20-minute train ride from Antibes. After a couple of phone calls (from a pay phone), I got an interview. I was wearing the only nice outfit I had, which just happened to be a woollen jacket with a matching woollen mini skirt. It was the middle of summer and 30°C. Unbeknown to me, the train that I hopped on was an express train to Italy (and remember — this was before the days of the EU). With no passport and no fluency in Italian, I had to convince the Italian border guards that I simply needed to get back across the border to my appointment. Many hand gestures later, I was back on the train and off to my interview.
I arrived in the beautiful village of Villefranche-sur-Mer. I had a moment of bliss, soaking up the surroundings; then I realised I had an hour's walk in my woollens to the quay where the boat was berthed. The bliss turned to big drops of sweat and throbbing feet. Miraculously, I arrived on time, dripping in sweat from head to toe, to meet the captain. I'm pretty sure he didn't offer me the job because he felt sorry for me in my ridiculous attire and with my red, sweaty face. I believe it just may have been the tiny, white porky pie that came blurting out of my mouth: 'I have enormous yacht experience. I'm from Melbourne!' Suddenly, my money troubles were over. I now had accommodation, food and a job as head stewardess, all in one fell swoop. And after all, I was from Melbourne, and I had seen plenty of yachts.
The boat was 74 feet. Think of a three-storey house with four bedrooms, a guest area and a further four bedrooms for crew. Now think of a cupboard — that's the cabin I shared with the other stewardess. The space in the cabin was about 1.5 × 2.5 metres and it was at the front of the yacht, so it was pointy in shape. It had a bunk bed about half the size of a normal single bed and the ceiling height was about 2 metres. And we had to share the tiniest wardrobe you have probably ever seen. Despite traveling in a cupboard, I was in heaven — I was on the French Riviera, cruising in a multimillion-dollar yacht. I had gone from dodging police, a potential rapist and a drug-addicted flatmate to floating in paradise.
Sex, drugs and rock 'n' roll
Six weeks after I started on the Deneb Star, David Bowie (yes, the David Bowie) bought the yacht. I was sailing the Mediterranean on the luxurious boat of a bona fide celebrity. Bowie was an amazing, down-to-earth, great bloke. He spent an enormous amount of time with the crew and we were very much part of his 'gang'. He took us to parties and was generous with his time. We cruised with him and many other rich and famous people to such events as the Cannes Film Festival and Monaco Grand Prix, and across the Atlantic to the Caribbean. We even stayed in his Bali-inspired house on Mustique Island in the West Indies.
David travelled with an entourage that included his financial adviser, Bruce Dunbar, his son, Joe, Joe's nanny, David's girlfriend and a couple of others. Just to name-drop, here are a few passengers who came on board: Robin Williams, Mick Jagger, Eric Idle and Michael Caine. This time in David's life was family time; there were actually no drunken parties, drugs or sex. There was, however, a lot of rock and roll. At the time, Bowie was starting a new band called Tin Machine, which meant a great deal of time was spent practising. During the day-to-day routine of life on the yacht, I would honestly forget that he was the David Bowie. Though one day he was warming up with Space Oddity and my mouth just dropped. I then said out loud, to no-one in particular, 'That is David Bowie!' (For the record, Bowie was a beautiful person who kept his feet on the ground. And if you want to know why Bowie has two differently coloured eyes, it's because he and his best friend were in a fight at school and the damaged eye was the result.)
Working on David's boat sounds glamorous, and at times it was, but it was also really hard work. We would have back-to-back charters for four months, which meant that you worked those months without a break. I needed to be available 24 hours a day and the job involved everything from cleaning silverware and the toilet to organising the helicopter to take guests out to dinner. While it was very glamorous to fall asleep in France and wake up in Monaco, the sea sickness was not. At times you wished someone would throw you overboard. And I won't go into the gory details regarding a very large man who managed to destroy the toilet, leaving whatever had just left his body all over the walls and roof — aargh, not glamorous.
Some of the wealthiest people in the world hired the yacht, and I wasn't too sure what to expect from them when I first started. I knew from my upbringing that people with money were 'not us'. My Gran experienced the Great Depression and worked as a cleaner — in her mind, if we got a job at Myer, we were doing exceptionally well. (Years later, when Boost started to get off the ground, she couldn't get her head around her granddaughter running the business. Gran was convinced that the part-time bookkeeper was the woman I worked for.)
But meeting the rich and famous was great fun and a significant learning experience, especially about people (see lesson 3 at the end of this chapter). Most people who came on the yacht were lovely, like David Bowie; others thought they were superior to the rest of the human race.
On board, we had guests whose attitude ranged from 'show us where the fridge is and leave us alone', to those who would send a boiled egg back because it was too hot. We once had a group of Americans on board and their kids were obnoxious (see lesson 4). They thought they were better than everyone and treated all the staff like dirt. On the flip side, we had one of the wealthiest men in Kuwait as a guest, and his son was a lovely young man. The father asked me to type up a list of expenses for his son who was off to college in the United States. I was expecting to read that his son was allowed a fortune. To my surprise, his expenses were moderate. In fact, for the son to survive, he would have to get a part-time job (see lesson 5).
Jeff says...
When I reflect on what characteristics Janine showed early on, the biggest one is being a great problem-solver. She travelled around the world with tuppence in the bank, she was a mum at 25, and she didn't whinge — she just got on with it. She is a real can-doer.
There is no doubt we were attracted to each other through our drive to succeed at whatever we were passionate about. Early in my life, I was passionate about assets, so I bought my first house at 19. Janine was passionate about travel, so she circumnavigated the world on a rock star's boat (slight exaggeration but within the realm of reality!).
After two years and a great deal of fun, I left the Deneb Star. I was seeing the engineer on the yacht at the time and we both left to work on another yacht with him as captain (this yacht was anchored in Monaco). We purchased a property in Valbonne, a lovely village just outside of Antibes, paying way too much for the house because we had no idea what we were doing — and it didn't help that my French was far from perfect. A few months later, I found out that I was pregnant. Sadly, I realised that I wasn't in love with this engineer; I knew that he was not my future. Though the pregnancy was not planned, I gave it a couple of years to see if I could learn to love him. But he just was not 'the one', so we discussed it and I told him that it was time for my son and me to leave. It was as amicable a separation as you could possibly want. We had a beautiful friendship and he is a lovely man; he was just not my man.
In 1993, I turned to the first love of my life, my two-year-old son, Samuel, took his hand in mine and headed back to Australia. It took me 35 hours of travel and I had nothing but the clothes in our suitcases. Financially, the house we had purchased was not worth what we paid for it, leaving me without a cent to my name. My dear friends in France lent me the money to return to Australia. I felt like a failure — I was 27 years old, going home to live with my parents until I got myself back on my feet.
The story so far... lessons learned
Travelling teaches you skills you don't learn anywhere else, or not as quickly anyway. I pride myself on my problem-solving skills and I put that down, in part, to the travelling that I did. You learn to look a little way into the future and see danger before it hits — an invaluable tool in business. When you have only yourself to rely on, you learn to trust your instincts and find resources within yourself you never knew existed.
Here are some specific lessons I learned while in my early twenties:
1 You don't have to put up with upsetting or unlawful behaviour at your workplace, and you shouldn't.
2 Do not set your teams up to fail, even if they think that they're ready. The worst thing you can do is promote too early.
3 All people, even the rich and famous, are just people with their own fears, dramas, happiness and sorrows.
4 You can make kids revolting if you spoil them.
5 Some people seem to think that because they have money they need to act in a certain way, like a young boy going through puberty trying to act like a man. Similarly, some women think that to be successful in business they need to act like a man. The most important thing: be yourself.
Chapter 2
Smoothly settling down (not!)
Back in Australia after my travels, and feeling like a failure, it seemed the party was well and truly over. Finding a secure job and supporting my son was now my biggest priority, even if I had to finesse my CV a little to come up with relevant skills. Sink or swim? I swam like crazy. Working hard and raising my son, I didn't have much time for love — until a well-meaning friend took up the matchmaker role and introduced me to Jeff.
Not only did we fall in love, but Jeff also unlocked the entrepreneurial spirit within me.
No sex, drugs or rock 'n' roll
During my time on David Bowie's boat, I met a film producer named David Puttnam (referred to now as Lord David Puttnam, one of the many films he produced was Chariots of Fire). At the time he was a director on the board of Village Roadshow. He told me that he knew Graham Burke at Village and that if I ever needed a job to contact him and he would set up an interview with Graham. Little did I know that Mr Burke was the CEO!
Regardless, I did get an interview with Graham and he was delightful — and I found myself with a job as a junior manager at Village Cinemas Knox City, not far from where I grew up. At the time, it did not occur to me to question why on earth they hired me as a manager — I obviously had no management experience. I did think it was strange that they didn't read my beautifully presented CV, or notice that I may have exaggerated the emphasis on my 'leadership' skills. I took the job happily and worked my butt off. I owed it to David Puttnam and Graham to prove that I was worth the punt (see lesson 1 at the end of this chapter).
(By the way, years later at a basketball match with Graham and his wife, I asked why they had hired me back then. Graham laughed and said at the time he thought I was sleeping with David Puttnam and that he was doing David a favour by hiring me! For the record, I never slept with David Puttnam. It did, however, finally make perfect sense as to why they had hired me with no management experience.)
But even though I had never been a manager of anything before, as it turned out, I was good at it. At Village Cinemas Knox City, I worked with a small management team of three. I ran the marketing, a woman named Robyn headed up accounts, and Sylvan was operational. Between us, the cinema did exceptionally well. We did so well that after six months I was transferred to run my own cinema in Frankston. That was a real eye-opener — the cinema was dark and smelly, and the curtains were infested with spiders. It was a challenge to say the least; completely unloved when I took it over, the cinema could not have been in worse shape. My first priority was to clean the place up; after that, there were bigger issues to tackle.
Spiders aside, one of the scariest aspects of the job was the accounting system. At the nice, new, shiny cinema I was used to, everything was automated. I could push a button and the accounts would magically appear. When I got to Frankston, I didn't even get a handover. I was presented with a key to the front door, a manual ledger and that was it — I had to just figure it out. Sink or swim? I decided to see it as a fantastic learning experience. Again, I got to work and within four months the cinemas turned a profit for the first time in years; it was exhilarating.
The Frankston cinema was an excellent development ground for my marketing skills because I was so unconstrained there. I could try pretty much anything, and I did. I set up a movie club, sent a staff member out each week to put up as many posters as possible, used promotional material to create competitions and established loyalty programs. It doesn't sound too extraordinary now, but at the time no-one else was doing it, so it set us apart. It was like running my own small business in a regional area. Instead of seeking permission to do things, I simply went ahead and did them. There was no assistance, no manuals, no occupational health and safety policy — absolutely nothing. It was challenging but definitely rewarding.
I was, however, on a very minimal salary. I had recently bought a tiny house in Ringwood East (very tiny — it was built behind another house), borrowing money from the bank to do so. I did my sums and, on my salary, I could just afford the mortgage — and it would only take me a mere 25 years to pay off. After completing my budget, I discovered if I was very tight with my money, I could save $50 per month.
The house was close to my mum's house, so I could drop Samuel off in the morning and then make my way to work. With driving an hour each way to and from work, the time spent with my son during the week was a quick morning rush and a cuddle at night — thank goodness for weekends. So along with the long hours came the guilt. It was a new era for me. Gone was the freedom of letting life take me wherever it wanted. I was now responsible for another human being and the weight of this responsibility was never far from my thoughts.
I'd worked for 14 months with Village in Australia when an opportunity arose in Singapore to assist in growing the cinemas there. With my three year old under my arm and a bewildered look on my mother's face, I went to Singapore to start another adventure. When I returned home 12 months later, I was a basket case. I was burned out to a crisp. So what went wrong?
It turned out that the standard working week was six days and I worked between nine and 12 hours a day. Given my work schedule, one of the biggest challenges I faced was finding suitable care for Samuel during the day. After hearing horror stories about some of the local nannies, I ended up ringing my cousin Rachel, who was 19 at the time. The company flew her over from Australia to be my son's nanny. Even with her there, I was still doing two jobs — working for Village and raising my son, Samuel, without the support of other extended family like my mum.
I also hadn't done my research about my finances. I was so flattered by the opportunity, I didn't realise that I would be even more financially constrained living in Singapore than in Melbourne. The stress and the hours simply took its toll; I became an emotional wreck. Also, I wasn't prepared for the isolation I felt in Singapore. The expat community can be a wonderful support network, or it can make a place feel like the smallest town in the world. Everyone knows your business and feels they have a right to an opinion on you.
It was tough but, having said all that, at the same time it was exciting doing business in another country with all the differences in cultures. And Singapore taught me an enormous amount and was a great grounding for my future with Boost. Often in business and life the lessons you learn from your negative experiences have more of an impact than the positive experiences. Take, for example, my direct boss in Singapore, who was not as warm and welcoming as she could have been. Or my senior boss who, upon first meeting, said my shirt was inappropriate for the workplace. We were making massive improvements and increasing profit, yet his only comment was a derogatory one about my choice of clothing (which was, by the way, a business-style, sleeveless shirt). I vowed that day I would never judge people for what they wear, only what they can deliver to the business (see lesson 2).
Through networking in Singapore I landed a job back in Melbourne, as a publicist with United International Pictures (UIP). I wanted a role where I wouldn't have to work nights and could have my weekends back to spend with Samuel. I'd never had a job in public relations — like all the jobs I'd had thus far, I wasn't qualified for this one either. However, my marketing background was strong and my portfolio of promotions work showed the UIP interviewers that I had the necessary skill set, even if I'd never had the title (see lesson 3). So, one year after moving to Singapore, Samuel and I returned to Melbourne. This was a great time to be at UIP and, as on David Bowie's yacht, I was once again surrounded by movie stars.
Working with movie stars and finding my soulmate
During my years at UIP, I was indeed surrounded by movie stars, but my professional life took something of a backseat because it was overshadowed by a big shift in my personal life. In 1995, I met Jeff Allis. We had been set up by my girlfriend Rachel, who arranged for us to meet at the Melbourne Skyshow. I would certainly not say it was love at first sight. Jeff was late and by the time he arrived I had somewhere else to be; it was one of those days. And I thought he had bad teeth and an attitude to match. Jeff remembers not liking the jeans I was wearing. He also thought I would have been better looking from the description my friend gave him. (In all fairness, my friend told him I looked like Elle McPherson! Jeff told me later, I was attractive, but no Elle. You will never die wondering what Jeff is thinking.) We said hello and went our separate ways. And that was it — or so I thought.
After the failed 'date' my girlfriend continued her campaign about how terrific she thought Jeff was. She kept talking about how great we would be together and, after about ten days, I caved in and called him. Jeff hadn't really impressed me, but my friend was nagging me and I figured I had nothing to lose. At the time, I was working on the promotion of the movie Rob Roy and Jeff was Program Director for the Austereo Network's Fox FM, so I rang him on the pretext of picking his brains about publicity opportunities. We arranged to go to dinner that week, but he rang the morning of our date and cancelled, telling me something about Adelaide and a sister giving birth. What was he thinking? I was this man's future, for heaven's sake! He didn't even reschedule when he returned. He was not exactly giving off keen signals, and I was wondering whether I should just move on. I have always believed in the idea of a soulmate and I did not intend to settle for anything less. Jeff finally got around to ringing me back, however, and rearranging our first date. And that's when things clicked. Though it was not love at first sight, it was love on first date. Conversation flowed, we laughed easily and we realised we both had the spirit of adventure. I remember thinking I really, really like this guy. I'm not sure if I can call our getting together destiny, but a few dates later when we had our first kiss, the earth did move and I did see fireworks. Even today we talk about that first kiss. After our dinner we saw each other every day for six months.
It was not all smooth sailing dating Jeff. When I was publicising the movie Clueless, starring Alicia Silverstone, she took a few of us and our partners out to dinner to say thank you. Alicia was touring and promoting the movie with her mum, and was a real delight to be around. Arriving at Alicia's dinner, I could see that Jeff was in a 'do I really have to be here' mood as he began to sip on the champagne on offer. We were taken to our seats and Jeff was seated next to Alicia with me opposite him. Polite conversation started and all was going pretty well when Jeff turned to Alicia and asked her what other films she had been in. She politely told Jeff that the last movie was Excess Baggage. Jeff was excited because he had seen the movie. I looked across at him thinking that his mood had improved and all would be fine. Unfortunately, we had not been dating that long and I didn't quite get the read right. Jeff told Alicia it was the worst movie he had ever seen and the only movie that he had ever walked out of! If a pin had dropped, we would have all heard it; the whole table fell silent. All heads turned quickly to Jeff. I jumped in and said, 'Oh, he didn't mean because of your acting,' (uncomfortable giggle). 'It was the plot he didn't like.'
Jeff, as Program Director at Austereo, was used to securing celebrities for promotions and programs, and so was also used to dealing with some of the very high egos that exist in radio. In other words, he should have known better. When I was on David Bowie's boat, I learned very early exactly how delicate the ego of artists is. Once I said to David that I loved him in the 1983 movie The Hunger, telling him the make-up was amazing. He quickly turned, looked pointedly at me and said, 'It was, in fact, the acting that was convincing'. As he turned away, I realised what a career-limiting comment was.
Having no idea that he had just insulted our generous hostess, Jeff continued to open his mouth and insert his foot. He candidly said he had not even seen her new movie Clueless. Alicia's mother turned to Jeff and politely, but in an ice-cold voice, said, 'You would not like it', and turned away. Needless to say, our discussion on the way home was not pleasant. Over time Jeff apologised for this night. And, like all understanding spouses, I bring this story up as much as possible, ensuring his punishment continues well into our marriage.
Another story I like to bring up is one that displays the 'whatever it takes' attitude Jeff had during his radio days. While he was working for Austereo's Fox FM, everyone there was at war with the rival station Triple M — it was a ratings war, and, in their minds, it truly was rate or die. Jeff and his colleague Sean had heard that Triple M was launching its new major promotion for the year and that all the top advertising agencies had been invited to the launch. Triple M was throwing a massive party with no expense spared, and with all the personalities, glitz and glamour that only the 1980s could provide.
Jeff and Sean knew they had to find out what Triple M was launching, so they asked a make-up artist at Channel 7 to make them look like advertising executives. Then, with their new moustaches (it was the 1980s), baggy suits and ponytails down their backs, they attended the launch. Jeff recalls shaking hands and talking to arch enemies, all while trying to keep a straight face. As soon as Triple M completed the presentation, Sean and Jeff snuck out through the back door, giggling like schoolgirls, drove to the Fox FM offices and formulated a plan to ruin Triple M's launch.
The Triple M promotion was based on a space theme, so Jeff and Sean worked through the night and created a promotion that was based around a sex in space theme. They then launched the promotion prior to Triple M, meaning Triple M would look like a copycat if it went ahead with its promotion. The cost of the launch for Triple M was in vain, and the potential spend that it was hoping to receive never materialised. Needless to say, Jeff was not that popular.
Over the first few years of our relationship, whenever people from Jeff's past told me they knew Jeff from his work in radio, I would hold my breath. What they said next could be really, really good or really, really bad. Either way, there was a general respect for this amazing man, who did achieve the unachievable in radio (more on this in the next section).
Aside from a few hiccups and stories such as these, our relationship went from strength to strength, and Jeff and I had decided we would be together forever. However, I was still 'patiently' waiting for him to pop the question. Even fate sometimes needs a little nudge, so for several weeks I took Jeff to every romantic spot I could think of. Finally, it was the Yarra Valley and Domaine Chandon (Möet & Chandon's Australian winery) that did it. This was my final ditch effort to 'set the scene'. I made sure he had a couple of courage drinks and then we went for a quiet walk, through the vineyard then down to the beautiful, blue lake at the bottom of the hill. We all sat to look at the white swans in the distance: Samuel, Jeff and me. Just when I was giving up all hope of him asking me and running out of ideas for romantic rendezvous, I turned to see Jeff on one knee. My heart jumped up into my throat as he asked me to marry him. He had with him a beautiful engagement ring to seal the deal (so had obviously cottoned on to my machinations). Once I said yes (surprise, surprise), Jeff turned to Samuel and asked him if he could be his Dad and gave Samuel a ring. Jeff is a true romantic. I will always remember that day, even if I did 'help' to set the scene.
Things had moved quickly with Jeff and continued to. He moved in with me six weeks after our first date, we were engaged after four months, married after eight months and I was pregnant with our son Oliver after 12 months.
Dormant DNA
Having learned an enormous amount at UIP about the power of PR, I left after two years to have our son, Oliver, and freelance. Six weeks after Oliver was born, I was doing the publicity to launch Triple M's new rock, sport and comedy format. (Triple M had been bought by the Austereo network and the new format was Jeff's strategy to revive the failing station.) At the same time, I set up the marketing and publicity for a comedian who was touring with Stealth Productions — a business Jeff ran with his mate Sean. When I found Jeff, he unleashed that elusive entrepreneurial spirit within me; I never went back to work for someone else again.
Jeff says...
When we first married, Janine played the wonderful wife — happy to play second fiddle to my meteoric executive rise in the radio world. She was building our home, raising our boys, playing her netball and being a great wife. We had fun — picnics and all the normal young Aussie-family things, complete with a Magna station wagon, dog, cat and a house renovation.
Then, of course, I ruined everything by trying to cash in on my wife's expertise in PR. In my little radio world we had just relaunched Triple M in Melbourne. I was given the task of resurrecting a station that was rock bottom of the ratings. I had hired every big name I could think of to launch a brand-new rock, sport and comedy format; for me, it was a make or break career move. The first rule I knew — surround yourself with the best. We had a great team on the air and now we just needed to get the word out. Hmmm... I'm married to the best PR person around (with a high care factor of me succeeding). Sure, she has just had a baby six weeks ago, but I'm sure she'll be fine.
Janine took the role with great relish and the station was everywhere — seriously. It was in the press and magazines for seven months — it was the most successful rebirth of a station in history. She showed me that real tenacity she became famous for.
And now the beast was off the leash! Some women like to be stay-at-home mums, some work, some like both; there is no right or wrong. Janine needs the mix, and is a great and devoted mum when she is home. But to lock her in a house all day with toddlers? Not a chance.
It was also during this time that Jeff and I tried our first joint business venture — a novelty book called Love Cheques. We had spotted a similar book in the United States, and we hoped we could convert the concept, put our own local slant on it and bang! Have the next big thing on our hands.
The book contained cheques that you used as little gifts — an IOU message, that kind of thing. I thought Love Cheques would be the beginning and we would have Kid Cheques, Mum Cheques, Dad Cheques and so on. Love Cheques did okay, but the series I had dreamed of never materialised. We also published a book called The Asian Mind Game, by Chin-Ning Chu. We thought this was going to be another winner but, after a book tour and launch, there was very little to show for all our efforts. With two ventures under our belt and no money, we decided that publishing was not for us. We learned a lot and that's one thing Jeff and I have never been afraid of. So what if we've never tried it before? We'll learn (see lesson 4).
Spinning my wheels
Nine months after the birth of Oliver I found myself pregnant again with our son Riley. Riley was only five months old when an associate and his friend contacted Jeff with a 'great idea'. These men were professional corporate types complete with the appropriate degrees but no retail business experience. They wanted Jeff to join them because of his marketing expertise — but it seemed to me that, to them, I was just 'Jeff's wife', so I was not mentioned. I was just someone who had babies and probably didn't have a business bone in her body. This was not the first, nor the last time, I experienced 'the female is not as good as the bloke' syndrome. This was just the first time it was thrown in my face.
Jeff's associates had seen the juice bar concept in the United States, where the industry had been established for around seven years and was quite large. They were pretty sure it would work well in Australia. Jeff was heading off to the States for a radio trip, so I went with him to investigate the juice bars. I personally really liked the category of health and was a big juicer at home. When shopping with my kids, none of the takeaway options offered anything that was healthy to eat or drink; I saw a huge opportunity in the marketplace, if we could offer a truly healthy option. So the juice and smoothie market interested me, and the great news was we found the industry was thriving in the States, so all the equipment (such as industrial juicers and blenders) was easily available. (The bad news was I would have to import everything from overseas.) However, I was not a huge fan of the existing concepts that we saw in the United States. So we had the spark of an idea, but an otherwise blank page.
When we got back to Australia, after careful consideration I said we should do it, but stipulated that our concept needed to be healthier and brighter than the US versions, and needed to define and demonstrate what we do. This was all just gut reaction. I had never started my own business — I was simply going on what I would want as a consumer. I asked myself, How would I like to be treated? How would I like the concept to look?
We started working in partnership with the two men, but right from the start I had little to no voice. I felt that nothing I said was taken seriously. Even the name we settled on — Sejuice — was not something I was happy with. This was my first experience in partnerships and it was not going well. The 'boys' were more interested in the colours than the nuts and bolts of how the store would work. We hired a designer for the store and she wanted a red ceiling. We had a 5-hour discussion on whether or not to go with this colour! Decisions were made by committees and everyone tried to have a say, instead of people sticking with what they knew best or had real experience in. Our associate wanted to get involved in the colour scheme, the other partner wanted to have a say in the marketing because he'd once done it as a subject at university. This unorganised approach was not how I was used to getting things done and it was driving me mad!
However, this did not stop me from putting the store together from scratch. While everyone else was just talking about it, I organised everything from the ground up. I sourced the equipment, I created and tested the recipes at home with my blender, I negotiated the deal with the raw ingredient distributors and the yoghurt contract, and I hired the staff and ordered the uniforms — everything. And I kept the process on track and on budget.
After nine months of planning, working, negotiating and putting the store together, it finally opened on Chapel Street, St Kilda (in Melbourne), and it was a massive relief. Certainly not my vision on the look or the name, but I was proud of the products I created, the early systems that were put in place, and the fact that I survived opening a new business with the partners that I had. I loved the journey of putting the business together, solving the problems and creating something from thin air. The store opened to strong sales and we made a profit from day one. The future looked bright.
However, the nine months of preparation, and particularly how I was treated by the other partners, took its toll on my relationship with Jeff. I did not feel supported by any of the associates and nor by Jeff, who seemed to be siding with the others. I just could not seem to get through to him. Jeff was a powerful executive in the radio industry but had never been a director of a business before. He liked the idea of having his own business and sitting in a director position, but didn't really want to get involved in the day-to-day organisation. And he thought surely the highly educated associates would know more than his wife.
In his role within the corporate world, Jeff was surrounded by lawyers, accountants and other corporate types. These people were all highly educated and were constantly doing courses to improve every aspect of their work. The radio industry was tough and ruthless — unless you rated, you did not survive and the ratings come out every six weeks. So it was probably natural at the time for Jeff to assume that the 'boys' knew more than his wife. But it did not stop me feeling unsupported and underrated. It was disappointing enough to not get the support of the other partners, but it was the lack of support from Jeff that hit the hardest. Jeff and I have talked about this time at great lengths. He is not proud of himself for how he acted then (see lesson 5).
The reality was that I did not have any of the partners' respect. In their eyes, I had no experience; I was a mother of three children and I had never employed anyone. Though this was semi-true, the store's success was all coming together, regardless of my 'lack of experience'. None of the partners had had any idea of all the moving parts required to start a business from scratch, including myself. I was so proud of what I had put together. I started to gain confidence and was looking forward to fulfilling the business vision. My style was based on common sense and solving each problem as it occurred. (I once read an Albert Einstein quote that said he didn't consider himself a genius; he just stayed with problems longer. I am certainly no Einstein, but I too stay with problems until I find a solution.)
We had our first board meeting not long after the opening. I remember that day as if it were yesterday. Everyone was shaking hands and congratulating each other when the discussion of who would run the business and be the CEO began. This came as a complete surprise to me; I had assumed that because I had put the business together, I would continue to run it. My name was submitted by one of the partners; the remaining associates looked at each other and laughed. They laughed as if that was one of the funniest things anyone could say. I was mortified, shocked and offended by the way they were so open about their lack of confidence. One of many thoughts at the time was, Hey, excuse me; I am actually in the room! To Jeff's credit, he looked at them, then at my mortified face, and got up, grabbed my hand and we both walked out. By the time we reached the car there was not a doubt in either of our minds — we wanted out of the partnership. Jeff finally saw what was going on and realised what I was capable of. The next day we got our initial $25 000 back and they gave me some money for my 'time'. We never looked back. I have never loved my husband more than that day, as I finally felt supported and heard. When you marry, you expect to be in a partnership in every way; it felt like our marriage was just about to start.
The remaining partners thought they had received the deal of a life time. They got the store and it only cost them about the same money as we had already put up. Sejuice continued for another couple of years, but the big plans of expansion and being a dominant player never materialised. The store finally closed and we never heard from the partners again.
Story so far... lessons learned
Here are some of the lessons I learned at Village Roadshow and in the early years with Jeff:
1 If you're ever given a job through someone you know or are related to, make sure you work ten times harder to prove you are worth it.
2 Look at what people do, not what they wear. At Boost we don't have a dress code in the support centre at head office. Having said that, I'm aware that not everybody shares my philosophy, and common sense sometimes needs to come into play. If you have important meetings, dress appropriately. While you might not be judged within your own company, you can't rely on the same attitudes existing outside your work environment.
3 You may overlook the fact that skills are transferable; instead, think laterally about how to apply your expertise and experience.
4 One of the great shames in business is when budding entrepreneurs give up when they do not immediately succeed. The shame is not in their failure; it is in the fact that, had they kept trying, they would have learned so much and their next venture might have been a success.
5 In life we are all on a learning curve; it's all about learning from the past and improving as a person.
Chapter 3
Holding all the apples
After splitting with the Sejuice partners, one thing was clear — if Jeff and I were going to create a successful business, we would need full control over all decisions. We started with our ideas for Boost, and strapped ourselves in for the ride.
Boost or bust
The same day we walked out of the Sejuice partnership, Jeff and I sat at a local cafe; tears were welling up and spilling over my face. I was crying out of pure frustration and probably no sleep. At three months since the opening, I had spent 12 very long, stressful months focusing on the Sejuice store. At the same time, I was disappointed because, aside from the partnership issues, I loved every minute of the actual set up and running of the store. Jeff and I wondered out loud what was next. We had the intellectual property in our heads — why couldn't we start again? Yes, they had the store, but they had no idea how to put all the required parts together or how it ticked. I had just finished putting myself through the best business university in the world — a real life start-up.
Jeff and I have always loved the name 'Boost'. We pushed hard for that to be the name of our first juice bar with the Sejuice partners; luckily, they didn't listen. We decided, fewer than 8 hours after walking out of the board meeting, that Boost would be created and launched.
It was the year 2000 and the GST had just been introduced; we were off and running, but this time we would do it differently. I started by purchasing a copy of QuickBooks, an off-the-shelf accounting package. And I arranged to get a QuickBooks expert into my home to teach me how to use the bloody thing. I had no real idea about accounting, but I was determined to know all the technical aspects, so I would know my business inside and out.
After our ill-fated partnership, we were anxious to start again and not repeat any of the same mistakes — we would have full control over all decisions, including the site for the store. Jeff's 'real job' as Program Director for the Austereo Network took him interstate two nights a week. This enabled him to scope out possible sites in other states, and he found a site he wanted to explore on King William Street in the CBD of Adelaide. It was an old building that was heritage-listed. He convinced his father, who lived in Adelaide, to sit at a table in front of the potential store and count the number of people who walked by and record specific details; he then broke this information into categories: men, women and age group. Happy with the flow, Jeff soon signed the lease and then called me with the news. This single act of signing a lease in a state that I did not live in truly shows the naivety that we had in starting a business. To be honest, I was so thrilled to get started without those partners, and create Boost the way I knew it should be done, that Jeff could have signed a lease on Mars. And it ended up being an advantage — having the business in a different state forced me to work on the business instead of in it (see lesson 1 at the end of this chapter).
We now had a site and a name, but not enough capital. We had just finished reading The E-Myth, a book about what entrepreneurs should focus on when starting a business, which discusses the idea of doing the work but getting others to invest their money. (See lesson 2 for more on what I think about this idea in hindsight.) After being burned before, we wanted to be very careful about whom we got involved with the business, and decided we only wanted silent financial partners. We created a business plan to raise the capital. Looking at that business plan today, it was really more of a marketing plan with some token figures at the back. But it did the trick, and it was Jeff's radio friends who came to the party.
Vision to fruition
The first days of starting Boost were so exciting — this was our baby and we were about to see what other people thought of our juice and smoothie concept. Having just come off the back of opening Sejuice, opening our first Boost store was not as daunting. I'd learned a lot along the way with Sejuice, and knew what I wanted to do differently — so now it was time to put all that knowledge into Boost. I used most of the same suppliers as I had for Sejuice, so simply got the same terms (without the hours of negotiation). I was having fun, doing the business the way I wanted to do it and not having to go through committees to get decisions made. Every process was so much easier, from designing uniforms to choosing product names to the creation of the products themselves (see lesson 3).
We still made some mistakes in those very early days. We worked at the logo and deciding on the look and feel of the first store — but, I can admit, we got this terribly wrong. The colours we chose were not what you see today — the store looked more like something the Adelaide Crows might choose instead of a vibrant juice bar. (This colour scheme was mostly Jeff's doing — but more on that later!) Jeff's negotiating of our first site was also a bit flawed. The store had no air conditioning, in a state that regularly has days over 40 degrees. And because the building was heritage-listed, we couldn't make any structural changes. So we spend the first summer running around getting portable air conditioning units so that the smoothies and staff wouldn't melt.
But this all came later — at the opening, all I felt was excitement. Our first Boost store opened on King William Street in Adelaide at 11.15 am and, to my great delight and shock, over 50 people were waiting to come in. We had queues going out the door! With no marketing! (That was planned for later.) I could not get the smile off my face. I couldn't believe the number of people, so I asked one of the customers how she'd heard about Boost and its Grand Opening. She asked, 'What opening?' and then explained that there was a bomb scare next door and the building was evacuated, so ours was the only cafe on the street not affected! I laughed so hard. A bomb scare is no laughing matter, but we had one of the strongest launches ever in our very first store!
For the next 12 months, I was forever on a plane to Adelaide and Riley was still only just over a year old. I visited Adelaide once a week, then less frequently as the business got up and running. Nine times out of ten, I took one of my children with me, while Jeff was at home with the others. I was very fortunate that my mum was also there to help out — she would come over to our house to look after the children. Without her consistent help, I just would not have been able to cope. I wanted it all — kids and a career. I made sure I got it, but had to work hard for it.
We were also extremely lucky to find a great manager for that first store. A real diamond in the rough, Sharryn did not have any retail experience, but she had the passion and drive we needed and this was clear even when we interviewed her. When looking for a store manager, most people hire someone with an enormous amount of retail experience, and so I was often questioned why on earth I hired a person who had never worked a day in retail in her life — not to mention while I was living in another state. My reason was simple: she had that fire in her stomach that we needed. She had determination and she understood what we wanted to achieve, and, like us, she did not have a history of bad retail habits. A sign of her determination emerged when she told us she was a champion speed waterskier. You need enormous mental resilience and courage to be successful in her sport, and these were just the skills I was looking for. At 6 am, Sharryn and I were up promoting smoothies and wheatgrass at SAFM, the leading radio station in the state. We also collected email addresses to use to increase our brand awareness. Back then, this wasn't used that much and so was powerful, because people did not get a hundred marketing emails each day like they do now.
Young businesses are usually hungry for cash so, like most young businesses, cash was usually in short supply for us. This meant we were always looking for the most cost-effective option, for everything. We needed to look at what resources we had available that might help, and that did not cost the earth. We were lucky that at Austereo Jeff used to get free CDs from the music companies, and these ended up becoming our prizes for joining The Boost Club.
We did everything we could to get our brand out into the marketplace. But we also wanted to use The Boost Club to create a sense of community around the brand. Every month, we would have great offers and competitions, as well as providing health and fitness tips, and this continued as our brand expanded. And I was involved at every level — I personally typed our first 10 000 Boost Club names into the database.
When you're starting something from thin air, you have to oversee every little detail: from the distance the blenders should be apart (so they don't blow up) to making certain the managers have a checklist. This kind of attention to detail was not Jeff's strength — it was mine. I was in my element but I still felt a tremendous amount of pressure at a very micro level to get everything perfect.
My biggest mission was 'the customer experience' and nothing was too much trouble. If the customers did not like a drink, we would change it. I wanted Boost to be the business other businesses strived to be regarding customer service. My dilemma was how to find out if we were not delivering on the customer experience, so we could then change what we were doing. We needed the right mechanisms in place, so we started with the idea of detailing on every store wall the experience the customer should have. We called this the Boost Guarantee — every store still has one, and it covers everything from the kind of ingredients we use, and our focus on friendly service and healthy living, to giving people a reason to smile.
We then invited people to tell me, personally, if we did not get it right. This gave us a chance to change a negative customer experience into a positive one. Aussies generally are not big on complaining, but I invited them to do so and made it easy for them, thus allowing us to find out how we were measuring up. With every single complaint, it became my personal challenge to convert that customer into a raving fan. I did it, every time — by thinking as a consumer and keeping it simple. At the end of the day, people know that things can and will go wrong. What made the difference? We acknowledged any mistakes, and then fixed the problem. Right a wrong — simple. My customer experience mission gave our customers a reason to choose Boost and still keeps them coming back today.
From the beginning, we were never going to be happy with just one Boost Juice store; we thought big from the start. Thinking bigger made us act bigger, and this influenced suppliers and landlords to believe in our vision, give us good prices and take us seriously. It turned out that starting the first store in Adelaide was brilliant. We were able to get the concept right without the eyes of the larger cities on us.
After the first store, I took total control of the brand look and feel. I realised the first store's design was terrible because I was using other people's views, mainly Jeff's. The new look for the brand actually originated from a massive picture of sliced tomatoes I saw in a store in Singapore. I know it may seem odd, but the picture was beautiful and it really showed the life essence of this fruit. So we experimented with other fruits — I had our photographer cut oranges, lemons, watermelons and so on. He then stuck the images on glass and let sunlight come through. The effect was amazing — the simple beauty and life of the fruit was captured in the photos. We used these photos as the core of the design concept; we chose the colours from the fruit and the early stores all had 3-metre images of sliced fruit all over them. The design was a winner — and, needless to say, Jeff will now openly tell you that he is not the one to talk to about design. (Again this relates to lesson 3.)
I gained many lessons setting up our first Boost stores and during the first 12 months after — see lessons 4 and 5 for more.
Jeff says...
In the early days of starting Boost, Janine was a young woman with all the ability but no experience. With that came a lack of confidence in some areas, but she took the 'Einstein approach', and just stayed with the problems until she figured them out. The personal confrontations with suppliers or shop fitters were difficult at first for her; I often got dragged in to make the hard phone call or appear at a face-to-face meeting. And, admittedly, being a self-obsessed radio guy, I no doubt had little knowledge of how hard the situation was for her — the obstacles she was overcoming or the day-to-day problems.
Janine was scared, but at the same time she was relishing the challenge. I knew she couldn't fail, but doing so was her biggest fear (as it is with most young, driven people). I was busy with my 'high-powered' radio position and not being overly supportive, and she was starting to build her own team of young, driven, motivated women that believed in her and her dream. Eventually the inevitable happened: Janine became more confident and busy, and I became less important and received fewer phone calls; I wasn't needed half as much as I would have liked. The tables had begun to turn.
Making the decision to franchise
Jeff and I decided to franchise quite early, though at the time neither of us knew much about franchising or how it worked. Our vision was always to grow the brand and we realised we had a small window to do so before bigger players came into the marketplace. We just wouldn't be able to hire enough quality managers to expand that quickly — and that's where franchising offered us the solution we were looking for.
Through a friend of a friend we stumbled onto Rod Young, who was just opening a franchise advisory business after having worked in franchising and business his entire professional life. We were his very first clients in his new venture and he became a small shareholder in ours.
Our first meeting with Rod was before we had completed a full year of trading. I remember this because Rod explained we would need figures for a full 12 months before we could begin. Even so, we were clear on our direction. It was 2001, and this would be the road we headed down.
Keep in mind, I was learning on the go and had three young boys at home; every day presented a new set of problems to solve. Thank heaven for my mother! Not long after we had made the decision to start franchising, Jeff (who looked after leasing) came home one night with a smile like the Cheshire Cat. I knew that grin; it comes out when he has done something that he knows will freak me out — and he had! He had just signed an 18-store deal with Westfield, complete with a $5 million liability in our names! Like most early (okay, mid–) thirtysomethings with a young family, we had no money. After a seven-second calculation, I knew the equity in our house was not worth a tenth of this figure. We needed to open 18 stores within 18 months and, at the time, we only had two stores open. This deadline didn't just give me a little kickstart; this exploded me into the world of franchising — without a parachute.
But, despite my initial shock, franchising did work for us, massively. We had so many franchise enquiries we could barely manage the load. Fortuitously, we'd recently hired a very young head of human resources (HR), Jacinta Caithness. (On her first day, Jacinta was greeted at the front door by Molly, our massive Great Dane. I could see in her eyes that she was wondering whether she should stick with the job or make a run for it. Thankfully she didn't run, and I immediately stopped letting Molly greet recruits.) While Jacinta started as our HR manager, she quickly became our franchise manager, and she did an amazing job recruiting the right people for our franchise businesses. On the leasing side, I also worked closely with Kristie Piniuta, a lawyer who would later come to work for Boost.
Kristie says...
It was 2002 — I had just finished my articled clerkship and was working in a mid to large law firm. I specialised in retail leasing, acting on behalf of tenants, which some lawyers thought was boring and repetitive. I, however, liked it. I especially enjoyed negotiating the best lease possible for the 'underdog'. I picked it up well and often won my battles. My success wasn't because I was any cleverer than the other lawyer, I just genuinely 'cared' — a trait lawyers are often accused of lacking.
I am pretty sure that is why J (Janine) picked me. We hit it off straightaway, despite only knowing each other from conversations on the phone. Boost was meant to be my 'pet project', with four shops open and a few more expected that year as the company started franchising its concept. Well, my 'pet client' opened 50 stores that year. I loved it! J and I discussed leases not like they were just a legal document to get right, but rather as the basis for the success of a Boost business.
Before I met J in person, I was told that, although she looked like a hippy, she was tough and not easy to please. However, as soon as we met each other we connected. We talked shop and no-one else at the lunch felt like they could intrude. Shortly after, I was seconded to Boost's support centre and within one hour of that secondment I was asked to join the Boost team — and I jumped ship with enthusiasm. I felt alive and valued when I worked with Boost. Everyone was so engaged. It was fun and yet constructive — in 30 minutes, it seemed we could solve every problem on the table.
Surrounding yourself with greatness
The business grew from strength to strength, and we were able to employ some other key people to help run it. I still proudly take my hat off to our young team in the early days, and I'm amazed at everything we achieved together and the amount of daily problems that we solved. The success of the business was a real credit to them. I was in my thirties and my team was in their early twenties: Kristie Piniuta, a lawyer; Naomi Webber, an accountant and our savvy CFO; and Jacinta Caithness, the franchise manager.
Kristie had a large care factor and a thirst for knowledge that helped keep us out of trouble. When I first met her, Kristie was a junior lawyer working in the leasing department of a law firm. I would spend hours with Kristie going through the leasing contracts line by line, asking her to explain exactly what every clause meant. After she came to work at Boost, Kristie went from only looking at property leases to needing to know everything related to business law. She was in her element. Through diligent research, she ensured that every decision made was the correct one, creating order and building a strong foundation on which this fast-moving beast of a business could grow. Very early, we had good corporate governance, unusual for such a small business at the size we were. We have made many mistakes in our business, but, largely because of Kristie, fulfilling our legal requirements was one thing we got right. And this was a critical part of the foundation of the business.
Naomi was presented with an absolute mess to fix (which she did). Naomi was a young accountant recommended by Geoff Harris (more on him later in this chapter) and we hired her as the CFO. The accounts at the time were in a terrible state but, to Naomi's credit, she built up a strong team and after many months of all-nighters she got the accounts back in order. Accounts tends to be that boring area that entrepreneurs tend to think of as unimportant, but not having correct numbers and clarity on what your business is doing means you're running your business in the dark. The numbers tell you exactly where to put your focus, which team members are thriving or struggling, and, more importantly, give you a solid business in which you can trade. Naomi helped give us that clarity.
And Jacinta, a woman with no franchising experience, learned all aspects and helped make the franchising tactic a success. Jacinta had great tenacity in achieving the required goals, no matter what it took. I remember her telling me once, 100 per cent seriously, that she did not understand why people missed deadlines. She rationalised that if your head was going to blow up if you missed the deadline, you would make sure you met it. So, really, no-one could have a reason for missing a deadline. From that time onward, we had a saying that something was a 'head-exploding deadline'. The other saying that got us through those times was 'eat that frog', from the book of the same title. Every day it seemed there were hard calls to make, and no-one likes making hard calls, no matter how tough that person seems. So we often referred to those days as 'eat that frog days'.
Seeing Jacinta develop over those years was incredibly rewarding. While at Boost she achieved the AFR Boss Young Executive of the Year award as well as the Telstra Australian Young Business Woman of the Year award, both of which she deserved in spades. I have travelled many frequent flyer miles with Jacinta over the years, setting up Boost international. From meeting with sheikhs in Dubai to looking for sites in the snow in Estonia, it has been a remarkable journey.
Together, Kristie, Naomi, Jacinta and I worked out the problems as they occurred. All of us were learning and doing things for the first time, but we had an enormous care factor to get it right. It truly was girl power! And in the high-pace growth of Boost, we never would have achieved what we achieved without the strong girl power from these three young, smart passionate women. We often laugh that a year at Boost was like working five years anywhere else. It was both scary and exhilarating pulling the business together and we had a ball. Many nights were spent with pizzas working into the early hours of the morning. Sitting at a round table with these women, there was always a feeling that any problem could be solved, and when we all went in our different directions in the business we all knew that no-one was going to drop the ball — we would achieve what needed to be achieved.
Kristie says...
Was I in over my head? Yes! Every minute of every day. But that was Boost's secret — we were all in over our heads but that kicked us into survival mode and we performed at our best at all times. We were all young and inexperienced but knowing that made us more determined to get it right. It made us appreciate and value one another. There was no place at Boost for negativity, cynicism or doubt. Everyone at Boost's support office embodied the company's mantra — being passionate, healthy and determined to be the number one franchise in the world!
Moving to Boost at such an early stage in my career shaped me as a lawyer and J was a mentor for me. J's minimum expectation of you was that you knew what you were talking about. You couldn't bluff with J. She was always happy to nut something through with you but only if you understood the problem, openly explored solutions and had a 'can do' attitude. What could be better than being challenged to perform at your best? Mistakes weren't tolerated — mistakes can let others in your team down — but to not try at all was inexcusable at Boost.
J had three 'right hands': Jacinta (the gate keeper), Naomi (the quick-minded and bold CFO) and me. I think if I was given a nickname it would be something akin to 'protector'. We were all in our early to mid-twenties. We worked incredibly well as a team and we achieved so much together.
Everyone in my small initial team played numerous roles. We had to — we didn't have a team of people sorting out the various solutions to problems. We all wore various hats: the accountant, the secretary, the publisher, the negotiator and the cleaner. We did everything behind the scenes. For the first two years, I worked from the kitchen table at home, while my first two employees (a PA and a part-time bookkeeper) used the dining room.
Having the business operations in my home also allowed me to be around for my three boys. I have always been a great believer in the idea that children should be in your life, not you in theirs, and they will have a richer life because of it. That is how I resolve the guilt that comes from being a working mum. But I also had a secret weapon (then and to this day) — my mum, Joan. I honestly could have never achieved the level of success I have without her. People call her 'Saint Joan' for good reason — it's with her help that I manage to maintain a balance between home life and the passion for my businesses.
Almost every day, Mum drove from Boronia to my house in Malvern East, a 60-kilometre round trip, to help me with my children. Not only did she do the drive and dedicate her life to helping me, but she also did this without trying to produce any guilt in me whatsoever. When I told her in a moment of guilt that I was asking too much of her, she told me that she loved every minute of it and it made her life complete.
I remember a day when I was in Sydney and Mum called to ask what time Jeff was getting home to look after the kids. I didn't know, so I told her I would call her back. I eventually got hold of Jeff to discover that he was in Brisbane for two days. (Nothing like great communication between the two of us.) I then called Mum back to let her know about my incompetence — that neither Jeff nor I were within 1000 kilometres of her and I was not going to get mother of the year that year. Mum just laughed and told me the kids would be fine.
Mum also had that ability to not cross from the grandmother role into the mother role. The second I walked in the door, she would defer to me for everything. She is the perfect grandmother and, for me, the perfect mother. She has eleven grandchildren and has a special bond with each and every one of them. So much so that at Christmas every single one of them flies from interstate to Melbourne, bringing their current boyfriends and girlfriends with them, for Mum's Christmas lunch. Mum didn't know what she got when I was born. Even now, she openly wonders where I came from. But she has been the most amazing support for me, and I love her from the bottom of my heart.
If my mum is the perfect grandmother, my father is the perfect grandfather. I take my hat off to him to be able to sit for hours and hours playing games with his grandchildren, letting them paint and even plait his hair. He has a great attitude to life, he is 83 and still umpires cricket and plays golf twice a week.
The business would also not be the success it is without Jeff — he has been with the business every step of the way. I lean heavily on him for advice and guidance. Particularly in the first couple of years, I was terrible at firing or counselling people, so I used to go to him for anything that was confronting. His greatest attribute was his absolute confidence in what we were doing and in my ability to pull it off. When I walked in the door completely stressed, he would calm me and tell me everything would be fine. This was largely because he was such a 'big picture' guy he had no idea of the day-to-day problems or cash flow. His full-time job allowed him to only keep his thumb on the macro picture, and sometimes stepping back and looking at this bigger picture was exactly what I needed. Seventeen years on, he is still my best friend. Together we make a wonderful team, in business and in life. Jeff unlocked many things in me that helped create Boost.
Running at full tilt
At the end of 2001, we had survived our first year of trade. There were four Boost stores, including one in Melbourne's Jam Factory (a popular shopping and entertainment complex). Boost had reached the point where the business was truly taking over the house. I was using the kitchen and dining room as offices, our master bedroom was the CFO's office, and Jeff and I were sleeping with the boys in their rooms. Jeff used to complain that the only action he got was me doing the laptop dance, as I typed until all hours of the night. I remember walking past the dining room one day, looking in the room and realising I'd reached the point where I hated not getting away from the business. I was working 17 hours a day. For my sanity and for my family, I decided Boost would have its own proper home. Up-and-coming, young businesses need a great deal of cash, so moving from my home to an office was a huge step financially, but it was also a big decision emotionally. While the move meant my boys would no longer be running under my legs while I was talking to suppliers or working out a solution for a customer, I had really enjoyed still being so close to them — and there is nothing like a child's hug anytime of the day.
In 2002, we thought it would be a good idea to join forces with our competitor, Viva Juice. They had four stores and we had four stores. At the time, I was feeling things were getting over my head. The business was taking over my life and I needed some of the work taken off my hands. We met with the owner of Viva and discussed a deal. Perhaps not surprisingly, they wanted more than what we thought was reasonable; in hindsight, though, not being able to merge the two businesses was the best thing that could have happened and it was a real turning point for me. I realised I had no-one else to turn to — the net didn't exist. It was up to me to nail this business. Jeff was great with securing new sites and helping me develop the marketing plan, but the nuts and bolts were all on my plate and we had everything on the line. I loved what I was doing and the adrenaline that came with running a new business. I was not always 100 per cent confident in what I was doing — okay, that's an understatement, I was not even 50 per cent confident in what I was doing — but the reality was I was the biggest expert out there in this specific area. I had no-one else to approach and I just had to work it out along the way. (More to come on the Viva story later.)
It was also around this time that the media really started to get interested in the Boost story. Basically, it felt like I was two people — I had Janine, the founder of Boost Juice, and me, the person who was employed to get PR for Boost Juice. I had to see 'Janine' as a tool to use to get people to understand what Boost Juice was about. Through my experience at UIP, I had sat through dozens of hours of interviews, and one thing I learned was that you have to be yourself — you cannot fake it. So that is what I did; I was just me, in all the interviews. I was always honest and transparent, and told the truth about Boost's journey and any mistakes along the way. What made talking about Boost easy was that I was (and, of course, still am!) genuinely so passionate about the company and the brand, so it was easy to talk about my favourite topic.
As time goes by, and with each problem solved along the way, you cannot help but evolve into a more confident business person. During those early years, I made sure that I understood every aspect of every decision I made. To me, the fact I cared so much about the business justified my behaviour. I painstakingly took the time in every area to get it right, from dealing with the franchising and trademarks to working on supplier relations. I was obsessed with the business. We rarely used outside companies for areas such as franchising, legal (where possible), marketing or advertising. I wanted to make sure everyone who worked on something for Boost had 100 per cent focus on Boost at all times. I was a total control freak, needing to know everything. I found it hard to trust that the job would be done well by other people. The reality at the time was no-one on the team, including myself, had been in the business long enough to know exactly what to do all the time. Back then, it would stress me to my core if I went on holidays because I thought the business would fall apart. Clearly it did not; we had great people doing great things (see lesson 6).
Jeff says...
The company was flying and the team was obsessed with their objectives; they didn't have all the answers, but we had a great board of wise-heads to help keep it all on track strategically. Janine moved out of the 'scared little rabbit in the spotlight' position and took the lead with gusto. The company demanded a true leader, and it got it with Janine. Along with her band of three young, female executives (who all fed off each other), she achieved the impossible.
Around this time, competing against 44 other start-up juice bars, it became apparent we needed an edge. Most of the other juice start-ups were set up by business guys who put someone in to run it, or other male entrepreneurs having a crack. Janine was unique, and having just read Anita Roddick's The Body Shop Book, our simple question was, 'Who was Australia's leading female entrepreneur?' Poppy King — the 'Lipstick Queen' — came to mind, but she had just fallen off the business perch. No-one else sprang to mind, and certainly no-one with three kids and a needy husband. I bounced the theory around with Janine, she readily agreed and we decided to make use of her PR background. The story of the woman with the three young kids who saw a need in the Australian market for a healthy alternative became a reality — and, wow, did it take off. We were in the hottest category. Janine was perfect media fodder: likeable, funny and beautiful, and she showed that a woman could do anything. She was an inspiration to the woman who wanted it all.
So, reiterating — the company was flying. Her team was amazing and now the media was all over her with very positive stories. She could have seriously become unbearable. In truth, this was the massive confidence boost she needed to build self-belief in herself. Not a fake facade of confidence, but a true core belief that she was good at her craft — which was building a company. Of course, at home it occasionally spilled into a bit of tension, with both of us being self-important. We had to rework the relationship model and as we did, right before my eyes, I saw an amazing transition in Janine's personality.
Risking it all
Any new business is hungry for cash, and Boost was no different. In 2002, we needed more money to grow and we had two choices: get other investors into the business or find the money ourselves. We decided that we didn't want to sell down by taking on additional investors, because it would be like working for someone again and that was the last thing we wanted. However, this didn't change the fact that we needed cash and fast.
The banks wouldn't touch us with a 10-foot pole because our only asset was our family home, so we had to find money some other way. My greatest fear was losing the house that Jeff and I had worked so hard for. (Admittedly, Jeff worked really hard to buy our house. While I was gallivanting around the world, Jeff was saving money. He purchased his first house as a 19-year-old — who does that? He was saving for a house and I was sailing around the world with David Bowie. The 'Gods of Yin and Yang' must have had a good laugh when they put us together. But he had assets and I had debt — a perfect match in my opinion.)
In the end, we risked it all. We sold our only asset, the house, and invested all the money into the business. We packed up the kids and moved into a rental for two years.
Picking the right partner
By the end of 2002, we had opened 15 stores and were going strong. There were 50 stores opened by the end of 2003. I could see a permanent frown on my brow — it seemed to have cut deeper into my forehead every morning. I was learning as quickly as I could. I did not have mentors; in fact, I did not have friends. I did not have time to sit down for a coffee let alone a chat. One morning in 2002, I was sitting at my desk when I saw a note to call Geoff Harris. I had spent most of my adult life abroad so I certainly was not up with the 'who's who' in business (and these were still the days before you could simply 'Google' someone), but it turned out this Geoff person wanted to meet and discuss the business. Geoff was not the first person to show interest in the business, and we were very guarded about who we wanted to 'play' with. We had already rejected many, many offers from people to get involved.
However, Jeff and I decided I should meet with Geoff Harris at a cafe. We sat down and he showed me the latest Business Review Weekly Rich List. (The BRW is the Australian business bible.) Upon reading his name in the Rich List and his worth, I spilt my entire coffee onto his lap and note pad. Not the best start to a relationship. Now, you may be thinking, What a show off, but he simply wanted to show us that he was not a tyre kicker. Geoff was someone genuinely interested in us and our business. I quickly learned what I probably should have known already — he was the co-founder of Flight Centre, one of Australia's greatest success stories. And, for the record, you could not find a more generous, kind, loyal and considerate man on the planet.
Geoff says...
I was starting to wind down my day-to-day duties at Flight Centre and was looking for a young brand where I could assist the owners in growing their business. After reviewing a number of brands in 2001, I came to a Boost Juice store in Melbourne's Doncaster Westfield. I quickly recognised that the brand was a potential winner in the 'healthy to go', sector just as people were starting to realise how unhealthy the traditional burger and chicken 'to go' outlets could be.
I phoned Janine and talked about how she was going and the challenges she was facing with a young family and a fast-growth business. Janine and I met at a local coffee shop to further discuss the business to see if we were 'culturally' compatible and our business goals were aligned. It was a great meeting and it was clear that culturally it was a fit. Janine was so enthusiastic about her brand and business that, in a gesture of enthusiasm, she split the entire contents of her coffee all over my newly acquired leather diary. The stains and the stuck-together pages were a constant reminder of that meeting and of Boost. After that meeting we agreed to meet again, this time with Jeff. After meeting both of them, I was excited about the prospect of becoming involved with this retail start-up.
There was no question in my mind that Janine was a 'young gun' on the go with plenty of energy and a great concept, and totally honesty. She just needed some rounding at the edges and a sounding board for her growth as she ramped up the business across Australia (and later overseas).
The other key element in my deciding to invest was that Jeff Allis was part of the deal — and that was vital, because his marketing and ideas 'grunt', combined with his radio background, added a unique element to the partnership, as did his backup and support for Janine.
Over the next four months, Geoff gave us 'precious gems' of strategic business foresight and never asked for anything in return. By the fifth month, we were ready for him to get involved.
Some months prior to this decision, I had stopped doing the accounts and hired a CFO. I quickly discovered this recruitment decision was a mistake and I learned the first lesson in hiring the right people (see lesson 7). The CFO I hired was previously employed by a business that had gone belly up. I'd assumed this would have given her hard-learned knowledge on what not to do; I was wrong. The figures we presented to Geoff Harris to review seemed to be all wrong. When his accountant said not to move forward, because there were problems with the integrity of the figures, I was alerted to our CFO problem. It wasn't that she wasn't trying; it was just that the job was too big for her.
Geoff did eventually buy into the business and we hired the right person to get the accounts balanced. I was thrilled to have him become a part of Boost. Geoff's buying into the business was simple; we agreed on a price, he handed me a cheque and that was it. I know lawyers are a necessary evil but if business deals could be done based on a handshake and someone's word, profit margins would certainly be a lot higher.
Geoff says...
After a short review of the numbers and many meetings later, we agreed that I would invest in the company and I acquired 24 per cent equity in Boost Holdings. A handshake deal cemented the relationship and this formulated a level of trust that was vital in the early, stressful days of growth. Both I and Janine were co-directors, and we wore the risk of offering personal guarantees, and took on the pressure from the banks and financiers/landlords/staff and so on. But, as the saying goes, without risk there is no reward. We both powered ahead with complete trust in one another.
Geoff had a goal to utilise his knowledge and share it with another start-up. He then met Jeff and me and, thankfully, it was a perfect fit. Geoff quickly became my mentor; he was so generous with information and his experiences. In the coming years, we worked closely together and the direction of Boost changed in many positive ways. I spent many, many hours on the road with Geoff, looking at stores and picking his brains. We would meet at least once a month, and during these meetings I would always bring out my long list of questions about various issues that I was having at the time. Geoff is one of the good guys; he is honest, loyal and a true Aussie bloke if ever there was one. Geoff expanded my personal business knowledge dramatically (see lesson 8).
Adding a Boost to Viva
Two years after first approaching Viva, in 2004 we ended up purchasing the Viva Juice business. At that time, we had over 80 stores and they had 24 stores, all owned by Viva and not franchised. They were the only real competitor we saw in the marketplace, and the owner had secured some great sites in the Melbourne and Sydney airports, which prevented us from getting into these positions.
The acquisition was a monumental learning process on all levels, because it was the first business that I had ever bought. I needed to know the difference between a share sale and an asset sale, for example, because if we got this wrong it could cost us thousands in tax. Getting solid advice and working with consultants and lawyers that we trusted was critical. The legal arrangements were extraordinary, and the process was painfully long and detailed — in all, the negotiations lasted six months. In some respects, however, the process was aligned with my strength of being detail-oriented. And even though it was stressful, it was invigorating to complete.
My girl power team (Kristie, Naomi, Jacinta) came into their element through the Viva Juice acquisition. As I mentioned, in the beginning none of us really knew what to do — we were all doing it for the first time. However, we all cared enough to make sure that we got it right, which we did. The research and pulling of favours from all of our contacts ensured that we made the acquisition a success. Kristie was beside me until the final sign-off on the deal. I have now watched Kristie go from being a keen and passionate young lawyer to being a married woman with two beautiful children and her own law consulting firm, and I am so proud of what she has become.
Kristie says...
J is smart, and hers is the kind of intelligence that's always refreshing to be around because it's the 'raw kind' that stems from a curious mind.
Whenever I was chosen to be J's 'right hand' at a meeting, I enjoyed watching her level the playing field. If we met with someone with expertise in a chosen area, often the meeting would start with the 'expert' (lawyer or otherwise) lecturing J on what this person assumed J did not know. The expert's assumption was often right
and J would openly admit it, which would make this person act even bolder or sometimes more arrogant. Of course, that's when J would 'turn the tables'. She may not have known what the expert was talking about at first but, by asking a few simple questions, she would very quickly understand the topic. And then with one swift statement she would turn the tables and leave the 'expert' dumbfounded.
On the very last day of the Viva deal, I was called into the legal office to finalise a number of minor points. I had been having dinner with some friends, so I remember arriving at 8 pm. The Viva owners were in the other room to go through the points, and I was promised it would only take an hour or so. Issues started to go back and forth between the rooms, so we decided to get into one room to finalise these points. We left the room at 11 am the next day. In utter disbelief, I clearly remember watching the sun come up; we had been negotiating all through the night. Strategically, this was a great win for us; mentally it was OMG! We were already growing at a store a week, and now we'd thrown in converting an additional 24 stores and getting the Viva staff on board — it was a great lesson in change management. I remember hearing the Viva owners cracking champagne and celebrating the sale; all I could think about was what I had to do next to make this work.
A tired mess after the all-nighter, Kristie and I found a local gym to have a shower. Jeff met me at a cafe in the city and, I admit it, I had a bit of a sulk to my husband that day, and may have demanded that he buy me something that 'blinged'. After four years of growing the business and realising what was on my horizon with the additional stores to bring on board, I was beyond tired and emotional. I must have had a furious look, as Jeff went straight into a local jewellery store and indeed bought me something that 'blinged'.
Business Woman of the Year
As I mentioned earlier in this chapter, franchising worked for Boost. At the start of 2004, we went out for dinner to celebrate because the business was now turning over $1 million a week. And 2004 continued to be a massive year for us, with the Boost machine of training, building and marketing in overdrive. We were opening a store a week, and every day I seemed to be creating another spreadsheet for a system or process. The people who reported to me called me the 'Task queen'. (I had discovered how to use the Task tab on Outlook and it was my saviour.) I could now effectively track the millions of moving parts that were Boost.
In 2004, we were also in 'The Top 7 Businesses' in BRW's annual list of the 100 fastest growing businesses in the country. The hysterical thing was that same year I was also in BRW's Young Rich list. The reason this was so funny? We had not taken a cent out of the business; every dollar made was put back into the business. For the first three years, I didn't take a salary. In year four, I did and it was $35 000. I was one of the lowest paid staff members at the time. I went shopping the day the article came out. When Jeff saw all the bags and raised an eyebrow at me, I smugly said, 'Have you not read BRW? Apparently I can afford it!'
And this was the same year that I won the Telstra Australian Business Woman of the Year award. I was absolutely thrilled, surprised and honoured, and the award was a pivotal turning point for me. The awards ceremony was the first time in over four years that I networked. I had sourced out businesspeople here and there for lunch and their advice, but never in a larger group. With this award came the opportunity to meet some of Australia's most amazing and inspirational women. One was Launa Inman, who was the managing director of Target Australia at the time. Her journey from South Africa to becoming one of the leading businesswomen in retail in Australia is profound. I have enormous respect for Launa — not just as a businesswoman, but also as a friend. The other person I connected with was Judith Slocombe, who started out as a vet. She had her own pathology business that was purchased by Gribbles and, while she ran this business for a number of years, she's now the CEO of The Alannah and Madeline Foundation. What she has personally done for this foundation is quite extraordinary.
Both of these women have also won the Telstra Australian Business Woman of the Year award. They too are mothers and wives facing the similar challenge of balancing their lives with their love for business. All of us enjoy what we do. We have a passion for creating and driving forward this think tank we call business. Between the three of us, we have 15 children (Judith has nine). Though we do not catch up as much as we would like, we meet at the National Telstra Awards Dinner annually and are on the end of the phone whenever needed.
Story so far... lessons learned
Here's what I learned in the first years of starting Boost:
1 Make working on your business your priority. Many people who start a new business spend so much time working in their business that they never get a chance to grow the business.
2 Be careful about who you take into your business. Hold as many shares of your business as you can, but be generous with getting people on board who can add value to the growth of the business.
3 You need a clear, single vision to create something truly special. Doing everything by committee doesn't work — if too many people have their say, you end up with a diluted version of the original idea. The store design with Sejuice was horrible because it was designed by committee. This was an example of my early lack of confidence; I was a pleaser. Have confidence in your vision and let this vision guide you on everything about the business.
4 Be resourceful. I do not have a business degree. What I had when I started Boost was the ability to think outside the box, because I didn't know there was one, and learn quickly. I knew I could, and would, figure out what was required. I went to great lengths to gain the business knowledge I have now. When people ask me what my background is, I remind them that I didn't go to university, but I had a hell of a teacher — Boost.
5 Remain true to yourself and your management style. Two years into running Boost, Jeff sat me down and told me how I should change to become a better CEO. He had strong ideas on what type of person should lead an organisation, and thought I wasn't dealing appropriately with difficult situations. I knew even then that you had to be true to who you are to be a good leader, and told Jeff so. A couple of years later, he told me how proud he was of me for sticking to my beliefs. He acknowledged that his advice many years prior was totally wrong — that my 'style', which is unique to me, worked.
6 Know when to let go and allow people the opportunity to thrive.
7 A business's success is all about the people — get the people wrong and it will be detrimental. For example, getting your accounts wrong can cause numerous problems. It's impossible for investors to come into the business if the figures are in question. Making the right decisions within your business is also difficult if you do not accurately know what the business is doing. One of the most important people in any business is the bean counter (okay, the accountant).
8 It's vital for people who are successful in business to pass on their knowledge to those who can benefit from it. Both Rod Young (our franchising expert) and Geoff Harris did this for me and, in a nutshell, it's why I've written this book.
Chapter 4
Keeping the juices flowing
I'd just been named Telstra's Australian Business Woman of the Year and made BRW's Young Rich list (even if I hadn't actually seen any of these riches). We seemed to be flying — and that's when the wings started to fall off. Internal and external problems emerged, so we had to start doing things differently, thinking proactively and cutting expenses (and rocking quietly in the corner occasionally). With our wings back in place, Boost started to soar again.
As Boost got back on track, my focus started to shift — epiphanies were to be experienced with another baby and starting a Zoo.
Cracks in the foundation
We were hot! Revenue was pouring in. Store sales were increasing year on year by nearly 30 per cent. Obviously, all the partners were happy and making money. That's when the cracks started appearing.
The first crack was something that I did not even see coming. The success of the brand meant franchise opportunities were in hot demand — and our early franchise partners knew it. They were onselling their businesses, sometimes for five times more than they paid for them. We could not legally then (and nor can we today) tell people how much they could sell their businesses for. However, problems emerged because the banks were lending to the incoming new Boost franchisees, helping them to cover the premium sales prices. The consequent enormous repayments were making it difficult for the new franchise partners.
I sourced out Lesley Gillespie, one of the founders of Bakers Delight (a company that had also used franchising to expand), and she told me this was a common problem. I was a massive fan of Bakers Delight and still think they are another true Australian success story. Lesley is a down-to-earth, no-nonsense woman who right from our first meeting was warm and likeable. Bakers Delight had been around for 25 years at the time, so they had gone through many of the same issues we were now facing. She shared some of the solutions that worked for them, such as the system they used for franchisees to report their financials (she was kind enough to actually give me the spreadsheet as well as permission to use it), and how they trained their incoming franchisees.
A number of uncontrollable, environmental factors also put enormous strain on the 'new' franchisees, in addition to some of the large loans. The second crack in our success started when A Current Affair (ACA) ran a story on juice bars, attempting to show they weren't such a healthy option. Remember — Boost isn't just about selling smoothies and juices. It's about offering a whole experience that ends with, 'I feel good about myself for choosing Boost'. For example, you walk into a store, the music is playing, there are bright, fun colours, and happy people and delicious fruit are all around you. Ideally, you're served by a smiling, happy, young person, and you walk away with a great experience and a healthy, great-tasting product. Now, if we get any of these things wrong, the concept will not work (see lesson 1).
ACA stated that juice bars were adding sugar to their juices, and that one juice or smoothie was equal to a Coke or a Big Mac. Our first reaction was that no-one in their right mind would believe the story — that we physically added sugar to our juices or that having a soft drink or a burger was equivalent health-wise to a highly nutritious juice. To our surprise, some people actually did believe it. To add to the drama, Today Tonight, the main competitor to ACA, did a story paralleling the claims about juice bars. This really got the public questioning whether Boost added sugar to our juices and smoothies, and wondering if our products were indeed as healthy as we said. I was horrified. I could not believe that people had started to doubt us. I thought if they saw how rigorously we vetted every product, if they realised how much time and effort went on taste, health and delivery — they would never believe these claims.
And then, as if this predicament wasn't bad enough, the Australian Competition and Consumer Commission (ACCC) got involved. With over 50 competitor juice bars opening their doors, some claiming all sorts of health benefits, the ACCC stated that they would be investigating all juice bars and their claims. The problem was this: Boost was the largest juice chain, so people just assumed we were the 'juice bars' the ACCC were after. The ACCC investigation was never directed at us, because we complied with all requirements and rules; our health claims on each product are 100 per cent backed up and documented.
Kristie says...
One of my first projects at Boost as their in-house lawyer was to advise on their obligations regarding food law. At this stage, many of the standards were new, and not a lot of commentary or information was available. All I could do was to rely on my interpretation skills as a lawyer. So for the first time since university, I went to the law school library and returned to basics. I remember sitting on the law library floor late one night and reading the Food Standards Code. I then began my audit on Boost, which resulted in me having to present to the board of directors the changes I believed were necessary to bring Boost in line with best practice and the Food Standards Code.
I soon realised that an in-house lawyer was expected to advise not just on the law but also on the flow-on effects to the business and, of course, the costs to the business of compliance. I had to tell the board members that in my opinion they would have to change the names of two of Boost's best-selling smoothies. Sales could be affected and a full panel of each menu board in Boost outlets would have to be replaced. Was I nervous? Very! But I had the support of J. She always backed her team and, to my surprise, the board listened, questioned me but then agreed that this was necessary for the business.
The board then huddled together and started putting together a plan to minimise the costs this would have on the business and to its franchisees. Years later, when the ACCC and local councils investigated juice bars for non-compliance with the Food Standards Code, Boost could be confident they were compliant — and, I have to admit, this made me feel very proud.
The icing on the cake (if you will) was when Bondi Council in Sydney blamed Boost for the litter on the beaches in their area. I found myself on radio station after radio station defending our honour against the ACCC and Bondi Council. Over and over, I explained that we had never and would never put sugar in our juices — and surely people were responsible for putting their own rubbish in the bin.
All of this attacked our core principle of 'I feel good about myself for choosing Boost'. Sales went from exceeding forecast, to flat, and then to negative growth. When your business starts to go in the wrong direction, it takes everything you have to stop the slide and turn it around. The worst thing: people believed what was written about us. It was simply wrong and unjust. I was crushed. What I needed to do was 'get over it', start to look within the business for solutions and not be a victim.
During this insane time for Boost, Jeff was amazing to have on my side. Having been in the tough, ruthless world of radio for 22 years, he had learned a thing or three and enjoys a good battle. (I think Jeff has read every war book ever published.) He has a very strategic view and sees the 'big picture' quickly.
The first thing we did was hire a lawyer for advice. We believed what the networks did was misleading. We contacted both networks and worked out an agreement, and they then assisted us in getting the correct message out in the media.
The second battle was the perception of our packaging and, by extension, its environmental cost. Was foam truly the best takeaway product for Boost? We contracted an environmental firm to compare and contrast. At the end of this report (and a dozen other reports we checked), it was conclusive — foam was indeed the best product for takeaway packaging. Foam is mostly air, it's 3.6 times more energy-efficient than the wax-coated equivalent paper packaging, uses fewer resources to produce, is recyclable, contributes fewer toxins to the water and creates less waste. Not to mention that foam is the best substance to hold a cold product. We published the full report on our website for all to read. Foam gets a tough rap from its early days, but it is a good product.
The third thing we did was hire well-known nutritionist Shane Bilsborogh and hop on the PR train to get our message out: 'A smoothie or juice is always a healthy option!' Our philosophy on health is really simple; in fact, Dr John Tickell explains it the best. He says people would not have a weight problem or a health problem if they did not worry about the low-GI diet or the Atkins diet, but instead followed the low-HI diet, or low-human-intervention diet. In other words, a diet that includes food that is as 'close' to the tree or ground as possible, and contains little or no processed foods. Let's not kid ourselves — if you read the back of a packet, can or bottle and find a ridiculous amount of numbers and words that you cannot pronounce in the ingredients list, you know that's not low-HI food.
By implementing these strategies, we were on the road to fixing some of the root causes of the external cracks at Boost, not just bandaging them. I could finally breathe.
Proactive boosters
All of the solutions mentioned in the previous section were great and they worked, but our biggest success came through picking up a mirror and looking hard into it, to see where the really major problems were. We had gotten complacent, arrogant and reactive. Growing to over 100 stores in four years meant we had also started to show some cracks internally, including in our staff training and systems. Our stores were looking tired, and so was our team. We just were not attacking every part of the business — so that's exactly what we did.
Geoff Harris's view was that we needed to keep the business compartmentalised, which would then make people accountable for their own areas and expenses, and reward people for their successes. So that's what we did. We broke each part of Boost into simple pieces. We began to get back on our toes and think proactively, reviewing and changing how we reported, cutting $2 million in expenses, and building a strong profit centre mentality in the business. No longer was there a black pit of expenses, as we reviewed small things, such as printing double-sided and only in black and white, as well as larger areas. We renegotiated everything we could, from our auditing and accounting costs to our raw ingredient contracts. We worked harder and smarter. The business became better because we took a hard look at ourselves.
Part of the marketing campaign to get back on track was hiring the hot hunk Tom Williams as our brand ambassador. NSW was struggling, so Jeff also came up with the 'Week Day Sucks' campaign. We invested heavily in PR and ran tactical promotions on TV, press and radio (see lesson 2 at the end of this chapter).
On the HR side, I again spoke to Lesley (from Bakers Delight). She was generous not only with her time but also with her tools, which never in my wildest dreams had I expected. During our meeting, we discussed all sorts of topics, including the challenges that she had faced growing a franchise network. Most of our discussion came back to the same thing: people. The horror stories were people-based and so were the success stories.
Systems and processes are the heart of a franchise network, and from my meeting with Lesley I realised Bakers Delight were light years ahead of us in their systems and knowledge. Lesley was more than happy to give us any system that would help and also allowed our executives to meet with her executives to see what each department could learn from the other. (In fairness, we were doing more of the learning than they were.) Many of the systems that we use today are a result of those meetings with Lesley.
After meeting with Lesley, we reviewed our training regime and HR strategy, because we realised who we needed where had changed as our business changed. These improvements at Boost were so effective they reduced our overhead by nearly $1 million.
We then looked at the stores that were looking tired, and started to upgrade both our company and franchise stores. This investment worked tenfold — each store that was upgraded immediately had an increase in sales.
From these few, meaningful process improvements came a dozen more; from this dozen came a dozen more and so on. This was the tipping point and it spread throughout Boost like wildflowers (see lesson 3).
This was a pivotal time in the business. The positive outcome and lessons learned made Boost the strong business it is today. Once we were back on the road to success, looking back and remembering all the stress and uncertainty that this period brought made me realise the downturn actually made us a stronger, more systematic and a substantially better company. When people ask me what our worst time was, my answer is, 'Boost is what it is today because of the worst times'.
The experience also taught us a great deal about ourselves. Jeff was the 'it will be okay' person, while I was the 'problem solver', running on fear and adrenaline. So I knew the risks of everything we did and all the 'what ifs'. Jeff was big picture, and did not share my fear of failure. His attitude and calmness (probably based on not knowing the details!) kept me sane. The stress that you go through when running a fast-growing business is enormous; Jeff had that ability when I walked in the door to make me feel everything would be all right. Without Jeff's calm energy, I would have fallen into a heap during this time. I know I mention Jeff a lot through this book, but that's because he has had the biggest influence on me. (So if there's a little repetition, it's because he deserves it.) Jeff has helped shape the woman I am today. He had more faith in me than I had in myself, and for that I am eternally grateful.
Most of the executives on the Boost journey have also made a huge difference. I've already mentioned just some of the many people who were a part of the initial team. I cannot say this enough: if we had hired the wrong people at the start it would have been very difficult, if not impossible, to get Boost to where it is today (see lesson 4). Mark, our current CFO, has created the systems and processes we have now, enabling us to truly grow. There are also many franchisees along the journey whose passion and brutal honesty helped us to continually improve.
Hitting the wall
After the domestic improvements, the international side of the business had also picked up significantly and I was growing that side of the business. We had a GM running the domestic business and Jeff was overseeing him. Travelling around the world creating deals was a blast! The intricacies of contracts and the differences in the countries and cultures suited my background. We were dealing with people from all walks of life, in places I had never dreamed I would visit. From Arabia, with men in their long white robes, to Estonia, where we had to walk in the middle of the road so we were not killed by falling snow from the rooftops.
I was obsessed with getting the international side of the business up and running, but this kept me away from my family for three months a year. My work–life balance pendulum was angling all the way to the work side. I had developed into a strong, confident businessperson with significant knowledge on how to start a business and make it a success, and had formed very strong bonds with people in my team; however, I was losing touch with my husband and family.
Going at this ridiculous pace, it was inevitable that I would get to a point when there was nothing left in the tank. Trips to a health retreat every year had helped, but there comes a time in every business that the reins need to be handed over. This brings new ideas, experience and passion to the table. I believe it is vital for a business to have fresh ideas and enthusiasm to continually grow. In 2006, I met with Geoff and Jeff for coffee and we started to look at a succession plan; two years later, I was still holding the reins and the business was still growing at a massive pace. We wanted the best person and we were prepared to wait for him or her.
Jeff says...
I decided at 40 to step down from my role as group program director of Australia's biggest radio group. Boost was going from strength to strength and financially we were okay, but at this stage not through Boost, because we still had not taken any money out of the business. I had signed a three-year consultancy/golden handshake deal, so I decided it was time to have a year off and take stock. How bizarre — I decide to relax, take a year off, and it almost costs me my marriage.
All of a sudden, the scales were tipped: Janine was in high-powered, global-expansion mode and I was in total unwind mode. Her self-importance, her three months of travel overseas a year, her long hours, her lack of consideration for family time, and her 20 texts each hour (bantering about work while we were playing with the kids in the park) — annoyed the shit out of me! Truth is, she wasn't that bad. However, taking a step back from my perspective, it was her self-obsessed world of focusing only on Boost — making everything else second — that made me question if this was the woman I wanted to be married to.
Thankfully, when the crunch came (and it was a big 'bang'), we both acknowledged our shortcomings and promised to work on the issues — and we never looked back.
Almost immediately, Janine made significant changes in her work–life balance. Clearly not working wasn't working for me, so I went full-time into the business as the CEO, while she concentrated on building our global push. At the time, Janine started to realise there was more to life than Boost; it had absolutely taken everything out of her and she had given it everything. She started getting her life back beyond Boost. Outside of Boost she had no friends, no hobbies and no other passions. That's when she had her epiphany.
On a business trip to South Korea everything changed. I arrived a day early to get rid of jet lag and decided to have a massage. It was an over-the-top, great massage. I even had mud gloves on my hands to get rid of toxins. To this day, it is the best massage I have ever had. I can still see myself lying there. And it was during this massage that I had the most amazing epiphany (though I don't think it was due to the mud gloves). I am not sure I have ever experienced such a strong, absolute feeling, and I still find it difficult to explain it. In that moment, I just knew I wanted to have another baby. There was absolutely no logical reason for having this feeling. Was it because I knew my time was up? Was it too many days spent entrenched in a 'man's world' and they were taking their toll? At the time, I was a director in the Hawthorn Football Club, an Australian Rules club, and in this and my other business groups I was one of very few women. Perhaps what I felt was an unbelievable desire to nurture, or simply it was a great excuse to step away from the business.
I was so excited after the massage I could hardly contain myself. I had to speak with my husband and tell him. I knew I faced a few problems before I even started. First, it would be impossible to reach Jeff because he was doing some work in Fiji. Second, per my request seven years ago, Jeff had had a vasectomy. So I did what any modern woman would do in the electronic age — I emailed him. I simply wrote, 'I want another baby'. My husband is a man of very few words and within an hour I had a reply: 'Sure!' I don't know what was going through his head at the time. He was probably laughing at me, because the vasectomy must have crossed his mind, but 'sure' was enough for me. My goal was to have a baby; now I just had to solve all the problems that were stopping me achieve this goal (see lesson 5).
Five minutes after getting the 'sure' I did a Google search on vasectomy reversal. I came across a microsurgeon who was based in Sydney and who just happened to be the surgeon who did the first hand transplant. He was also part of the team who performed the first face transplant. South Korea was approximately on the same time as Australia so, after a few wrong attempts, I got to speak to the doctor. He said there was normally a six-month wait but he'd had a cancellation on Wednesday and could do the surgery that day. 'That's fantastic — book him in', came flying out of my mouth. Far away, my unknowing husband was sunning himself in the beauty of Fiji. He was completely unaware of the plans I was putting in place. Of course, he had said 'sure'.
I arrived back to Australia on the same day Jeff arrived back from Fiji. Timing is everything with Jeff. I made sure I was prepared and waited for the right moment, which came after we had put our three boys to bed. We were relaxing with candles and wine. I had the lighting at just the right level. Jeff casually referred to my email. 'Oh that', was my attempt to downplay my hours of frantic research and phone calls. Suddenly I couldn't hold it in and blurted out all the facts and data; the romance went out like the candles. I was speaking so fast, but Jeff just sat back, smiled and said he would do it. He suggested that we start the research on reversing the vasectomy. My eyes widened, 'No need — I have found a surgeon who is world-renowned'. Jeff didn't say anything, so I pressed on. 'We are sooooo lucky because he can do the surgery this Wednesday.' This took Jeff back a bit and, after a long pause, he said, 'That would be my birthday!' Needless to say, I did not win wife of the year — although I did spend more on his birthday present that year than I had in all the years we had been together.
Jeff says...
I was in Fiji at a conference and Janine had just had a massage in South Korea. An email came through from her: 'I want to have another baby'. My reply to this was short and sweet: 'Sure!' Janine had been on loan to Boost for the last seven years and this was a sign she was rediscovering some balance; I was going to get my wife and family life back. Don't get me wrong — Janine was still working day-to-day in the business, but to have one other thing to focus on (and problem to solve) was gold for her.
Jeff's vasectomy reversal was a success in theory; however, three and a half months and many failed tests later, none of the sperm were alive. We also interviewed for international adoption. Almost a year from the date of my epiphany, we were still no closer to having another child.
This baby continued to 'tap' me on the shoulder from wherever she was, telling me to make it happen. Back to research my options. The next path was IVF and, though this road is complicated and emotional, I felt for me it was relatively painless. However, the first four attempts failed. All this was happening while Boost was growing leaps and bounds. We began to look into domestic adoption when a friend suggested a doctor in the United States who'd had amazing success. Because we were always travelling for Boost, we thought, Why not? After a further few unsuccessful attempts — finally, we were having another baby! I was 42 years old and going to be a mother for the fourth time; I could not have been happier. Jeff had been so supportive of me and this decision from his first 'sure'. Watching him with his daughter and witnessing the love he has for his 'princess' makes all the effort of all those years totally worth it.
Building a Zoo
Even with my focus on bringing the very much wanted Tahlia into the world, there was no question that we still had to grow — all businesses do. But growing any business and developing a brand is extremely difficult. Having successful systems and processes in place makes it easier, but people still tend to underestimate how hard it is to actually grow.
We realised Boost had started to reach its capacity of growth within Australia at around the 200-store mark. We understood it was not about growing Boost anywhere we could put a store; we were very strict on where we believed the Boost brand could be successful. At the same time, international growth was amazing, with 12 countries signing a master franchise agreement. Signing agreements to open Boost stores overseas is one thing, however; opening stores in these countries was slower than expected. Then it hit us — we were just so focused on growing the Boost brand domestically and internationally, we were not exploring other options.
We have always believed in the value of holding strategic, working retreats with the team, to evaluate where we have been and where we need to go. It was during one of these off-site retreats that we asked a single, simple question: 'What are we good at?' A few hours of brainstorming and numerous white-board scribbles later, we discovered what we were good at wasn't our fluffy bits of marketing; it was our boring back-end. We were great at business, franchising, IT, design and development, legals and finance. These were our cornerstones. These back-end departments gave the business the strong infrastructure required to grow. My annoying focus on every detail and refusal to outsource these departments (see lesson 6) were the keys to building these cornerstones into the foundations of our business. At the end of this exercise, it all seemed obvious; we'd identified what we were good at and would utilise that to grow.
Geoff says...
The cultural wiring of the company was a 'support centre' that was team-based and focused on providing a 'fee for service' — in other words, people in every department thought of themselves as their own little business. A 'profit culture' was reinforced strongly throughout the group. From each department's members to all the company and franchise shop teams, a non-hierarchical structure was vital for the company's growth. We also introduced reward and recognition nights, and annual conferences and award nights to recognise key achievers, and welcome new franchisees and new ideas into the Boost family.
After much discussion our strategy was to create a new parent company — Retail Zoo — of which Boost would be the foundation brand. Retail Zoo would facilitate the growth of strong small businesses, using our proven back-end and marketing skills to make these concepts successful. The brands we would look at growing had to be pre-existing and had to show evidence of being successful models. Often the entrepreneurs who started these kinds of concepts struggled to take their businesses to the next level. Largely, this was due to lack of expertise and the resources required to do so. Retail Zoo, however, had both the financial and intellectual resources to take a four- to ten-store business model and grow it within Australia and around the world. What you discover when you speak to businesses of all sizes is that strategy is 10 per cent of the success — execution of that strategy is the other 90 per cent. We had the know-how to execute a concept into a successful brand and business.
Once this strategic step was made, everything fell into place. To be honest, after me growing Boost to where it was at this stage, the business needed new legs to continue the journey. Jeff was fresh after a year's break and we were ready to swap roles; he was chomping at the bit to make his mark on the company. Jeff became the CEO, Retail Zoo was born and we started the journey of buying up-and-coming brands.
Jeff came across a brand in Chadstone Shopping Centre (in Melbourne) called Salsa's Fresh Mex Grill. I've always been a fan of authentic Mexican food and the variety of dishes it offers — while I don't like Mexican food that's stodgy, heavy, dull and uninspired, I love Mexican food that's made with fresh ingredients such as coriander, red and green chillies, tomatoes and limes. We have our own Aussie version once a week, and it's such a great way for the family to put together their favourite combination.
Like the smoothie and juice industry when we started Boost Juice, when we first looked at Salsa's Fresh Mex Grill there was not really a 'fresh Mex' category in Australia, or not one that was well known. In the United States, Chipotle Mexican Grill was growing a market in this fresh Mex category, after early investment from McDonald's. Salsa's Fresh Mex Grill had four company-owned stores and they were operated by Lawrence Di Tamasio — a wonderful man with a real passion for his food. The financial numbers coming out of the stores were amazing, thus proving Lawrence had found a successful fresh Mex formula for the Australian palate. We invested in the company and then, a year later, purchased the whole company. Since then, we have taken Salsa's from four stores to 52 stores in two and a half years (at the time of writing). The Mexican wave is here, with a number of contenders now in the marketplace — again, a very similar trend to that of the juice market in 2004. Like the juice market, the fresh Mex market will settle down to a couple of dominant players. Lawrence is no longer in the business; he was handsomely rewarded and is now building his next concept.
Salsa's has been an excellent addition into the Retail Zoo family. In any business, change is one of the hardest things to manage and Retail Zoo was no different. Retail Zoo went through its own transition period, moving from everything being about Boost to a situation where people needed to think, and systems needed to be bigger and broader. Managing this change led to one of Jeff's greatest skills: his ability to hire great people. We have two different leadership styles — I am more hands-on, while Jeff is more hands-off. His looser reins allow his executives to thrive, and his new ideas and approach have proven to be incredibly successful, taking Retail Zoo to the next level.
Pausing to reflect
The birth of Tahlia gave Jeff and me time to reflect on what was important in our lives. After riding this incredible 50-foot wave of business, we asked ourselves, 'What do we really want out of life and what does the future look like?' We were over 14 years into starting Boost and, even after creating Retail Zoo, we still really had all of our eggs in one basket. We were also aware that we needed more expertise in the international markets — though we were having some success overseas, we knew we could do it better.
Since the beginning of Boost, we have owned various percentages of the business during different times. I had never been overly concerned about the percentage we held — I focused more on the value of that percentage, and having enough percentage to ensure that we maintained control of the direction of Boost. Other than that, having the right partners was always the more important aspect of this balance, and making certain that all partners could contribute within their field of expertise.
Throughout the whole journey of Boost Juice, we had talked about ways of growth, and going public was discussed a number of times. We spoke to a friend of mine who was an expert in the field and who described the various options. One was private equity (PE). We had never considered PE before, because we had heard some horror stories from people who had gone down that path. If you have an open mind, however, you never know where this can lead, and that is what we adopted. We began to research all of our options and realised the right PE partner would provide us with capital out of the business for personal use and put capital into the business for growth, and we could also find someone with strong experience in the international marketplace.
We retained the services of my friend's company Wingate and once again embarked on a new learning curve — selling part of our business. The important thing for us was that we maintained real control of the business — that is, we would still control the day-to-day operations. Where our chosen PE partner came in would be to assist us in our international expansion, use their expertise at a board level and be a good supportive partner as we continued to grow Retail Zoo.
During the initial interview process we met PE companies that were 'shockers' — some of which we wouldn't ever consider doing business with. Thankfully, we were introduced to Riverside, an American company, and, after meeting all the partners in Riverside, we were quite comfortable they could meet our objectives and would be a good fit both personally and for Retail Zoo.
We started discussing a possible deal with the partners at Riverside. Selling part of my fifth child was an excruciating experience for me. However, I knew it was the right move to make on a very personal level, enabling us to take hold of opportunities that we have always dreamed of (see lesson 7).
We maintained over 25 per cent of the business, and made certain that we controlled areas of the business that were critical to us. The reality of any decision is that you never know for sure how it will play out. There is always a 'honeymoon period', when everyone is on their best behaviour. What would it be like when something did not go our way? Fortunately, it turned out that our new investors are great to work with. The honeymoon period is over and they are focusing on what they do best — finding new acquisitions for Retail Zoo — and leaving us to do what we do best.
Geoff says...
Looking back, I believe that the two key growth elements in the business were first getting to $100 million in turnover, and then to a $10 million profit. Both economic indicators proved that Boost had arrived as a very serious business indeed in Australia and was now on the international radar for investors.
After resigning from the Boost board in 2010 and selling down to a 6 per cent holding in the company to Riverside, l have a real feeling of being part of something special. A brand that is indeed one of Australia's premier brands, and one that has achieved its success with real expertise, class and verve from Janine and Jeff as the founders and main shareholders. They both deserve immense credit for their vision and business skill.
Boost Juice is a vastly different company than when we started — as it should be (see lesson 8). All of our executives now have shares in the business, allowing them to participate fully in the success going forward. The future is truly exciting for Retail Zoo.
Often I think how lucky we are that both Jeff and I can run the business. Though we have different approaches and it hasn't always been easy working this closely, we have an enormous respect for each other's talents. In any and every battle, I want Jeff beside me. For now, we have completely switched roles. I see the helicopter view and Jeff is knee-deep in the details. Boost is a part of me and always will be. It would be difficult to give it up forever. I am there for advice, keeping the brand DNA on track and always asking the right questions. For now, I am where I should be. I seem to prefer the roller-coaster in life more than the merry-go-round — it's just that little bit scarier and so much more fun.
Jeff says...
Janine today is the best she has ever been, and I think the best she ever will be. She is totally self-confident; she is at peace with where she has taken the business. She has let others come in and do what needs to be done, she has a beautiful daughter and three sons (who love her to bits), and she has unleashed her high-achieving, obsessive, improvement capacity on the wonderful world of yoga.
Funny enough, we made a deal at the start of 2012: she would get into horses (which is my obsession) if I would get into yoga (her obsession). The result? We're both getting 'obsessed' with each other's suggestions, which is great.
With Janine, I can honestly say what you see is what you get. She is a beautiful woman inside and out. A girl from the 'burbs who worked her guts out, while simultaneously scaring the crap out of herself, to build a new and exciting brand in Australian retail. I have been a very lucky partner to share the journey with her. We made it out the other side stronger and more in love than ever.
The day-to-day hard grind and day-to-day Boost journey is over for her, though she will always be associated with the brand and will always be the founder of Boost Juice. And all I can say is wow! What an incredible journey... And you will never meet a better chick.
The secret
What's the secret to it all? There isn't one. I believe that my personal success is due to a combination of many factors: naivety, my certainty that there is a solution to everything, and the great people who surround me. I now believe in myself and, if I'm given a challenge, I don't question that belief.
I believe Boost has been such a phenomenon because it's a great product and it's marketed well; every store acts as a billboard for our brand. Beyond that is my unshakeable faith in the necessity of doing the right thing, of having the right people and looking after them, and of understanding your customers and giving them what they want.
I'm passionate about health and I want to do everything I can to help counter the terrible toll that the fast-food lifestyle is having on our society — particularly on our children. The desire for a healthy alternative for my own kids was a large part of my initial interest in juice bars; that desire hasn't wavered. Passion can't be faked. Our sincerity is one of the reasons our customers have taken to Boost Juice.
My philosophy about the importance of sharing knowledge and experience is the reason I have written this book. In the following part, you'll find the key ingredients that make up my particular recipes for success. In this part, you've read a bit of my life story — and if I can do it, so can you!
Story so far... lessons learned
Here's what I've learned in the last few years with Boost:
1 You need a fully fleshed-out concept for a brand, and you must provide the whole package for your concept to succeed.
2 Marketing is important. Often when a business is in trouble the owners or managers look for areas where they can save money — the worst area to take from is marketing. During our downturn, we added $1 million to the marketing budget.
3 People can manifest their reality from either fear or love. In a franchise network, if the sentiment is negative, it spreads like a disease. When you can inspire and get people to believe in something positive, this is what spreads.
4 Surrounding yourself with the best people is vital to your success. No one person can do it all. The reason that Boost is Boost, and not one of the other juice bars that no longer exists, is the people who I choose to have around me. I think of them as my personal board members. When you get your people right, it makes everything flow. When you get it wrong — get it right quickly!
5 Identify your goals and then how to solve the problems stopping you. At Boost, there have been many problems where I truly believed a solution would not surface; however, I would not give up until I found one.
6 Keep important business areas in-house. Over the years we have tried to outsource some of Boost's departments; however, in the end, there is nothing like working with a team of people who are passionate about only your business. When you outsource, you are one of many; I feel these providers often lack the enthusiasm needed. Keeping important areas in-house means everyone is on the same page and working together.
7 Having money in the bank does not make you happier; however, it does create the financial freedom to do what you want, not what you have to do.
8 I look at businesses as children: in the early days, they need every part of you; as they grow and become more mature, they need you in different ways.
Part II
Thirty recipes for success
In this part, I provide what are indeed the 'recipes' for our success. These are what we run the business by, and we use them to remind us what we should be doing when making business decisions, both day to day and long term.
I've reiterated the key points for each recipe with a list of important ingredients. In some of the recipes, I've also included super supplements for you to try yourself to really give your journey a boost.
Chapter 5
Executive juices
The recipes in this chapter are all about you — building such aspects as your integrity and confidence, and finding your motivation and passion. So start squeezing — and really get your juices flowing.
Integrity
A lot of people talk about 'integrity' — do you know what it really means?
These days, people talk a lot about 'integrity' and 'values'. They've become buzz words, used to impress. It's true that if your company is known as being one of integrity, you will attract and keep like-minded people — honest, reliable, moral people. Yet ideals often seem to disappear if the bottom line is affected. The words 'integrity' and 'values' mean nothing if they're not backed up with solid hard work and the right decisions.
So, what is integrity? To me, integrity means always telling the truth and acting in a way that is right for the business, no matter how hard that may be and no matter what the cost. You can't please everyone all the time. There may be some people who believe you haven't acted with integrity — whether it's an employee you had to let go, or franchisees who think it's your fault they haven't made more money. You can't always help what other people think, but as long as you know you've done the right thing, you can feel at peace. Having true integrity has made Boost what it is today.
Janine says...
We had a senior executive in the business who was an amazing talker. When he spoke, his words and tone instilled confidence, and he always seemed to have the answers to all the hard questions. However, this man lacked true integrity — it was all about his ego and making sure that he looked as good as he could. So he would often lie and just tell you what you wanted to hear.
As we came to discover, these can be the most dangerous types of people to have in your business, because it takes a long time to realise what they are really like and, in that time, they can do a lot of damage. By the time we uncovered what this man was doing, his area of responsibility was in a terrible condition and we had started to get a high level of resignations — and, worst of all, the culture of the business had changed. Your business's culture is just like fitness — it can take months to get your fitness level up and only weeks to completely lose it. We realised we had (admittedly unknowingly) gone against one of our biggest rules: only hire people with the utmost integrity. At the time this man was hired, we thought he had integrity but we were sadly mistaken. Often mistakes make the business better, and this is true in this case — we now have better systems in place to make sure this does not happen again.
Key ingredients
Here's how to act with integrity and ensure it in those around you:
• You can't create a workplace culture of high integrity without hiring people with high values. Be single-minded in seeking out the right staff.
• Always aim to keep your word; if you have to change your mind, be honest about the reason. Be straightforward, accept that you are wrong and don't make excuses.
• Seek consistency between what you think and what you say. People can see through lies, and you cannot fake sincerity.
• Yes, it does sometimes cost money to do the honourable thing. Look at it as a short-term financial loss that can be absorbed. A loss of trust can never be regained.
Super supplements
Add a boost to your integrity with the following:
• Integrity breeds integrity. The holder of the values in any company or business is the CEO and/or owner. If you are a leader of high integrity, it will flow through to the rest of the staff. With that said, the leader should be surrounded by some ruthless minds to ensure balance.
• The core leadership team of a business must be beyond reproach. The people who run a company set the cultural tone. If they maintain high standards on what's right and what's wrong, a culture of great integrity will develop. You need to be rigorous in your expectations of staff and make a hard call if someone lets you down. If you don't, you will lose the people who matter most.
• Avoid people who have low standards of integrity, but don't judge someone on reputation alone. While a doubtful reputation always makes me cautious, I prefer to make up my own mind. Never ignore your gut instinct or intuition.
• Always support your staff in front of people outside your business. If you don't agree with a colleague, have the discussion behind closed doors. Ensure that your staff and peers do the same for you.
• Know your limitations and do not bluff; if you need expert help, get it.
Motivation
I will, I will, I will, I will — you need all the will in the world to get where you want to go.
I have five words for you: move it or lose it. Unfortunately, I can't teach you how to find the will to do so. Motivation, inspiration — call it what you like — is different for each of us. To succeed, you must unlock that deep personal energy. So, what motivates you? If you can't answer that question, it's time for you to find out. Without that knowledge, you'll never experience the total joy of achievement.
I can't give other people motivation, but I can provide a culture where their achievements are recognised and rewarded in a way that is meaningful to them. When it comes to motivating staff, I look for their 'hot buttons'. For some people, a small gesture is enough — a metaphorical pat on the back when all seems lost. Others may take more convincing and may appreciate incentives, such as time off work or a small gift of appreciation.
I believe that motivation is born from working with great people and achieving great things together. The bigger your purpose, the more fulfilled you'll be at the end of every day.
Janine says...
Many things motivated me on my business journey, but the biggest one in the early years was fear. I had everything on the line (as I talk about in part I): we had sold our family home, all of the cash was in the business and we were renting. Not to mention the home doubled as our office. At 34 years old and with three children, I truly did have everything to lose. There is nothing like putting yourself in this position to give you motivation! The business had to work no matter what so, one way or another, I had to find the solution to every problem.
Fear was there as an underlining factor on the journey but something else also popped up — and that was passion. I found that not only did I love the world of retail and business but also, to my surprise, I was actually quite good at it. My brain worked in systems and in high detail. Whenever something went wrong, my mind immediately went to solutions, and then to ways of ensuring a problem of this type never happened again. Suddenly, the business didn't seem like work anymore — I loved it! And when you love something, motivation to continue is a by-product of the journey.
Key ingredients
Keep yourself and others around you motivated with the following:
• Remember to say thank you. My own son works in a cafe and he said to me once, 'Truly nice people are nice to waiters. How hard is it to say thanks?' People work incredibly hard at Boost and I know how important it is to acknowledge that. The biggest complaint most people have in their workplace is that they are not appreciated for the work they do.
• Always celebrate your successes. When you're incredibly busy, stopping to acknowledge success can seem like a chore. However, it shouldn't be optional, because it's vital for ongoing morale. Imagine if we hadn't allowed our staff to sit back and enjoy the fact that we'd launched bottled Boost Juice, or that we'd opened our first international store in Chile, or purchased a competitor and converted every single one of their stores to Boost stores in ten days. It's hard to find ongoing motivation if you feel like all your work is for nothing.
• Keep going, even when others give up. Having the will to do so can serve you well at home and at work.
• Great people thrive on working with great people. Continue to lift the bar and create a place to work where people strive.
Super supplements
Here's how to get yourself super motivated:
• Initiate a structured reward program for staff. We reward our customers with a loyalty incentive, so why not do the same for staff members? This is one area where simple is best. For most people, the biggest motivators are ownership of tasks and acknowledgement of achievements; successful completion of tasks should be recognised and/or rewarded.
For example, every three months we take all our staff out for a Boost Night — a night of reward and recognition, where we celebrate achievements and being together.
• It's difficult to be inspired if the company culture is negative. Provide a stimulating environment in which people can achieve personal goals, as well as great outcomes for the company.
• Your key executives must be self-starters, problem-solvers and leaders. If motivation can be judged on a scale of one to ten, look for the elevens — those people who have shown high levels of achievement throughout their lives, in different areas. Give them an environment in which to thrive... and look out!
Passion
Having passion is the easiest way to make money, but the hardest thing to find.
You can teach anything and train anyone, but you cannot instil passion in someone. Passionate people are few and far between. To me, they are like gold. They stand out. People are attracted to them and believe in them; people want to be them and invest in them. Do you have to be loud, extroverted and opinionated to be passionate? No. You can be quietly passionate. This may mean you're overlooked — but not for long. Along with integrity, commitment and a strong work ethic, passion is always recognised.
So what do I mean by a passionate person? Essentially, passion is tied up with loving what you do. To achieve a goal, you need to throw your heart and soul into it. If you don't, you're in trouble.
When we have a problem at Boost, everyone on the team is focused on the solution. Their eyes sparkle with the challenge; they will do whatever it takes to sort it out — whether it's getting on the next plane or working through the night. When Boost first began, the business took over our lives. Jeff and I were surrounded by work 24 hours a day, seven days a week. That level of commitment gets tiring very quickly if you're not passionate about what you do. When you are passionate, it doesn't feel like 'work'. To be passionate, you need to be doing something that means something to you. You can't fake it. If you do, you won't make it.
Janine says...
I have done an enormous number of media interviews over the past 12 years, and the key thing that I'm often told afterwards is that I ooze passion. When people ask me about Boost, my eyes start to sparkle, my body gets taller and then I'm off — spouting all sorts of information about this business that has been so much a part of my life.
What I also have found is that passion is relatively rare — and so I'm lucky that my husband has it in spades as well. Sometimes, I would say that a fine line exists between passion and obsession. (These days, when you ask my husband about a horse, his whole face lights up, and he can happily tell anyone who is listening the pedigree of any horse you care to mention.) But this energy and obsession is why others like passionate people and want to be around them — they're addictive and others want some of what they have.
Anyone who has achieved incredible goals has passion, without exception — whether it's a top sportsperson or someone who is an amazing musician, passion and commitment is the common bond.
Key ingredients
Here's how to find your passion and act passionately:
• What do you love to do? If you can identify your passion and make it your life's work, you can't fail.
• Passionate people love going to work because they love what they do. Can you imagine a better feeling than getting paid to do what you would do for free?
• Don't overlook the quietly passionate. You don't need to make a lot of noise to get a lot done!
Super supplements
Increase your passion with these tips:
• Passion is like a drug. When you achieve things that other people do not, passion grows. Like the endorphins created when you exercise, passion fills you as you meet goals — both big and small. What are you passionate about?
• Find out everything you can about whatever you're passionate about. As you discover more and your knowledge grows, so too does your passion. Passion means that learning is no longer a chore but, instead, it becomes a joy and something that fulfils your soul. Passion and purpose is what we all look for at Boost.
Confidence
Self-doubt is your greatest enemy.
Confidence is everything. If you believe in yourself, others will believe in you too. Even if you have doubts, hide them! Your confidence is your shield — it will protect you and your team from that highly contagious disease known as defeat. (This is different to pretending you have all the answers when you don't — something I would never recommend.)
When Boost went from being very small in scale to a medium-sized concern, I found myself questioning whether I'd be able to manage the growth. I was worried that I'd had no formal business training or prior experience. The fact that the expansion happened practically overnight, because Jeff had secured 18 sites with Westfield, didn't help! How did I overcome my concerns? With common sense, a clear vision and by looking at the people who believed in me. I realised these people weren't stupid — and if they backed me all the way, who was I to question their judgement?
Sometimes, questioning your own ability makes you work that much harder; having great people around you is an invaluable safety net. And it turned out the keys to running a business were not as complicated as I thought. If you simplify everything, are sensible when making decisions and look for the solution that exists for every problem, you'll go far.
Janine says...
Don't think I'm kidding myself — when I started Boost Juice, I certainly didn't have the confidence that I have today. In the early days, I used Jeff as a crutch and, if anything was too hard or too confrontational, I would turn to him. Jeff, of course, had no issues with telling someone how it was.
For example, early on I invested over $5000 in a cash register for the new store; however, I soon discovered that it was an absolute lemon. The salesman I'd dealt with was a pig and basically told me that it was my bad luck. We had hardly any money and the $5000 would not be easily replaced, so I went for my fallback response: calling in Jeff. He called the salesman, who again said he was not refunding the money. So Jeff told him that he was sending over a man and he expected him to give this man the refund in full.
Jeff hung up the phone and put on a dark suit, his shoes with the biggest heel and some sunglasses, and went to the showroom for the money. He introduced himself as Jeff Jackson and that he was here to collect. The salesman was clearly shaken and told Jeff he would have the money in 30 minutes. Jeff went into the car to wait when the phone rang — it was the salesman, who told Jeff that he was not impressed that he had sent in a thug to collect his money. Jeff calmly told him that Jeff Jackson was not leaving until he had the money in total. Needless to say, 'Jeff Jackson' had a full refund returned to me that afternoon.
I was always conscious that I hadn't gone to university, and believed that most professional people who I came across would know more than me. However, one thing I did have was a curious mind — I wanted to know more and I wanted to not have to rely on anyone but myself. The other thing I came to realise, and which Geoff Harris had instilled in me, was that no-one could know my business like I did, so I shouldn't follow advice blindly.
So I started asking more questions, not caring if the questions clearly showed how little I knew. I started questioning some of the documents and discovered that common sense and logic were really the main skills you needed in business. By asking questions, I also discovered how often so called experts actually make mistakes, and that they didn't always know as much as I gave them credit for. Slowly, I weaned myself off Jeff and started to take on more of the difficult problems myself — even if, in my mind, I always had Jeff Jackson 'on ice' for another day. However, I'm happy to say that we never had to use Jeff Jackson again. It was my money that I was using, so I became a tough negotiator, making sure I was always over-prepared for every meeting. I'm not sure exactly when the change took place, but I do remember Jeff commenting that I wasn't using him as much anymore — in fact, I think he was a bit put out that his services as the 'tough guy' were no longer needed.
Education is important, but you should never lack confidence because of any formal education you feel you've missed out on. Education comes in all different forms, and people learn in dozens of different ways. I can confidently tell you that I know more about local and international trademarks than most lawyers — not because of a course, but by actually working out the issues in practise and learning along the way.
As you get older, you do get more comfortable in your skin. I still, to this day, listen more than I talk, and I will continue to ask and ask my questions until I believe I truly understand. Knowledge gives you confidence.
Key ingredients
Here are some tips for building your confidence:
• Having the confidence to use your knowledge is critical. Knowledge is pointless unless you can back it up with decisive action.
• Confidence is contagious. I focus on solutions — solutions that my staff believe in. Their belief means they also catch the 'I am, I can' attitude.
• Instil confidence in others through delegation. Make sure they feel your total trust and support. Give them the tools they need to complete the task and ensure they have all the knowledge they need. Allow them the space to ask questions without losing face if they don't know something. Make yourself into the kind of boss who inspires loyalty and hard work — your staff members will naturally feel that they don't want to let you down.
Super supplements
Make sure your confidence is unshakable with the following:
• Confidence is everything in a team because it eliminates uncertainty. If people in management are insecure, it will suck the confidence out of those around them. You will know people who lack confidence by the following traits: they hold onto power by not sharing their knowledge with colleagues and staff, their moods are as changeable as the weather, they lack communication skills (which leaves their colleagues wrong-footed most of the time), they manipulate and/or turn people on each other, they find it hard to make decisions, and they may out-and-out lie. There are varying degrees of this behaviour, but if someone is guilty of these traits, you have no choice other than to remove the offender. Remove people like this quickly, as they do enormous damage that can take months to repair. These insecure behaviours are cancer in your organisation.
• People who feel good about themselves have confidence. Here is a simple exercise to boost self-confidence and build a positive team environment. Get your people together and have everyone write their name at the top of a sheet of paper. Hand the papers around, and ask everybody to write an affirmative statement about the person whose name is on the top of each sheet. Each person ends up with a list of positive attributes. As an extension, you can have each sheet framed and presented to your team members. On those 'all too hard' days, they will be able to get out that list to help reinforce their self-belief.
Reputation
Would you do business with you?
Many doors will open or close for you depending on your reputation, which will precede your presence every time. The people with whom you're doing business with will have made it a priority to learn about you before you meet. What they discover — usually through the filter of other people — will either give you an edge or create early difficulties for you in your dealings with these new associates.
A 'firm but fair' reputation is the best one to have. What is yours? Does it need work? Try this exercise: Write down five attributes that you'd like people to say you have. Stick that list in the front of your diary and refer to it often. Are you achieving your desired reputation?
I strongly believe that the presenter is more important than the presentation, the messenger more important than the message. If you have a good reputation, guard it with your life. However, if it has been tarnished in some way, now is the time to work on it. The only way to get it back on track is through action, not words. Don't tell me that you've changed — show me.
I remember dealing with an advertising agency that promised Boost the world but didn't deliver. In fact, it dumped us at another client's request. The ad agency rep claimed that he wasn't responsible, but he should have been honest about the process. If he'd kept me informed about what was happening, instead of blaming others when it was over, the situation wouldn't have been such a disaster. That rep is still trying to win me over. I'm listening, but I'm cynical. He'd have to do (emphasis on doing, not talking) something amazing to win back my trust. All this is not to say that I don't believe in giving second chances. After all, everyone makes mistakes — just make sure you get it right the second time around.
Janine says...
Every person and company has a reputation for something. It is the core of what you are. Jeff had a reputation for being a tough guy in radio and, even when he became a father and got a bit older and wiser (and a bit less tough in his core), he kept up his tough-guy behaviour because it suited his role at the time. I have worked with many businesses and suppliers over the years and have found it's usually no surprise why companies have the reputation that they do. What type of business it is and what type of reputation it's creating is usually highlighted when something goes wrong.
If it's a great business, managers and staff will go above and beyond to fix the problem, regardless of who is at fault. These types of businesses focus not on short-term losses but at the long-term reputation that they're developing — and so they thrive.
The other side is the type of business where, when things go wrong, managers and other staff put their heads in the sand and go into blame mode. This is such a short-term view and often these types of businesses are the ones that don't survive — and the owners are left wondering what they did wrong.
Your business reputation is everything — it is why you will get repeat customers and referrals. A personal reputation is the same. I believe that my reputation is firm but fair — if you commit to me that something will be delivered by a certain date and you do not deliver it without any communication, I definitely have a reputation for not accepting this for too long.
Key ingredients
Guard, or improve, your reputation with the following:
• Treat your reputation as an important part of your business arsenal. It will get you through doors — and also close them on you.
• If you've made a mistake in the past with someone, fix it. Win back this person's trust before a tarnished impression of you loses an important opportunity in the future.
Discipline
A lack of restraint can create big problems.
Self-discipline is the most common trait of great and successful leaders. Commit a few undisciplined acts in politics or business and soon you're being asked to drag yourself around the chat-show circuit.
Take alcohol, for example — just one alcohol-fuelled stupid instance at a work function and you could lose hard-won respect. In other words, if you drink and work, you're a bloody idiot! Nobody gets smarter after a few drinks. As a rule, I never have more than two drinks at a work function. Whether you're drinking with colleagues or suppliers, anything you say, can and will be used against you the next day.
Set aside time each day for exercise. If you have the discipline to look after yourself, you'll feel better, look better and function better. As an added bonus, fit-looking people seem to garner greater respect from others, probably because others see that personal discipline will carry over into business.
Being self-disciplined doesn't mean being self-denying; it's not about working harder, it's about working smarter. Self-discipline is getting to your children's sports days, being home when you told your partner you would be, keeping appointments and being on time. Self-disciplined people aren't the ones in the office from dawn to midnight — they're organised enough to get their work done within a reasonable time frame.
Janine says...
One of my first jobs was working for an advertising agency as a media assistant. Back in the 1980s, the Christmas party where I worked was legendary. One year, a senior executive was absolutely blind drunk. He made a pass at anything in a skirt, he threw a typewriter through a window and assaulted another client. He was fired the next day. The shame of it was that this guy was a good, solid executive who was simply a nasty drunk. If he'd had the discipline to respect that it was a work function, he could have had an amazing career in advertising.
Key ingredients
Here's how to choose the right elements for self-discipline:
• In all aspects of life, control yourself before you influence others.
• If you are serious about career growth, say goodbye to alcohol and drugs and focus on your goals. You'll be amazed at how much more clearly you're able to view opportunities.
• You've heard it before — work smarter, not harder. Disciplined people aren't distracted during business hours, which means they get the job done in time to have a social life. How much time do you waste each day?
• Keep fit. Not only will you have more energy and be ready for any challenge, you'll evoke a great response in others.
Super supplements
Here's how to take your self-discipline up a notch:
• Do you know your weaknesses? List the top five character traits that you feel may hold you back or be your undoing. Keep this list in your diary and tackle one issue at a time.
• Are you a slave to your email inbox? Set aside an hour or two each day to organise and respond to your email, and stick to it. Don't let it distract you outside that designated time. If you do, you will only spend each day reacting to emails.
Listener
'An essential attribute of a good conversationalist is to be an equally good listener.' \- Errol White
No-one learns anything while talking. Everyone has ears and using them to listen might be the most underrated quality of all. It goes hand in hand with asking questions, which is one of my favourite pastimes.
Are there people you admire? People you consider successful? If you get the chance, talk to them. If they're not immediately accessible, perhaps send an email to their corporate website or attend a public presentation they're giving.
Asking questions of successful people helps to remove the aura of achievement that surrounds them. You could ask them how they got started. (Usually you'll find the start point was a very basic opportunity, perhaps something available to many — only successful people have the foresight to grab opportunities with both hands.) You might ask how they find the best people, how they motivate their staff, and whether there is anything they'd do differently if they had their time again. See if they have one or two pieces of great advice they could share. Generally, most people like helping others — why not let them help you?
When I started out, there wasn't one particular person whose success I aspired to emulate, but there were two people who helped me. Geoff Harris has had a profound effect in assisting Boost's growth. After he came on board, the business was able to go to the next level. Lesley Gillespie from Bakers Delight has also been very generous with her time and insight. She is an amazing woman who has been a great help to me.
Janine says...
We're constantly looking for great talent on every level. Recently, we were in search of a great board member (and we still are, for reasons that will become clear). We felt that we needed more experience in international expansion on the board, and were recommended a man who had run a successful franchise business in the United States. We decided to try him as a board member. We paid for his first-class flight over from America and put him up in a five-star hotel. We then put him on the road with our senior company-store and franchise managers, so he could understand and learn about our business and get a feel for where he could contribute.
The feedback from our senior managers was horrific. They told us that all he did was talk — telling them how good he and his business were. He did not ask any questions on the financials, the product or the Australian market, he just talked and criticised. When I was told this, I initially thought that they may have been exaggerating a bit — until I got into the board meeting with him. The meeting opened and so did his mouth, and it just kept going. People were polite for a little while, but pretty soon I'd had enough and told him to shoosh. Yes — I said, 'Shoosh. That is enough'. He looked at me in shock and told me his wife tells him the same thing — at which I suggested he listen to this wise woman.
When he was quiet enough that we could ask questions of him, he didn't have any answers. We asked him how the Australian model and his model compared, what the differences in the systems and processors were, how our labour rates compared with his, and what he'd discovered during his time on the road. He did not have one answer to the questions that we put forward — because he hadn't cared to find out or to gather enough knowledge on our business to be of any value. He was shocked and mortified that we did not add him as a board member.
As I said, you never learn anything from talking.
Key ingredients
Improve your listening skills with the following:
• Listen to other people, and take notes. It's a great feeling to capture someone's full attention.
• If you can, create an opportunity to talk to someone you admire. Find out what worked for this person in the early days, and what didn't. Listen carefully to the advice. Although not everything this person says will be appropriate for your situation, much of it will transfer. Business is business, whether you are selling insurance, sofas or juices.
Super supplements
Here's how to become a true listener:
• Make a conscious effort to ask more questions than you usually do and to say 50 per cent less for a day. Evaluate what you learn from that exercise.
• Don't assume you have all the answers, even when in a situation you're familiar with. And don't think you're the most interesting person in the room. Make it your goal to find out one new thing about another person each day.
Solutions
'It's not that I'm so smart; it's just that I stay with problems longer.' \- Albert Einstein
You may not have all the answers to every problem you face. You just have to believe that you have the resourcefulness to find the solution to any problem — rather than asking your boss to come up with the answer.
If you're going to shine in the business world, you must be the person who presents the solution, not just the problem. I like it when a staff member comes to me and says, 'We have a problem and I think we should do this to solve it'. Like solving a puzzle, finding the right solution is a big reward in itself.
We all have problems, but not everyone tries to come up with the answers. The employee who just sits there, looking like a puppy that needs rescuing and waiting for me or someone else to save the day, will not win Employee of the Month. In fact, if employees do that too often, I might start questioning if they're the right people for the job.
Janine says...
Don't you find that people tend to spend far too much time talking about the problem and not enough time finding the solution? People moving into 'protection' mode is one of my pet hates — when a problem occurs, they care more about not being blamed for it than about finding a solution and then putting a system in place so that the problem does not occur again.
In 2002 we were opening a store in Hornsby, New South Wales. The store was built in South Australia, shipped by truck and then put together in the shopping centre. Every store created is specially designed for each site, so we can't just move one store to fit another site if there's a problem. The Hornsby store had costs us about $100 000 to build, and this was very early in the business when funds were at their tightest. Sharryn, my manager at the time, was on site making sure everybody was doing what they were supposed to. She called me at 10 pm on the night that the whole shop was meant to arrive and told me that the shopping centre had made a mistake and that they would not be able to get any services to the position, ever. If we could not open this store and had to write off $100 000 in shop fit, the business would be in a financially stressed position — one that we may not be able to recover from.
To her credit, Sharryn was on the phone to everyone and anyone for a solution. She managed to raise the centre manager and the construction manager for the site — and just would not take no for an answer. The store arrived and the so called impossibility of getting services to the spot suddenly evaporated. We had services and all was good.
This is one of the biggest thought processes to learn in business — there is always a solution to every problem; you just have to stay at it longer. The person with the answers is you!
Key ingredients
Here's how to stop looking at problems and start finding solutions:
• Believe in yourself and consult with others to confirm your thoughts.
• Knowledge, experience and research are the secrets to finding solutions. I have worked in every position at Boost Juice from the ground up. I use what I know about the business to come up with the answers; I then use the people around me to make sure they are the right answers.
• Don't fake it. If you're a manager and you profess to have all the answers, but really don't, take cover! People may listen while you're in the room but, as soon as you exit, the knives will be out. Instead, open up the problem for discussion and get everyone working together on a valid solution; your staff will respect you for it.
Honesty
Honesty is the bravest path you can take.
Honesty is an essential element within a company's core integrity. It takes a brave person to be honest, in work and in life. Nobody likes to hurt another person's feelings, but most of us hate finding out that someone has been dishonest with us.
Often people don't mean to tell lies; instead, they filter the truth to justify their actions. This means that the picture you get is somewhat distorted, but the truth is in there — somewhere.
If you make a mistake, admit it. In most cases, trying to cover up an error will simply result in more problems. Trust is the first casualty of dishonesty. As a boss, I find that I get over someone's mistake very quickly if the responsible person owns up. We can then work together to find a solution. Once there's a question mark over someone's honesty, however, trust can take a long time to rebuild.
Janine says...
Scott, one of our best executives, has had a dramatic rise through the company ranks, and this is mainly because of his weekly work-in-progress reports (WIPs). These reports break down responsibilities by department, outlining who is responsible for what and when. They are one of our core strategies in keeping everyone accountable and on track.
In Scott's WIPs, I know he is giving me the whole story, without any sugar coating. We discuss the report and enjoy the positive side of the company's performance, but equally look at the dark side and try to figure out solutions. When Scott stuffs up, as we all do from time to time, he is always first to 'fess up. As a consequence, we have total faith and trust in him because we know we are getting the truth on every issue, no matter how confronting it may be to the company. Everyone answers to someone, no matter how high they are up the ladder. Even as owners, we answer to our customers.
Honesty is the most refreshing part of business — a problem shared should lead to a problem solved.
Key ingredients
Here's how to keep yourself honest:
• When you lie to someone, you're putting the relationship at risk. Is it worth it?
• The more honest you are in your business and personal dealings, the better you will sleep at night. Don't think for a moment you will be able to avoid your conscience constantly reiterating the guilt of having done the wrong thing. A clear conscience means a clear head — which means more room for new and exciting strategies.
• Most gains from being dishonest are short-lived. You can learn a valuable lesson from being honest about a mistake and then moving forward.
• Don't be afraid to admit you're wrong. I have seen general managers who are respected and loved by their staff because they are not scared to stand up and say, 'Sorry, I stuffed up!' Being human is endearing; being bravely honest reinforces the character of a true leader.
Super supplements
Power up your honesty with the following:
• You don't have to be brutal to be honest. By carefully listening and fully understanding the issues, you will be able to tell others the truth skilfully, in such a way that they are left grateful for your honesty. Try it.
• Are you truly honest with yourself? Do you let yourself not do things you said you would by making excuses? Do you ignore 'problems' within yourself rather than find solutions? Look at yourself honestly to discover if something is holding you back — and then find the solution to this problem.
Partners
Make sure the person sharing your dreams is with you all the way.
Financial success does not create emotional prosperity. The truly rich life is a balanced one — unless you keep your work and family in harmony, your economic gains will be meaningless.
Having the right partner at home is vital. If your partner supports your dreams, your path will be a lot easier and, ultimately, success will come sooner.
The person you rely on may be a life partner, a friend or a family member. He or she may not necessarily agree with you, but this person must support you 100 per cent in your goals. The last thing you need is someone whispering negatives in your ear all the time.
Jeff and I are Yin and Yang — different people, but a perfect fit. We each have our own nuances and our own individual methods of operation, but we're highly effective as a team. He is the perfect complement to my style: an entrepreneur and the man with big ideas, but not always the attention to detail to make them reality. I am the planner and doer, the person who makes things happen. We often laugh that if it was just me in control of Boost, we'd have ten amazing stores. If Jeff were in charge, we'd have a thousand — though they might have gone broke. His backup has been integral to my success.
Janine says...
When Riverside Private Equity bought into Boost in 2010, one of their primary concerns was having a husband–wife team running the company. It was their experience that such a team can quite often be a compromise of skills and expertise, so they gave us one of those long fandangled personality tests to see where we both sat. Based on the results of this test, we now have in perfect graph form what we knew intuitively — we are the perfect business couple. Every area I'm weak in, Jeff is strong in, and vice versa.
Your partner is critical in your journey. I see lots of couples where the wife is completely underrated and the roles are set in stone — the husband is the main breadwinner and the wife takes care of the family. This set-up all seems to go well in the early years, when the kids are super-needy, but it starts to crumble as the family evolves. Hubby is still mentally stimulated at work, the kids are in their teens, or later, and are more independent, but the wife is — hmmm — lost.
We all need a purpose, we all need to be challenged; however, in this situation what does the wife's purpose become? Women aren't just there for their husband's and children's needs.
Just as you should seek support from your partner, you should also encourage your partner to explore and fulfil as many of his or her skills and dreams as possible. These skills and dreams could be anything, but the earlier you start supporting your partner, the better your relationship will be over the long term. And hey! Just like our Boost story, you never know where you could end up. I can't imagine what my life would be like now if I had fallen in love with a man who wanted a stay-at-home wife, but I think it's unlikely I would still be with him.
I have become the best I can possibly be because I had a husband who encouraged me to have a crack at anything and believed in me to take on the world of retail.
Don't let your ideas be squashed for the sake of your family — you can fit everything in. And remember — happy wife (or husband), happy life.
Key ingredients
Here are some components for perfect partners:
• To truly achieve your best, you need support from your partner or, even better, a partner who can contribute to your plan.
• Keep your partner informed and involved as your plans and dreams evolve.
• Look for a life partner who will bring out the best in you.
Positive thoughts
'Many of life's failures are people who did not realise how close they were to success when they gave up.' \- Thomas Edison
Rome wasn't built in a day. Every cloud has a silver lining. If at first you don't succeed...
Looking on the bright side is not just for Pollyanna. Being positive is crucial to achieving success. Decide that you are going to be successful, and then be unfailingly upbeat in your pursuit of that goal. A firm decision makes you unstoppable — no 'maybe', 'could've' or 'should've' allowed.
You will get knockbacks along the way. Problems may arise from every corner. How do you remain positive in the face of constant negativity? By knowing there is a solution and that you will find it. If ten people tell you your idea will not work, how do you keep believing that the eleventh person will love it? Well, it depends on who those first ten people are. If Jeff or Geoff Harris were to disagree with me, I'd probably have to reconsider my position. If it were a person with no retail or business experience, I simply wouldn't listen. So many people have missed out on attaining financial freedom because they couldn't handle the knockbacks. If someone says no to you, don't close the book on the idea. It just means that the idea or request was not right for that person at that time.
Key ingredients
Release the power of positivity with the following:
• Adopt a 'can-do' philosophy and resolve to chase your goal to the end.
• Don't listen to just anyone. Choose a few key people whose opinions and expertise you trust completely, and shut out everyone else.
• Don't take knockbacks personally. Meet challenges head on and learn from rejection. If you can understand the reason for a setback, next time you can find the path to a 'yes'.
Super supplements
Take your positivity to the next level with the following:
• Affirmations are a great tool for positive thinking. We all have an inner voice; get yours in a constructive frame of mind by choosing five uplifting sayings or quotes and repeating them to yourself daily. Perhaps have them stuck to your bathroom mirror and start each day by reciting them. Or have them on your phone as wallpaper — wherever you have them, just make sure that you see them every day.
• When negative thoughts creep in, stop them in their tracks and change them with positive thoughts. For example, I'll never know how to do this can become If I ask the right questions, I can learn or I will try to achieve it can become I will achieve it.
Negative thoughts
Negativity is like a disease that can bring down the best and the brightest.
The average person can have 200 negative thoughts every day. How exhausting! If positive thinking is one of your greatest weapons, negative thoughts are like a disease. They will bring you down, no matter how good you are.
From my perspective, a service-oriented company is at the mercy of its staff's emotional moods. It's vital to recognise underachievers or dysfunctional personnel and either retrain or remove them. Some people simply do not like dealing with people. This doesn't mean they're bad, just that they shouldn't be in the service industry.
As the boss, my mood has a huge impact on my team, and it can dictate the atmosphere of the environment around me. Every person on the team can also have the same effect on those around them. One negative person in an office can upset the whole apple cart!
We all have bleak moments — we wouldn't be human if we didn't. But we also have the power to choose how we feel about the situations we encounter every day. Not every situation will be great, but you will see a lot more abundance and opportunity with the 'glass half full' way of thinking.
Janine says...
I have mentioned many times in the book that I was driven by fear of failure, and in some ways it was a great motivator. This fear, for example, made me double- and triple-check everything I did to ensure that I did not miss anything. Fear kept my adrenaline at full speed, but there was a massive downside — you can manifest what you fear, simply because that's what you're focused on.
Now I'm not sure if I truly believe in books like The Secret, but I do believe that if you have a positive mindset and you are looking for the good, then the good follows you. The more you focus on something (good or bad), the more likely it is that what you're focusing on is going to happen.
In the early days of Boost's success, I was a complete stress-head. Anything and everything would send me off, and everything was a drama. I was a terrible wife and mother and I was terrible to myself because I was so stressed. It got to a point when I stopped and realised that I actually hadn't eaten anything for three days. I was as thin as I have ever been and, while my mind and body was all about making sure the business was a success, I was not living Boost's 'love life' philosophies, not even a bit. I was negative in my thoughts and my health.
This was a massive wake-up call, so I shipped myself off to a health retreat to get myself back on track. Once there, I came across a 'healer' (as you do at health retreats), who recommended I read Conversations with God. I'm not against religion, but it has never been something that I have wanted to be a part of. I do believe that I am quite spiritual, but I think spirituality comes from within, not from a church. The title of the book put me off straightaway, but I had nothing else to read so I gave it a go. After the first few pages, I was ready to throw the book in the bin, because the author was basically saying that he was writing 'through God's words'. I persevered, however, and in the end I could not put the book down.
I have no idea whether or not the book really contained 'God's words', but I did like the messages. One key message was that you should not try to be better — you should just be better. Do not want to be a type of person, simply be that person. And the second key message for me was there is only love and fear — if you live in fear, you will manifest this; if you are grateful for what you have and be positive (or live for love), that will be what you manifest.
I do believe that books and people come into your life at different times for a reason, and this book (and the others in the series) did help me change the way I thought. I let go of fear and took more control over my stress, and that helped change the way I thought. There are hundreds of books that tell you the same message. Forget about what you have or have not done in the past — all that matters is what you do now and into the future. Be grateful for what you have and don't focus on what you have not.
Key ingredients
Here's how to get rid of negativity in your life:
• Don't be a cynic. The best way to miss opportunities is to carry around thoughts like, 'Nothing good ever happens to me'. Having a victim mentality will kill your chances of achieving financial freedom.
• Don't waste time overanalysing what other people might think of you. Assumptions, theirs and yours, are generally wrong anyway.
• Having an 'off' day? You can choose how you feel and react to things; it's all in your perspective.
I was a seriously fat, happy baby and I did not walk until I was about 15 months old. There was just too much blubber to carry around!
For some reason I tended to get all the modelling gigs with animals. This job was advertising merino wool. The animal hated me and every opportunity it would try and charge me, nearly charging me into the local lake!
Rae and myself as bridesmaids on our sister Lisa's wedding day. I was 17 at the time.
Travelling the world, aged 21, hiking the Grand Canyon. Smart people did this on horseback or by helicopter, but on my budget I had to hike it the old fashioned way and use my feet. I was fit at the time, but 10 kilometres all the way up nearly killed me!
This was our first store on King William Street in Adelaide. The place was a mess and Jeff and I painted the floor with polyurethane to save money. We basically did as much as we could as we were on a tiny, strict budget. There was no air-conditioning so we had to bring portable units in just so we could operate.
This is Sharryn, our first ever employee. She grew up in the business and was a great asset. This was a concept we tried to get off the ground called 'Tossers'. We loved the name but the landlords did not want their building known as the 'Tosser' building, so it did cause us some grief. At the end of the day the concept failed due to a lack of focus. It was a good early lesson for us.
With Jacinta in Lisbon visiting our master franchisee partners.
We ran a Boost competition for the Boost Olympics and asked our staff to come up with tricks and ways of having fun in the store.
This photo was taken at a Boost conference that we do every year. Geoff Harris (far right), was just a great ambassador for the Boost business. He attended every event and dinner. I could not ask for better support from a shareholder/director.
This was the booth that we set up in Washington DC. It was an international franchise conference. We did not sell any franchises but we learnt an enormous amount on how to go about launching internationally, and we had a blast!
Boost give away a store! This was an amazing promotion that Jeff made up. It was a huge marketing success for us.
My great team. Without them I would have never won the Telstra Business Woman of the Year. From the left: Naomi, my gun CFO; Jeff, the love of my life; Kath, our PR officer; myself; Kristie, our strong-attention-to-detail corporate counsel; Brooke, the marketing manager who did an amazing job with understanding the Boost brand, and creating a great vibe in the office; Jacinta, our talented franchise manager; and of course my mum and dad, who were very proud parents that night.
With my mum at the Telstra Business Woman of the Year awards. I was thrilled she was there with me on the night.
Our annual Boost conference.
Taken from our television advertisement 'squeeze what you love'. Jeff wanted me to squeeze one of our team members, which I wasn't sure about, so they put my son Riley in. It made a difference as we just played around together, making the advertisement work.
From left: Samuel, my save-the-world, kind-hearted hippy; Jeff, my partner in crime; Tahlia my wonderful midlife crisis; Riley my son who I am sure has been on this planet before; and Oliver, my handsome, stylish and sporty son.
Chapter 6
Staff smoothies
The recipes in this chapter focus on getting your team right and really getting the most out of your staff — from delegating appropriately to picking out protégés. Get the mix right, blend and relax — if all goes to plan, you'll find yourself out of a job in no time.
Teamwork
Getting the right team balance leads to goals being scored.
I was fortunate to be the first female board member of the Hawthorn Football Club. I loved the years I worked with the club, and found the business of football fascinating. A key 'ah ha' moment for me was when we were discussing getting the right team to win a grand final. It was exactly the same formula as creating a winning team at Boost.
The 'recipe' for a winning football team was as follows: you need three to four superstars to start, and then you add your up-and-coming superstars. Next, sprinkle heavily with the solid team members (the ones who get their job done with no fanfare), while cautiously blending in the last group — those who need to move forward or move out.
The bottom line is this: a great team is a solid mix of different personalities, all working together to even out each other's weaknesses. A great team is healthily competitive, yet comfortable enough to truly celebrate each team member's individual wins. The right team can achieve the unachievable.
A good team mix could look like this:
• The leader: efficient, focused, ambitious, confident, honest, strong-willed and someone who can inspire. Leaders may, however, demonstrate little patience, a tendency towards bullying and a desire for personal success over team success.
• The thinker: analytical, concerned with detail, unassuming, precise, well organised, rational and a good listener. On the downside, thinkers may be perceived by others as aloof and negative — a killer vibe at the best of times!
• The 'doer' or worker bee: hard-working, patient and keen to get the job done without much fuss. Worker bees can, however, be easily manipulated.
• The emotional creative: social, energetic and competitive, but prone to the odd tantrum, especially under pressure, and may lack the necessary follow-through. In other words, creatives can be high maintenance if their astrological planets are not aligned!
A good team mix includes all elements from the preceding mix, and manages to reduce the risk of any of the negative attributes associated with each element emerging.
Taking all of that into account, how do you pick the best team? It's important that you also recognise all the dominant traits in your team — doing so will help to ease any frustration. You may have too many analytical types among your key personnel, for example, which means no-one will ever make a decision. On the other hand, if there are too many drivers and leader types, you're on a road that will be heavily paved with conflict.
Janine says...
Your first employee can really make or break a business. This time is when you are learning and focusing on growing the business, and confidence in your concept is what makes you grow. If our first employee had been a disaster, the growth of Boost Juice may have been very different, because timing in business is everything — at
that stage of growing Boost Juice, we needed to quickly become the first option in the mind of the consumer. Luckily, we found Sharryn for our first employee.
Sharryn was the Australian speed-waterskiing champion — she had muscles on the muscles on her arms and, even though she could not be taller than 5 foot 2, she could scare a man twice her size. Sharryn had never worked in retail before but she had the drive and the passion that we were looking for in our new concept. She understood what we wanted to achieve and was passionate about achieving it.
Sharryn found herself running not one store but a number of stores very quickly. She moved from store manager to area manager and then to project manager, running the design and development team in opening new stores around Australia in just two years. She had no experience in this area either, but took on each challenge with enthusiasm. Sharryn salary-sacrificed her early wages to obtain a share of the business, which paid off handsomely many years later. She and her husband and their three children now live in Mildura, where she runs a successful cafe.
Key ingredients
Here's how to perfect your team:
• If you're having difficulties with a team member, ascertain not only the team member's personality traits but also your own. This will help you learn the best way to deal with your colleague and the best way to manage your response to them.
• The nicest people (or those most easy to manage) are not necessarily the ones who will produce the best results. Every organisation needs balance, and having a bit of 'mongrel' in a group is essential.
Super supplements
Finetune your team with the following:
• Develop a core team for your business that consists of four or five people who work well together and with whom you trust. Under good leadership (yours) this group will become a cohesive unit, creating a synergy that will make your company or department unbeatable.
• If you're not confident enough to conduct personality tests, consider hiring an expert to evaluate your staff.
People
'Imagine being paid to work here — it almost feels wrong.'
Words spoken by a member of my staff, and among the best I've ever heard.
Your people are your company's greatest asset — reward them and reap the rewards for your business. There's no doubt that if you get the right group of people together, anything can be achieved. I believe that staff members need to share in the rewards. If you take the time to acknowledge a person's contribution, you will be repaid by their loyalty and hard work a thousand times over.
Public acknowledgement can be extremely motivating. The Boost Awards are an annual event that we use to both reward and recognise exceptional achievements by employees. The awards have been developed to inspire individuals and teams to aim high. They are open to everyone in the company, regardless of position. As well as offering rewards for high performance, it's equally important to show genuine understanding and kindness when a colleague or employee is at a low ebb. Compassion will also be repaid by trust and loyalty.
There was a time in 2010 when generation Y was getting some really bad press and being blamed for all sorts of negative things. If you did a search for 'gen Y' online, you would find millions of results on how to deal with this apparently 'difficult generation'. We passionately did not agree with this perception, because we saw some amazing, talented and focused generation Ys in our organisation. We felt so strongly about this we did an advertisement attempting to change the perception of gen Y. It was a risk for the business, but it paid off; the research was amazing. And when an ad agency did a spoof on our ad, my husband and boys thought it was hilarious. (Have a quick look here for that spoof: www.youtube.com/watch?v=QlvYjIkk4e0. It's hard to find something funny when others are taking the piss out of you. However, I admit the spoof was clever.)
Jeff says...
One thing I am very, very good at is picking people — hey, I picked Janine as a wife, didn't I?
Once you absolutely and totally commit yourself to the idea that you are there to hire people smarter than you, and who will eventually take your role, the rest is easy. In every company, there is only a small number of people who can control whether the company is steered in the direction of success or failure — those people must be amazing.
My best-ever example of the 'people' rule occurred during my time at Austereo. I had just started my radio executive career at Sea FM (on the Gold Coast in Queensland) as program director. I hired Guy Dobson, a bloke who is slightly left of centre, super-intelligent and has a real presence to him. We were both in our late twenties at the time. We had a function on a boat and, over a beer, I asked him what he wanted to do in his professional radio life. He said he was going to be the best radio programmer in Australia. I laughed and said that was my goal too.
Then I told him I was a year or two further down the track than him so if he helped me get there first, he could then take over the mantle. Done deal — from that day on, we looked after each other. He was outstanding, but his performance only made me look good too. We kept rising through the ranks until I got to 40 and stepped down from the head programming role in Australia to take a year off and then work with Janine. Guy was promoted into my role as group program director and then continued from there to take the role of CEO of Austereo. Total faith and total trust got us there.
The truth is, I always end up having to figure out my next career step well before I'm ready to go because the people I surround myself with are so good that they put me out of a job. Your job as a leader is nothing more than clearing the path and helping your people have a clear run at hitting their marks.
Key ingredients
Here's how to get the most out of your people:
• Consider instigating a formal recognition and/or award program, in line with your company values, that will give your employees meaningful rewards for exceptional achievement. Acknowledgement is an excellent motivational tool.
• Recognise the fact that people respond to compassion as quickly as they do to encouragement.
Relationships
'A real friend is one who walks in when the rest of the world walks out.' \- Walter Winchell
Learning to manage the various relationships you have should be a fundamental part of your ongoing strategy.
On a day-to-day basis, like most managers, I spend more time with my core staff members than I do with my family. Close bonds have grown between us over time; it's acknowledged that I would do anything for them, and vice versa. Together we have created and achieved amazing things, with plenty of hardship, stress and celebration along the way. In this kind of situation, you do develop a bond that's stronger than a mere employer–employee relationship. Not slipping over into complete friendship mode is a fine line, but one that I believe we now tread easily.
These days, I have very little free time to spend with friends. I make sure that those people I do see are the ones who make me laugh, allow me to feel good about by myself, have my back and are honest. In business, you can be spoiled by the wealth of mental stimulation you receive from the people you meet. But in life, it is not what someone has or hasn't achieved that makes them interesting; it is who they are as a person. Friendships are vital to everyone. Nothing revitalises me more, or makes that bad day not so bad, than a good old belly laugh with trusted friends. Choose your friends wisely.
Janine says...
Scott is the day-to-day operational leader of the company and, as such, he has plenty on his plate. He has hard calls to make every day. He said to me recently, 'I really don't have any friends in the company — as in mates I would socialise with'. And I thought Cool. That means you're doing your job right.
The higher up the corporate ladder you go, the fewer 'mates' you will have in the office. Life and work is not a popularity contest — great people often have to make tough calls, and these calls are made tougher if a friend is involved. Sure, early in your working life, having workmates who are also friends can be super-important, and going out and being able to live and breathe the day-to-day dramas of work is all part of it. However, climb higher and get older and, trust me, you need to keep them separate — doing so is rejuvenating for both aspects of your life.
Key ingredients
Here are some important elements for business and personal relationships:
• Business relationships are about give and take. If you give a lot of information to those who impress you, most will return the favour.
• When you're passionate about your work, it's easy to neglect friends and family. Don't! There are only so many people in the world with whom you can have a close relationship — keep in touch. Good friends are the best tonic for bad times.
• The world is full of people who believe that the glass is half empty, that they are owed a living, that they are hard done by and that nothing is their fault. You do not need these high-maintenance people in your business, or your life. You owe it to the others around you to get rid of those negative influences.
• The best friendships are equal friendships. Look at your relationships: are they two-way streets? Do you feel good about yourself when you are with your friends? Make sure those around you bring out the best in you, and vice versa.
Super supplements
Build even stronger relationships with these tips:
• Take the time to be curious about other people's lives and their wellbeing. Genuine interest will have a large impact on that person. Token measures will get token results.
• If you believe that someone has let you down, step into the other person's shoes for a minute. Could this person believe that you have let him or her down? Perception changes everything.
• We all have different energy levels. Find people who share yours and avoid high-maintenance types who will drain you of your life force.
Protégés
Start training your next-in-line now.
Your job as a business owner or an executive in a business is to continue to grow strong people so that, eventually, you are out of a job. Insecure businesspeople hire below them so they have control. This never works. Hiring the right people and training those people so that they can one day take over your role is critical to your success.
How do you identify a protégé? The right person will have core values that reflect your own. He or she will have a willingness to follow your philosophies, even if in a different way to your own. Learn to recognise leaders among your staff. In my case, it's not necessarily about actively hiring the right person — rather, candidates for the job emerge and evolve. Don't be threatened by an up-and-coming talent. I once worked for someone who made sure he hired incompetent people because he wanted to run things his own way and didn't want to be outshone. That's stupid! The smarter the people around you, the smarter and better you are.
Encourage your protégés with training and counselling. Challenge them by occasionally throwing them in the deep end — this will help them to discover their own strengths and weaknesses, enabling them to work on their shortcomings. It will also allow you to examine their potential. Remember, however, that there is a fine line between giving your protégé opportunities and setting them up to fail.
A piece of coal placed under the right amount of pressure for the right period of time will transform into a diamond. Of course, it doesn't always work out. There have been times when I've felt very confident about an employee and it hasn't worked out. As a business grows, some employees can be left behind, unable to evolve with the changing business. In those situations, you have to consider your options and, sometimes, no matter how hard it is, it is best for you both to part company.
Key ingredients
Start recognising and building protégés with the following:
• Don't be afraid to replace yourself. It's your only opportunity to grow. By beginning early and training someone thoroughly, you will be placing your company in a better position for the long term.
• Identify and nurture your protégés; harness their enthusiasm to help them achieve great things.
• Good leaders surround themselves with smart go-getters. Bad managers choose inferior employees in an effort to bolster their own position or ego — which never works!
Leaders
'The difference between a boss and a leader: a boss says "Go", a leader says "Let's go".' \- E.M. Kelly
Leaders may be born, but leadership is learned. What type of leader do you want to be?
My style of leadership has evolved by learning from my strengths and weaknesses. My business has evolved the same way. All leaderships must evolve, but the fundamental philosophy should remain the same.
The role of a leader is to inspire extraordinary performances from ordinary people. If you're the boss, your level of enthusiasm will be reflected throughout your company. The more people you can influence daily, the more power you will gain.
How are you going to do this? Study how to be a greater leader. Define yourself, your values and your attributes. Play up your strong points and work on your weaker areas. Importantly, you must avoid the need to be liked. If you have a great personal life and feel secure within yourself, why should you desire affection from everyone? This need makes you vulnerable and weakens your decision-making ability; it has no place in building a successful business empire. Instead, you should aim to gain your employees' respect; you want them to respect you more than like you.
It's also essential to recognise leadership qualities in others, and these won't always be immediately apparent. That timid worker whom you've always perceived as slightly introverted may roar like the king of the jungle when put to the test. In the same way, someone who comes across as self-assured and competent may buckle under pressure.
Janine says...
Often people say that one person cannot change the world. The reality is great leaders can — the great and evil things that have been done in the world always start with one person. Just think about all the great leaders out there, such as Mahatma Gandhi, President Kennedy, Nelson Mandela, Martin Luther King Jr, Abraham Lincoln, and the Dalai Lama — as well as leaders who people followed but were far from great, like Adolf Hitler, Attila the Hun and Joseph Stalin. What they all have in common is the passion and drive for what they believed in; their personalities are addictive, and people want to be them and be with them. They all had a very clear vision and would commit everything to achieve their vision.
The leader dictates the culture and, ultimately, the profit and growth of any business.
Key ingredients
Boost your leadership skills with the following:
• Leadership is usually an innate ability, but your style of leadership can be chosen.
• Don't be the kind of leader you think you ought to be — be the best leader you can be. If you choose a style that's natural to who you are, you will be successful.
Delegation
Do the words 'control freak' mean anything to you?
Once upon a time, people who worked long hours and spent most of their weekends in the office were seen as dedicated — great examples to other employees. Not anymore. An extreme working day is no longer seen as a badge of honour. Now, it's more likely to be thought of as showing poor time-management skills, a lack of organisation or a poor approach to delegation. For a leader, a lack of delegation shows bad management skills. Do not misunderstand these comments; there are times that you may need to put in long hours or pull an 'all-nighter', but this should be the exception not the rule.
When Boost first started, I did everything myself. I had to — there was no-one else to do it. There are still days, I admit, when I think, It would be quicker if I did this myself. However, I know that I simply don't have the time to do everything. Who does? If I tried to do everything, I'd succeed at nothing.
Of course, I have high expectations of those to whom I delegate. I never choose someone simply because I feel I should, or because that someone thinks he or she deserves a chance. I always try to hand a task to a person whom I believe will do a better job than I could do myself.
I also keep a close eye on how people respond to being given responsibility. Do they deliver? Do they keep me informed? Is the project completed on time? It's all about their actions, not their words. I don't want to be told someone can do it — I just want them to do it!
If you find that you're not getting the desired results when you delegate a task, find another person to do the job. Don't feel bad and don't play along to save someone's feelings. Never reward mediocrity.
Janine says...
Much to the horror of all of the people who reported to me (and there were 16 direct reports at the time), when things were really ramping up at Boost I discovered the Task icon in Outlook. For those people who are like I used to be (and don't know it even exists), how the Task function works is that you click on the icon and you add in the instructions to the person to whom you're delegating your task. You can add in reports, emails — anything really. You then add the date for the task to be completed by and the person receives a copy of the delegated task. The person doing the delegating also keeps a copy of the task. I even colour-code tasks based on the people they're delegated to... but perhaps that's the OCD in me.
The Task function was my lifesaver — prior to discovering it, I would have so many balls in the air that deadlines were often missed, there was often confusion on which tasks had been delegated and to whom, and the not-so-great employees benefitted from the lack of follow up. I started to use the Task system for all my meetings. The rules were simple: I send you a task with a date that it has to be completed by; you had 24 hours to come back to me if the task was not achievable; if I heard nothing, I would expect the task to be complete. I meet with all my reports weekly and we always started with the task list first.
Great people loved the system because it kept everyone on track and on the same page; average people hated it because there was nowhere to hide. If someone missed a deadline, it was there in black and white (actually, in red, because the task goes red when you miss a deadline).
It was not uncommon for me to have hundreds of tasks active. With everything involved with opening over 70 stores in one year, it was critical that everyone kept on track and knew what everyone else was doing. Okay, my obsession with the Tasks function was the standard joke at Boost, and there was even a Christmas skit on my tasking, but it kept me sane and, more importantly, on track.
The key to good delegation is clear communication, with everyone understanding who is ultimately responsible. Setting deadlines is also critical. But when you delegate a task, let the person run with it. Being a control freak, I'm sure that I was probably a bit overbearing in the early days because I wanted everything to be just so, which was not necessarily the best way to go. If you hire the right people in the first place, you can empower them to do the job.
Key ingredients
Here are the components of good delegation:
• If you hoard tasks, you set yourself up for failure. Remember, if you have chosen your employees well, you are not the only person capable of completing that task.
• Lack of delegation does not show devotion to the job — it showcases bad management skills.
• Always give deadlines when you delegate tasks. Ask people if they believe the timeframe is reasonable and, if they agree, set up an appointment where they will report on the completed job.
Meetings
OMG! Not another meeting.
Meetings can be powerhouses of ideas and actions, or they can just be a group of people sitting around a table putting off decision-making. The difference between the two comes down to tactics.
Always ask yourself if you really need to have a meeting. Once you've established that a meeting will indeed be the fastest and clearest way to communicate with staff, follow these guidelines:
• Set a start and finish time, and keep the process efficient. Allocating time prevents those rambling, open-ended discussions that are time wasters.
• Set an agenda. This will help keep the meeting on track and on time. Those attending should be given the opportunity to list their own points for discussion.
• Brainstorm and write up ideas or key points. Flip charts or whiteboards are invaluable for getting everyone involved. Remember, there is never a bad idea. Encourage input from all attendees.
• Write down clear actions to be achieved, and next to each point write the name of the person chosen to take care of that task, along with a deadline. By the end of the meeting there should be a consensus of what needs to be done. Minutes must be taken at the meeting, and should be typed up and distributed as quickly as possible. Refer to these minutes at the next meeting to ensure all action points have been completed.
• Send each person the list of action points as a gentle reminder to complete the tasks before the next meeting or approaching deadline.
• Get to the point. Respect other people's time if you are giving a presentation; make it slick and make it short.
• Take it offline. If a discussion is between only two members of the meeting group, the two can meet on that issue after the meeting; do not waste everyone's time on issues that do not concern them.
Janine says...
Meetings are great in some ways, because you can get together face to face and you have a great chance to get everyone on the same page. Especially now that our world is so focused on electronic devices, you can find yourself doing much of your job without talking to anyone. But everyone has experienced people misunderstanding the written word, so meeting face to face can reduce these misunderstanding.
But you can overdo meetings and they can be a huge waste of time if done incorrectly. I remember meeting with someone who was selling me something. I cannot for the life of me remember what he was selling, but I do remember he came into the meeting without an agenda but with a 40-slide PowerPoint presentation. He handed me a copy of the presentation and off he went — each slide was like a novel and he was just reading off the slides. I had to stop him. I then flipped to the last page of the presentation and got the point.
The meeting would have taken almost two hours if I'd just sat there politely; however, at the end of the day, time has to be seen as money. He was a lovely young man, but he did not follow the rules of a meeting. When you're presenting you have to be respectful of people's time. He also thought he was prepared, but it was in all the wrong areas. And he shouldn't have handed me a copy of the presentation. (Never do this — people stop listening to you and start reading.) You need to be clear in meetings and ensure they go for no longer than 45 minutes.
Key ingredients
Here's how to get the most out of meetings:
• Before you call a meeting, decide if it's really necessary. There is such a thing as too many meetings. If you're spending more time talking than you are doing, you may need to reassess your work practices.
• Meetings can be a quick and easy way to disseminate information or they can be a long, drawn-out discussion that goes nowhere. Follow a strategy to keep your meeting fast and effective.
Chapter 7
Power blends
The recipes in this chapter are all about giving yourself that extra edge, and really boosting yourself and your business. I talk about bigger picture issues such as principles, respect and credibility, as well as the two most important factors in building your brand — communication and customers.
Assumptions
Are your feelings getting in the way of the facts?
You've probably heard that to assume makes an ASS out of YOU and ME. Believe it. When you make assumptions, you let emotions colour your view of a situation. Emotions can blind the smartest people. Some of the assumptions you make about others might be wrong.
Check yourself: are you an emotional person? Do you react to situations, or do you respond? If you react, don't!
A reaction is explosive. There have been times when I've been told a story by one person and reacted by ringing another to blast that person. Then, when the second person's side of the story is put forward, I'm left feeling foolish. People don't necessarily mean to lie; they may only give you their version of events — a filtered view. When you're handed the objective facts of a situation, or told both sides of the story, a knee-jerk reaction is often out of place. A response is more measured. In this case, you wait until both sides of the story are in front of you before choosing how to reply.
There's no doubt about which of these actions is more professional. Be a person who responds, rather than reacts. Learn to sleep on situations. Keep that angry email in a draft folder for a little while before you hit send. You must also ask questions. It comes back to my philosophy that no question is silly. Asking questions means that you are not making assumptions. It also means that you will have all the information you need to make (hopefully) the correct decision.
Janine says...
Come on... You have to admit that at some point in your life you've jumped to an assumption and, in your mind, have built a situation up to something it is not. And then you discovered that what you'd assumed was not actually quite right. I had an executive who used to tell me various things that people were saying or doing in the business. As I trusted this colleague at the time, I would often get angry and act on this information — to then find that I had been told a half-truth. I learnt quickly that you should always wait until you have the whole story before you act. Acting on wrong or twisted information is a fast way to ruin that great reputation that you have worked hard to develop.
Key ingredients
Here's how to stop yourself jumping to assumptions:
• Think before you speak. Yes, the story you're hearing may seem outrageous, but you may not be hearing the whole story.
• Try not to make assumptions of any sort — whether you're assuming that someone else will fill the empty printer tray or that a staff member knows what you're thinking.
• Try to keep your negative emotions out of the office, particularly if you are the leader. As the boss, my emotions set the daily tone for the entire office, so it's vital that I keep positive.
Mentors
Everybody needs guidance — yes, even you!
As the journey of Boost continued and I was trying to manage the incredible growth, I discovered that there were things I didn't know. More correctly — everything I was doing was for the first time! Fortunately, I had help along the way, and still have a number of business mentors.
The first is Geoff Harris, founder of Flight Centre. Geoff called me one day because he was looking for a business opportunity; he'd heard that Boost was a company that was going places. Over the next few months, Jeff and I got to know Geoff and his ideas and philosophies better. His stand-out attribute was the generosity he showed us in sharing his amazing knowledge, even though at that stage he had no financial involvement in the business. Geoff has enormous integrity and honesty. His philosophy of empowering his staff, of giving them ownership without giving them the farm, was an enormous lesson for us.
The others are people who may not be long-term mentors but who have been generous with their time and advice — people like James Fitzgerald from Muffin Break, and Lesley Gillespie from Bakers Delight.
All these people have given me a wealth of invaluable knowledge. I've found that those who have been in business for many years are often very open about sharing their experience. It's refreshing when this happens, and it has always encouraged me to make sure I help other people where possible.
Is it any wonder that I firmly believe everyone should have a mentor? After all, nobody has all the answers all the time. Sometimes you just need help from someone who has 'been there, done that'.
Ask someone you respect to be your mentor, your personal sounding board, but don't have too many expectations. As with all things, the timing must be right. The person you approach must be in the right circumstances to give you his or her time. Geoff Harris, for instance, was looking for an opportunity to help someone. Be incredibly respectful of that person's time. You should realise that you're encroaching upon this person's space. If the person you approach agrees to be your mentor, you must allow him or her to set up the means of communicating and amount of time given. Hopefully over time, your mentor may be happy to increase dealings with you. Always respect your mentor by following through on any advice provided.
If the person you admire doesn't have time for mentoring at the moment, look at other ways to study this person's success. Read about him or her, and read any books he or she may have recommended — there is always a lot to learn.
Janine says...
As I mention in this section, Geoff Harris was an important mentor for me. Geoff and I were the perfect pair: he received enormous joy out of teaching and I was a sponge that took everything in.
Geoff's message on culture fit was also loud and clear — in any meeting when we spoke about acquisition, the first topic that was discussed was always whether the possible acquisition was the right cultural fit for our business. One thing Geoff, Jeff and I agreed on was that we wanted to do business with people we liked and who had the same integrity and honesty in business that we had. If people did not get the tick, we would put a line through the business and move on.
I am thrilled to call this man my friend.
Key ingredients
Keep in mind the following about mentors:
• A mentor can be an invaluable source of experience and wisdom. Sometimes the best thing a mentor can offer is a pair of objective ears.
• Be respectful in your approach if you're seeking someone's mentorship. Understand that a 'no' is probably not a rejection of you personally, but rather a reflection of the person's time constraints.
• If the person does agree to mentor you, don't expect too much in the beginning. Allow the relationship to develop over time. Also, prepare well for each meeting so you maximise your and your mentor's time.
• There are professional mentoring networks in place. By contacting one of these, you might be paired up with a suitable person who has decided he or she has the time to give something back.
Principles
Frankly, you can't run a business without principles.
Set clear principles for your organisation and stick to them. What does your organisation stand for? What is its moral fibre? If every staff member knows the answers to those questions, your company will have a united front. Of course, your mission statement will evolve over time. As your company grows, so too will your perspective; however, your core values should stand firm.
What do your goals and principles look like? Write them down; the power of doing this is phenomenal. The following outlines how our goals and values look at Boost.
The goals of Boost Juice bars:
• Make certain everyone operates by the same principles.
• Have a clear and concise mission statement.
• Build leaders rather than managers.
• Hold to the highest possible standard of ethics.
• Have a clearly defined organisational structure.
• Encourage the development of character.
• Develop a broad award program to highlight achievements.
• Build morale to foster enthusiasm, devotion and esteem.
• Respect real-world experience; respect the team on the front line.
• Have an ongoing formal training program in place.
• Inspire positive action in suppliers, franchisees, staff and customers.
Boost Juice believes:
• Decisions must be made by individuals, not committees.
• Executive staff should present solutions, not just point out problems.
• We must analyse exactly what the consumer wants.
• A business plan is a road map from here to there.
• Issuing orders is 10 per cent of work; execution is 90 per cent.
• Having clear standard operating procedures reduces confusion.
• Any written communication should be kept simple.
• All briefings should be brief and to the point.
Janine says...
Business is like a living beast: you may think you have the right culture and everyone is living by the right principles and then a number of things go wrong you realise you are off track. This has happened many times in my business — as it has with raising our children. I find with my kids that all seems to be going great and then one day they are out of control. I've realised that the thing that has usually changed is not my children — they've not suddenly become feral — but me. I had lost focus and hadn't followed through on the principles that I live by as a parent and leader of a business. You quickly find yourself surrounded by chaos when you lose sight of the vision. You need to continue to check yourself to make sure you are consistent with your message and your actions.
Key ingredients
Here's how to make sure your principles are consistent and maintained:
• Every organisation should have clear principles and goals, and these need to be communicated to all staff. These principles and goals ensure that your team members understand what's expected of them.
• Periodically, you also need to check that your and your staff's behaviour matches up to your core principles and goals. Sometimes everyone can get caught up in the day-to-day chaos, so it pays to step back and assess from time to time.
• Make your principles and goals clear during the hiring process. Companies often spend a fortune on outside experts on culture, trying to get staff to have the same principles as the company. You're wasting your time and money if you have not hired people with the right principles in the first place.
Super supplements
Here's how to really keep yourself and your staff in line with your core goals and beliefs:
• Write down your top five goals and put them in the front of your diary. Goals that are written down are more likely to be achieved.
• Also write down your core beliefs that work in with your goals and are going to help you achieve them. The mind is a powerful thing — you choose your core beliefs, so choose good ones. You also choose whether you're going to follow your goals or not, so simply choose to follow them — it is no more complicated than that.
Respect
Respect is much harder to earn than dollars and it can be more valuable.
Respect cannot be bought, sold or traded. You must earn it. In my opinion, having the respect of peers and staff members is the most valuable goal to which a manager can aspire. Your employees don't necessarily have to like you; however, if they respect you, they will listen, understand and cooperate. They will trust in you. If your staff believes what you say, they will follow your instructions. If your customers believe in your product or service, they will buy it.
Showing respect for others is equally important, and the more you do so the more others will give you respect in return. Being an attentive listener is the first way to do this. Never interrupt or mock people when they have found the courage to speak. They believe in what they are saying and you owe them the space to air their thoughts. Try to learn something from them or something about them. By encouraging people's opinions and ideas, by sharing in their successes and not blaming other people when failure occurs, you will earn their respect. Also, if you value other people's time, they will acknowledge that your time is important too.
Always put yourself in the place of the people you're dealing with, and treat them as you wish to be treated. If you give them the respect you believe you deserve, you will find that others begin to treat you in the same manner.
This is particularly important with creditors and debtors. Pay on time and keep the wheels of commerce rolling! You want to be paid quickly, don't you?
Janine says...
We'd worked with a particular supplier from the very start — the relationship was great, we liked the company and its staff went above and beyond to ensure we always had supply. This was until the owners sold the company. After this, the relationship started to sour — the new owners were inflexible and did not return calls. The relationship got to the point that we needed to cancel the contract simply to guarantee supply. A meeting was called and a very tall man came in. My hope was that we could either resolve our problems or part company with respect. His strategy was to threaten and use heavy-handed tactics.
From the moment he sat down, there was no respect in the room shown by him for me, which in turn ensured that none was shown for him. It inevitably ended with the only winners being the lawyers. The shame of the whole episode was that if both parties worked together with respect and with a firm commitment to resolve the problems, it would have ended in a win–win.
If all negotiations are based on respect, even if you think that you got a raw deal, you know that in the future it will all work out. I would much rather be respected than liked. Great leaders are respected. If you are always looking to be liked first, you will find that respect will eventually dwindle.
Key ingredients
Here are the components for building respect:
• If people respect you, they will believe in you. This is the cornerstone of good leadership.
• Respect is a two-way street. Treat people with respect and they will respond in kind.
• Attentive listening is an important part of showing someone respect. If you are distracted, people can quickly pick up on that fact and it will make them feel unimportant to you.
Confrontation
Yes, I'm talking to you!
Most of us will go a long way to avoid a confrontation and I used to be exactly the same. Now, however, I've learned to deal with unpleasant situations. I face up to difficult problems at the beginning of each day and get them out of the way. At Boost we call it 'eating that frog', a phrase one of my executives picked up from a book by Brian Tracy.
You won't earn respect for being inactive and pussyfooting around an issue because you don't want to clash with someone. I used to hand some of the more difficult situations at Boost to my husband (who thrives on confrontation). Now, I see that confronting these issues can be an amazing and cleansing experience. The key is choosing the right way to fight.
A calm manner gives you the upper hand in any argument — if you show control of yourself, you will be in control of the situation. Keep your voice level, your eyes directed at the person and speak clearly and concisely. It's very difficult for your opponent to speak or act aggressively towards you in the face of such composure.
Make sure that you have all the facts you need at your fingertips. I will only get into a confrontation if I have right on my side. If that means I need time to prepare, I will avoid having the discussion until I am ready.
And remember, confrontation does not have to be an argument but rather can be a discussion about different points of view — and it can have a positive outcome! Bully tactics may win a particular battle, but they will lose the respect of all those witnessing or involved in the discussion. There are two sides to every argument. Try to understand the other point of view because, believe it or not, you could be the one in the wrong.
Jeff says...
Janine asked me to write about confrontation because I am supposedly good at it. Well, we are all different and if you're keen to grow in business you have to adapt to your strengths and find points where you can cut through and stand out. I guess a strength of mine has always been to tackle issues head-on — an approach probably exacerbated by the number one mission I was given when I first arrived in Melbourne radio: 'Go in and kick a few doors down'. This mission was given to me by the radio's senior management, who were frustrated at the amount of 'grey' they were seeing. ('Grey' includes ineffectualness and wishy-washy actions and leadership, as well as the inability to make a firm call on anything. Grey allows people to say that they were confused and uncertain about what was required. It is a weak person who lives in grey.)
There really is so much grey in the world and it often seems much more practical to confront a person on an issue and get it sorted then and there. In practice, of course, it very rarely works out that well in the short term, because you catch the person on the hop and he or she goes into massive defensive mode. Very little may get resolved at this point — but don't be fooled into thinking nothing has happened. The confrontation usually has a massive aftershock.
I once had to confront an executive over a pretty serious matter. During the meeting he was amazingly calm and collected, even ambivalent over some of the issues I was confronting him with. So much so I was almost questioning my facts. However, I then tracked his movements after the meeting — and he went into overdrive, with eight phone calls and six emails on the subject within an hour. So, yes, the point about what needed to be sorted did get across.
The other great thing about confrontation is that it's often only under pressure that you find out the type of person you are truly dealing or working with. How different people respond to confrontation can be a real eye-opener. Obviously, the best type of people take it on the chin, work out solutions and move on.
While I'm all for tackling something head-on, if you are going to confront a person remembering all of Janine's rules from this recipe is critical. You must be in a position of power with regards your knowledge of the facts. The initial electric volt that starts the confrontation will give the person a shock but you need to be able to follow up with facts to show you know your detail.
Confrontation should be seen as cleansing for everyone — it shouldn't be a shouting match but should (hopefully) be a calm (and sometimes awkward) delivery of something that needs to be addressed. Once you've addressed the issue, you can all move on.
The reason appropriate confrontation is so effective is because so many people avoid it — so issues are allowed to grow and fester through your business. Good confrontation stops or at least diverts the stream of bad practices you are trying to prevent. So start to think about the things and people you need to confront. Obviously, bringing up issues is much easier if you're in a position of power, but the truth is if you confront issues for the improvement of the business with people above you on the chain, you'll likely be looked on with admiration for the courage that you've shown.
Confrontation is certainly not for everyone, but if you can learn to enjoy it and use it effectively, you will certainly stand out from the pack. And remember — the quicker you tackle something head-on, the better.
Key ingredients
Here's how to deal with confrontation:
• Aim to ignite and extinguish an issue in one meeting. Have the confrontation face to face, and keep coming back to the facts to support your point of view.
• Always attack the problem, not the person. If you can avoid injuring egos during the exchange, the relationship will always recover much more quickly.
• Don't take it personally. I've always found this difficult, as I am a passionate person, but I've learned to follow this advice. You cannot respond in a calm and intelligent manner if you take things to heart.
Super supplements
And here's how to take your confrontation skills up a notch:
• Take notes during your business dealings. If a confrontation brews, you can avoid a 'he said–she said', mudslinging match by showing supporting evidence that backs your position. Remember — you cannot have enough written support.
• Avoid email fights because they can be misread — and they could come back to bite you later.
Money
'Loan oft loses both itself and friend' \- William Shakespeare
William Shakespeare knew what he was talking about, and nothing much has changed in 400 years. It's always easier to lend money to a friend than it is to get it back. Don't do it! Learn to say no.
If you're running your own business, also learn to say no to any unnecessary consumption in the short term. For the first three years of running Boost, I didn't take anything out of the business, not even a salary, and put everything I could back into the business. During those years, we lived very basically.
Key ingredients
Here are some things to keep in mind about money:
• If you build a reputation for never lending money, it will be easier to say no to requests.
• Offer your time and advice to those seeking help; those forms of assistance are much more valuable than cash in the long term.
• If running your own business, try to forgo short-term gratification for the long-term success of your business. Ask yourself whether certain personal purchases really need to be made or whether the money would be better invested in the business. Every cent spent should return double. Ask yourself: is this expenditure really necessary right now and will it return me profit?
Negotiation
A negotiation is not successful unless both sides feel they've won.
There are two elements that are vital to a successful negotiation:
• Information: Do not go into a negotiation with only your point of view. Understanding the other party's needs is equally as important as understanding your own. Ask yourself 'What do I feel is reasonable?' or 'What would I want if I were them?' This will allow you to counter the opposition's arguments before they are raised.
• Lack of emotion: The only way to negotiate a great deal is having the ability to walk away. The only way to know if it's a great deal is to listen, listen and listen. Never let your heart rule your head in negotiations. If you are emotionally attached to something, you will give away too much. This holds true whether you're buying a house or making a business deal. Try not to take the proceedings personally. It's difficult, but try to think of yourself as a third party.
Negotiations take an enormous amount of courage and a very clear head. You should always ask for more than you want and then negotiate down. Don't give away your minimum requirements — you may end up with even less. Also, don't favour the same weapon over and over; it will become less effective each time you use it. The more options you have, the more power and control you have.
I remember my first major negotiation — we took on a small juice chain in Queensland that had started up with the name Juice Boost. In the end, we paid for them to change their name and we bought their trademark. It was a win–win situation. We had to pay, but it was worth it to retain the purity of our brand. We had right on our side and got the best out of the deal!
Janine says...
I have negotiated all sorts of things — including bills from lawyers who in my opinion have overcharged me (perhaps not surprisingly, this happens a lot), multimillion-dollar master franchise agreements and sales of businesses — and the biggest thing I have found is that no negotiation is the same. The greatest skills you need in any negotiation, however, are emotional intelligence and the ability to be a really (really) good listener. If you listen more than you talk, you will hear what the other party wants. The other great skill in negotiation is putting yourself in the chair of the person you are doing business with. (Remember — the best negotiation is when everyone wins. If you cannot put yourself in the other person's shoes, you should not be negotiating.)
In 2005 I was in Dubai with Jacinta, negotiating a master franchise agreement with a party we thought would be a great company to open Boost stores in Dubai. We wined and dined with their sheikhs, we listened to their needs and we presented our business with the passion and enthusiasm that we have for the brand. The business spoke for itself — it was a sexy brand, and was in the wellness category from a country that is perceived to 'love life'. They had the contacts and experience to launch the business in the market and were ready to move into the wellness category. We negotiated the contract while we were there, constantly working on the memorandum of understanding in our hotel room. I remember being with Jacinta in the airport on the way home when it was all over, not quite believing that we had completed an amazing deal to launch Boost in Dubai with a party who had never seen a store.
Key ingredients
Here are the core elements of great negotiation:
• Learn as much as you can about the other party. If you're negotiating with a public company, you'll find that everything you need to know is easily available — from information on the shareholders and senior executives to the company's profit forecasts. Information is your key weapon.
• Try to stay detached during the negotiation. Emotion has no place when you're making a deal.
• Put yourself in the opposition's shoes. This will help you to counter their arguments and provide a win–win solution.
• Know what you are prepared to accept, but never give away your minimum. Aim high and negotiate down — never the other way around.
Success
'I don't know the key to success, but the key to failure is to try to please everyone.' \- Bill Cosby
There is a price to be paid for success. No-one achieves their goals and dreams without giving up something. But can you afford not to try?
What have I given up? A hell of a lot of time! Is it worth it? For me, yes. That's because the results are not just financial success — I also get fun, stimulation and the chance to follow creative pursuits. I've given up a lot of time and energy, but I wouldn't do so if I didn't want to. Sometimes when I'm torn between work and family, I feel like I'm robbing Peter to pay Paul, but it comes down to what's worthwhile for you.
If you want to follow your dreams, you need to be prepared to make sacrifices; you must decide what you're willing to put in. Many people expect luck to deliver them a fortune — well, I've got news for them and it's all bad! Success does not just come about magically — you have to make it happen, and that means giving it everything you've got. With any luck, your hard work will pay off. It has for me and I have no regrets. I have a great life and nothing to complain about. Can you ask for more than that?
Janine says...
Success means different things to different people. My sister Lisa is 21 months older than me, and I remember being in the car with her when she was 17 years old. She said then that what she wanted in life was to find love, get married and have a family.
Lisa has never not got a job she went for, and at every job she gets they love her. However, even now her family is her success — I look at her three girls and her husband who she loves and I really see success. Happiness is success, not dollars. I am sure that Lisa has never looked at me and my business success and wanted my life, because she is already happy and successful. I am successful because I have a husband I adore, and kids who are all individual and amazing — because that is what makes me happy and that is my success.
Key ingredients
Here are some things to remember when chasing success:
• Be prepared to give your all to achieve your dreams.
• Something will have to give. Decide now what you can and are willing to give up.
• There are no fairytales — there is just damned hard work!
Credibility
Credibility is easy to talk about, but difficult to come by.
Credibility is vital in today's business landscape. In this electronic age, with the information and awareness that is out there, you can't fake it.
What is credibility? It is a general opinion held by others that you 'do the right thing'. Your credibility should be part of your reputation. How does a young entrepreneur or business attain credibility? Well, there are a few operational basics.
First, you must demonstrate that there is real organisation behind your company. Highlight the expertise you offer and the services you provide. Show potential customers or associates that honest and trustworthy people stand behind your brand. Make sure your company is accessible, transparent and easily contactable. Ensure that your business premises have a professional appearance and your employees a professional manner.
The next step seems simple but is often more difficult: follow through on your pledges. In my position, I get promises every day. Initially, I take people's word that they will do what they say — and then I wait and see what happens. Do they get back to me on the day they said they would? Do they deliver the expertise they've presented to me? If not, their trustworthiness immediately starts to wane. Your business will achieve credibility through actions.
Janine says...
I expect a few young guns are reading this book, so here is the best piece of advice I can give anyone who is under 35: you are just playing at business. You cannot hope to understand all the complexities and issues because many of you will be facing them for the first time.
Don't get me wrong — the older business heads love your energy and enthusiasm to attack the day with vigour and purpose. But don't for a second think your smooth-talking, fast-thinking ways have anyone fooled — unless, of course, you have complete credibility.
I have been in many board meetings where hot young executives come in and present their big plan that's going to change the world. The old heads listen intently, enjoying the show. They nod approvingly and the young execs leave — and then the old heads tell the truth about their impressions of the performance. And trust me — it can be more scathing than the judges on reality talent shows. The success or failure of the young gun will come down to one word: credibility. Was the person and, more importantly, the information and evidence of success presented credible? Remember, to be incredible you must be credible.
Key ingredients
Here are the most important components of building your credibility:
• Credibility grows from honesty. If you start with a measure of integrity and throw in some hard work, delivering on your promises, positive results will begin to flourish.
• Don't overcommit. When you're starting out, it is often difficult to say no to work or to clients. Unfortunately, this can mean you take on so much that you can't deliver the right results to anyone. Your credibility will be shot to pieces before you even begin! Promise only what you can definitely supply — or find a way to make it happen!
• Credibility must emanate from every level of a business. Once again, it comes down to finding the right people for the job. If you believe that an employee is not delivering and is causing holes to appear in your reliability, be rigorous in fixing the problem.
Communicating
Everyone talks about good communication... but how do you go about it?
When we think about 'communicating', most people tend to have talking in mind. However, communication comes in many forms, including social media, email and the web. Given that we spend so much time emailing and texting, it can be overwhelming when you're faced with the prospect of speaking to a group of people. There's no opportunity to 'backspace' or 'delete' if you blush, stutter or stumble through a speech.
Good presentation skills offer you the opportunity to leave a positive and lasting impression on others. You don't want to be remembered for the number of times you flicked your hair or adjusted your tie. If you are, your message is hopelessly lost. Make no mistake — the audience will judge you on your performance.
Do you need some work in this area? Not sure? Try taking a video of yourself in action. Is that the image you want to project or are you cringing at your performance the whole way through? (Don't worry if you're a bit embarrassed by the sound of your own voice at first — most of us are.)
Confidence is the key to a good presentation and you can gain the poise you need by practising and refining your skills. There are plenty of courses in public speaking available. They cover all the essentials — dealing with nerves, projecting your voice, cultivating the right image through your appearance, delivering your key message and the secret of the 'pause' — as well as other skills you can nurture and adopt.
Most tutors will advise that your natural style should not be changed completely. The best course of action is to refine and improve your inherent ability. If you attempt to adopt a totally foreign persona, you will come across as insincere.
Of course, these days much of our business communication is non-verbal, or at least not face to face. By investing in the latest technology for your office you will speed up the flow of information. Embrace new technology and make it work for you.
Ask your colleagues, associates and contacts how they like to communicate and work within their preferences. Some people like the effortless speed of an email; others favour the more personal touch of a phone call.
Regarding email, be sure to investigate correct etiquette and read your emails thoroughly before clicking the 'Send' button. (Be wary of 'Reply all'!) Be courteous and remember that the ramifications of an email can be with you a long time; email can be subpoenaed in a court of law!
On the subject of phone calls, don't do anything else while talking to someone on the telephone — it degrades the conversation. The sound of your fingers typing on a keyboard is a sure sign to the person on the other end of the line that you place little importance on the person and the conversation.
Janine says...
What I have learnt about communication and developing your communication style is to be true to yourself. It's fine to take tips from other people, but still do it in your way.
I have been asked to tell the Boost story a number of times and, in the early days, I could never quite do the story justice — the way I told it just wasn't quite right and I knew I wasn't communicating the ideas clearly enough. Then I saw a presentation by Simon Hammond, who had put us in his top 20 list of wonderful brands. He is one of the few people we have come across who truly understands the power of the brand.
We heard he did a great presentation on brands so we asked him to present at our annual conference. His presentation was wow! It was a show, complete with music and interesting snippets from the internet on marketing ideas. He made us laugh and almost cry with the emotion of his presentation. I was so impressed that I asked him afterwards if he could help me formulate my story into a great presentation. We spent weeks (and weeks) pulling it together and getting to the truth of the story, and also the true essence of what I am naturally like as a presenter.
We found my presentation could vary greatly from day to day, depending on the audience. In essence, I was a confidence player if I felt like I had engaged with the audience — if they laughed and nodded, my presentation was pretty good (if I do say so myself). But if I didn't feel like I had connected with the audience, it was normally a disaster. Simon assisted me with understanding who I am and what my natural style is, with the added help of a few videos and props to ensure my communication is consistent. The main point has always stayed with me — whatever you do, be you.
Key ingredients
Here's how to polish your communication skills:
• A professional speaking or presentation course is an investment in your business future.
• Be professional in all your business communications, be they typed or spoken.
• Learn to use technology to your advantage.
• Give all your attention to the person to whom you're speaking, whether it's face to face or on the phone.
• Treat email communication with care. It may be an instant medium, but the message can come back to haunt you.
• Poor communication is the number one reason for conflict.
Customers
'Your most unhappy customers are your greatest source of learning.' \- Bill Gates
As Boost has grown, I've found that I've gotten further away from the 'on the ground' customer experience. As the manager of a growing company, you ultimately move on to the next level of development in your business, delegating tasks to others. You become more removed from the day-to-day running of the company — and you can miss the simplest problems. Try not to make this mistake and never be a stranger to the front line. Don't ignore the very hand that feeds you.
I work hard to counter any movement away from the front line. It's vital to continue to tap into the root of your business, and for me that means going into a random Boost store and queuing up with the other customers. Or when I'm in the office I might answer a ringing phone — any phone. If it's a customer wanting to vent frustrations or even give positive feedback, I talk to that person one on one. I ask questions about the company and the level of service received; I ask for people's opinions. It's a simple task, but an extremely valuable exercise because it allows me to derail potential problems that may be quietly simmering away.
One tool we use to encourage feedback is our Boost Juice guarantee. Every store displays this guarantee, and you can also find it on our website. If you do not have a good experience, let us know and we'll fix the problem. And we absolutely do fix it, every time. What's more, we offer customers a number of easy ways to get in touch with us, and we employ two full-time staff to respond to customer feedback.
Resolving customer complaints immediately and effectively is critical; our policy at Boost is to respond within 24 hours. Customers are usually so grateful (and surprised) to receive a response, their problems are easily resolved. This is a vital and mostly unseen part of our marketing strategy.
Janine says...
There are whole books dedicated to the subject of creating and keeping loyal customers, so all I am going to say is this: love them; truly love them. And if you are the leader of your company, make sure that every day you find out what your customers are thinking and wanting; this knowledge will flow through to influence all your behaviour.
Get angry when your customers aren't treated well, and fix every customer problem with vigour — they are your life support system. Hire front-line staff who like people. Every Boost store has a multitude of talented people, specifically chosen for the role they need to play in the customer's experience. We've even given each role names. 'Eva', who is bright and bubbly, is on the front counter greeting the customer. 'Ian', introverted but super-diligent and process-orientated, is making the smoothies. And finally 'Beth', a real extrovert, is on the last station — pour up. This person is the last impression customers get of the brand, so (hopefully) she hands over the finished product with a big smile. We don't always get this process right, but it is indoctrinated into our belief and it is what we strive for.
Make the love of your customer an absolute pillar of your company's beliefs and you're on the way to success.
Key ingredients
Here's how to focus on your customers:
• No matter what your position in a company, never take your customer base for granted. Keep in touch with your market and respond quickly to its needs.
• Small problems can become large if not dealt with quickly. Put systems in place to ensure customers never feel ignored.
• Customer liaison is vital, and should be part of your marketing strategy. Word of mouth is your best friend — and can be your worst enemy if you don't address issues.
Part III
Expanding skills and overcoming obstacles
This part of the book was interesting to write because it covers areas where my learning curve was large and lengthy. These areas are also where other individuals and businesses tend to struggle. Because the point of this book is to share with you my lessons learned, I emphasised these particular skills, hopefully helping you avoid some of the pitfalls I found myself in.
In the final chapter in this part, I've also provided some insight into some common obstacles to success — and how to avoid them.
Chapter 8
Blending a great team
In case I haven't reiterated it enough throughout this book, let me make it clear again here: the right people are integral to the success of any business. This chapter covers hiring and firing staff.
Hiring the right ones
How do you find the right staff? What do they look like? If I've discovered anything along my journey, I have learned that the most important thing employees can have is the right cultural fit. They have to fit in with your team. They must understand what it is that you're trying to do.
Secondly, they must have the right core fit. This is their attitudes: work ethic, ambition, self-motivation, passion, honesty and whether they're team players. Everything else is just mechanics — everything else you can teach them.
If you don't get the right cultural fit for the business, it just won't work. Unfortunately, in an interview situation, people will often tell you whatever they think you want to hear. It can be incredibly difficult to break down barriers to really get to the essence of the person.
So, how do you sort the wheat from the chaff? We generally know within the first 60 seconds of an interview whether a person is going to fit in at Boost.
Firstly, it's important to ask the right questions. The culture at Boost is energetic, honest, passionate, sometimes funky, fun and always high performance. You can't fake those traits, so we don't need to ask an enormous number of questions to ascertain whether people will fit in — they either have it or they don't.
We have a rigorous selection process for employment at Boost's head office and the 'cultural fit' interview is the last part of the process. Within this interview, we have specific questions that we ask candidates to answer and we look for specific traits within these answers. The following are the questions we ask; underneath are some of the traits we are looking for within their answers.
Q: What do you know about Boost?
We want to employ staff who frequent our stores and are familiar with the concept. It's important to employ people who are attracted to our philosophies. You can't work at your optimum level if you're working for a company that doesn't fit with your own beliefs, or a business that produces something you're not passionate about.
Q: What is your ideal working environment?
The Boost workplace is fast-paced, dynamic and vibrant. Staff need to be comfortable working in offices featuring more colours than Play School and being around people who are, for the most part, bouncing off the walls with enthusiasm. If you work best in a library-style setting, where there's no buzz and no fun, you're not going to be comfortable or productive working at Boost.
Q: Why do you want to work for Boost?
We look for enthusiasm for Boost as a company, for someone ready to fully endorse the product range. Obviously, this relates back to the first question — you'd find it difficult to answer if you know nothing about us.
Q: Our slogan is 'Love Life', and we live by the values of honesty, integrity and passion. How do these philosophies apply to you and your current lifestyle?
It's difficult to lie about your way of life. While you might be able to fake enthusiasm enough to answer the previous questions, it's much harder to answer this one without revealing a little of your true nature. This is another way in which we determine whether the values at Boost align with the candidate's values.
While several of the preceding questions might seem to be asking the same thing, it's important that we throw all of them in during an interview. It's a very old trick to ask the same question in different ways, and it's up to the potential employee to answer these questions consistently.
We look for people who fit the Boost culture, but it's also fair to say that the majority of employees in our head office are ambitious and self-motivated. This isn't a coincidence. At Boost, we're achieving twice as much as other franchisors in half the time; to keep up in this fast-paced environment, our employees regard their role as more than just a job. It may come as no surprise that many of our staff members are triathletes or are passionate about a particular sport; high achievers are drawn to Boost.
To keep attracting high achievers, we make sure that each candidate's core fit matches ours. We look for passion, ambition, self-motivation and drive. You can tell how much potential employees have of each of these traits by asking them about their 'achievements' in past roles, or in their personal life.
Listen for the particular traits you value within your potential employees' answers. An obvious starting point is to say, 'Tell me about one of your greatest achievements'. You then just have to listen for the traits you value. If they match your 'core fit' — bingo! A follow-up question is, 'What do you wish to accomplish in the available position?' If the traits that are in your core fit exist within that candidate, it's generally easy to hear. However, honesty, integrity and high standards are harder to determine in an interview. For these you often have to go on instinct or past employers, or from the candidate's CV.
Super supplements
Ever had to audition for a job? At Boost, we've devised an interesting and unusual way to interview for counter staff. Every six months or so, depending on our employment needs, we hold a Boost audition, usually in a different capital city. We advertise in the papers and often hundreds of people turn up. These are not your ordinary job interviews! The applicants actually audition: they play games, they dance, they sing, they perform. We're looking for the right spirit for our business, and we use these exercises to single out the leaders and those who work well in a team — the right people for Boost. It's not always about hiring extroverts. You need a mixture of personalities on a team.
Firing the wrong 'uns
Even with our well thought out approach to corporate hiring, we do sometimes get it wrong at Boost. Some people will tell you what you want to hear to get the job. It can be difficult to get past that before it's too late. Alternatively, people's particular idiosyncrasies may only come to the fore once they're working in a team on a daily basis, or they simply may not 'get' what we want. I've learned that in these instances it is best to act quickly — not tomorrow, not next week, now! And act with a rigorous, not ruthless, philosophy. I received great advice once from a respected businessman, who said, 'Hire slowly; fire quickly!'
I've never been great at firing people, though I have gotten better with time. It's not an enjoyable process. Essentially, you're sitting down with people and telling them, for whatever reason, that you don't want them working for you anymore. This inevitably affects their ego and incites that terrible fear of wondering where their next job is. In short, it's a horrible thing to do. That being said, it's also a good thing to do. You'll find that in the long run it's probably the best thing for them as well. If you don't act quickly to remove the wrong people, you stand to lose the right ones. As a company with high standards, we have to uphold those standards, or other staff members start to wonder. Why put up with mediocrity?
In 2004, as you read in part I, we had to make changes in the business. At the end of the year, we asked the heads of each department to assess their teams and decide who they wanted to take into 2005 with them. Unfortunately, there were a group of people who didn't make the cut. Were they bad people? Far from it. They were just not right for Boost at that stage in its growth. As a leader, you must have courage; even when you hate doing what you know is the right thing for the business.
What are fireable offenses? Obviously, dishonesty is a big one. Accountability is another. I do not want to hear about why your stuff-up is not your fault; if you try to make excuses, I will immediately lose an enormous amount of respect for you. I like people who agree that they've made a mistake and then tell me how they're going to fix it. Even better, I like people who bring a mistake to my attention, even if they know I'd never find out about it, and give me the solution.
There are other reasons people end up losing my respect. I heard a clever acronym years ago, describing a particular mentality, and it has since become a part of my professional vocabulary. This acronym is VERB, or Victim, Entitled, Rescued, Blame.
At Boost, we don't like a VERB mentality. We don't want people who see themselves as victims — we don't want to hear 'poor me'; we want to hear 'can do'. In addition, there is no such thing as entitlement at Boost. I believe that people should be rewarded appropriately for what they do. I hate hearing someone say, 'That's not my job' or 'I don't get paid for that'. We also want people who find solutions — rather than feel they need to be rescued. We want people to come to us with answers, not problems. Lastly, and this is a pet hate for me, we don't tolerate people who blame others. It's true we all do it at different times in our lives — and we wouldn't be human if we didn't — but don't be a serial offender.
The choice you can make is to instead use the SOAR approach, or Solutions, Ownership, Accountability and Responsibility. This approach is the opposite of VERB. If you use SOAR in everything you do, you will suddenly find things going your way.
You can find the solution to your problem, just stay with it. Take ownership of everything in your life and business, and soon you will see the power it gives you. Accountability is there to remind you to fix no-one else and, finally, take responsibility for everything you do — if everything is your responsibility, you can fix it. I challenge you to SOAR — and find staff who SOAR with you — and see the difference.
Of course, an actual firing never really comes as a complete surprise. It's not like one minute you're telling staff members they're doing a great job and the next they're out the door. That doesn't happen. At Boost we follow the law to the utmost extent, ensuring that everyone gets a fair go. However, people have different levels of what they believe is acceptable.
You can't afford to have people who sit around twiddling their thumbs; you can't have one department drowning in work and another department leaving on the dot at 5 pm. Of course, as I've mentioned previously, staying late is not a badge of honour. But you do need people to deliver on what's required to get their job done. I don't enjoy letting people go, but it is a necessary part of my business. A situation will get worse if you don't do anything about it.
Chapter 9
Mixing up your marketing
It is no accident that Boost Juice achieved 94 per cent brand awareness in Australia in just five years. After all, there's no point in having a terrific product if nobody has ever heard of it!
Much has been made of Boost marketing and, the truth is, it should be. We weren't the first juice bar, but we were the first in most people's minds. At one point, there were 47 different juice bar brands, vying to be in the top two to survive. Boost was the brand that made it through.
It may be old school but the philosophies of Ries and Trout, covered in their book The 22 Immutable Laws of Marketing, were followed to a T. This is marketing defined simply as find out what customers want and give it to them — but it's only half the story. The other half of the story for Boost was find things that they could talk about and give it to them. So consumer desire and innovation became our mantra.
Early on, so we had a flag for our team to point to, we came up with the 'Boost JAM Factor', which still stands today. JAM stands for Juxtapose, Assimilate and Make your mark, as follows:
• Juxtapose: If the other guys do it, we don't!
What can we do because we are small and don't answer to a marketing committee? Things like the 'What's ya name' game (where customers receive a free smoothie if we call their name), or 'Boost Vibe Challenges' (where customers do crazy stuff for free smoothies). Another promotion that was a huge success for us was the 'Give away a Boost store' promotion. We ran this in conjunction with a major radio station, and it formed the station's major rating promotion for the year. The concept was a bit like MasterChef — we short-listed 100 people and then worked with these contestants and, eventually, through a series of workshops and interviews, reduced the list to ten people. We then reduced the list even further, until we got to a final three, all of whom were given a key — and the person with the key that opened the door to the store owned it. (We were worried that the winner may end up not being satisfied with the win — after all, not everyone is suited to being a business owner; it's hard work and not everyone understands the reality of running your own business. So we had a fallback — in six months, the person could choose to keep the store or take $50 000.) The exposure from the promotion was incredible — sales in the state in which the promotion was held went up 20 per cent and we received over $1 million in free advertising. These are all things that the big guns of the world couldn't do, and they're innovative and new food retail ideas.
• Assimilate: Find ideas everywhere and from anyone at any time.
Tweak these ideas, twist them, and blend them into another 'by Boost' idea. Never stop searching and assimilating ideas from the Boosties (staff who work in the stores).
• Make your mark: If you do something, do it big!
Don't be included in the 'and thanks to...' list with eight other sponsors. Do one big thing great, for maximum cut through.
Using this basic formula, we smashed into the minds of Australian youth, and they loved it. These Australians are now in their thirties and still drinking Boost.
Why our marketing strategy worked
Here are some of the reasons our marketing strategy worked (in no particular order):
• Vision: 'Love Life' was our soul. If something could make people smile, we had licence to do it.
• Loyalty: Our Boosties believed in us and were proud to work in our stores. Because of this loyalty, ideas were well executed in store.
• Saying yes: We told staff to not think too much about an idea, just have a crack, and say yes! Some of our ideas may not have become a reality if we had put it to a vote.
• Radio: We couldn't afford television advertising, but radio was incredible for building our brand. We still only buy breakfast spots on high rotation.
• Pillars: We know what supports our brand: taste, the experience, health and me. If even one of our pillars gets a crack in it, we put energy and focus on it and fix it.
• Brainstorming: We allow for the energy flow of an idea, and let even the craziest idea have its space to grow and develop into something magical. When one of the team suggested doing a kids' cup promo with The Wiggles six years ago, it was so 'off brand' for our hip little Boost brand that I had to try very hard not to blurt out, 'No, stop!' The team member worked on it, came up with a vision on how it could all work in harmony, and bingo — we nervously proceeded into one of the best things we still do today.
It's important to push the boundaries philosophically. It's my objective that Boost will have a huge impact on the health of our society — a society that's becoming increasingly overweight. Research indicates that the increase in obesity in children in Australia is in line with the growth in fast-food outlets. I think I'm in a very exciting position to make a difference in the future. If I can create a trend in health and make Boost a phenomenally successful healthy product, perhaps others will see that there's money in health, and they will make more healthy products. I think it's already happening. By promoting our 'Love Life' idea, we're driving home not only brand extension, but also life expansion!
Chapter 10
Picking the franchising fruit
I never even considered having just one juice bar. I always knew that Boost was going to be big (which shows just how naive I was, and lacking in any true knowledge of what was involved to make this prediction a reality).
In the very early days, I also hadn't considered how exactly I was going to create all these stores. I simply thought I'd have another and another and another... Unfortunately, as the brand expanded, I simply couldn't hire the high-quality people I needed as well as expand. I realised early that if we wanted to grow quickly, people in the company needed to have ownership. There's nothing like having everything on the line to make a business work well. Enter franchising.
I'd never had anything to do with franchising, and didn't really understand how it worked. We met Rod Young, a franchise industry consultant, through a friend of a friend, and approached him to assist us in setting up the Boost Juice franchise. I always think that if you're going to do something, you should do it right, and getting Rod on board was definitely the right thing — at the time, he was the best in the business.
Another important factor was getting the right franchise manager. I've always seen our franchise manager as the gatekeeper of the Boost Juice name. I imagine her standing in a knight outfit (with a very sharp sword) on a plank outside the Boost castle, and I know that she will not let the wrong people into my business (or our business, as she would say). We do not tolerate mediocrity. Believe it or not, the best decisions we have made involve the people who are not with us!
The Boost franchising method
For Boost Juice, franchising has been a wonderful way to get amazing people into our business and to create ownership in individual Boost stores. We call our franchisees 'Boost partners' and in a sense that's what they are. Franchisees pay Boost an ongoing royalty — a percentage of turnover — for the right to use our brand on a single store. They are buying into the brand and the support we give them. The statistics are very straightforward and, overall, you're more likely to run a successful business if it is a franchise business; however, buying a franchise business is not a blanket guarantee.
As the franchisor, we get two major benefits from the relationship. The first is growth of the brand. We were listed in BRW in early 2005 as Australia's fastest growing franchise network. The second is the quality of the people you bring into your business to run the individual stores. Both the franchisor and franchisee have the same motivation to be successful, and both parties have everything on the line to make it work; this is a great motivator.
It's not easy to become a Boost franchisee. We are known within the franchising world as being incredibly selective — and we're proud of that. Our existing partners, in particular, love this reputation because they know that any new partner has undergone rigorous assessment. We approve roughly 11 per cent of applicants as Boost partners — and we get around 50 to 100 applications every month.
We select franchisees based on the philosophy, 'It's not necessarily about where they've been, but where they want to go'. We have some partners whose previous retail experience has made their transition into our stores quite smooth, and yet many others with no previous experience who have taken it all in their stride and ended up operating very successful stores. The only common stream among the partners is passion for the brand, following the system and understanding the concept.
We often have applicants who are interested in Boost only because of the financial reward they believe they'll get from owning a Boost franchise. While we appreciate that all applicants are interested in a financial return from investing in Boost (after all, they're not going to go into a business to lose money), this cannot be the only reason. They need to have a passion for healthy living and amazing customer service skills.
The culture of our company is the most important thing to protect — if we compromise on the quality of the people, we compromise on the brand. One of the reasons we are so successful is because we have incredible partners fostering our culture and giving our customers the phenomenal Boost experience every day. If we dropped our standards, they would drop them too.
I have heard a lot of stories about partners who have cried tears of joy at the news they've been approved as a Boost franchisee and, on the other hand, stories of unsuccessful applicants who have phoned to rant and rave. (Including one person who sent Boost a doctor's invoice because his wife had to be sedated after hearing the news of their fate.)
Boost's recruitment process is transparent. On our website, we've posted answers to around 90 per cent of the questions we're most frequently asked, right down to how much it costs to become a franchisee. It's quite an intense process. Applicants have to fill out an eighteen-page questionnaire and send it to us with an application fee. We then call them for a chat. That's step one. There are another four stages to pass before approval is granted. The final stage is an interview with the CEO of Boost.
Once franchisees are in the system, they take part in a three-week training program. They work in their store with one of our experienced people for support. They have full access to our team and we do our best to ensure they succeed. To help keep things on track, we have audit programs as well as mystery customers who visit every store every month to make sure the service and quality is as high as it should be. The reason we do all this is that we are not prepared to settle for mediocrity; this process helps us to get the right people in our business, and keep those people acting in line with our values. To us, the brand is everything; we do everything we can to uphold its integrity. It is one of Boost's greatest strengths.
Super supplement
What to franchise? When to franchise?
It seems that everyone with an idea is trying to franchise it these days. So how do you know if your idea is a good bet? When should you franchise?
First, you need at least a year's worth of figures that are profitable. You also need to make sure that you're in possession of a full operations kit (a 'how-to' guide with tools, forms and information) so that everyone knows exactly what they're doing. And you need to be certain that your intellectual property rights and trademarks are set up. And I'd recommend you hire a solid accountant and lawyer to ensure you get it right.
How do you know if your business is a good franchise option? Work out if your concept is a profitable model that can be transferred to another location. I have a friend who's a naturopath, and she is so successful that she was considering franchising. Unfortunately, in her situation it's not possible — she is her business. If she could find 100 clones of herself to run the franchises, yes, it would work, but that's not going to happen! In my friend's case, she's now considering creating a product line instead. Franchising may not be for you, but there are always other ways to expand your business.
Chapter 11
Loving life
Ever feel like a mouse on a wheel? You know, running and running and going nowhere... fast. Sometimes you need to get off that wheel, for the sake of your mental and physical health. This chapter is all about ensuring you're happy with your work–life balance, and taking care of you.
There's more to life
Do you feel like there just never seems to be enough time in the day to fit it all in? If you're anything like me — a parent, with a business to run or a job you're passionate about, and a partner whom you love to spend time with — the days probably fly by. Sometimes the pace of my life is enough to make me wonder whether somebody has actually dropped an hour or two out of the standard twenty-four-hour day without telling me.
Work–life balance is something that's talked about a lot these days. But what does it mean? To me, it is all about finding happiness and contentment. To get enormous stimulation from my work, yet be equally happy to sit down and play Monopoly with my 14 year old for two hours. Sometimes I do find it hard to drag myself away from work — when you're really passionate about your job, it can be all-encompassing — but I'm getting better. I'm learning to switch myself off from the office and enjoy the other aspects of my life as well. I think that we will all be better wives, husbands, mothers, fathers, employees, leaders and friends if we find that happy medium.
Work–life balance is about prioritising what's important to you, and having equal space in your life for your job and your family. Unfortunately, a lot of employers only pay lip-service to flexible work hours and 'family time'. The reality is that you get paid to do a job and the job has to get done. Having said that, work–life balance is becoming recognised as an important strategy in creating a healthy corporate culture. The best companies are certainly making great steps in that direction and they're finding great improvements in productivity in the process.
When Boost Juice first started I worked from home, juggling my three sons and a growing company: the two little ones were still at home, and the eldest was at primary school. It was important to me to be home when my oldest son walked in at 3.30 pm. Eventually, my business baby (Boost) began taking over room after room of my home, and more and more hours in the day. I ended up despising my house, and particularly the rooms used for workspace, because I never got away from work. It was a relief to everyone when Boost moved out of the family home and into corporate offices. Things became much easier for me once I had a definite demarcation between work and my personal life. The advantage was that home became my sanctuary again. The disadvantage was that the kids couldn't see me when they walked in after school. So I made sure my kids also felt at home at the Boost offices. One of the first things I did when the business moved was to buy a large toy box and a television for the premises, and I also made sure there was a spare computer so the boys could come in and do their homework. These days, my kids are part of the furniture in the Boost offices.
Another step I've taken is to set up 'non-negotiable' dates. These include dates such as school sports days and concerts — if it's important to the kids, I'll be there, no matter what's going on at the office. I'll walk into one of these events and see my son scanning the room for me. When he catches my eye, his face lights up. It means everything to my children that I'm there for these events, and that matters to me.
You may be thinking, That's great Janine, but you live within two minutes of your work and you have your own business. My work won't allow me to do that. You're right, I do have a lot of freedom, but that's because I've created my freedom and I've done it through hard work. It's only now that my business has the executives in place that I do not need to be there 24/7.
I'm not stepping away from the business; I love it too much. But I have realised that I cannot allow it to consume my life forever. Any good businessperson should be constantly hiring people to do themselves out of a job.
Whatever choices you make, ensure they are working towards you having a full life. While I'm no expert in this area, having made many mistakes, the following are a few things I've learned along the way.
Prioritise
You will never get what you want if you don't know what that is.
It's time to stop seeing work as something that finances the rest of your life, and start thinking about doing what you love and making money from it. I know I've reiterated the idea of passion several times in this book, but that's because I feel so... well... passionately about it. If you're still not sure how to define your goals and dreams, seek professional help in the guise of a life coach or a career counsellor.
My kids are my priority, but I believe that children are in your life, you're not in theirs. Admittedly, this gets easier as kids get older. I work killer hours, there's no doubt, but I fit them in around quality time with my children. When I'm at home with them in the evening, it's their time. We do normal things like reading and watching television until they have to go to bed. Then it's my time again, the laptop comes out and I do what I need to do. Yes, it's a long day, but I wouldn't do it if I didn't love it, and there's no way I'm going to compromise family life by letting my work take over completely.
It takes two
Jeff and I are both passionately involved in Boost. Do we ever see each other? Do we ever talk about anything other than business? The answers are yes and yes. I must admit we do have corporate meetings over dinner, but both of us are aware of the importance of letting each other switch off. We understand when one of us arrives home from work and simply doesn't want to talk about business. We also make time to go away together. We have a good, open relationship (which is vital) and enormous respect for each other. Even with that kind of partnership as a base, you will have your ups and downs; Jeff and I always talk through our differences.
Get help
Who said that you have to do everything? Women in particular suffer badly from the 'I must' syndrome: 'I must put in a full day's work, organise the family, clean the bathroom, collect the dry cleaning, bake for the school fete' and so on. Get over it! There are terrific people working hard to make a living doing those things you don't want to do. Let them. Whether it's a cleaner, a gardener or a caterer you need, call in the professionals and save your energy for chasing your goals.
Work out how much you need
There's a tendency today for us all to be working at making as much money as possible — without really knowing how much we need. You don't have to curtail your financial ambitions, but if it's money that's driving your workaholic tendencies, it might help to sit down and cost your ultimate lifestyle. Once it's written down, you may find you don't need as much cash as you think. Or you may find the motivation you need to start overachieving!
Enjoy extracurricular activities
It is healthy to have a hobby or social activity outside of work that helps you relax and forget the problems of the day. I feel very strongly about this. I do yoga five days a week, and this is a time where I can focus on nothing else but me. Whether it's a drink with your mates, a visit to the gym or whatever — don't get too busy for something that helps you cope with life's pressures. You can always find time for the things that are really important, so make this important.
Super supplements
Here's how to really focus on improving your work–life balance:
• If work is the only thing in your life making you feel worthwhile, it's time to take action. Sign up for a course that gets you out of the office by 6 pm, for example.
• Find a weekly or regular activity you enjoy — join a running club or organise a regular meeting with friends. Or perhaps a book or investment club is a good way for you to get started.
• Create some boundaries between your work and home life, and put some emphasis back on health, social life and relationships.
• Clean up your desk. You know where everything is, but do you really need everything? It sounds simple, but you will feel much better when you're not working in chaos!
Taking care of yourself and your health
Taking care of you is taking care of business. You can't win in business, or in life, unless you look after yourself first. When you're chasing a goal it can be easy to overlook simple things like sleeping, exercising and eating right. All I can say is don't. Working around the clock makes you inefficient and unfocused. Sitting on your backside all day makes you sluggish and, let's face it, overweight. A diet of caffeine and fast food will leave you low in energy and not looking or feeling your best. It does not matter how financially successful you are; if you do not have your health, you have nothing.
As I've said before, I'm passionate about eating right and I don't miss my yoga for anything. I also do my best to get a decent sleep every night. But I am human — I understand what it's like to be so stressed that you physically can't eat. The key is balance.
We all have heard the saying, 'You are what you eat — and drink'. Boost defines 'eating for success' as eating naturally, and avoiding artificial colours, flavours or preservatives. The emphasis is on fresh and nutritious. It's a theme repeated in one of my favourite books on the topic: Laugh with Health, by Manfred Urs Koch, which focuses on health, diet and natural foods. Of course, I'm not suggesting you suddenly adopt a macrobiotic diet — it's more about doing what you can do within the confines of your busy life.
What do I do? I eat as much unprocessed food as possible — I eat a lot of fruit and veggies. If I pick up a packet of food and there are numbers and words I cannot pronounce on the ingredients label, then to me this is not food. Yes, I have a smoothie or juice (sometimes both) every morning. I'm a big fan of all our juices; the simple ginger, carrot and apple juice is my 'go to' juice. Lunch is usually a big sandwich packed with salad, although it can sit on my desk for a long time before I get to eat it! I can't remember the last time I cooked a big dinner. When I get home, I tend to eat whatever I can put together in ten minutes, still using fresh ingredients and lots of veggies. (Jeff doesn't like eating after 5 pm, because he's always watching his weight.) I am lucky, because most of the time when I get home my wonderful mother has cooked for my kids — meat and three veg every night (just like I had when I was a kid).
This is how I eat most of the time; however, life sometimes gets in the way and I find myself not eating the way I should. When this happens, I feel lousy and look lousy — and I know I need to get myself back on the health wagon. I could not do what I do if I did not eat healthily.
I'm very conscious that what I put into my body is the only fuel it has to run on. Given the pace of my life, I do my best to make sure it's efficient. I could always do better, but I do what I can with what I've got. I'm lucky that juices and smoothies are such a fast, nutritionally packed option! (I know it sounds like a sell job, but it really is true.)
Janine says ...
Your secret weapon? Juice, of course! Why am I so passionate about juices and smoothies? Because they work! There are no short cuts to good health, but there are steps you can take to help you on the journey. If you're anything like me, there aren't enough hours in the day to plan and cook the right meals to fit five serves of vegetables and two serves of fruit into your diet every day. That's before you even consider the raft of vitamins and minerals that are essential to efficiently run the human body, and the veritable mountain of food you need to get through to ensure you get your quota of each.
Each year, Boost Juice uses over 6 000 000 bananas, 5 000 000 oranges, 3 000 000 apples, 3 000 000 carrots, 100 000 celery stalks, 500 000 pineapples, 800 tonnes of watermelon, 19 000 000 strawberries, 1 000 000 mangos, 8 000 000 raspberries and 40 000 000 blueberries in our smoothies and juices. (These figures are based on 2011 Boost statistics.) That's an enormous amount of fresh fruit!
A glass of juice can be considered an excellent snack — it's healthy and it will fill you up. An average glass of orange juice contains around five oranges. While it's true that by juicing you do lose some of the benefits of the whole fruit (a certain amount of fibre), there's still no doubt that it is a great way to mainline a cocktail of vitamins in minutes. Imagine sitting down to eat five oranges. You'd need a whole soccer team to get through them! It always puzzles me that people only turn to fresh food when they have a health problem. Why wait?
Fruit and vegetables are an excellent source of vitamins, minerals, phytochemicals and antioxidants, which are all essential for a healthy body.
Vitamins are organic compounds, and are classified as either fat-soluble (such as vitamins A, D, E and K) or water-soluble (the B-group vitamins and vitamin C). Fat-soluble vitamins are stored in body fat while water-soluble vitamins are carried in the blood and excreted in urine so are needed in small, frequent doses.
There are 13 recognised vitamins, each with a special role to play in the body. They do everything from regulating your metabolism (how food is digested, absorbed and used within the body) and helping to produce energy, to assisting with the growth and repair of body cells. Apples, bananas, beetroot, carrots, celery, ginger, lemons, mint, oranges, pineapples, raspberries, strawberries and watermelons are all good sources of vitamins. Conveniently, they also combine well in juices!
Minerals are pure, inorganic elements required in small amounts for good health and growth. Think zinc, magnesium, potassium and other tasty-sounding substances. Minerals form the bone structure of the body, play an important role in the chemical reactions that keep your body going, regulate water balance, and assist in controlling nervous responses and muscle contractions. There are sixteen minerals considered essential to our health. While it can be difficult for the body to absorb minerals, it's important not to over-consume them (for example, in concentrated tablets) because they can be toxic in high levels. Good sources of minerals include bananas, beetroot, carrots, celery, ginger, lemons, mint, oranges, pineapples, raspberries and watermelons.
Phytochemicals are plant chemicals that contain properties that may aid in disease prevention. There are thought to be more than 900 different phytochemicals in the foods we eat, and it is believed that they tend to act together, rather than in isolation. Phytochemicals are found in tomatoes, spinach and other leafy greens. They can also be found in apples, carrots, citrus fruits (such as oranges and lemons), ginger, mint and watermelons; there can be dozens of them occurring in any one serve of these foods.
Finally, antioxidants — a major group of phytochemicals — act as a bodyguard, limiting the activity of free radicals in the body. They include vitamins A, C and E, selenium and the carotenoids.
Freshly squeezed juice is one of the optimum ways to reap the benefits of all the goodness in fruit and veggies. For even more nutritional benefits, try including some of the ingredients we use in Boost's super supplements. There are, of course, natural sources for all of them, but when you can get them all together in one power-packed cocktail, it makes sense. (Remember — all our supplements comply with the guidelines from Food Standards Australia New Zealand.)
Super supplements
If you do only four things to help keep your health on track, make it these:
• Drink more water — very few people drink enough of the liquid gold. Always have a bottle or glass nearby, and use it.
• Eat as much natural, wholefood as possible, and eliminate highly processed foods from your diet.
• Do not eat too late; it is terrible to eat a big meal and then go straight to bed.
• Do not beat yourself up if you have eaten poorly in a day. There is always tomorrow. As long as poor eating is the exception not the rule, you will be fine.
Chapter 12
Throwing away the pulp
Even if you think you're doing fine on the fast track, there are dangers lurking to trip you up. Whether you work in an office environment, a retail outlet or a work site, the traps can be the same, even if they wear different disguises. They're not always the most obvious problems — some of them might be considered assets under different circumstances. This chapter is all about knowing how to recognise problems — and overcome them.
Remember to harness your positive energy and there will be no such thing as an obstacle!
Handling conflict with your colleagues
At Boost, we place an enormous emphasis on 'cultural fit' during employment interviews — and for good reason. It's not just to save us the hassle of employing a person who won't fit in; it's also to save that person the unhappiness of being a square peg in a round hole.
If you are in an environment that isn't right for you, you will feel isolated. You'll never know what's going on, you will never hear about the best opportunities for advancement and going to work will be an absolute chore. If you're the kind of person who likes peace and quiet, for example, and you find yourself in an open-plan office, full of creative types who like to bandy about ideas, you'll never be able to show your full potential.
If you're having a personality clash with one or two other people in a company of a hundred, well, that probably just makes you normal. On the other hand, if you find it tough to name even one or two colleagues who you actually like, you probably need to take a hard look at your current position. Most people spend more time with their work colleagues than their family, so it is important that you enjoy your co-workers.
Solution
Find a job that suits your personality. Remember, if you're passionate about something, you'll be good at it! Passion can be quiet, but you need to be in a position where you feel confident about getting your point across. When you go for job interviews, ask a lot of questions about the 'culture' of the company — you want to make sure it's right for you.
Setting realistic expectations
When was the last time you woke up and thought, I'm going to wow them at work today! A lot of people trudge off to work and sit down at their desks, prepared to do nothing more than what's necessary to get through the day. Low expectations like those could be killing your career. To get ahead you need to be more positive! I go to work every day determined to learn something new — and every day, I manage to do just that. There's always something you don't know.
If you've been killing time at your desk for a while now, you may find that other people have fallen into line with your low expectations. Has your boss given you any interesting new projects of late or suggested you attend a training course? If your answer is no, chances are your boss has noted your attitude. We get back the energy that we put out — if you're giving off bad vibes, nobody is going to go out of their way to give you opportunities.
Solution
It's time for an attitude change. Try surrounding yourself with positive mantras — write them down and put them where you'll see them in the morning.
Go to work with a smile on your face and a can-do attitude. If you lift the expectations of yourself, you'll find that others will too. If you look ready to receive more, you'll be given more. Yes, that might mean a heavier workload but, really, aren't you bored just sitting there watching the clock?
Also consider whether you're expecting too much too quickly. We live in a 'now' era. You only have to look at the spiralling credit card debt in our society to see that this is not a generation of people willing to wait patiently for things to happen. Too often young people sabotage their positions by demanding too much, too soon. Ask yourself if you're being realistic in your expectations. So you're not managing a department at twenty-two — very few people are. Having said that, many of my employees may be considered young for the roles they hold, but they've worked very hard for their success.
I have never been a fan of those who believe they're entitled to things. Yet I've seen people take entry-level positions and then, within what feels like three minutes, demand pay rises or ask for promotions just because 'I have a degree', 'My friend gets x dollars' or 'I've been here three months'. I encourage ambition in my staff, but progression within a company has to be earned.
Some people manage to progress very quickly; for others it takes longer. If you feel you've been passed over several times, it may be time to speak to your boss, but overall you need to look at your career as a long-term prospect. Promotion may not happen overnight, but if you put in the hard yards and aim for respect from management and your colleagues, it will happen. Remember that respect and reputation are very difficult to earn and incredibly easy to lose. Don't forget about the big picture in a fit of pique about what's happening now.
Learning to say no
Do you find yourself saying yes to everything? A last-minute report needs doing, the photocopier needs fixing, you're loaded down with a big project, but you can't turn down the new recruit when she asks for help with her expense sheet. Every office has someone like you. Good old dependable you.
The trouble is that it's very difficult to remain dependable when you're being stretched in every direction. Sooner or later all those balls that you're juggling will start to fall — and you'll be too busy doing something else to catch them.
If you've found yourself in this role, you know wiggling out of it can be difficult, particularly if the word 'no' rarely passes your lips. You must assert yourself better. Taking up the slack in your office or department is not helping anyone — either someone else is letting you pull their weight or you're simply understaffed. Whatever the reason, your boss will never be able to manage resources if he or she can't get a clear picture of what's going on. It's time to stop working those incalculable hours; let others in the office stand on their own two feet. You don't have to be everything to everyone to be a valuable member of the team.
Solution
Learn how to say no. If you find that you're being piled up with work, under the assumption that you always find a way, it's time to sit down with your boss and discuss your deadlines. If your colleagues are taking advantage of your good nature, help them find the solutions for which they should be responsible by suggesting they put their own ideas together and discuss them when you get some time. There is always a way to let people down gently. If you don't start to do this, the only person heading for a fall is you.
Respecting others
You may call it being cool, or creative. Maybe you simply don't think about it at all. I'm talking about the little things that show a lack of respect — like being consistently late for work or not ready for meetings, throwing things together at the last minute, and passing the buck to others because you've got 'other things to do'.
What these things add up to is a big lack of respect for your colleagues — and never imagine for a second that they won't notice. They may not say anything at the time, but those slights are being filed away and will count against you in the future.
At Boost, we value respect immensely, but it has to be earned. It also has to be given. When you wander in late, fail to give 100 per cent, or you let others pick up your slack, you send a message to the rest of the team that they don't count. That doesn't make for a very happy workplace. I believe that happy teams make the best teams, and I do my best to weed out any negative influences. Any smart boss will do the same.
Solution
The old adage 'Do unto others as you would have them do unto you' has never held truer than in today's workplace. The best way to get ahead is to show your colleagues respect. They will not help you or make you look good unless you show them that you value their contributions. If one person performs badly, it lets down the whole team — don't be that person!
By the way, if you happen to be the person who's picking up the slack for someone — that is, having a heap of stuff dumped on you because your colleague 'doesn't have time', you're not doing yourself or your colleague any favours (refer to the preceding section).
Acting the part
You only need to read part I of this book to realise that I'm the last person who will ever judge people on their appearance. I don't believe that you need to present any other image than your own to be successful. But — yes, there's always a but — that's within reason.
Most workplaces have a dress code and it makes sense to work within this code as best you can. Advertising agencies and other creative environments may seem to be freer with their fashion, but their unspoken rules of cool can be as set in stone as rules on attire in a corporate accountant's office.
Why would you bother with these rules? Two words: your comfort. If you're uncomfortable, you can't be at your best. If you stand out in your workplace, like Britney Spears at an IBM convention, your fabulous ideas may never get the audience they truly deserve.
Which brings me to my next and most important point: actions speak louder than words. You can be the best talker in the office, full of amazing plans and strategies, but if you never get past the talking, you won't get far.
Solution
It can be difficult trying to sell your ideas and solutions to people who aren't looking past your belly-button piercing. You'll find it's easier if you take the time to work out the best way to express your personality within the boundaries that are acceptable in your workplace.
Also, once you've sold your ideas, don't forget the most important part of the equation — the follow-up. I don't want to hear about how well you can do something, or how quickly — show me. There are very few people who talk their way to the top (no matter how things may sometimes look). The truth is that solid hard work needs to follow up any burst of hot air.
Eliminating the fear
Whether you have a fear of failure or success, drop it! Everybody makes mistakes. There — the truth is out. Unfortunately, there are some people who would rather not make any impact at all than risk making an error. You only need to watch an episode of a reality television show to see these people in action. In this situation, they're hoping to go unnoticed so they don't get voted out; however, even in this example you can see this tactic only works for so long. Soon the numbers get so few that those hiding in the shadows are forced into the spotlight.
In a work environment, those people who never speak up, never volunteer and never commit will also never get ahead. They may get by without making waves for a while, but sooner or later their workmates will notice a certain void where that person should be. Nobody ever said that every member of a team has to be an extroverted leader. However, every member of a team does need to commit to the group. If you're too afraid of failure to even have a go, you create imbalance within the team.
Making an error will not be the end of your career; write that down in your diary if you must. You need to step up to the plate and have a swing if you hope to hit a home run. The key to learning is having a go. As far as I'm concerned, learning something new every day is the key to a successful career. If you make a mistake, fine — own up to it and do your best to rectify it. I have more respect for people who do this than for those who skate through their working life, never really making an impact. After all, how can your boss see what you can really do if you never take a swing?
On the opposite side of the coin, many people seem frightened of the changes that success might bring to their lives. They feel they are imposters — that somehow the opportunities that have come their way were not really meant for them. They sit waiting for someone else to uncover the awful truth. I've seen talented staff who haven't gone for promotions because they're worried about ending up as the boss of one of their friends. Others are concerned about how their lives might change, and whether their partner will be able to handle their success. Many people go about their ordinary lives, ignoring great opportunities because they don't fit into the scheme of those ordinary lives. It's the few people willing to take a chance who find that success comes their way.
Solution
Face your fear. Try my 'worst-case scenario' approach: what's the absolute worst thing that might happen should you take a chance on success? Perhaps the partner you have right now won't make the jump with you — if that's the case, is that person really right for you? Perhaps you will end up your friend's boss — have you talked to her (or him) about it? How close is your friendship anyway? The point is this: can you live with the 'worst case'? Once you've identified the absolute worst that can happen, it usually just doesn't look as scary.
There are a million reasons we can use to convince ourselves that we shouldn't do something. There is only one reason that we can use to convince ourselves we should: 'Why not?'
As for feeling like an imposter — and I've seen it with so many people — never underestimate the effort you put in, and accept that you deserve everything you've achieved. There are times when I think, 'How did I get here? Me, who knew nothing about business?' Then I remember — 'Oh yes, I'm here from a lot of bloody hard work!'
Avoiding burnout
There were days not so long ago when I started to wonder what the hell I was doing. I'd spend 10 hours in the office, go home to spend a few hours with my family, then sit down at my computer to do another five hours' work. Of course, I'm not the only one who has worked that hard at some time in their life. The workplace is full of people spending way too much time in front of their computers and under fluorescent lights. Unfortunately, if we don't see the warning signs and put the brakes on in time, we will burn out. The fastest way to derail your career is to lose track of the importance of downtime.
Burnout is what happens when we don't get our work–life balance right. If you focus on just one area of your life, to the detriment of all else, it's no wonder things begin to go pear-shaped.
Stress can only be endured for so long. Long hours can only be tolerated for a little while. Being the first into the office and the last to go home will only get you brownie points with management for a short period. Once that wears off, they will simply begin to question your time-management skills.
In short, if you're beginning to feel that your workplace simply couldn't function without your presence, it's probably time to take a break. Holidays are a vital part of working life — we need them to refresh our minds and re-energise our bodies. Never underestimate the importance of taking a breather, and stopping to do nothing.
Solution
When was the last time you had a break — a proper one, without the laptop and the mobile phone? Trust me — the office will not fall apart in your absence. This may be a little deflating for the ego, but the health benefits will more than make up for it. When you return from your holiday, rested and rejuvenated, you'll be able to make that charge up the corporate ladder at full speed.
Conclusion
You've now read all the parts in this book, so know all about my journey and my recipes for success, and my tips for expanding your skills and overcoming obstacles. Hopefully, you've also gained some insights along the way. Thank you for taking the time and choosing to read a book about my passion — Boost.
Every day I learn something new. Every day I try to apply the knowledge that I've learned, incorporating it into my work and home life. I hope that you take some of what I've written and apply it to your own life. If I can help a thousand people on their business journey — fantastic. If I help only one, I will have reached my objective for writing this book.
Good luck to you in your adventures, and remember — life is a journey; seize yours.
The Boost Juice timeline
2013: Acquire the 23-store Italian style chain, Cibo Espresso
2012: Operating 39 Salsa's Fresh Mex Grill stores and 285 Boost stores
2009: Open 14th Salsa's Fresh Mex Grill store, and open first Boost store in South Korea
2009: Operating over 250 stores in Australia and a further 50 stores in 16 countries
2008: Purchase Salsa's Fresh Mex Grill (four stores)
2008: Winner of National Retailer Associations Awards
2008: Open first store in Thailand, Lithuania, Germany, China and Malaysia
2007: 200 stores open worldwide
2007: Open first store in Portugal, Macau, South Africa and Singapore
2007: Open first store in the UK (this is now the highest turnover store in the world)
2006: Open first store in Chile and Indonesia
2005: Finalist for International Veuve Clicquot Business Woman Award
2005: Awarded AMEX Franchisor of the Year
2005: BRW Fastest Franchisee growth
2005: BRW seventh Fastest Growth Business
2005: My Business People's Choice award finalist — retail
2005: Launched Boost retail juices into Coles
2004: Winner of 2004 Telstra Australian Business Woman of the Year
2004: Winner 2004 Telstra Victorian Business Woman of the Year
2004: Open 150th Boost store
2004: Finalist Ernst & Young Entrepreneur of the Year awards
2004: Open 100th Boost store
2004: Acquire the Viva Juice business (24 stores)
2004: Open first store in New Zealand
2004: Boost Juice turning over $2 million per week (late in year)
2004: Boost Juice turning over $1 million per week (early in year)
2003: Open 50th Boost store
2002: Open first Boost store in Queensland
2002: Open first Boost store in Western Australia
2002: Boost Juice turning over $1 million per year
2002: Open first Boost store in NSW
2001: Open first Boost store in Melbourne (Victoria)
2001: First franchise granted in James Place, South Australia
2000: Open first Boost store (in South Australia)
1999: Trip to US to research new concept
| {
"redpajama_set_name": "RedPajamaBook"
} | 7,750 |
\section{Introduction}
Sense of touch is essential for humans. We use it constantly to interact with our environment. Even without vision, humans are capable of manipulating and recognizing objects. Our mastery of dexterous manipulation is attributed to well developed tactile sensing~\cite{Howe1994}. To give robots similar skills, researchers are studying use of tactile sensors to help robots interact with their environment using the sense of touch. Furthermore, different studies show the importance of tactile feedback when applied to object manipulation \cite{dahiya2010tactile}\cite{yussof2009grasp}.
Specifically, in the context of object recognition, tactile sensing provides information that cannot be acquired by vision. Indeed, properties such object texture and softness can be better investigated by actively interacting with the object. In order to detect such properties, different approaches have been proposed. Takamuku et al.~\cite{takamuku2007haptic} identify material properties by performing tapping and squeezing actions. Johansson and Balkenius~\cite{johnsson2008recognizing} use a hardness sensor to measure the compression of materials at a constant pressure, categorizing the objects as hard and soft. Psychologists have shown that humans make specific exploratory movements to get cutaneous information from the objects~\cite{lederman1987hand}, that include, pressure to determinate compliance, lateral sliding movements to determinate surface texture, and static contact to determine thermal properties. Hoelscher et al.~\cite{hoelscher2015evaluation} use these exploratory movements to identify objects based on their surface material, whereas other researchers have focused on how to exploit them to reduce the uncertainty in identifying object properties of interest~\cite{xu2013tactile}.
\begin{figure}[t]
\includegraphics[width=0.8\linewidth]{iCubObjRec.jpg}
\centering
\caption{The iCub humanoid robot carrying out the object recognition task.}
\label{fig:icub}
\end{figure}
All these approaches carry out exploratory movements using a single finger and assume that the object does not move. Conversely, other works recognize an object by grasping the object, putting less restrictions on the hand-object interaction. Schneider et al.~\cite{schneider2009object} propose a method in which each object is grasped several times, learning a vocabulary from the tactile observations. The vocabulary is then used to generate a histogram codebook to identify the objects. Chitta et al.~\cite{chitta2010tactile} propose a method that, using features extracted while grasping and compressing the object, can infer if they are empty or full and open or close. Chu et al.~\cite{chu2013using} perform exploratory movements while grasping the object in order to find a relationship between the features extracted and haptic adjectives that humans typically use to describe objects.
However, most of these approaches do not deal with the stability problem and assume that the object is laying on, or are fixed to a surface such as a table. When the object has to be held in the robot's hand, stability problems such as preventing it from falling, make the task of extracting features through interactions more challenging. Kaboli et al.~\cite{kaboli2015hand} recognise objects using their surface texture by performing small sliding movements of the fingertip while holding the object in the robot's hand. Gorges et al.~\cite{gorges2010haptic} merge a sequence of grasps into a statistical description of the object that is used to classify the objects. In a recent work Higy et al.~\cite{higy2016combining} propose a method in which the robot identifies an object by carrying out different exploratory behaviours such as hand closure, and weighing and rotating the object. In their method the authors fuse sensor data from multiple sensors in a hierarchical classifier to differentiate objects.
In these approaches the stability is typically managed by performing a power grasp, that is, wrapping all the fingers around the object. This means that in general, the final hand configuration after the grasp is not controlled. It strictly depends on the way the object is given to the robot. Due to this, the tactile and proprioceptive feedback suffer from high variability. \lorenzo{This requires a larger number of grasps to be performed and negatively affects the performance}. Moreover, performing power grasps may limit further actions that could help in extracting other object features such as softness/hardness.
In this work we propose a novel method for in-hand object recognition that uses a controller proposed by Regoli et al~\cite{regoli2016hierarchical} to stabilize a grasped object. The controller is used to reach a stable grasp and \lorenzo{reposition the object in a repeatable way}.
We perform two exploratory behaviours: squeezing to capture the softness/hardness of the object; and wrapping all of the fingers around the object to get information about its shape. The stable pose achieved is unique given the distance between the points of contact (related to the size of the object), resulting in high repeatability of features, which improves the classification accuracy of the learned models. Differently from other methods, we do not put any restrictions on the objects.
We validated our method on the iCub humanoid robot~\cite{metta2008icub} (Fig.~\ref{fig:icub}). We show that using our method we can distinguish 30 objects with \lorenzo{99.0\% $\pm$ 0.6\% accuracy}. We also present the results of a benchmark experiment in which the grasp stabilization is disabled. We show that the results achieved using our method outperforms the benchmark experiment.
In the next section we present our method for in-hand object recognition. In section \ref{sec:experiments} we describe the experiments carried out to validate our method, while in section \ref{sec:results} we present our results. Finally, in section \ref{sec:conclusions} we conclude the paper and provide future directions.
\section{Methodology}
\label{sec:methodology}
Here we present the method used to perform the in-hand object recognition task. We use an anthropomorphic hand, but the method can be easily extended to any type of hand that has at least two opposing fingers. \lorenzo{We use the tactile sensors on the fingertips of the hand~\cite{jamali2015new}, which provide pressure information on 12 taxels for each fingertip. An important assumption in this work is that the object is given to the robot by a collaborative operator, in such a way that the robot can grasp it by closing the fingers. The remaining steps are performed by the robot autonomously, namely:}
\begin{itemize}
\item grasping the object using a precision grasp, that is, using the tip of the thumb and the middle finger,
\item reaching an optimal stable pose,
\item squeezing the object to get information about its softness,
\item wrapping all the fingers around the object to get information about its shape.
\end{itemize}
We start by giving an overview of the grasp stabilizer component. This is followed by a description of the feature space, and then we give a brief overview of the machine learning algorithm used to discriminate the objects.
\subsection{Grasp stabilization}
\label{sec:stabilization}
Grasp stabilization is a crucial component of our method for two reasons. First, it is needed to prevent the object from falling, for example, when executing actions like squeezing. Second, reaching a stable and repeatable pose for a given object improves the classifier accuracy. We use our previously developed method to stabilize the object~\cite{regoli2016hierarchical}. In the rest of this section we quickly revise this method and explain how we apply it to our problem (details of the controller can be found in~\cite{regoli2016hierarchical}). In this paper we use two fingers instead of three, namely, the thumb and the middle finger. Figure~\ref{fig:controller} shows the controller, which is made of three main components:
\subsubsection*{\textbf{Low-level controller}}
it is a set of P.I.D. force controllers responsible for maintaining a given force at each fingertip. The control signal is the voltage sent to the motor actuating the proximal joint, while the feedback is the tactile readings at the fingertip. We estimate the force at each fingertip by taking the magnitude of the vector obtained by summing up all the normals at the sensor locations weighted by the sensor response.
\subsubsection*{\textbf{High-level controller}}
it is built on top of the low-level force controllers. It stabilizes the grasp by coordinating the fingers to a) control the object position, and b) maintain a given grip strength. The object position $\alpha_o$ is defined as in Fig.~\ref{fig:objectPosition}, and it is controlled using a P.I.D. controller in which the control signals are the set-points of the forces at each finger, while the feedback is the object position error.
The grip strength is the average force applied to the object. It is defined as:
\begin{equation}
g = \frac{f_{th}+f_{mid}}{2}, \\
\label{eqn:gripStrength}
\end{equation}
where $f_{th}$ and $f_{mid}$ are the forces estimated at the thumb and the middle finger, respectively. The target grip strength is maintained by choosing set-points of the forces that satisfy (\ref{eqn:gripStrength}).
\begin{figure}
\includegraphics[width=0.7\linewidth]{objectPositionNew_resized.png}
\centering
\caption{The object center $C_{o}$ is defined as the halfway point between the two points of contact. The object position $\alpha_o$ is defined as the angle between the vectors $\vec{OC_{o}}$ and $\vec{OB}$. $A$ and $B$ are set at the base of, respectively, the thumb and the middle finger, while $O$ lies at middle distance between $A$ and $B$.}
\label{fig:objectPosition}
\end{figure}
\subsubsection*{\textbf{Stable grasp model}}
it is a Gaussian mixture model, trained by demonstration. The robot was presented with stable grasps using objects of different size and shape. The stability of a grasp was determined by visual inspection. A stable grasp is defined as one that avoids non-zero momenta and unstable contacts between the object and the fingertips. We also preferred grasp configurations that are far from joint limits (details are in~\cite{regoli2016hierarchical}). Given the distance, $d$, between the fingers, the model estimates the target object position, $\alpha_o^r$, and the target set of non-proximal joins, $\mathbf{\Theta_{np}}$, to improve grasp stability and make it robust to perturbations. The target $\alpha_o^r$ is used as the set-point of the high-level controller, while the $\mathbf{\Theta_{np}}$ is set directly using a position controller.
\begin{figure}
\includegraphics[width=1.0\linewidth]{controlSchemaObjectRecognition.png}
\centering
\caption{Grasp stabilizer control schema. When grasping an object, the distance, $d$, between the points of contact is used by the Gaussian mixture model to compute the reference values of the non-proximal joints, $\mathbf{\Theta_{np}}$, and the object position, $\alpha_o^r$. In order to reach $\alpha_o^r$ and $g$, the high-level controller sets the appropriate force references, $\mathbf{f^r}$, of the low-level controller for each finger. The low-level force controller, in turn, sends voltage to the motors actuating the proximal joints to compensate the force error. The actual object position and the actual forces at the fingertips are represented by, respectively, $\alpha_o^a$ and $\mathbf{f^a}$. }
\label{fig:controller}
\end{figure}
\subsection{The Feature Space}
Once a stable grasp is achieved, the robot manipulates the object to capture its softness and shape by performing two exploratory behaviours: a) squeezing the object between the thumb and the middle finger, and b) wrapping all the fingers around the object. The softness of the object is captured both by the distribution of the forces in the tactile sensor and the deflection of the fingers when the object is squeezed between the fingers of the robot. The shape of the object is captured by wrapping all of the fingers of the robot around it.
As mentioned earlier, the grasp stabilization implies a high degree of repeatability of the achieved pose, independent of the way the object is given to the robot. Thereby, the features produced during the exploratory behaviours exhibit low variance between different grasps of the same object. Which, in turn, increases the accuracy of the classifier.
\subsubsection{Tactile responses}
the distribution of forces in the tactile sensors is affected by the softness of an object. A hard object will exert forces that are strong and concentrated in a local area. A soft object, in contrast, will conform to the shape of the fingertip and exert forces across all tactile sensors. The tactile sensors also capture information on the local shape of the object at the point of contact. We use the tactile responses from the thumb and the middle finger, $\mathbf{\tau}$, in our feature space, since the objects are held between these two fingers.
\subsubsection{Finger encoders}
the finger encoders are affected by the shape and the harness/softness of the object. When the robot squeezes the object, a hard object will deflect the angles of the finger more than a softer object. Since we use only the thumb and the middle finger during the squeezing action, we use both the initial and the final encoder values for these fingers --~$\mathbf{\Theta_{grasp}^{init}}$~and~$\mathbf{\Theta_{grasp}^{fin}}$, respectively.
To capture the shape of the object, the robot wraps the rest of its fingers around the object. We also include the encoder data, $\mathbf{\Theta_{wrap}}$, of these fingers in our feature space.
\subsection{The learning algorithm}
\label{sec:learning}
In order to train the classifier, we used as features the data acquired during the grasping, squeezing and enclosure phase, as described in the previous section. We simply concatenated the collected values, obtaining the feature vector [$\mathbf{\Theta_{grasp}^{init}}$\,$\mathbf{\Theta_{grasp}^{fin}}$\, $\mathbf{\Theta_{wrap}}$\,$\mathbf{\tau}$] composed of 45 features, 21 related to the encoders and 24 related to the tactile feedback.
As learning algorithm we adopted Kernel Regularized Least-Squares using the radial basis function kernel. For the implementation we used GURLS \cite{tacchetti2013gurls}, a software library for regression and classification based on the Regularized Least Squares loss function.
\section{Experiments}
\label{sec:experiments}
To test our method, we used the iCub humanoid robot. Its hands have 9 degrees-of-freedom. The palm and the fingertips of the robot are covered with capacitive tactile sensors. Each fingertip consists of 12 taxels~\cite{jamali2015new}.
\subsection{The objects}
We used a set of 30 objects shown in Fig. \ref{fig:confusionMatrices}, of which, 21 were selected from the YCB object and model set~\cite{calli2015benchmarking}. Using a standard set helps in comparing the results of different methods. The objects were selected so that they fit in the iCub robot's hand without exceeding its payload. The YCB object set did not have many soft objects fitting our criteria, hence, we supplemented the set with 9 additional object with variable degree of softness. We also paid attention to choose objects with similar shape but different softness, as well as objects with similar material but different shapes.
\subsection{Data collection}
\label{sec:dataCollectionMethod}
The dataset to test our method was collected using the following procedure (depicted in Fig.~\ref{fig:taskSteps}):
\begin{enumerate}
\item The iCub robot opens all of its fingers.
\item An object is put between the thumb and the middle finger of the robot. The robot starts the approach phase, which consists of closing the thumb and the middle finger until a contact is detected in both fingers. A finger is considered to be in contact with an object when the force estimated at its fingertip exceeds a given threshold. To capture variations in the position and the orientation of the object, each time the object is given to the robot, it is given in a different position and orientation.
\item \label{it:stabilize} At this point the grasp stabilizer is triggered with a given grip strength. The initial value of the grip strength is chosen as the minimum grip strength needed to hold all the objects in the set. The method described in section~\ref{sec:stabilization} is used to improve the grasp stability. When the grasp has been stabilized, the robot stores the initial values of the encoders of the thumb and the middle finger.
\item Then the robot increases the grip strength to squeeze the object and waits for 3 seconds before collecting the tactile data for the thumb and the middle finger. At this point the robot also records the encoder values for the thumb and the middle finger.
\item Finally, the robot closes all of the remaining fingers around the object until all fingers are in contact with the object. At this point, the robot collects the values of the encoders of the fingers.
\end{enumerate}
These steps were repeated 20 times for each object. To test our algorithm we use a fourfold cross-validation. That is, we divide the dataset into 4 sets. We hold one of the sets for testing and use the other three to train a classifier. This is repeated for all 4 sets. We compute the accuracy and the standard deviation of our classifier using the results of these 4 learned classifiers.
\begin{figure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{init_CROP.jpg}
\caption{}
\label{fig:taskStep1}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{stable_CROP.jpg}
\caption{}
\label{fig:taskStep2}
\end{subfigure}\\
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{squeeze_CROP.jpg}
\caption{}
\label{fig:taskStep3}
\end{subfigure}
\begin{subfigure}[b]{0.22\textwidth}
\includegraphics[width=\textwidth]{wrap_CROP.jpg}
\caption{}
\label{fig:taskStep4}
\end{subfigure}
\caption{The steps accomplished to identify the object: approach (a), stabilization (b), squeezing (c) and enclosure (d). \lorenzo{Notice that the controller repositions the object irrespective of its initial pose. As discussed in the text this greatly improve repeatability and, consequently, recognition performance.}}
\label{fig:taskSteps}
\end{figure}
\subsection{Benchmark experiment}
To test our hypothesis that reaching a stable pose improves the classification results we carried out an experiment in which we disable part of the grasp stabilization. As described earlier and depicted in Fig.~\ref{fig:controller}, the grasp stabilization consists of three modules: the low-level force controller, the high-level controller and the stable grasp model. We only disable the stable grasp model. The other two components are needed to stop the object from slipping and to control the grip strength.
The stable grasp model produces two outputs: a) the target object position, $\alpha_o^r$, and the target set of non-proximal joints, $\mathbf{\Theta_{np}}$. In the benchmark experiment we calculate the value of $\alpha_o^r$ and the $\mathbf{\Theta_{np}}$ when the thumb and the middle finger make contact with the object. That is, the alpha is set to the current position of the object and the theta is set to the current joint configuration. Apart from this difference, the high-level controller and the low-level force controller are still active, controlling grip strength and maintaining a stable grasp. However, without the stable grasp model, the grasp is less robust to perturbations.
Henceforth, unless stated otherwise, when we mention that the grasp stabilization is disabled, we mean that we only disable the repositioning based on the GMM. Hence, we collected the data for the benchmark experiment following the same steps as described in section~\ref{sec:dataCollectionMethod} where the grasp stabilization was disabled.
\section{Results}
\label{sec:results}
In this section we present the results of our method and show how each of the selected features in our feature space helps in capturing different properties of the objects, namely, the softness/harness and the shape of the object. This will be followed by a comparison between our method and the benchmark method in which the grasp stabilization is disabled. When reporting the results for brevity we concatenated some of the features: $\mathbf{\Theta_{grasp}}$ = [$\mathbf{\Theta_{grasp}^{init}}$\,$\mathbf{\Theta_{grasp}^{fin}}$], and $\mathbf{\Theta_{all}}$~=~[$\mathbf{\Theta_{grasp}^{init}}$\,$\mathbf{\Theta_{grasp}^{fin}}$\,$\mathbf{\Theta_{wrap}}$].
\begin{table}[t!]
\centering
\begin{tabular}{cccccc}
\toprule
Features &$\mathbf{\Theta_{grasp}^{init}}$ & $\mathbf{\Theta_{grasp}}$ & $\mathbf{\Theta_{all}}$ & $\mathbf{\tau}$ & All\\
\midrule
Mean & 80.5\% &93.3\% &96.3\% &95.0\% &99.0\% \\
Std & 2.0\% &0.8\% &0.7\% &0.8\% &0.6\% \\
\bottomrule
\end{tabular}
\caption{Classification accuracies using our method with classifiers trained using different set of features.}
\label{tab:methodAccuracy}
\end{table}
\subsection{Finger encoders}
To study the effectiveness of the encoder features, we trained a model using different combinations of these
features. Table~\ref{tab:methodAccuracy} reports the results of these experiments. We notice that using only the initial encoder values, the accuracy is already quite high, 80.5\% $\pm$ 2.0\%, while including the final encoder values of the thumb and the middle finger after squeezing it increases to 93.3\% $\pm$ 0.8\%. This is because the fingers will move considerably if the object is soft, thereby, capturing the softness of the object. Figure~\ref{fig:confusionMatrices} shows the confusion matrices for the experiments. We notice that several pairs of objects such as the tennis ball (11) and the tea box (30) or the sponge (26) and the soccer ball (28) are sometimes confused if only the initial encoders values are used as features, while they are discriminated after the squeezing action.
Finally we analysed the results of including all encoder data, that is, including the data when the robot wraps its fingers around the object. This improved the classification accuracy to 96.3\% $\pm$ 0.7\%. From the confusion matrices we notice that adding such features resolves a few ambiguities, such as the one between the soccer ball (28) and the water bottle (22) and the one between the yellow cup (24) and the strawberry Jello box (19). Indeed, these pairs of objects have similar distance between the points of contact when grasped, and cause similar deflections of the fingers when squeezed, but have different shapes.
\subsection{Tactile responses}
As discussed earlier the tactile sensors are useful in capturing the softness of the objects as well as the local shape of the objects. In Fig.~\ref{fig:accuracyBars} we can see that using only the tactile feedback we get an accuracy of 95.0\% $\pm$ 0.8\%, which is comparable with the 96.3\% $\pm$ 0.7\% obtained using the encoder values. Although they have similar classification accuracy, studying the confusion matrices reveals that objects confused by them are different. For example, the classifier trained using only the tactile data often confuses the Pringles~can~(1) and the tomato~can~(7), since they are hard and share similar local shape. Conversely, due to their slightly different size they are always distinguished by the classifier trained using only encoder data. This means that combining the two feature spaces can further improve the accuracy of the learned classifiers.
\subsection{Combining the two features}
Finally, using the complete feature vector we get an accuracy of 99.0\% $\pm$ 0.6\%. We also notice that the standard deviation in our experiments is decreasing as we add more features. From the confusion matrix we can see that several ambiguities characterizing each individual classifier are now solved. A few objects are still confused due to their similar shape and softness, namely the apple (5) and the orange (6), and the apricot (16) and the prune (10). Less intuitively, the classifier once confuses the apricot with the SPAM can (21), and once it confuses the apricot with the brown block (18).
To explain the confusion between these objects, we notice that there is a particular way to grasp them such that the joints configuration is very similar. \lorenzo{This happens when the middle finger touches the flat side of the apricot, and the little finger misses both objects}.
\begin{figure}
\includegraphics[width=1.0\linewidth]{accuracyBars.png}
\centering
\caption{Summary of the results comparing our method with the benchmark method for different set of features. It shows that our method outperforms the benchmark method with statistical significance. The error bars are standard deviations.}
\label{fig:accuracyBars}
\end{figure}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{confMat_init.png}
\centering
\label{fig:confMat1}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{confMat_initFinGrip.png}
\centering
\label{fig:confMat2}
\end{subfigure}\\
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{confMat_allEnc.png}
\centering
\label{fig:confMat3}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{confMat_tactile.png}
\centering
\label{fig:confMat4}
\end{subfigure}
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{confMat_all.png}
\centering
\label{fig:confMat5}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[t]{0.43\textwidth}
\vspace{-6.7cm}
\includegraphics[width=\textwidth]{objectsWithNumbers_resized.png}
\vspace{0.2cm}
\label{fig:confMatObjects}
\end{subfigure}
\caption{The confusion matrices obtained using our method with different sets of features. At the bottom right, is the object set used for the experiments. It is composed of 21 objects taken from the YCB object set (left), and additional 9 objects of various degree of softness (right).}
\label{fig:confusionMatrices}
\end{figure*}
\subsection{Comparison with the benchmark experiment}
Figure~\ref{fig:accuracyBars} shows the results of running the same analysis on the data collected in the benchmark experiment where the grasp stabilization was removed. The results show that the proposed method performs significantly better than the benchmark experiment, achieving 99.0\% $\pm$ 0.6\%, compared to the benchmark experiment which achieved an accuracy of 69.9\% $\pm$ 1.4\%. This is because the stabilization method proposed in this paper increases the repeatability of the exploration, which makes the feature space more stable. \lorenzo{Indeed, the initial position of the object in the hand strongly affects the collected tactile and encoders data. This variability is reduced using the grasp stability controller.} Note that the accuracy of the benchmark experiment increases as more features are added, showing that the feature space is able to capture the object properties.
We run a further analysis to study the effect of increasing the number of trials in the training set. In this case we always trained the classifier with the complete feature vector and considered 5 \lorenzo{trials} per object for the test set, while we varied the number of trials in the training set between 3 and 15. Figure~\ref{fig:accuracyTrends} shows the results of this analysis. The results show that the proposed method boosts the accuracy of the classification, requiring less samples to be able to distinguish the objects. The trend of the accuracy obtained using the benchmark method suggests that it may improve by increasing the number of samples in the training set. However, this is not preferred because it makes it impractical to collect data on large sets of objects, adversely affecting the scalability of the learned classifier.
\begin{figure}
\includegraphics[width=1.0\linewidth]{accuracyTrends.png}
\centering
\caption{The accuracy of our method and the benchmark method as a function of the number of training set samples. Our method obtains high accuracy even with a much lower number of training samples.}
\label{fig:accuracyTrends}
\end{figure}
\subsection{Results using objects form the YCB set only}
\begin{table}[t!]
\centering
\begin{tabular}{cccccc}
\toprule
Features &$\mathbf{\Theta_{grasp}^{init}}$ & $\mathbf{\Theta_{grasp}}$ & $\mathbf{\Theta_{all}}$ & $\mathbf{\tau}$ & All\\
\midrule
Mean & 85.0\% &91.4\% &95.0\% &94.1\% &97.6\% \\
Std & 3.1\% &1.5\% &1.8\% &1.6\% &0.5\% \\
\bottomrule
\end{tabular}
\caption{Classification accuracies using our method on the YCB objects only. }
\label{tab:ycbResults}
\end{table}
Finally, in table \ref{tab:ycbResults} we provide the results of our method using only the object from the YCB object set, in order to let researchers having the same dataset compare their results with ours.
\section{Conclusions}
\label{sec:conclusions}
In this work we proposed a method for in-hand object recognition that makes use of a grasp stabilizer and two exploratory behaviours: squeezing and wrapping the fingers around the object. The grasp stabilizer plays two important roles: a) it prevents the object from slipping and facilitates the application of exploratory behaviours, and b) it moves the object to a more stable position in a repeatable way, which makes the learning algorithm more robust to the way in which the robot grasps the object. \lorenzo{We demonstrate with a dataset of 30 objects and the iCub humanoid robot that the proposed approach leads to a remarkable recognition accuracy ($99.0\%\,\pm\,0.6\%$), with a significant improvement of $29\%$ with respect to the benchmark, in which the grasp stabilizer is not used.}
\lorenzo{This work demonstrates that a reliable exploration strategy (e.g. squeezing and re-grasping) is fundamental to acquiring structured sensory data and improve object perception. In future work we will employ an even larger set of objects and explore the use of other control strategies and sensory modalities.}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,415 |
\section{Introduction}
The \textsl{IRAS}\ satellite discovered a population of dusty Ultraluminous Infrared Galaxies \citep[ULIRGS;L$_{\rm IR}>10^{12}$L$_\odot$;][]{1987ApJ...320..238S,1988ApJ...325...74S} in the local Universe that we now know to be significant at high redshift as well. Followup studies of sub-millimeter galaxies (SMGs) found in SCUBA and MAMBO surveys \citep[e.g.][]{1997ApJ...490L...5S,1999ApJ...515..518E,2002MNRAS.331..817S,2003MNRAS.344..385B,2004MNRAS.354..779G} have shown that ULIRG activity peaks near $z\sim2.2$ \citep{2005ApJ...622..772C}, with a space density about $10^3$ times higher than locally. While these distant galaxies play a central role in the course of galaxy evolution, limited photometry at wavelengths where they emit most strongly restricts detailed characterization of their stellar, nuclear, dust, and evolutionary properties.
This is due in large part to a lack of sensitivity by instruments operating at Mid/Far-IR wavelengths. While \textsl{IRAS}\ and \textsl{ISO}\ were able to detect a number of \mbox{$z\ga1$} sources, many fall into the {\it Hyper}-luminous regime (HLIRGS; L$_{\rm IR}>10^{13}$L$_\odot$), of which a notable example is IRAS-F15307+3252 \citep[z=0.93;][]{1994ApJ...424L..65C}. This high luminosity tail is dominated by systems with an active galactic nucleus (AGN) \citep[e.g.][]{1999ApJ...522..113V,2001ApJ...552..527T}, but there may be a selection effect at work here; the fraction of hot dust due to an AGN can be much higher than in starbursting galaxies, meaning that the emission peaks at shorter wavelengths, closer to the \textsl{IRAS}\ bands. Indeed, followup studies of these systems with SCUBA have shown them to be faint or invisible at long sub-mm wavelengths \citep{2001MNRAS.326.1467D,2002MNRAS.335.1163F}. In contrast, sources at higher redshift selected in the sub-mm by SCUBA, although harboring an AGN, probably have extreme luminosities driven by intense star-formation \citep{alexanderb}. Again this may be due to a selection effect, since hot AGN at high redshift will be biased away from the long-wavelength SCUBA channels.
In both cases, the luminosities and dust temperatures are based on assumptions on the shape of the Spectral Energy Distribution (SED), which are drawn from a handful of bright \textsl{IRAS}\ selected galaxies in the nearby Universe (and hence not necessarily representative of high$-z$ galaxies). Altogether, we are left with an incomplete picture of luminous dusty sources between $z\sim1-2$, where \textsl{IRAS}\ and \textsl{ISO}\ detected sources taper off, and before SCUBA sources peak. The problem is compounded by the fact that many such sources lie in the so-called `redshift desert', which optical spectrometers have difficulty probing.
One goal of the {\it Spitzer Space Telescope} \citep{2004ApJS..154....1W} is to fill in this redshift interval with unbiased samples of luminous IR galaxies. Combining the increased sensitivity afforded by the Multi-band Imager for \textsl{Spitzer}\ \citep[MIPS;][]{2004ApJS..154...25R} with wide area Legacy and GTO imaging surveys, we now have large samples of Far-IR selected galaxies. Ground-based sub-mm observations in the 350/450$\,\mu$m\ windows are a particularly valuable complement, since at \mbox{$z\gsim1$}, they probe at or near the peak of the SED (typically at rest frame $\sim100$$\,\mu$m), allowing tighter constraints on luminosity and dust temperature. To that end, we have initiated a program of 350$\,\mu$m\ sub-mm follow-up of MIPS selected sources using the SHARC-II camera \citep{sharc} at the Caltech Submillimeter Observatory (CSO). Here we report a striking detection of a starburst dominated HLIRG at $z=1.325$.
In the discussion that follows, any calculation requiring cosmology assumes $\Omega_{\rm M}=0.27, \Omega_\Lambda=0.73$, and H$_0$=71\,km\,s$^{-1}$\,Mpc$^{-1}$.
\section{Field Description and Selection Criteria}
Through a \textsl{Spitzer}\ GTO program (PI: Soifer), 24, 70 and 160$\,\mu$m\ images of the NOAO Deep Wide-Field Survey \citep[NDWFS;][]{1999ASPC..191..111J} Bo\"otes\ field were obtained using the MIPS camera in February 2004. The 9.3\,deg$^2$ survey reached $1\sigma$ sensitivity levels of roughly 0.1, 7, and 15\,mJy at 24, 70, and 160$\,\mu$m\ respectively.
This field boasts an impressive collection of multi-wavelength data: radio imaging for part of the field, including the region around the source presented here, is available from a moderately deep ($1\sigma\sim15\mu$Jy) and high angular resolution (1.5$^{\prime\prime}$\ FWHM) VLA 1.4\,GHz observation \citep{2005ApJ...626...58H}. Another GTO program surveyed the entire field with the IRAC camera, providing deep imaging at 3.6, 4.5, 5.8, and 8.0$\,\mu$m\ \citep{2004ApJS..154...48E}. Combined with the NDWFS optical data, and our own $JHK$ photometry obtained from the WIRC camera at Palomar for the target discussed here, the wavelength coverage is sufficient to fully characterize the rest frame $B-$ through $K-$ band SED up to $z\sim3$. The third \textsl{Spitzer}\ GTO component to the Bo\"otes\ observations involves IRS spectroscopy of 24$\,\mu$m\ selected sources (PI: Weedman), which included the object presented here.
To select sources for 350$\,\mu$m\ SHARC-II observations, we used a list of MIPS sources that were detected in all three bands and were confirmed to have unambiguous optical counterparts. We further culled the list by selecting only those that appeared compact and red in the optical images (compared to the field). These additional conditions are expected for SMGs at higher redshift \citep[e.g.][]{2004MNRAS.355..485B}. For commonly used ULIRG SEDs (i.e. Arp 220 and HR10), $S(350\mu{\rm m})/S(160\mu{\rm m})$ varies from $\sim0.3$ at $z=1$ to $\sim1.0$ at $z=2$. Our goal was to select $z=1-2$ targets that could be detected at $>4\sigma$ confidence with SHARC-II in under 1 hour of good weather conditions, and hence we used a flux cut of $S(160\mu{\rm m})>200$\,mJy. These conditions imply that any source found should have a luminosity near $10^{13}$L$_\odot$. This paper concentrates on the target from this list which has the brightest 160$\,\mu$m\ flux: MIPS\,J142824.0+352619. Although the NDWFS standard is to use the $R$ band to derive the position and name\footnote{The NDWFS catalogs, and a description of the naming conventions are available at \texttt{http://www.noao.edu/noao/noaodeep/index.html}}, we use the $I-$band image which has the highest signal-to-noise ratio of all the available data on this object. For completeness, the J2000 coordinates are: 14$^{\rm h}$28$^{\rm m}$24\fs07 +35\degr26\arcmin19\farcs4.
\section{Targeted observations of MIPS\,J142824.0+352619}
\subsection{Sub-mm imaging and photometry}
Observations using the SHARC-II camera were carried out at the CSO on UT 2004 June 06. Atmospheric opacity was low, with $\tau_{\rm 350{\mu}m} = 1.7$. Pointing and flux calibration were performed on the nearby source Arp~220, and data reduced using the {\sc crush} software package \citep{crush}. Instrumental fluxes were measured within 20$^{\prime\prime}$\ apertures (corresponding to the $3\sigma$ width of the SHARC-II beam), and scaled by a factor derived from the same aperture on the calibration images. In Fig.~\ref{fig:booimg}, we plot the 350$\,\mu$m\ contours on a false-color optical image.
\begin{figure}
\plotone{f1.eps}
\caption{The main panel shows SHARC-II 350$\,\mu$m\ flux contours at [3.0,3.5,4.0,4.5]$\sigma$ overlaid on a 30$^{\prime\prime}$$\times$30$^{\prime\prime}$\ square NDWFS false color image of the field. Axes are aligned along cardinal directions, with North facing up. The lower of the two smaller cutouts shows the IRAC 3.6$\,\mu$m\ image of a region around MIPS\,J142824.0+352619. The upper panel presents a closeup view of the $I-$band image with VLA contours at [0.2,0.3,0.4] mJy levels. Although there at least three faint red sources in the vicinity of MIPS\,J142824.0+352619, the radio position clearly favors the central bright object.}
\label{fig:booimg}
\end{figure}
The SHARC-II detection probes near the peak of the SED, but to constrain its shape further into the Rayleigh-Jeans portion of the spectrum, longer wavelength observations are required. On UT 2004 August 28, one hour of photometry data at both 850 and 450$\,\mu$m\ were obtained in good weather ($\tau_{850\mu{\rm m}}=0.3$) using SCUBA \citep{scuba} on the James Clerk Maxwell Telescope (JCMT). The data were reduced using {\sc surf} \citep{surf} and flux calibrated against CRL 2688. Since the source was observed while setting late in the semester, these data were obtained before and during sunset when calibration uncertainties at 450$\,\mu$m\ (due to the sensitivity of the dish to temperature) render that channel's data unusable. At 850$\,\mu$m, this is not a problem, and the measured flux is accurate to about 5\%.
\subsection{Spectroscopic observations}
\citet{vandana} present the \textsl{Spitzer}\ IRS spectrum for MIPS\,J142824.0+352619, and determine a redshift of $z=1.34\pm0.02$. Despite the many strongly detected PAH features, the relatively low resolution of the spectrometer prevents the determination of a more accurate redshift. With this spectrum as a guide, we used the NIRSPEC instrument on Keck II to search for H$\alpha$. Although the conditions were not photometric (overcast skies and a seeing of $\sim1$$^{\prime\prime}$), we are able to clearly detect it (Fig.~\ref{fig:spec}) and derive a redshift of $z=1.325\pm0.002$ by fitting Gaussians jointly to H$\alpha$\ and the flanking [NII] lines. The H$\alpha$\ line width is $530\pm160$\,km\,s$^{-1}$\ (FWHM), but we caution that this may be an overestimate since it may contain contributions from the [NII] lines which were difficult to de-convolve due to the low signal-to-noise of the spectrum.
While the Near- and Mid-IR spectroscopy find an object at $z=1.325$, optical observations uncover something else. A spectrum taken with the Keck-DEIMOS instrument on UT 2005 May 06 reveals the presence of a $z=1.034\pm0.002$ galaxy identified by MgII(2800), the Calcium H and K lines, as well as common blended features (see Fig.~\ref{fig:spec}). These features suggest it is an elliptical galaxy. No line features from the $z=1.325$ galaxy are seen, although there is generally positive flux near $8669$\AA\, where [OII](3727) would be expected. However, the spectrum is much noisier on the redder end, and we cannot say with confidence whether this feature is a real line, or simply a noise excursion. The $z=1.034$ galaxy is likely lensing the background source, and is a complication we address in \S\ref{sec:lensing}.
\begin{figure}
\plotone{f2.eps}
\caption{LEFT: NIR emission lines detected at Keck using the NIRSPEC instrument on UT 2005 March 17. The exposure time was 2400s, and used the moderate resolution setting with a 0\farcs76 slit. To improve the signal-to-noise ratio, the spectrum is smoothed to 0.02$\,\mu$m\ bins. Vertical lines denote the positions expected for H$\alpha$\ and the flanking [NII] lines at $z=1.325$. RIGHT: Keck-DEIMOS spectra of MIPS\,J142824.0+352619, smoothed to highlight several absorption features which suggest the presence of a $z=1.034\pm0.002$ foreground galaxy. The thick line at 7400\AA\ denotes the divide between the red and blue arms on the spectrometer. The A and B band telluric lines are denoted by the symbol $\oplus$. We also indicate where [OII] would appear at $z=1.325$. Although this does appear to coincide with an emission line nearby, the difficulty in removing atmospheric lines at the red end renders it inconclusive. }
\label{fig:spec}
\end{figure}
\section{Determining the SED of MIPS\,J142824.0+352619}
We present a summary of the measured photometry in Table~\ref{tab:photom} and Fig.~\ref{fig:sed}. Concentrating on the Far-IR portion, we find that the MIPS, SHARC-II, and SCUBA observations are well fit by a modified blackbody of the form \mbox{$B(\nu,T/(1+z))\nu^\beta$}, where \mbox{$B(\nu,T/(1+z))$} is the usual Planck function, and $z,T,\beta$ describe the redshift, dust temperature and emissivity respectively. The $T/(1+z)$ degeneracy \citep[e.g.][]{2003MNRAS.338..733B} reflects the fact that a hot dust source at high redshift has the same SED shape as a colder one at lower redshift. From our multi-wavelength spectroscopy, it is clear that the Far-IR emission is coming from the $z=1.325$ galaxy, and hence we are able to derive a dust temperature of $T=42.7\pm2.7$ and emissivity $\beta=1.5\pm0.2$. The Far-IR luminosity, integrated between rest-frame 8 and 1000$\,\mu$m\ is ${\rm L = 3.2 (\pm0.7)\times10^{13}\,L_\odot}$, placing it squarely in the hyperluminous classification. Of course, this could be complicated by lensing, which we now discuss.
\begin{deluxetable}{lcl}
\tabletypesize{\scriptsize}
\tablecaption{Multiwavelength photometry of MIPS\,J142824.0+352619.}
\tablehead{
\colhead{Wavelength} & \colhead{Flux} &
\colhead{Instrument\tablenotemark{a}}
}
\startdata
445\,nm$(B_W)$ & $0.25\pm0.03\,\mu$Jy & MOSAIC-1 \\
658\,nm$(R)$ & $2.02\pm0.08\,\mu$Jy & MOSAIC-1 \\
806\,nm$(I)$ & $6.21\pm0.11\,\mu$Jy & MOSAIC-1 \\
1.22$\,\mu$m$(J)$ & $31.8\pm5.2\,\mu$Jy & WIRC \\
1.63$\,\mu$m$(H)$ & $38.4\pm5.2\,\mu$Jy & WIRC \\
2.19$\,\mu$m$(K)$ & $72.4\pm9.4\,\mu$Jy & WIRC \\
3.6$\,\mu$m\ & $250.9\pm7.5\,\mu$Jy & IRAC \\
4.5$\,\mu$m\ & $290.6\pm8.7\,\mu$Jy & IRAC \\
5.8$\,\mu$m\ & $198.3\pm16.6\,\mu$Jy & IRAC \\
8.0$\,\mu$m\ & $211.2\pm14.2\,\mu$Jy & IRAC \\
24$\,\mu$m\ & $0.72\pm0.07$\,mJy & MIPS \\
70$\,\mu$m\ & $34\pm6$\,mJy & MIPS \\
160$\,\mu$m\ & $430\pm90$\,mJy & MIPS \\
350$\,\mu$m\ & $226\pm45$\,mJy & SHARC-II\\
850$\,\mu$m\ & $21.9\pm1.3$\,mJy & SCUBA \\
20cm & $0.937\pm0.039$\,mJy & VLA \\
\enddata
\tablecomments{Optical fluxes are taken from directly the NDWFS catalog, and are measured in 5$^{\prime\prime}$\ apertures. IRAC fluxes are measured using the same sized aperture, and and are corrected to total magnitudes by using the IRAC PSF.}
\tablenotetext{a}{MOSAIC-1 is the wide-field optical imager on the Mayall 4m at KPNO, and WIRC is the NIR camera on the Palomar 5m.}
\label{tab:photom}
\end{deluxetable}
\begin{figure}
\plotone{f3.eps}
\caption{The SED of MIPS\,J142824.0+352619, shifted to the rest frame at $z=1.325$. Overlaid on the MIPS+sub-mm points is the best fit grey-body. The radio portion of the SED assumes a spectral index of 0.7, which is commonly used to describe starburst galaxies \citep{1992ARA&A..30..575C}. Squares and upper limits represent the rest-frame SED of HR10.}
\label{fig:sed}
\end{figure}
\subsection{Lensing of MIPS\,J142824.0+352619}\label{sec:lensing}
The radio emission should be a good proxy to the location of the Far-IR emission \citep[e.g.][]{2000AJ....119.2092B,2001ApJ...548L.147C,2002MNRAS.337....1I}, and we find that the peak of the radio emission is only 0\farcs1$\pm$0\farcs3 away from the $I-$band centroid. Furthermore, \citet{iono} use the SMA to obtain a $\sim2$$^{\prime\prime}$\ resolution image of this source at 890$\,\mu$m, and find that it is within 0\farcs4$\pm$0\farcs3 of the optical position. Hence we argue that the alignment between the source and lens is very strong. Adding the VLA/SMA uncertainty in quadrature with the optical astrometric accuracy, we estimate that the observed positional offset between the source and lens is $\la$0\farcs5.
The foreground elliptical is not the source of the Far-IR emission, but how much does it contribute in the observed optical--Near-IR? MIPS\,J142824.0+352619\ is an extremely red object (ERO), having an $R-K$ color of $5.9\pm0.2$ (Vega). Since the ERO criterion matches the observed colors of ellipticals at $z\sim1$, and since the DEIMOS spectrum is consistent with a K+A galaxy, it is possible that the foreground $z=1.034$ elliptical dominates the observed Near-IR emission. However, bright submm galaxies can exhibit similar colors due to extreme extinction \citep[e.g.][]{2002ApJ...577L..83W,2004AJ....127..728F,2004ApJ...605..645W}, and hence the lensed source is also a good candidate.
To see if we can place constraints on the relative contributions of the source and lens, we employ a Singular Isothermal Sphere (SIS) to represent the foreground elliptical. In this simple model, the relevant angular scale is roughly twice the Einstein radius ($2\theta_E$), which is the separation between the two images of the lensed source (in the case where the {\em unlensed} separation between the source and lens is $<\theta_E$) or between the lens and the single image of the source which results when the unlensed separation is $>\theta_E$. The $\sim2$$^{\prime\prime}$\ resolution of the radio and SMA images makes it impossible to distinguish between the two lensing scenarios, but we can use the 0\farcs5 upper limit between the SMA/radio position and the optically detected source as a limit on $2\theta_E$. This in turn places a limit on the velocity dispersion of the lensing elliptical: the SIS model gives $\theta_E=0$\farcs$23(\sigma_L/220$\,km\,s$^{-1}$$)^2$ and hence we derive $\sigma_L<230$\,km\,s$^{-1}$.
Using the Near-IR Faber-Jackson scaling relation (Fig.~11 of \citet{k20}), the absolute rest-frame $K-$band magnitude for an $z\sim1$ elliptical with $\sigma_L=230$\,km\,s$^{-1}$\ is M$_K=-26.0\pm0.2$. In flux units, this corresponds to 33$\mu$Jy. Since the rest-frame $K-$band at $z=1$ corresponds directly to the IRAC 4.5$\,\mu$m\ channel, we see (via Table~\ref{tab:photom}) that the foreground elliptical contributes significantly less in the Near-IR than the background source, by at least a factor of $\sim6$.
This agrees, at least qualitatively, with the observed optical-IRAC SED, which does not show strong evidence of being composed of multiple objects. Using the photometric redshift code developed by the Bo\"otes-IRAC team \citep{brodwin}, we obtain an estimate of $z_{\rm phot}=1.44^{+0.19}_{-0.15}$, in good agreement with the spectroscopic result for the background object. Note however, that the absorption lines seen in the optical spectrum are rather deep, hence the foreground galaxy is likely dominating the emission shortward of $\sim1$$\,\mu$m. This would suggest that the background source is even redder than what is observed.
The faint red source 5$^{\prime\prime}$\ east of MIPS\,J142824.0+352619\ (see Fig.~\ref{fig:booimg}) could not be a second image of the source under the SIS model since the brighter image (when two are present) is furthest from the lens. However, we note that the three highest signal-to-noise images, $I$, 3.6$\,\mu$m, and 4.8$\,\mu$m, demonstrate a flux ratio of 9, 10, and 11 respectively between MIPS\,J142824.0+352619\ and this companion. The uncertainty for each ratio is $\sim1.0$, meaning that it is conceivable that the objects have the same colors (and by inference are the different images of the same lensed source). Deeper imaging and spectroscopy would be required to verify this.
\subsubsection{Estimating the lensing amplification}
Although the well constrained Far-IR SED provides an accurate {\em apparent} luminosity, the value of the lensing amplification, $\mu$, is needed to derive its true brightness. For an ideal point source, $\mu$ varies from being infinite if the source is directly behind the lens, to $\mu\sim2$ at the 0\farcs5 upper limit to the observed angular separation between source and lens.
However, MIPS\,J142824.0+352619\ is not a point source. We can use the Stefan-Boltzmann law to estimate its physical size assuming that the object is a spherical blackbody. In practical units, the diameter of the galaxy using this relation is ${\rm D/kpc = 0.94(L/10^{12}L_\odot)^{0.5}({T/40\,K})^{-2}}$. This results in a source size of $4.6\mu^{-0.5}$\,kpc (0\farcs5$\mu^{-0.5}$), where we have now included the unknown lensing factor explicitly in the luminosity. \citet{1993ApJ...414L..13D} used this approach to estimate source sizes on the order of 400\,pc for typical \textsl{IRAS}\ selected ULIRGs. If MIPS\,J142824.0+352619\ is inherently similar, then the lensing must be very strong, with $\mu\sim100$. However, even 400\,pc corresponds to an non-negligible fraction of the Einstein radius (400pc = 47\,mas at $z=1.325$). This argues against a high lensing factor for MIPS\,J142824.0+352619.
Furthermore, ULIRG sizes on kpc scales are not uncommon. \citet{2004ApJ...611..732C} used high resolution radio imaging to show that $z\sim2.2$ SCUBA selected galaxies are extended over scales as large as 4\,kpc. If MIPS\,J142824.0+352619\ is indeed extended, then the overall lensing amplification is low, since the bulk of the emission would be due to parts of the galaxy that are not significantly lensed. In such a scenario, differential lensing would affect the observed properties of the source. For instance, the stars and dust may dominate in distinct parts of the galaxy, meaning that the overall amplification for each component may be different. However, the Far-IR and radio emission should arise from the same parts of the galaxy, so it is reasonable to assume a common lensing amplification and use those flux estimates to characterize the dust properties of the background source.
\section{Discussion}
The source of energy powering Far-IR luminous objects is typically attributed to AGN or starburst activity, and in this section we use the available data to determine the relative contributions from both processes to MIPS\,J142824.0+352619.
Our $\sim43$K dust temperature estimate is comparable to the mean $36$K derived via sub-mm observations of nearby \textsl{IRAS}\ selected star-forming galaxies \citep{2000MNRAS.315..115D}; the dust temperatures attributed to AGN are much larger ($>60$K). Furthermore, \citet{vandana} find that the mid-IR spectra is dominated by PAH features, which usually indicate a star-burst power source since they are easily destroyed by the hard radiation field of an AGN \citep[e.g.][]{1998ApJ...505L.103L}. Finally, we find that the `q' parameter \citep{1992ARA&A..30..575C} which relates the Far-IR and radio luminosity agrees well with values derived for star-forming galaxies in the local Universe : \citet{1992ARA&A..30..575C} derives $2.3\pm0.2$ for local galaxies while MIPS\,J142824.0+352619\ has $q=2.5\pm0.2$.
If the observed optical/Near-IR flux is indeed dominated by the $z=1.325$ source, we note that the SED lacks any prominent break near 4000\AA. This suggests that the optical spectrum is due to a young ($\la50\,$Myr) obscured stellar component, and hence supports the star-formation case. It would also be further evidence against a strong AGN component, as the stellar bump is clearly seen near 1.6$\,\mu$m\ (rest-frame).
Altogether it seems much more likely that the source is powered by vigorous star-formation. These results do not preclude the presence of an AGN in MIPS\,J142824.0+352619, but only suggest that starburst activity dominates the Far-IR properties of the source. Using our derived Far-IR luminosity and adopting the relation in \citet{1998ARA&A..36..189K}, we estimate a star formation rate of SFR$=5500\pm1000\,\mu^{-1}$M$_\odot$yr$^{-1}$. Note that the empirical luminosity-temperature (L-T) relationship presented in \cite{2003MNRAS.338..733B} for sub-mm galaxies shows that the average luminosity for a source of 43K is $\sim10^{12}$L$_\odot$. If MIPS\,J142824.0+352619\ obeys this trend, then the inferred lensing amplification is $\mu\sim10$, and the SFR still at least several hundred solar masses per year.
\section{Summary}
At $z=1.325$, MIPS\,J142824.0+352619\ allows us to gain insight on the population of luminous dusty galaxies between $z=1-2$. The agreement with the radio/Far-IR correlation and the lack of broad or high excitation spectral lines suggests that MIPS\,J142824.0+352619\ is a starburst-dominated galaxy. Although its observed extreme luminosity is likely enhanced from lensing by a foreground galaxy at $z=1.034$, the amplification is likely not higher than $\sim10$. Hence MIPS\,J142824.0+352619\ is a bright ULIRG not dissimilar to those selected by \textsl{IRAS}\ in the local Universe, or by SCUBA at $z\sim2-3$. As such, MIPS\,J142824.0+352619\ provides a unique opportunity to study in detail the properties of a dusty starburst galaxy at moderate redshift. This object is also interesting since the Near-IR and IRS spectroscopy verify that it lies in the relatively unexplored spectroscopic desert. For future studies, observations of the CO molecular lines would be valuable in determining the gas content in which the stars are forming. More urgently, high spatial resolution optical and Near-IR imaging is essential to understand and correct for the lensing characteristics of this galaxy.
\acknowledgments
We thank Shri Kulkarni and Derek Fox for generously observing this object with LRIS-B which inspired the more extensive optical spectroscopy effort with DEIMOS. Advice from Scott Chapman and Ian Smail greatly improved this manuscript. The work of MB, PE, and DS was carried out at the Jet Propulsion Laboratory, under contract with NASA. AWB is supported by NSF AST-0208527, the Sloan Foundation and the Research Corporation. This work was supported in part by the National Science Foundation through its support of the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA). We thank the NDWFS team for the NDWFS data products used in this work, and NOAO for supporting the NDWFS. The authors also wish to acknowledge the very significant cultural role that the summit of Mauna Kea has for the indigenous population of Hawaii.
Facilities: {Spitzer(MIPS)}, {Spitzer(IRAC)}, {CSO(SHARC-II)}, {Keck(NIRSPEC)}, {Keck(DEIMOS)}, {JCMT(SCUBA)}, {KPNO(MOSAIC-1)}
| {
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SET @app_user_id = (SELECT `id` FROM `person` WHERE `username` = 'static data' OR `user_reference` = 'Static Data');
SET @connected_equipment_id = 12;
INSERT INTO `person`(id, username, user_reference, person_auth_type_lookup_id, family_name, created_by)
VALUES(@connected_equipment_id, 'connected equipment', 'Connected Equipment', 1, 'connected_equipment', @app_user_id); | {
"redpajama_set_name": "RedPajamaGithub"
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Gerald Raymond Sowa, 69, of Beaver Island, died Friday, February 3, 2006, at the Hemet Valley Hospital in Hemet, California.
Jerry was born April 6, 1936, in Muskegon, the son of Leo M. and Mary (Ropotor) Sowa. He graduated from Muskeogon High School in 1954. He earned a Bachelor of Science in Education from Western Michigan University, enlisted in the United States Marine Corps and graduated from Quantico Officer's Candidate School. He served in the Marines for twenty-six years, which included two tours of duty in Vietnam, and another in Okinawa at the end of the war. He retired as a Major in 1977 at the age of 42.
After retiring, he learned to become a mechanic and play the guitar, took singing lessons, went to electronics school, became EMT certified, completely built two homes, and learned to build golf clubs. He also earned a Master's Degree in Secondary Education from Arizona State University. He made his home on Beaver Island for the past fifteen years, where he was a member of Holy Cross Catholic Church, On May 9, 1959, he married the former Shirley E. White in Muskegon, and she survives.
He is also survived by his children, John L. (Elaine) Sowa of Hemet, Calif., Michael A. Sowa of Beaver Island, Catherine (Joel) Meintsma of Coopersville, Lois M. (Kevin) Stipp of Hemet, Calif. and Beaver Island; grandchildren, Theresa, Melissa, Cory, Jeremy, Tara, and Sowa, Aubreanne, Kyle, Kameron, Joleen and Shelby Meintsma, Jacob and Jessica Stipp; great-grandchildren, Cody and Amanda Stipp; brothers, Robert Sowa, James (Linda) Sowa, Gregory (Teresa) Sowa, Lawrence Sowa all of Muskegon; sisters, Teresa (Chuck) St. Amour of Muskegon, Mary Kreisel of Quartzite, Ariz.
A Rite of Christian Burial will be 11:00 AM, Saturday, February 11, at Holy Cross Catholic Church on Beaver Island, the Reverend Patrick Cawley officiating. Burial will follow at Holy Cross Cemetery. A Rosary will be recited at Holy Cross Church at 7:00 PM Friday, where friends may meet the family. Memorial contributions may be made to the Holy Cross Altar Society (Beaver Island). The family is being assisted by the Winchester Funeral Home in Charlevoix. The family will miss the sound of his laughter and his wit, but mostly they will miss his tremendous love.
The AMVETS Post #46 Ladies Auxiliary will again have an Easter Egg Hunt at the Beaver Island Community School on Saturday, April 15 at 1:00 PM . Have the little ones bring their baskets or bags to collect the eggs with all the goodies.
The Beaver Island MI Post #46 AMVETS and Ladies Auxiliary will host a USO Dance on Saturday, June 24 at the Holy Cross Hall, Beaver Island MI with music from the 40's and things to be expected at the USO, etc. Whether you know how to Jitter-bug or not, come join the fun.
Movin' on up is the word this week as the beginnings of the second floor take shape. The huge trusses were moved to the building site so expect to see them in place soon.
This weekend games were played on Mackinac Island and so the Islanders flew there on Friday afternoon. Friday night a wholloping blizzard hit Michigan and as a result the teams and coaches spent Saturday and most of Sunday there also.
Friday night the Islanders basketball fell to Mackinac Island 41 - 49 but came back on top Saturday morning with a score of 67 - 58. Thanks go to Kerry Smith for the scores.
Thanks go to Coach Connie Boyle for sending me the information on the Lady Islanders. | {
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{"url":"http:\/\/hal.in2p3.fr\/in2p3-00163870","text":"# Inclusive $D*pm$ meson and associated dijet production in deep-inelastic scattering at HERA\n\nAbstract : Inclusive D*\u00b1 production is measured in deep-inelastic ep scattering at HERA with the H1 detector. In addition, the production of dijets in events with a D*\u00b1 meson is investigated. The analysis covers values of photon virtuality 2 \u2264 Q2 \u2264 100 GeV2 and of inelasticity 0.05 \u2264 y \u2264 0.7. Differential cross sections are measured as a function of Q2 and x and of various D*\u00b1 meson and jet observables. Within the experimental and theoretical uncertainties all measured cross sections are found to be adequately described by next-to-leading order (NLO) QCD calculations, based on the photon-gluon fusion process and DGLAP evolution, without the need for an additional resolved component of the photon beyond what is included at NLO. A reasonable description of the data is also achieved by a prediction based on the CCFM evolution of partons involving the kT-unintegrated gluon distribution of the proton.\nDocument type :\nJournal articles\n\nhttp:\/\/hal.in2p3.fr\/in2p3-00163870\nContributor : Sabine Starita <>\nSubmitted on : Wednesday, July 18, 2007 - 4:46:49 PM\nLast modification on : Saturday, October 3, 2020 - 3:18:21 AM\n\n### Citation\n\nA. Aktas, V. Andreev, T. Anthonis, B. Antunovic, S. Aplin, et al.. Inclusive $D*pm$ meson and associated dijet production in deep-inelastic scattering at HERA. European Physical Journal C: Particles and Fields, Springer Verlag (Germany), 2007, 51, pp.271-287. \u27e810.1140\/epjc\/s10052-007-0296-5\u27e9. \u27e8in2p3-00163870\u27e9\n\nRecord views","date":"2020-11-25 20:35:04","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\": 0.4002344608306885, \"perplexity\": 5750.4398276148}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141184123.9\/warc\/CC-MAIN-20201125183823-20201125213823-00265.warc.gz\"}"} | null | null |
Q: Bash - Remove domain/subdomains from file B, if main domain is in file A (or piped) I would like to be able to use a piped input or reference file of domains (file B) to remove each domain and it's subdomains from file A
I can't use grep "bbc.co.uk", for example, as this would include entries such as cbbc.co.uk.
I have tried to use a while read loop to iterate through file B, running grep -E "^([^.\s]+\.)*${escaped_domain}$" fileA to identify both domains and subdomains but this is very, very slow with the amount of comparisons required.
Is there a better way to do this? Perhaps using awk?
File B (or piped input)
~30k lines
bbc.co.uk
amazon.co.uk
doubleclick.net
File A
~150k+ lines
123123.test.bbc.co.uk
123434.rwr.amazon.co.uk
ads.bbc.co.uk
adsa.23432.doubleclick.net
amazon.co.uk
bbc.co.uk
cbbc.co.uk
damazon.co.uk
fsdfsfs.doubleclick.net
test.amazon.co.uk
test.bbc.co.uk
test.damazon.co.uk
Desired output:
cbbc.co.uk
damazon.co.uk
test.damazon.co.uk
Current method (different input with grep/regexps)
# Convert input: address=/test.com/ -> ^([^.\s]+\.)*test\.com$
regexList=$(cat fileB |
sed 's/\./\\./g' |
awk -F '/' {'print "^([^.\s]+\.)*"$2"$"'})
while read -r regex; do
grep -E $regex filaA
done <<< "$regexList"
A: $ awk '
NR==FNR {
gsub(/[^^]/,"[&]")
gsub(/\^/,"\\^")
doms["(^|[.])"$0"$"]
next
}
{
for (dom in doms) {
if ($0 ~ dom) {
next
}
}
print
}
' fileB fileA
cbbc.co.uk
damazon.co.uk
test.damazon.co.uk
or with a pipe:
$ cat fileB | awk '...' - fileA
If fileB is small enough then you don't need an array you can just build up and test 1 regexp for all domains:
$ awk '
NR==1 { doms = "(^|[.])(" $0; next }
NR==FNR {
gsub(/[^^]/,"[&]")
gsub(/\^/,"\\^")
doms = doms "|" $0
next
}
FNR==1 { doms = doms ")$" }
$0 !~ doms
' fileB fileA
cbbc.co.uk
damazon.co.uk
test.damazon.co.uk
The 2 gsub()s in each script are ensuring that all regexp metacharacters in the domains are treated as literal characters instead. See is-it-possible-to-escape-regex-metacharacters-reliably-with-sed for details on why and how that works.
A: You can transform the first file into a set of regular expressions for what to remove:
sed 's/[][\\.^$*+?()]/\\&/g;s/.*/^([^.]+\\.)*&$/' fileB
The output is a sequence of regular expressions you can pass to grep -vE:
... | grep -vEf - fileA
There are limits to how much grep -Ef can keep in memory in one go, but 30k expressions is probably within limits on modern hardware. In the worst case, split fileA in half and run the process twice.
| {
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Annie John
by: Jamaica Kincaid
Chapter One: Figures in the Distance
Chapter Two: The Circling Hand
Chapter Three: Gwen
Chapter Four: The Red Girl
Chapter Five: Columbus in Chains
Chapter Six: Somewhere, Belgium
Chapter Seven: The Long Rain
Chapter Eight: A Walk to the Jetty
Annie's mother (Mrs. John)
The Red Girl
Mother-Daughter Relationships
The mother-daughter relationship drives the plot in Annie John and is its primary theme. The difficulties and tensions in this relationship stem from Annie's inability to accept the fact that she is a separate self. Kincaid paints Annie's desire to remain united with her mother as an emotion shared by most girls of her age. Annie's classmates all commiserate with her essay about her fear of separation. Furthermore, the girls befriend one another in an effort to find substitutes for the maternal love that appears to be dissipating. As Annie ages, she finds herself caught between love and hatred for her mother, which drives her to be both a good student and a disobedient child. Again, the rationale behind her adolescent rebellion seems to be proffered as an explanation for a general psychological trend rather than merely a specific fictional phenomenon. The dynamics of mother-daughter relationships take up a prominent place in Jamaica Kincaid's work and have frequently appeared in her other novels such as Lucy and The Autobiography of My Mother.
Colonizers and Colonial Education
Antigua was colonized by the British until 1967 and remained a commonwealth in 1981. As Annie John takes place in the 1950s, it remains in the colonial period. Kincaid explores the colonial relationship particularly through her discussion of the school that Annie attends. It is run as a British institution and all the materials taught in the school deal with English literature, history, and culture. The girls dress in a formal British style and they are discouraged from engaging in local activities, such as calypso dancing in the playground. Annie's musing on the failure of the school to discuss the negative history of slavery and her delight in the imprisonment of Columbus highlight the ways in which the school teaches the students not to question the history and social order that is being handed down to them. Annie excels in her school, which shows that she has learned all of the skills necessary to prove her intellectual and social worth in the colonial world. However, her spunky behavior behind the teachers' backs shows that her feisty Antiguan spirit still thrives within.
Although Annie's father appears a gentle and reticent man, he serves as a testament to the unequal gender relations in Antigua. Annie's father is about thirty years older than his wife. He had numerous sexual affairs before marrying Annie's mother and the women with whom he slept frequently harass Annie's mother on the street. Now that he has his married life secured, he provides for the family while his wife takes care of his domestic and sexual needs. While as a man Annie's father could philander, Annie's mother interprets Annie's mere discussion with a group of boys as inappropriate sexual misconduct and calls her a "slut." With these two standards, it becomes clear that the behavior expected of men and women in Antigua are quite different. Although the women who curse at Annie's mother appear unfriendly, even Kincaid's depiction of them is sympathetic. They, after all, committed the same sexual act as Annie's father, but have been left in the difficult economic position of raising their children without a husband.
Annie John: Character Analysis | {
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For our very first feature for The Regulars, we caught up with Italian DJ and producer Fabio Monesi. Many know him as the man behind Wilson Records, a label that has been pushing Fabio's own hardware-driven sound along releases from the likes of Chicago's Gene Hunt and Jordan Fields. He has also released music under his Hissman moniker on labels like Dog In the Night and his own Hardmoon London. Fabio has been a regular at the shop for a while now so we wanted to ask him about his experience at Phonica.
Do you remember the first record you bought at Phonica?
It was summer 2011 and I remember very well Nick was in that day. I was going to launch my first record label and Simon was so kind to help with distributor contacts. I remember buying two Secretsundaze records. One of those was by Two Armadillos plus many other New Jersey deep house records as at the time I was really into that sound.
What was your latest purchase?
I bought a lot of records just a few days before the Record Store Day, I only remember the Helena Hauff LP.
How do you think Phonica has influenced you over the years?
Phonica has always been the first record shop where I check the new electronic stuff. I have always liked the idea of supporting physical record stores, especially those where I met nice people working in it.
Fondest memory at 51 Poland Street?
Name a record or two that you associate with Phonica?
Wilson Records WLSLTD03 as I remember Simon messaging us many times asking for more copies as it was literary flying!
Tell us about your own projects.
If you enjoyed this, check out the interview we did with Fabio after the Wilson in-store, discussing his approach to DJing, producing and running his own label. | {
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journal of research in medical sciences pubmed
GET THE APP. … Journal covers all medical specialties including. The Journal of Research in Medical and Dental Science is a peer-reviewed medical journal which publishes articles from Health Sciences which bear relevance to the current research scenarios. Welcome to Journal of Medical Science And clinical Research is an international, peer-reviewed open access journal of medicine and health science published in English. International Journal of Innovative Research in Medical Science (IJIRMS) currently is crawled and listed by - Google Scholar. The PubMed journal list covers the entire span of MEDLINE, not just currently indexed journals. Citations may include links to full-text content from PubMed Central and publisher web sites. Modern prevalence of the Fredrickson-Levy-Lees dyslipidemias: findings from the Very Large Database of Lipids and National Health and Nutrition Examination Survey. 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The Journal's primary aim is to make every effort to contribute to progress in medical sciences. These journals are the medium of open access publication of novel medical research findings for the benefit of the entire scientific community. A bibliometric analysis was performed to reveal the trends in lead removal research in 1991–2015. 4 Sleep Laboratory of Ibn-e- Sina Hospital, Mashhad University of Medical Sciences, Mashhad, Iran. American Journal of Biomedical Science & Research (ISSN: 2642-1747) is an Open access online Journal dedicated in advancing the latest scientific knowledge of science, medicine, technology and its related disciplines. The Journal aims at publishing evidence-based, scientifically written articles from different disciplines of medical sciences. The Irish Journal of Medical Science is the official organ of the Royal Academy of Medicine in Ireland. 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Research Journal of Medical Sciences (RJMS) is a scientific journal which publishes research works in all areas of medical sciences to enhance understanding of medicine in general. The PubMed journal list covers the entire span of MEDLINE, not just the journals currently indexed.Non-MEDLINE journals include those whose content is deposited in PMC (PubMed Central). 3 Post Graduate Medical Student, University of Sydney, Sydney, Australia. Journal of Medical Science (JMS) is a peer-reviewed, Open Access journal that publishes original research articles and reviews which cover all aspects of clinical and basic science research The International Journal of Medical Sciences is a peer-reviewed open access medical journal published by Ivyspring International Publisher covering research in basic medical sciences.Articles include original research papers, reviews, and short research communications. IJMSE is a peer reviewed journal which is available online and in print format as well. List of Topics. International Journal of Medical and Health Research is a peer reviewed, indexed, open access journal, publishing high quality papers on all aspects of medical sciences. International refereed journal publishes leading research papers in all areas of medical sciences from basic research to clinical and experimental work. Established in 2007, RJMS has been publishing articles of researchers from all over the world. International Journal of Medical Research & Health Sciences (IJMRHS) ISSN: 2319-5886 Indexed in: ESCI (Thomson Reuters) Journal of International Medical Research is a peer-reviewed open access journal which focuses on original clinical and preclinical research, systematic and perspective reviews, meta-analyses, pilot studies and case reports, with every article accepted by peer review given a full technical edit to make all papers highly accessible to the international medical community. 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Research Interest: As a professional with intensive interdisciplinary backgrounds, his research goals are to understand the human diseases mechanisms through studying the clinical relevant animal models, in the fields of Neuroscience, Oncology, Orthopedics, Molecular Immunology/Hematology, Pathology, Genomics/ Genetics and Stem cells. 5 COPD Research Center, Imam Reza Hospital, School of Medicine, Mashhad University of Medical Sciences, Mashhad, Iran. IOSR Journal of Dental and Medical Sciences is one of the speciality Journal in Dental Science and Medical Science published by International Organization of Scientific Research (IOSR). Ophthalmology. The Journal aims at publishing evidence-based, scientifically written articles from different disciplines of medical sciences. The journal's publisher is the IGM Publication. Full text of published articles is archived in PubMed Central. About This Journal. International Journal of Medical Research & Health Sciences is an international, peer-reviewed, multidisciplinary, monthly, online journal published by Sumathi Publications. Indian Journal of Medical Sciences Review latest developments in the field of Medicine and Allied Sciences and Publication of Scientific Investigation. Established in 1832, this quarterly journal is a contribution to medical science and an ideal forum for the younger medical/scientific professional to enter world literature and an ideal launching platform now, as in the past, for many a young research worker. The data used in this study were derived from the Science Citation Expanded Index of Web of Science. The non-MEDLINE journals include those whose content is deposited in PMC (PubMed Central) .
Picture Of Scent Leaf, Homes For Sale In California, Do Coyotes Leave Remains, Remove All Ubuntu Desktops, Corymbia Ficifolia Orange Splendour', Oreo Chocolate Calories, Sankey Diagram Tableau, Quasi Star Vs Uy Scuti, Bosh Falafel Mix, Iron On Clothing Labels,
2020 journal of research in medical sciences pubmed | {
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THIS IS A BORZOI BOOK PUBLISHED BY ALFRED A. KNOPF
This is a work of fiction. All incidents and dialogue, and all characters with the exception of some well-known historical and public figures, are products of the author's imagination and are not to be construed as real. Where real-life historical or public figures appear, the situations, incidents, and dialogues concerning those persons are fictional and are not intended to depict actual events or to change the fictional nature of the work. In all other respects, any resemblance to persons living or dead is entirely coincidental.
Text, cover art, and interior illustrations copyright © 2017 by Whitney Gardner
All rights reserved. Published in the United States by Alfred A. Knopf, an imprint of Random House Children's Books, a division of Penguin Random House LLC, New York.
Knopf, Borzoi Books, and the colophon are registered trademarks of Penguin Random House LLC.
Visit us on the Web
Educators and librarians, for a variety of teaching tools, visit us at RHTeachersLibrarians.com
Library of Congress Cataloging-in-Publication Data is available upon request.
ISBN 9780399551413 (trade) | ISBN 9780399551420 (lib. bdg.) | ebook ISBN 9780399551437
Random House Children's Books supports the First Amendment and celebrates the right to read.
v4.1_r1
ep
# Contents
Cover
Title Page
Copyright
Dedication
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
Chapter 53
Chapter 54
Chapter 55
Chapter 56
Chapter 57
Chapter 58
Chapter 59
Acknowledgments
For all the girls looking to leave their mark on the world
Six stencils in and it's gone. Okay, the tag vanished by Stencil Number Two, but I have a point to prove. I'm not covering up your scribbled slur with just anything. I'm _making art_ here. I'm creating. I'm on fire.
I've never thrown up such an intense piece—I was worried I wouldn't be able to pull it off in time. My arm flies across the wall, pink paint striping across the last stencil. It looks like it's going to work out. I chuckle to myself. This is what it's all been for, the hours of paint-pen practice, filling up every inch of every sketchbook with tags and words and pictures. All my hard work has paid off, and it's all up here on the wall.
I know I shouldn't be tagging the school. _I know that._ But I wasn't the first, and that mess had to go. Jordyn told the principal that someone tagged the gym, she had to. The vandal singled her out, and word gets around real quick at Kingston School for the Deaf. But three weeks went by, and "Jordyns a SLUT" was still there on the back of the gym for all to see. And good ole Principal Howard hadn't done a damn thing.
No one gets to call my best friend a slut, especially not up on a wall, not on my turf. She asked for help, and I took matters into my own paint-stained hands. I designed a killer piece, cut out the stencils, shook up the cans, and got to work.
_I'm getting away with it._ I'm about to get up. On my way to becoming an all-city queen of street art. I rip down the last stencil, take a step back, and admire my work. It's killer. You're welcome, Universe. I check over both of my shoulders again, eyes on constant watch. I can't rely on my ears, so my eyes work overtime. It's nice and dark. I pretend I'm nothing but a shadow.
I'm so proud I just can't help myself and I text Jordyn a picture of the new mural on my way back home.
"You don't have any proof!" I snap at our principal.
"Don't lie to me, Julia. You'll only make it worse." His hands are big, with stubby fingers. He might be hearing, but he signs perfectly. He has to, or he never would have gotten the job.
"I'm not lying! You can't say it was me." I know there are no cameras on that side of Kingston. I know there won't be any footage to review.
"I have all the proof I need. Look at your hands!"
_I'm so stupid. I was being lazy._ I'm going to need to buy gloves. Lots and lots of gloves.
"This was from art class." I sign as fast as I can before dropping my hands out of view and into my lap.
"I'm going to give you one more chance to tell the truth, Miss Prasad." Mr. Howard seems more agitated than angry. He keeps sighing, looking at me with droopy, tired eyes.
"I don't know what to tell you. Sorry." _Let me go already, you've got nothing._ He stares at me, waiting for a better answer. I'm not giving it to him. I'm not confessing to anything, as much as I want to take credit for it. He hangs his head and pinches the bridge of his nose.
"Well, what can you tell me about this?"
My heart shakes up in my chest like a paint can as he produces a cell phone from his desk drawer, the case dotted with red cherries. It's Jordyn's. He slides it across the desk like some detective on _Law & Order._
I don't want to look. I don't need to. I know what's about to happen. And I know without looking that Jordyn, my best friend in the universe, sold me out. _How could she?_
"The paint on your hands, the picture on her phone. You can't tell me you didn't do it."
"Fine. But I was covering up—"
"That's not your job."
"Well, whose job is it? Because that nasty graffiti was up there forever."
"Not _your_ job. We had someone scheduled to take care of it."
"But mine is art!"
"That's not art, it's vandalism. I'm worried about you; you're not exactly showing any remorse here," he lectures. My face flushes hot with rage. He's not worried about me, he's relieved he has someone to pin it on. I wonder if the slut-shaming toy-tagger got the fifth degree, too. I doubt it.
"I don't understand what the big deal is! I didn't hurt anyone. I didn't destroy anything. I've tagged the girls' room dozens of times. No one cared then—"
"You _what_?!" His face is turning as red as mine.
"So now, when I try to make something worthwhile, _art_ even, you're up in arms, calling me a vandal?" Just tell me how much detention I have so we can all move on with our lives, and I can X-Acto–cut Jordyn out of mine. I wonder how long she had to sit here before stabbing me in the back. She's spineless, so she's always asked me to break the rules for her. Which I've done plenty of times, because I thought we were a team. I bet all Mr. Howard had to do was ask, and she rolled right over like a David Hockney dachshund. The light by Mr. Howard's door flashes, indicating first period is about to begin. All my anger fizzles away and I just feel weak, depleted at the thought of Jordyn heading off to her first class, no worries, all smiles, while I get interrogated.
Mr. Howard stands up and walks to his office door without saying a word. He opens it and my stomach flips; all my bravado turns bashful as he ushers my mothers into the room. It's one thing to piss off the principal. I can barely look at my parents as he tells them I'm expelled.
—
It's silent.
Who am I kidding? It's always silent, but this—I can _feel_ it. Like for the first time, I know what the word really means. It pounds in my head. Silence is the loudest sound. Ma doesn't scowl in the rearview. Mee doesn't sign a word.
I messed up. It was beautiful. Not a masterpiece but, I don't know, close? Didn't matter, got caught. Shouldn't have done it on school property and definitely shouldn't have texted anyone evidence; those were toy mistakes and I knew better. I stood up for Jordyn, tried to save her dignity. She cried and cried the day we discovered it. And when it looked like the school wasn't going to help her, I did. I helped her, and she ratted me out—I just don't understand. I get expelled and Jordyn gets what? Nothing.
The expulsion was an overreaction, if you ask me. But that was the "final straw" and "the school won't be responsible" for whatever "mayhem" (really?) I cause next. My first real piece and I'm expelled. And now I need a new tag. Go ahead, call me a vandal, say I'm some sort of delinquent, it isn't going to insult me. It's not going to stop me. Please. _This is what I live for._
Silence. I stare at the backs of my parents' heads, waiting for one of them to start in on me. Waiting for Mee's pointer finger to fly to her chin with that grimace she saves for special occasions.
_Disappointed._
—
It never comes, so I kick off my shoes and rush upstairs as soon as we're home. If they're not talking yet, I'm not going to be the first. I crash-land onto my bed face-first and grip the quilt in clenched fists. I pound the mattress. _What's! Wrong! With! Her?!_ Who would do something like that? She was the only real friend I had, the only one who knew me and my whole paint-splattered story. It eats at me, worming its way through my stomach and up to my brain. Neither organ can make any sense of it.
My phone vibrates in my pocket, and I'm hoping Jordyn has a damn good explanation for what she did to me. Because only one person I know would be texting me right now.
> JORDYN: Srry :(
>
> JULIA: щ(ºДºщ)
>
> That's it?
>
> JORDYN: They were gonna call the cops. On meeee!
>
> JULIA: ¬_¬
>
> JULIA: No. They weren't.
>
> JORDYN: Mayb.
>
> JULIA: They kicked *me* out!
>
> JORDYN: I didnt think they would really do it.
>
> JULIA: WHY
>
> JORDYN: Idk. I mean u did break the law and stuff.
>
> JULIA: Standing up for you!
>
> JORDYN: U didn't have to. I didnt ask u.
>
> JULIA: Are you kidding me?!?!
>
> JORDYN: It's not like u care abt getting in trouble.
>
> JORDYN: I did u a favor. Ur gonna be famous now.
>
> JORDYN: Don't be so mad.
I stuff my phone under my pillow. I don't care what else she has to say. Nothing can make up for what's already been done. Nothing.
I love gray days. Every tree, building, telephone pole highlighted against the gesso-colored sky. This past week has been especially overcast and it's a relief. I thought getting registered at a new school would take at least a month, that I would get to stay out of the educational system for a while. But with both of my moms at the helm, it only took four days. Now, three weeks in at Finley, the spotlight hasn't grown any dimmer. I welcome the clouds. Bring on the fog.
It's getting to be that time of year when it's still dark in the morning and the roads are empty. The drive to Finley is one of the few things I don't hate about the transfer. You would think the forty minutes it takes to commute from Queens Village would suck, but I love driving. Gives me time to think. I drive through the 'burbs of Greenlawn with the tree-lined sidewalks and traffic lights reflecting in the wet road. The leaves aren't turning yet, but they're about to. I spot a red leaf here and there, pilot lights to the season. Just me and my car, Lee.
Good ole Lee. I bought her off of Craigslist this summer for twelve hundred bucks, a 1994 Oldsmobile. She's older than I am, but she's got some moves left. When I got her, she was this horrible maroon color. Now she's perfect: black and white, with flecks of color here and there. Krasner meets Basquiat. That's Lee. She's the only real friend I have left, the only one who's never let me down.
I fish through my bag on the passenger seat, getting my morning ritual started en route. Pull out a can of Red Bull, hold it between my thighs (I'm an expert at driving one-handed), and crack it open. I hate coffee. It's either bitter or sour or chalky, not to mention the bad breath. Red Bull isn't the most delicious morning elixir, but a girl's gotta get a jolt from something.
Pulling into the parking lot of my new hellscape, I look for a spot up front in case I need to make a quick getaway. I haven't actually tried escaping from school yet, but you never know. Doesn't matter that the overly accommodating administrators reserved a spot for me next to the front doors. I refuse to park there. I can walk. Don't baby me.
I don't get the best spot this morning, but it's not a gym day, so I probably won't feel the need to flee. I reach to put Lee into Park when— _SLAM!_ —she lurches forward and my seat belt digs into my chest. I swivel around in my seat and look out the rear window.
Kyle Fucking Stokers.
He tried to park in my spot, not noticing that my car was already there. What a tool. He's one of those people who's unaware of anyone or anything else in his vicinity. Bow down to him, the only person on earth who truly matters. So of course this whole ordeal is about to be blamed on me. Doesn't matter that I was already parked, minding my own business. I exist, therefore I am at fault.
I get out, not bothering to put on my shoes. My socks are getting damp as I walk around on the wet pavement. Lee's bumper is okay, no real harm done. Tough bird. Some of my paint job has come away, but the maroon showing through isn't a tragedy. I'm the only person who would even notice. Before I can get a closer look, there's hands on my shoulders and Kyle spins me around to face him. He's yelling.
"What—-—-——-parking here?!"
There's always a moment when one of these kids asks me a question and I have to figure out if speaking is worth the risk.
I cross my arms.
"You—--aint—my bum—r!" he rages. It's not easy to lip-read when people are yelling at you. Despite what the distorted-face yeller might actually think.
I stare back at Kyle. He probably spent more time on his dusty blond hair this morning than I ever spend on mine. He has great eyebrows, but that's beside the point.
"Well?" He gestures to his car again and again, trying to drive his point home.
Walking over to his slick silver car, I spit on my sleeve. _I_ should be yelling at _him._ I should scream and say, "You ran into ME, dipshit!" Honestly? He's not worth it. I buff off the paint and gesture at the spot. _All better._ I raise my eyebrows and smile. He doesn't catch the sarcasm.
"Bitches shouldn't drive," he says slowly, deliberately. I catch every word. He turns and walks toward the school. I imagine throwing my keys at him, chasing him down, kicking his shins until he's on the ground. I slam my fists into his chest over and over and—
There's a tap on my shoulder and I snap to. Kyle disappears into school through the double doors.
"Julia! Where are your shoes?" Casey signs. She's looking at me like I'm crazy, not a hair out of place in her perfectly cut chin-length bob. Her eyes behind her black-framed glasses are magnified to a ridiculous size, like something out of a Margaret Keane painting. I point over to Lee.
"One minute," I reply. "See you in history." I shoo her away from me, because the last thing I need is Casey thinking she can solve all my problems _outside_ of class, too.
I get back in the car and peel off my socks. Great. Now I'm going to end up with blisters. Mee bought me new Doc Martens before the transfer. She winked when she gave them to me—a signal she reserves for when something is to be kept just between us. Ma would kill her if she knew Mee was buying me gifts now. Rightfully so; I know I don't deserve them. But they make me smile. They're yellow, my favorite color. Problem is, they're impossible to break in and twice as impossible to drive in, so I drive in my socks and put the boots on before school. I squeeze my size 10 feet in and lace them up loosely.
I reach into the backseat to grab my hoodie, but the one I pull out isn't mine. It's Jordyn's, all purple and pilly. It even smells like her. How long has it been in here? Sand spills out of the folds, and I remember that day on Coney Island when we shared a spicy mango on a stick. Like we always did. Like we never will again. Not any time soon. I shove it under the passenger seat. I can't stand to look at it right now.
I need my own hoodie, my trusty black-faded-gray-with-age armor. The sleeves and hem flecked with rainbows of spray paint. This is what I wear when I go out and tag stuff. I yank the zipper up to my chin, and I'm protected. The hood falls over my two loose black buns, down over my ears. I take my bag, open my second Red Bull, and drink it, heading toward the big blue building.
At first I thought transferring to Finley wouldn't be a big deal. School is school; I hated it at Kingston, I'd hate it at Finley. I mean, Jordyn is always going out with hearies and they seem fine, but it's not like I'm looking to make friends. I don't have time for that shit anymore. Not after Jordyn showed me what she's really made of. No one here would even notice me, right?
Casey took care of that right quick. Having an interpreter in every class is like having a giant neon sign hanging around your neck, blinking: _Freak Freak Freak._ I've been here three weeks and people are still confused about how it all works. It's not hard: teacher talks, interpreter signs, I understand. They act like Casey's conjuring black magic, waving her arms around, when really she's only blathering on about tariffs or decimal places.
I toss the empty Red Bull into the recycling bin and head for my locker. Mine is stuck in the freshman hall, even though I'm a junior, because it's one of the few left over from the start of the year. I open it up and all of a sudden I feel lighter. I take a certain pride in every tag, and I've done a good job claiming my space here. I know I shouldn't have tagged the inside of the door, but I couldn't help myself. A new color for every week, my tag, my sign: _HERE._
U.S. History isn't so bad because no one is awake yet, not even Mr. Clarke. It's the only class I'm glad to have Casey around for, because watching his wrinkly old-man lips collect foamy spittle in their corners makes me want to hurl.
Casey always has her nails painted some fun color. It's a huge interpreter no-no, but I won't report her; it's easier on the eyes than old-man spit-mouth.
"You found your shoes!" she signs, hitting her fists together.
"Yay." I wave my hands next to my deadpan face.
Casey stands up and I take the seat from her. She always gets to class early to "prep," which looks a lot like sucking up to the teachers. I think she's trying to prove her worth by getting me better grades. Ma hired her specifically; Casey shadowed her last year for some sort of college credit. Sometimes I wonder if Ma chose Casey so she could spy on me. Fresh out of terp school and shining with enthusiasm after finding her calling, Casey doesn't know why I was expelled, but she's on a mission to "improve my quality of life." I'm sure she goes home and talks to whatever friends she has about how brave I am. I didn't choose to be deaf. I have no idea why it makes me brave.
The underpass is what makes me brave. I don't know when I'll have time to go out there, but it had better be soon. I've had plans for it ever since I laid my eyeballs on it. I'm going crazy sitting around the house at night, scrolling through the Stencil Bomb forums until I pass out. My life has to be about more than the Refresh button. After my Kingston piece, with all the risk and the rush, painting in the basement isn't really cutting it anymore. I want to make art that makes my heart race. Art that demands to be felt, even if that feeling is terror.
"You have your homework?" Casey asks. _She_ asks, not Mr. Clarke. It's not _her_ job to ask. I've told her this before. The only thing she's supposed to do is interpret for the teachers and me—not police my homework, or scold me for doing poorly on a test. I'm about to lay into her again when I feel the girl sitting behind me staring. I take the worksheet out of my bag and hand it to Casey. At least I did my homework this time.
The girl behind me taps my shoulder. She's got glossy blond bouncing bangs and long feather earrings. I don't think she owns any jeans; I've only ever seen her in yoga pants. Thankfully not the kind you can see through, but she does seem to own a pair in every possible color. And animal print.
"Look, I go a new slines!"
I'm not very good at lip-reading to begin with, and that wad of gum she's gnawing on isn't making it any easier. I'm sure I look puzzled when she starts signing. "Friend. Family. I love you."
Ah, she knows the same signs every other hearing kindergartner learns. She's looking for her gold star, no doubt. I beam at her, and sign back happily, "Bitch. You don't. Know me." She gives me a thumbs-up and goes back to her work.
—
"Why did you do that?" Casey asks as we make our way down the hall after class.
"What?"
"Why did you say that to her?"
"Who, Yoga Pants? She couldn't understand me. I wasn't starting anything." I wonder how many of the students who stop and stare at our conversation end up being late to their next class.
"She was trying to be nice!" Casey's eyebrows angle together in annoyance.
"Nice? What should I have said? 'Wow! You know three signs, I can see we're gonna be BFFs! Oh, the late-night chats and...' "
"Come on! You're never going to make friends if you don't lose the attitude."
"Who said I wanted to make friends? I think—"
There it is, a glimpse of red plaid flannel, and I'm off charging after it like a bull. Only when I'm tapping Mr. Katz on his shoulder do I realize I made a mistake leaving Casey behind.
"Julia? What's wrong?" Mr. Katz asks me. I try signing first.
"Please." I rub my palm on my chest. He doesn't understand me. I switch my hands up to look like I'm begging. I plead with my eyes. I squeeze my interlocked fingers and suck air in through my teeth.
"My class is full. I'm sorry. Truly." Mr. Katz's brown eyes show real regret. He moves to leave, but I dash in front of him and open my bag. A few other students have stopped to watch me play my version of charades. I couldn't care less. I pull out my sketchbook (not my Black Book, which is full of plans that aren't superlegal) and put it in his hands.
"Please," I sign again. "I can't go a whole quarter with no art classes!" I don't care if he doesn't know exactly what I'm saying—I'm sure he gets the gist. I've been asking to get into his advanced-art studio class since my first day at Finley.
He flips through my book and I bounce on the balls of my feet, watching his face intently for his reaction. Mr. Katz handles my sketchbook thoughtfully. Even though we're both pressed for time, he holds it in his hands like a baby bird, gingerly turning each page and considering its worth. Sometimes he smiles.
I _need_ to be in Mr. Katz's class.
His focus is broken abruptly and he glances upward at the ceiling. It must be the bell. Great, I've made him late. He holds up a finger and takes a pen from his red flannel pocket. I wonder how many of those shirts he owns. I rarely see him wearing anything else. He carefully turns to the back page of my book and writes:
_We'll see._
I can feel Ma stomp her foot to get my attention before I leave the house. Should I make a run for it? She stomps again and I turn around.
"Your bag." She doesn't ask to see it, she _demands,_ and I fork it over. Most moms search for drugs and skimpy clothes and stuff. Mine is looking for paint pens and spray cans.
Ma and I look nothing alike. She's fair with green eyes; Mee calls her an Irish gem (try not to gag). I'm the spitting image of Mee: oil-black hair, big feet, brown skin. I'm her little jewel of India. Get it? _Jewel, Jewel-ia, Julia_ (more gagging, I know).
I love watching Ma's hands when she signs. Normally, you just watch someone's face while they're signing. But I can't keep my eyes off Ma's hands. I know it doesn't make any sense, seeing as she's not my bio-mom, but our hands are very similar. Sometimes when she's talking to me it's like I'm watching my own in the mirror. Except the mirror hands have a perfect manicure and wedding band and mine are—were—always covered in paint and Sharpie ink.
"I wish I didn't have to do this." She holds out my backpack, inspection complete.
"So don't." I zip it up and sling it over my shoulders.
"I want you back home by ten thirty, no messing around after work." More demands. One wave and I'm out the door.
The inspections haven't let up. Ma went through Lee the day of my expulsion and pretty much gutted the car of anything I could make a mark with. I'll admit I wasn't being careful. That's the problem when your own personal contraband comes in the form of _basically_ legal art supplies. I never thought I would have to stash anything. I was such a freaking toy. I've had to step my game _way_ up since then. I'm not the kind of person to make the same mistakes over and over again.
I feel my phone vibrate in my pocket: Jordyn, no doubt, looking to see where I'm at. She's so oblivious. She should have deleted my number the second she got her phone back from the principal. After that pathetic attempt at an apology she still likes to send me vapid, meaningless texts even though I only ever text back the same thing:
> JORDYN: Wrks boring without D.
>
> JULIA: k
Jordyn and I both work at McDonald's four or five shifts a week, thanks to Kingston's job-placement program. When I got expelled, I thought I might get fired, too. I guess they figured a delinquent like me would need a job, especially a punishing one that involves standing over a fryer for hours.
Jordyn has a cochlear implant. That's how she gets along so well with all of her boyfriends, and with everyone else. It also means she gets a better job at Mickey D's. Not only does she work the registers, she's also a _beverage specialist,_ making sure all the shakes and faux-Frappuccinos and sodas come out right. It's a stupid job, but it's better than fry girl. No one has to talk to the fry girl.
That's all I get to do. Fries go in, fries come out. Fries go in their little bags to go into bigger bags. Don't forget the salt. I almost got fired when I first got the job; the timer didn't have a light on it, and I might have burned a batch while I was distracted by Donovan's seven o'clock shadow.
Donovan Diaz. No one hates working at McDonald's more than Donovan. He has the worst job out of all of us: the drive-thru window. I can't quite imagine the stress of it, not fully, anyway. He stands in the window with his little headset on, grimacing at people who think they have to scream into the intercom. At least that's one thing I understand; Jordyn, too. People find out I'm deaf and think that yelling at me is the cure. I've seen Jordyn switch off her CI mid-conversation with that same pinched face. Rude, but less painful, I'm sure.
I have a front-row seat to all the drive-thru drama from my station. It's too bad Donovan faces away from me and I miss most of the details. I definitely stare too much. I've learned to read his body language as he leans on the window with those forearms of his—you know, the kind with perfectly smooth arm hairs—gesturing with his hands and stubby, chewed-up fingernails when he talks with customers, a slight bend in his knees. That's when I know he's on a good streak. The next time he turns around, he'll be smiling, a smile that has been known to pacify an enraged customer as soon as they see it shining through the window.
It's not like Donovan's ever done anything nice or cute for me. I don't know all that much about him, either. I don't know if he has any hobbies, or what his favorite movie is. I only know his work schedule and that he drives a 1997 purple Jetta. And he has wild black-blue hair that sticks straight up, like an Egon Schiele self-portrait.
There's just undeniably something about him. Maybe it's because he's never asked me stupid questions about my ears, or even cared that I can't hear him. We just exist in the same place at the same time and we're both fine.
He's hot, okay?
I'm not all that good at getting guys interested in me, let alone hearie guys. That's Jordyn. She gets whatever she wants, and she doesn't mind stepping on anyone to get it—and then flaunting it in their face.
—
Fries go in, fries come out. Fries go in, fries come out. Small, regular, large, extra-large. Fries go in, fries come out. Sweat drips down my back, my chest burning hot. I try not to scald my forearms when people slam into me, rushing between stations. Fries go in, fries come out. I am the siren call of McDonald's: smell the fries, you cannot resist. You want the fries. You need the fries. I hate the fries. I am the fries. Fries go in, fries come out.
After our shift I make myself a shake with extra Reese's Pieces mixed in. It's only about a billion degrees over the fryer, and all I want after work is an ice bath. The shake is the next best thing.
"I'm all hyped up! Wanna do something?" Jordyn flings her visor into her locker. All of her tight curly hair falls onto her shoulders.
"Seriously? I'm still on lockdown. Because of _you._ " I wouldn't go anyway. Treachery aside, work doesn't hype me up. It burns me out.
"Still? They painted over it. No one even remembers anymore."
My stomach turns over, a lump hardens in my throat. I figured they would patch over it, but to actually find out that no one remembers it? Needles at my skin. People have stopped talking about my grand departure already? I used six colors for that piece. It was a labor-intensive stencil job that I managed to get up in under twenty minutes. I thought the legend never dies, or some shit.
Jordyn takes forever to get her things together. She can't be that dense, that aloof. Asking me to hang out, like she never turned me in, like it wasn't a big deal. It was. It is! I used to be able to count on her. We had a deal, an understanding. I wouldn't judge the parade of boys that marched through her life, and she wouldn't judge my illicit afterschool activities. We were outcasts, and we leaned on each other. I thought we needed each other. Now, I need her to go. Far, far away. The last thing I need is for her to see what I have stashed in my locker. It's none of her business anymore.
I wish she would quit, and leave me to fry on my own. She could find a job anywhere. I'm trying not to resent Jordyn's CI, not to dwell on how easy she'd have it if she were mainstreamed, when my weeks at Finley have been so awful. People would love her immediately, since it takes no extra effort to accommodate her.
She gets both worlds. I'm totally and completely on my own.
"What's wrong?" she asks, reading my mind a little.
"Nothing," I sign one-handed, sipping the shake at the same time. _Nothing you'd understand anymore._
"Hey"—Jordyn straddles the bench and sits next to me—"I need to ask you something." She looks anxious, her eyes darting around the room. I know what she's going to say. She's going to ask if I'll forgive her, even though she's done nothing to earn it. At least she'll be making a better apology than just texting "srry." Like I'm not even worth buying a vowel.
"You aren't still interested in Donovan, are you?" she asks, and immediately looks down at the dirty tiled floor. I tap her, forcing her to look at me.
"What?" I scowl.
"I mean, I know you used to like him. But that was forever ago, wasn't it?"
"Why?" This is what she wanted to ask me? What does Donovan have to do with us? With what happened between us and why she snitched? That's what I want her to talk about, not some stupid boy drama. Even if I did see him first.
"I was thinking about asking him out. That's all." She stands back up and zips her jacket. "I mean, I waited for you to make a move. You never did. I assumed you were over him." I can't say anything, my hands are frozen in midair. Jordyn keeps on going. "I think he's into me."
"Oh, do you?"
"Yeah, and he just broke up with that girl, the one with the green hair. You know, she works the morning shift?"
"Sure," I snipe.
"So, do you mind?" The way she asks cuts at me; it's so obvious she doesn't actually want to know what my answer is. Like she doesn't even notice or care that I'm not at Kingston with her anymore. She's swept it all under some dingy rug as if it never happened, but I can still see the lump. Her phone, Jordyn's stupid phone, lights up and she starts texting. She waves it in the air and signs, "Later!"
"Yeah, later," I sign at her back, before flipping the bird at the closed door. And for the first time I'm glad I'm not at Kingston, glad I don't have to see her fake face every day. At least I can pretend I get paid to deal with her now. It wasn't enough that she got me kicked out of school, she had to take it a step further and go after Donovan.
Smoldering, I take the lid off the shake and chug the rest. The sweet coolness calms me, and I slowly begin to feel like myself again. I pop the lock open and slide it from the metal loop. I'm relieved she's finally gone. It's not that Jordyn would care if she knew what was in my locker, but I'm not going to take any more risks with her. Or anyone. I'd rather have everyone think that I put it all behind me. No more bragging, no more getting up. I'm not Julia.
I'm HERE.
I swing open my work locker. Inside are two backpacks: the one my mom checked and the one she didn't.
I leave most of it behind. I have no plans to go out writing tonight. I take out a Yakuza Yellow paint pen and my X-Acto knife kit. I'm pretty sure I can get away with my X-Acto, since Mee knows I use it to sharpen pencils. She doesn't know about the whole cutting-stencils bit. I'd like to keep it that way.
The lights flick off and on twice, my black bag hits the floor and one lone paint pen (Zombie Green) rolls across the tiles and collides with Donovan's boots.
We're frozen for a moment before I scramble to get my stuff back into my locker. By the time I look up at him, he's reading the fine print on the pen case. He shakes it, pops the lid off, and takes a deep whiff.
Donovan's lips are hard for me to read. I think it's equal parts his mumbling and my not being able to look at his mouth for too long without imagining, well, a lot of things. We don't talk much if Jordyn isn't around to interpret.
He caps the pen and holds it out. I can see that he's talking to me, but my heart is racing. _Get it together._ I hold both my hands up, more charades. Lately it feels like I'm some sort of caveman: _ME JULIA, YOU HOT BOY WORK DRIVE-THRU._ I bring my hands down through the air, nice and easy.
"Slow down," my hands say.
"Oh——-, sorry." He points to the marker with his other hand and gives me a thumbs-up. Then he points right at me.
"You," he mouths, "draw?" Donovan takes the marker and pretends to draw on his hand. At least he's trying. I place my hand out, palm down, and tilt it back and forth. I'm willing my lips to stay neutral but the corners are defying gravity.
"Cool." He hands me the pen and with a little wave goes off to start his shift.
There's only a million more things I wanted to say to him. Or, you know, have an actual conversation. But that's all I get with hearies. Hi. Bye. Thumbs up, thumbs down. Head nod. Friend, family, etc.
I talk with the Donovan that lives in my head the whole drive home.
"What're you doing here?" I ask coyly. "I thought you didn't have a shift today."
"I had to buy a new tire for Henry this week, so I had to pick up a night shift." His hands sign as deftly as a native speaker.
"Poor Henry. Give him my regards." Henry is the name I gave his car—not that he knows this.
"How's the new school?" he would ask.
"Rough going for now. I'm dying to get into this advanced art class, but it's full and the—"
"Advanced? I didn't know you were an artist."
"Oh. Well, you know, I'm okay. I guess."
"Can I see your stuff sometime?"
"Just drive under the overpass on Spring Road."
My eyes sting. That conversation would never happen, not even close. Who knows what he gets to talk with Jordyn about? A lot of things, I bet.
No tagging on a whim anymore, that's my new rule. But I need to feel like myself again; I need to be HERE. Screw it. _Thm! Thm! Thm!_ There it is: that heart-pounding, finger-numbing adrenaline that surges through my head and radiates through the rest of my body. _Let's do this._
I take Zombie Green out of my pocket. Normally I would use the yellow one, but I'm all about this pen now; it has a secret. This tag is a note, a journal entry. Today Donovan held this marker and asked me about art. I wasn't supposed to see him, and I did. He asked me about art, and I wanted to tell him everything, and since that's never happening, this will have to do.
I lace up my boots, pull on my hoodie, and hide the marker up my sleeve. I walk four blocks away from Lee, which should put enough distance between us, before I look around for the perfect spot.
I like to pretend I'm waiting for a bus when I'm scoping out a good place to write. I can't have it look like I'm wandering around without a purpose, but there's no bus stop here. The main street has more traffic than I would like, but tonight I want to feel the risk. There's a pizza place (closed), a drug store (closed), some mom-and-pop gift shop (closed), and the Dairy Barn (open). A car passes and a pedestrian-crossing sign is illuminated, all neon and yellow, in its headlights. _That's my spot._ The sign is directly across from Dairy Barn, but the drive-thru window faces the opposite street. It's a decent spot. I like pedestrian-crossing signs the best because of the little silhouetted people on them.
This pedestrian is Donovan, and he's saying my name. The art gods are on my side tonight; they know I need this and the roads are dead empty. I reach up and draw. When I'm tagging, the rest of the world blurs out. That can be really dangerous—another reason to keep it tight and work quickly. My arm knows what to do before my brain even tells it to. I have this weird connection between my brain and my limbs, kind of like when hearing people say they talk without thinking sometimes. I move without thinking. I close my lines.
I've drawn a word bubble, and inside: HERE
I add a few of Donovan's perfect arm hairs for good measure. I cap it and scram. I don't stand around admiring, I don't take an Instagram. I write and run.
W _HAM. THUNK._
That's me, getting hit in the head with a basketball, then hitting the floor. The ball is hurled at the back of my head, so I can't see it coming. Next they'll say I shouldn't be allowed to play. I feel like I'm back in middle school.
Let's get one thing straight: if I threw a basketball at the back of Kyle Fucking Stokers's head, he would hit the floor just as hard as I did. Deafness has nothing to do with it.
Gym is the one and only period I don't have my terp following me around, which makes it even worse. No one willing to play my version of charades with me, and there's no paper to write anything out on.
Casey is required to be in all my classes, including gym, but we struck a secret deal: I get some time without a chaperone, and she gets to pick up our lunches from somewhere less vile than the Finley cafeteria. Every time I have gym, I start to regret our arrangement, but then I see the processed meat slabs they serve at lunch and change my mind.
The floor is cold against my face. Nice floor, good floor. I must have really hit my head. I push against the nice, nice floor and roll onto my back. Ms. Ricker glares down at me, her whistle dangling over my face. She stands there, hands on her hips, doing nothing, while Yoga Pants comes running over to help. More gold stars for Yoga Pants. At least she looks more concerned for my well-being than Ricker, who is probably telling her to take me out back and shoot me.
I end up in the locker room, holding my thumping head in my hands. Yoga Pants takes a small bottle out of her gym bag and hands it to me: Extra Strength Tylenol.
"I—-headaches." She points to her head and squints.
"Thanks," I sign. I shake out two pills and gulp them down at the water fountain. Yoga Pants is talking to me, but the only word I manage to catch is _question._ She's skittish, pacing back and forth as she speaks. I reach out and place my hands on YP's rounded shoulders, and she finally stops talking. I point to my eyes, then to her mouth, and do my patented "slow down" gesture.
"Oh, oh! Sorry!" She actually uses the right sign for the word _sorry._ She looks at me straight on with her full, cherubic face. Her cheeks and heavy blond bangs remind me of that woman from Manet's painting _A Bar at the Folies-Bergère._ Pretentious, I know, but I can't help but see it. Paintings get stuck in my head in the same way I imagine songs get trapped in hearing people's ears.
"I ask you questions?" she says slowly, taking her time to mouth the words. I relent and nod my head. Right now I'll do anything so long as I don't have to go back into the gym.
"I'll make——like, yes-or-no ones. So you—-——nod for yes, shake for no, or whatever, okay?"
I just nod or whatever.
"Okay. So! Can you hear, like, anything?"
Shake.
"Whoa, nothing—all?"
Shake.
"You -----——-hearing-aid thingie?"
This is bigger than a yes-or-no question. I have hearing aids, but I hate them. They don't do anything but distract me, and I have awful tinnitus for hours after I take them out. I shake my head no.
"Was everyone deaf --——old school?"
I waggle my head, leaning toward yes. We had some hearing teachers, kids with cochlear implants, some people who were going deaf, some hard of hearing. A mixed bag. Not everyone who is deaf is profoundly so.
"Why did you leave?"
I look at my feet and pick at a cuticle. Should I tell her? Certainly not the whole story, maybe...
"Oh! Sorry"—with her fist on her chest she signs it again—"that's not a yes-or-no." I give her the benefit of the doubt. This is the first almost-conversation I've had at Finley with someone other than Casey, and it's all right. Even if her questions are the most basic hearie-meeting-a-Deafie-for-the-first-time ones. I point to myself.
"You...," she repeats.
I kick my foot out, like I'm kicking a ball.
"Kick?"
I nod and throw my thumb back over my shoulder like an umpire making a call at first base. See, I know some things about sports.
"Um..."
I kick and thumb again.
"You got kicked out?" Her ice-blue eyes grow wide. She's excited to get a bit of gossip from me.
Nod. I point to myself again and she follows along. The bell in the girls' locker room is so loud I can feel it in my chest. I sign "later" to her and then point to my wrist for extra clarity as the room fills up with girls rushing to get changed and get out.
Yoga Pants points to me, then to her head one last time. She doesn't talk when she signs, "You. Head." Thumbs up?
"Better _,"_ I sign back, swiping my chin. I assume she understands because the sign conveniently has a thumbs-up already built in.
—
"Good -----, Kyle. Great work." Ms. Ricker pats him on the back as he leaves the gym. _Great work?_ Was she not in the same class I was just in? That's some serious selective memory. I've had it up to _here_ with Kyle Fucking Stokers. I don't care how the game went, or how nice his eyebrows look. I'm standing here with my head throbbing from the ball he threw at me, and he gets an "attaboy"? I look over to Ricker and her eyes dart away from me.
"Hey!" I shout, not sure how loud I am.
"What?" Ms. Ricker scowls, confused. Good. I'm loud enough, then.
"He gets a pat on the back? What do I get?" The words scratch my throat as I yell them. The way he looks at me, like I'm a joke, takes my simmering blood up to a boil. He smirks, and one of his bros elbows him, chuckling.
"It was an accident." Kyle grins the fakest grin that ever graced his stupid face. "I'm sorry." He fakes the apology, so it's just genuine enough to be believable. I don't buy it for one second, but Ricker is obviously satisfied with his bullshit excuse. He almost winks at me on his way out of the gym.
"Don't walk away!" I shout, and chase after him. I grab at him and my fingers brush the back of his jacket. Yoga Pants steps between us, grasping my hands. I thought I was angry before, but this? Holding my hands still, trying to shut me up? I wouldn't dream of putting a hand over a hearing person's mouth while they're talking. I try to wiggle away but she's holding tight, her eyebrows arched with that all-too-familiar expression: pity. I am on fire, I am HERE. _Don't pity me._
"Let me go!" I scream and thrash until she releases my hands.
"Stop screaming!" Ricker finally butts in.
"You! Leave! Me—" I'm screaming and signing at the same time to everyone and no one in particular, until KFS leans in over Yoga Pants's shoulder.
"No—-—-understand you," he spits. "Get it? You're embarrassing yourself."
"Do————take——-to --- office?" Ricker separates us all. As far as I'm concerned, she's said the magic word. _Office._ God, what am I thinking? I can't handle getting kicked out of another school; I'm in enough shit as it is. Deep fucking breath. I back off.
"Okay——now get out——gym."
She hurries YP, Kyle, all of them out of the door. I lag behind in the gym entrance, my face flushed red and hot.
"Hey." Ms. Ricker hunches over slightly so our faces are on the same level. "I'm———wasn't—purpose. He said——-sorry. You're—-——-yourself—-favors." She doesn't look into my eyes as she talks to me, as if her ugly sneakers are more interesting than my face. "I——-it's great——you're————-school—-—-, but you're——-———try and fit in———-normal kids, okay? It'll make life————you." She pats me on the back, a stiff hard thump that moves me through the doors of the gym.
I seethe, my boiling blood turning to acid. I think about sprinting to my car, to the road, to the underpass, anywhere but this spot. And as I think about it my legs take over for me and start running, down the hall, through the cafeteria, my eyes seeing red.
Red flannel. Mini Mondrians woven all over his shirt. We collide and I almost fall to the ground for the second time today, but Mr. Katz steadies us both. He always seemed sort of gangly to me before, but he is able to catch me at full speed without toppling. A curl of black hair falls in his eyes as he studies my face for a moment. He exhales.
"Not a good day, huh?" he asks slowly enough for me to understand. No day here has been a good day. There have been days where I've slipped under the radar, but I wouldn't call those _good_ days. He motions for me to follow him and walks down the hall without looking back.
I follow. I imagine what we must look like, him gliding along like a proud trumpeter swan, me waddling along in his wake like some pissed-off goose. When we get to Room 105, he unlocks the door and holds it open for me.
This is the room I've been dying to get into ever since I landed at Finley.
There are no desks, just long tables set up around the perimeter of the room. In the center stand two big lamps on tripods, their light framing a carefully draped table. On top sit some empty wooden buckets, a teakettle, four apples, and a pear. Still life isn't really my thing, but after a month of bad English papers and full-contact basketball, I could turn this into a masterpiece. The walls are covered in sketches, some of the still life on the table, others of previous still-life arrangements, and random drawings, too. Comic pages, a drawing of Mr. Katz, whatever. Brushes are drying in big paint-stained coffee cans. A record player sits precariously close to the slop sink. You would think he would be more careful with vintage electronics. The whole room has that sharpened-colored-pencil smell—nothing is clean but everything is perfect.
Mr. Katz doesn't have a desk but a podium with shelving underneath it. He fishes out a yellow legal pad and starts writing. When he's finished, he holds it out to me. He's written:
_Stay and work awhile. It always helps me._
I take the yellow pad. The sunny paper shines up at me, saying this could be all right. I motion for his pen. It's one of those nice ones, the kind that feels heavy and expensive. I concentrate. Normally, I don't give a crap about my grammar. I got so sick of my old English teachers hammering away at us, saying we Deafies have to rise above the stereotypes, and I get it. I do. But I just don't care about commas and capitals and sentence structure. Just like I don't care about chemistry or U.S. history. I don't discriminate. I've never been one to set a good example anyway.
Still, I feel this pull, and a desire to impress him. I'm asking for a favor and I want to be understood. When it comes to art I don't like to be lazy. I take my time and write as clearly as I can.
_May I join your class?_
He scribbles a response, some excuse about it being the wrong time, and there's no room, and he wishes he could, and blah blah blah. I wave my hand to stop him and sign, "I know." He goes back to his first sentence and underlines the word _always._ Before I can respond Katz takes his pen, retrieves a tote bag from under the podium, and is at the door. He closes it behind him, gives a little wave through the window, and is gone.
He was right. How did he know that? That I would almost instantly feel better sitting here by myself, left alone to do what I do best. I decide to try my hand at the still life and take out my sketchbook. I don't feel right about helping myself to the supplies scattered around the room. I don't rack paint either, even though some writers would say that doesn't make me legit. I'm not a thief, I'm a vandal. That's why I work at Mickey D's: I buy my own supply.
I keep thinking about the pear: Why only one pear? Was there a sale on apples? Did he pick up the pear by mistake, thinking it was an apple? Or were the apples mistakes and he was sad to find out he got only one pear? I like this pear. I'll start there.
I begin sketching the base of the pear, where it's sitting. There is a nice little fold in the cloth underneath. Folds are tricky. I love the way they look, curvy and kind of sexy in a way. Which is probably why I'm not very good at drawing them yet. I'm better at hard lines and solid forms, like letters. My old art teacher told me I draw like a man. I've never forgiven him. I don't draw like anything, I draw like everything. I draw like me.
Folds in cloth and bumpy, organic-shaped pears make for a rocky start to my still-life adventure, but I'll be damned if I am going to walk out and give up this room. I'll get it right. The bucket in the background would have been no problem, all hard lines and wood. I fall in love with its little iron handle. Where do you even get a bucket like that anymore? I go back to shading in my pear. I'm not mad, not hungry, not anything. Just me in an empty room with my pencil and pad and pear.
I call it finished with about a minute to spare before either Mr. Katz comes back or the next class floods in, neither of which I want to be here for. I want to vanish like a ghost, savor this mood I'm in now. I carefully tear out my still life and pin it on the wall next to Mr. Katz's podium. It's from the last page in my sketchbook. His writing is in the corner, almost like a signature: _We'll see._
"Are you sure you don't want me to start coming to your gym class?" Casey follows me out to my car after school. Ricker must have told her what happened. I certainly didn't.
"I'm fine, don't worry about it."
"I do worry! Making some friends might help, too, you know." She's so genuine, I can't tell if she's making puppy-dog eyes or if it's just the magnification of her glasses.
"Don't have time for friends." And after Jordyn's dirty double-crossing sent me off to this place, don't want 'em, either. I'm meant to be an outsider. I highly doubt Banksy spends his weekends palling around with his BFF.
"Seriously, let me introduce you around to some of the kids in your lunch period. They aren't all bad."
I never thought I would be thankful to see Yoga Pants walking toward me, but here I am. I wave at her and smile. She barely nods and keeps walking. Shit, was I that harsh in gym? I wave at her again with a bit more enthusiasm. She heads over to us.
"This is my friend," I sign to Casey.
"Really. You're friends?" Casey asks YP without signing. YP looks over to me, her eyebrows screwed up and confused. I smile and nod, pleading with my eyes.
"Sure, why not." YP signs the word for _friend_.
I sign and Casey jumps at the chance to interpret for a peer: "Julia wants to know if you would like to go to Dairy Barn with her."
"Oh, Julia! That's your name?" YP blurts. _Come on now, play it cool, we're "friends," remember?_
"This is YP," I sign to Casey, introducing her.
"YP? What?" Casey signs with that puzzled look of hers.
"It's her name sign." Another lie, but whatever, it's a fine name sign.
"You gave her a name sign?" To YP: "She gave you a name sign?" _Crap._ Casey has been bugging me for one since I met her. I've kind of resisted on principle.
"What's a name sign?" YP asks us both.
"It's a sign for your name," Casey explains. "So you don't have to fingerspell your name every time you say it. You can't make up your own, you have to be given one by a Deaf person." Casey can't help but teach her.
"How do you sign yours?" YP asks Casey. I'm tapping my feet, trying to ride out the good feelings from drawing earlier. This is about to be a buzzkill.
"I don't have one yet." Casey looks so disappointed, and I feel the slightest pang of guilt. Look, I'm not going to just give a name sign to everyone who asks, as if it would make them some sort of official member of the Deaf Club. I wave to get YP's attention.
"My name sign is—" I'm signing to her, but I can see Casey translating.
I show her how to make the hand shape for my name. Mee gave it to me when I was little, and sadly it stuck. It goes along with her whole "my girls are precious jewels" thing.
"Let's go!" I point to the road, changing topics.
"Can we walk?" YP makes her fingers into little legs and wiggles them. "It's so nice out."
"Sure." I nod yes and head far away from Casey.
—
It's true, the weather is golden. It's one of those fall days where you start out wearing a coat but forget it on a bench or something because why would you have brought a coat, look how nice it is outside! We cross the street and head toward Dairy Barn.
We don't have a Dairy Barn in my neighborhood in Queens Village. I've never even gone to one before. Greenlawn has at least three of them. At home I'd just go to a bodega, but there aren't too many of those out here in the suburbs. I'm not exactly sure what the appeal of Dairy Barn is, but it seems like everyone goes and gets either chocolate milk or iced tea. I didn't even think about it when I asked YP. It's ingrained into the Finley subconscious: Want to hang out after school? Go to Dairy Barn, get iced teas.
"How's your head?" She points to the back of mine.
"Fine," I sign, mouthing the word for her. YP immediately copies me. This is going to be another long-ass beginner conversation. I take out my cell phone and mime for her number. I plug it into my contacts and text her.
> JULIA: Hey. Easier to chat if we text.
>
> YP: Cool!
>
> JULIA: Thanks for covering the "friend" thing.
>
> YP: What do you mean?
>
> JULIA: Just thanks for sayin that.
>
> YP: Yeah sure.
Both of us look at our screens, walking, neither of us really knowing what to say. She was nice enough in the locker room, but I'm not really sure we have all that much in common.
> JULIA: Your aspirin really saved my head.
>
> YP: Np - that sucked.
>
> JULIA: Kyle makes me ( ∩ )
>
> YP: Ha. Yea.
She looks up from her phone. Her shoulders droop as she exhales. What now? Something I said?
> JULIA: You OK?
>
> YP: Hes my ex.
I knew that Yoga Pants wasn't exactly the kind of girl who would run with outcasts like me. She could easily have been one of those thicker cheerleaders who holds the featherweights up in the air. She could be on the softball team, or not even play a sport, she's pretty enough to score a cute guy and be popular by association. However all that shit works.
But dating KFS? I was just starting to think she had more sense than that. He doesn't even try to hide his doucheness. Doesn't have to, I guess. It's a letdown, YP going for someone like that. The confession is a bummer, but she doesn't owe me anything.
> JULIA: His loss.
>
> YP: I guess.
>
> JULIA: I know.
She sighs and signs to me, pulling her hands far apart and then bringing them together like a book.
"Long story."
"OK," I sign. Probably best to drop it.
"Maybe I———learn—-alphabet. ———-help, right?" she asks.
> JULIA: U don't have to.
>
> YP: Why not?
>
> JULIA: ¯\\_(°_°)_/¯ Dunno. seems like a lot of work or whatever.
Every time someone offers to learn ASL, they bail. They realize that it's actually a whole language and give up when it gets hard. Suddenly I'm not so fascinating anymore and they move on to some other obsession. It makes me miss Jordyn, and Kingston for that matter. Just being able to talk without having to figure out some workaround. Jordyn and I would sit up and talk until my fingers hurt.
Dairy Barn is a funny place. It really looks like a little barn, painted red with a fake silo. There's nowhere to sit inside except for one stool for whoever's working there. The whole store is like a supermarket dairy aisle, except as a drive-thru. Donovan's nightmare, I bet. But they don't have an intercom: you pull up, tell the attendant what you want, and pay all in one go. I type out on my cell:
> TWO LARGE ICED TEAS
We walk up to the window and I hold up my phone and debit card. The man working at the window chuckles.
"Cat got your tongue?" he asks. It always baffles me when people think I'm just typing things out to be different. Or lazy. My new favorite is when they say it's my generation. _Damn millennials, never off their stupid phones!_ No, you ableist jerkwad. This is how I'm going to communicate with you.
I point to my ear and mouth the word _deaf._ This usually is enough to get my point across. His nose turns red, his eyes glance back and forth between me and my phone. He realizes he screwed up.
"Oh, sorry, sweetheart, -—-no idea."
I lift my arms as if to say, _Oh, well!_ YP looks mortified. I just wish he hadn't called me sweetheart.
"Here, it's on me," he says as he nervously hands us each a cup. Normally, I would complain and tell him not to treat me differently than any other customer, but hey, free drinks.
> YP: Whoa, he was pretty embarrassed huh?
>
> JULIA: Oh well! TB.
>
> YP: ??
>
> JULIA: Too bad
She gives me a funny look, but obliges anyway, and we "clink" Styrofoam cups. One sip of the famous iced tea and my mind explodes. Bye-bye, Red Bull—this stuff is like crack. It's insanely, sickly sweet, but you can still taste the lemon and tea flavors underneath. Unlike the stuff that comes from the school vending machine, this doesn't have a bit of sour aftertaste; it doesn't coat your tongue in that syrupy, chemical way. It's amazing. I'm so busy swooning over bliss-in-a-cup that I almost miss it: my tag on the crossing sign. My little love note to Donovan in Zombie Green.
Someone wrote over it.
It hasn't been buffed off or anything—someone's calling me out. They're dissing me.
My heart skydives into my stomach. So soon? I haven't seen any good graff around here, and already someone's trying to throw down? YP looks up at the sign with me and points.
"Huh,——-kind—cool, no?" I stop reading her lips. What is she even talking about? Cool? More like insulting. I'm not HERE? Where, then? Why not? Who are you to tell me what and where and who?
"Um...come on." She pulls me away from the sign and I let her, not wanting YP to catch on to me. Maybe it was some toy acting stupid. I'm jumping to conclusions. That's all it is. Some punk kid trying to step to my game, and making a fool out of themselves, frankly, because they turned my writing into a drippy mess.
As we head back to the school, I focus on the weather and the walk and the iced tea, concentrating on sipping and not chatting. It feels good to give my hands and mind a rest. They make reading lips look so easy on TV: every deaf character has absolutely perfect lip-reading superpowers. But in reality it's inaccurate, and _exhausting._ Not all of us are good at it. People don't get that.
"This is _your_ car?" YP dances over to Lee and puts her hands on the trunk. "It's amazing! I was————-whose it was. I———-—might be Mr. Katz's.—-into this———thing."
"Really?" I sign.
"Totally, it's so beautiful!" I guess she's talking loudly because two girls start watching us from across the parking lot.
"—-—-bad I don't have a car, I would—" One of the girls glares and snickers. They huddle closer, talking.
"Oh—" YP stops herself midsentence. "I should --- -----. See you in history." I place my hand on her shoulder as she's about to go and pull out my phone.
> JULIA: Don't worry...I'm used to it.
>
> YP: Theyre not laughing at *you*
>
>
All I can think about during dinner is what happened to my tag and what I should do about it. The takeout from Rajdhani's isn't much of a distraction. If some toy wants to come along and wreck my work, I'm just going to have to make it more challenging. It was my first tag in town and the paint was barely dry before it got done up. My chicken makhani is cold by the time I take another bite.
"I got a call this afternoon." Mee snaps me out of my fog.
"You haven't been...?" Ma looks shocked.
"No!" I sign with extra emphasis. Not anywhere the school would know about, anyway.
"No, no, this isn't about that." Mee looks over at Ma, then back at me. "What happened at gym class?" she asks, probably knowing full well what happened. Is this why she cracked and got us Indian again this week? She knows it's my comfort food, so she's trying to soften the blow of her planned confrontation? I guess the school finally figured out how to make a Video Relay Service call. I was kind of hoping they wouldn't, so my hearing teachers and Deaf parents would never, _ever_ communicate. I wonder who interpreted that call between an angry hearing teacher and a pair of Deaf parents. Did the terp take my side or theirs? Probably neither.
"Just some dude, mainstream-garbage stuff." I break eye contact and chew on a lukewarm slice of naan.
"Is that _all_?" Mee leans in, trying to squeeze info out of me like Mr. Howard. Do all adults use this tactic? They should just come out with what they know and let you apologize. Don't hand me a shovel and make me dig the grave deeper.
"I was upset, he was rude. They called you for that? I'm fine now. It's over."
"Casey says you yelled at a teacher," Mee continues. She's obviously very concerned. Her brown eyes droop and she brushes my hair off of my shoulder.
Ma places the clear plastic lid on top of her container of biryani, signaling she's ready for battle. "You did what?!" Ma signs. "You couldn't be peaceful and respectful, after everything that's happened?" Ma, a teacher herself, will always side with one. This is the worst thing for her to find out right now and it shows. Her hands, so similar to mine, sign with purpose. Every movement is sharp and swift. She chews on her bottom lip when she's especially angry. Like now.
"Ma, you don't understand!" Casey wasn't even there, and now she's calling my parents? This is way, _way_ out of bounds. Usually my mothers would see that, and they'd put an overstepping interpreter in her place. But of course, just as I suspected, they're using Casey as an extra set of watchful eyes on their little vandal. No wonder they didn't go through the system to find an interpreter. They have Casey in their pocket. Why should they trust me? I'm only their daughter.
"I do! I do understand," Ma says. "You knew going to a mainstream school would be difficult, but you brought this experience on yourself. You don't get to act out when things don't go your way."
"That's not what happened!"
"What am I going to do with you? We work so hard to—"
"Cara." Thankfully, Mee cuts her off. I've heard this speech way too many times since leaving Kingston. "Even though she's made a mistake, that doesn't mean she can't feel upset."
"There is no excuse for yelling at a teacher!"
I stand up, ready to fight. Why does it feel like I'm always fighting lately? But Ma's eyes are wide and alert; this isn't one of those nights where she's had a few glasses of wine and I can make a dent in her argument. Ma is on an unshakable streak, standing up for teachers everywhere. She won't hear me or Mee. Debating her will only make things worse. Once again, I'm forced to surrender.
"You're right. Won't happen again." I shove my chair against the table and head for the basement.
—
How am I supposed to unpack this day? Sweating, I yank my hoodie over my head and hurl it across the room. I start turning on my lamps one by one. I hate the harsh overhead fluorescent lighting we have in the basement. It was never meant to be a workspace, only storage. It's not one of those nice finished, hang-out-with-your-girlfriends, have-a-sleepover-type basements. Even so, I keep begging my parents to let me turn it into my bedroom. I spend 90 percent of my home life down here anyway.
I collected the lamps from some thrift shops. One is this old cracked faux-Tiffany glass thing; another is a bunch of illuminated plastic balloons, a little clown holding the strings. And a huge purple lava lamp, the catch of a century that Jordyn and I found in someone's trash. I remember having to carry it for thirty blocks because Jordyn didn't feel like waiting for the LIRR. She gabbed and gabbed the whole way home, but I can't for the life of me remember what about. I do know that I couldn't really respond with more than a nod, because this beast of a lamp is heavy as shit. I don't turn that one on. I don't know if I ever will again.
I sink down into my work chair. Mee bought it for me ages ago. It's my command center. I do all my work from here.
She said she doesn't care if I get paint or ink all over it, that it was mine to make use of. When I got expelled, I only felt guilty about one thing: letting her down. Mee's never told me not to draw or pursue art. Even when she doesn't understand my work, she tells me she loves it. When she found out what I did, though, a ten-foot-tall slab of concrete went up between us, and I dare not paint on it. Before, I kind of thought she might even like the fact that I was going public, but yeah, not so much. The boots were the one sign that forgiveness is possible.
I click on the last lamp that hangs over where I sit. One of those round crinkly paper lanterns, big and yellow, it shines over me like the sun. It's the only place I actually appreciate a spotlight, where I don't wish some fog would roll in and envelop me in obscurity.
I take out my X-Actos and a charcoal pencil from my lucky mug with all the strawberries on it and get to sharpening. I like really hard pencils and charcoals—5H is primo. I'll settle for HB if I have to, but anything softer than that, and I can only draw for sixty seconds before I'm sharpening again. Soft pencils just don't last, and when you buy your own supply, you need that shit to get you through more than one drawing.
As I hack away at the pencil, every crappy thing that has happened to me the past few weeks replays in my head. Over and over.
Getting expelled.
_Shunk._
Getting expelled over art that no one even remembers.
_Shunk._
Being assigned to the world's most annoying terp.
_Shunk._
Having to need a terp at all, at a stupid mainstream school.
_Shunk._
A school where people think I'm an idiot.
_Shunk._
Where I can barely talk to anyone at all.
And when I do they can't understand me.
And give me shit for it.
And throw things at me.
And get away with it.
And I get the blame.
_Shunk._
_Shunk._
_Shunk._
_Shunk._
_Shunk._
_SHIT._
Blood flows from my left index finger. I grab my hand tightly, double over in pain, and cry out. I slide off of my chair onto the floor, pencil shavings clinging to my black leggings. Sucking air in through my teeth, I let the pressure off my finger and assess the damage: it's still attached but it's dripping red. Holy shit, it felt like I cut the damn thing off. I stick my finger in my mouth and it fills with that weird metallic taste. Taking deep breaths through my nostrils, I try to calm myself down. I'm shaking. Blood and I don't really get along.
When my finger quits throbbing against my tongue, I take it out for a closer inspection. So much blood for such a minor cut. I might have a tiny scar, but it's really not too bad. In the open air it starts to sting and pulse again. I wrap the finger up as tight as I can in the hem of my shirt, willing the pain to go away. I thought I could handle it. I'm supposed to not care.
Tears find their way into the corners of my eyes but I refuse to let them fall. I bury my face in the cushion of my beautiful armchair, my command center, and scream. Over and over, my throat vibrating and crackling with fire. Nobody comes to see what's wrong. Nobody can hear me.
It's nine billion degrees standing over the fryer this morning. The Sunday morning shift might be my very favorite, though. I can tell the difference between french-fry smell and hash-brown smell, and I prefer the latter. It's a bit after eight and the actual work hasn't picked up speed yet, but the fryers are running full blast.
I'm going to Urban Café after my shift to make a paint order. I've had enough drama; it's time to get to work. I'm debating between hitting up that spot near the Little League field and doing a big pen piece in one of the bathroom stalls in the Greenlawn Diner. I don't feel ready for the overpass yet. I'll probably go with the field, more public, more—
There's a little pinch on my waist and I snap back to reality. Donovan flashes me a smile and points to my blinking timer. Whoops. I pull the browns from the oil and hook the baskets up so they can drip for a second.
"What's up?" he signs.
"Nice signs." I shake my hand, impressed.
"Jordyn taught me." Unfortunately, I read his perfect lips. _Thanks, Jordyn._
"More, please," I sign, my fingertips touching.
"Kiss you? Damn, girl." He plants a big one on my cheek and heads past me to his station. Suddenly it's fifteen billion degrees and I am but a puddle on the floor, no bones, just thoughts and feelings. He looks back over his shoulder at me and chuckles. I assume it's because the expression on my face must look something like a cross between HOLY FREAKING UNICORNS RIDING ON MAGICAL GLITTERING RAINBOWS and DEAD. And to put the cherry on top of my melted Mickey D's sundae of feelings, he actually winks at me. The bastard.
This is the sign for _more:_
This is the sign for _kissing:_
I can see why he got confused—the signs have the same hand shapes and all—but it's not really a hard one to mess up in terms of context. _What's up? Kiss me!_ Doesn't that sound like a bit of a leap? Jesus. I'm a brave person, but Jesus.
Speaking of Him, church must have just let out, because the lines start to grow, drive-thru gets hectic, and orders start flying up on the screen. _Fry! Fry! Fry!_ It's pretty cute when parents bring their kids after church, all pouty in their patent-leather shoes and polo shirts, exhausted moms and dads bribing them to behave during Mass so afterwards the kids can pray in their molded plastic pews to the gods of cheap toys and Happy Meals and french fries, amen.
I fill up a bunch of small fry bags, knowing how many Happy Meals are about to get ordered, and it hits me—of course he knows the sign for _kissing_. Why would Jordyn teach him anything else? The thought of them making out leaves a bad taste in my mouth. I thank God she isn't working today.
Something changes. Donovan raises his shoulders, his arms tense up. He starts pacing in what little space the windowed area allows. Checking the monitor, shuffling around the take-out bags. Oh, crap. Drive-thru drama.
The first thing I like to do when I notice things going downhill is put a big batch of fries down. The gush of hot steam, millions of bubbles rushing to the surface, means business. I know if an order has gone wrong or if someone is acting hellacious, Donovan likes to upgrade their fries, so it's best to overprepare at this point. Twenty billion degrees.
The next car pulls up to the window. It's a minivan with some sort of Little League team crammed in the back like sardines. The owner is pissed. She keeps checking her watch and screaming into the backseat. She also has one of those Bluetooth receivers wedged in her ear. How do hearies listen to so many things at once? It must be maddening. She peeks above her oversized sunglasses and snaps at Donovan. He's trying to calm her down. He hasn't even lost his temper yet when the manager, Arnold, comes and taps me on my shoulder.
"Bring——bagsover—-mepleasewouldja?" He hands me five stuffed paper bags and dashes back to the office, talking into his own headset. He talks way too damn fast. I pull the fries out of the oil with one hand and make my way to the window. I place the bags between Donovan and Mrs. Soccer-hockey-tee-ball Mom.
"Excuse me! No!" She must be yelling. She jerks away from the window.
"I don't want Muslim hands touching my food! Or the children's." She turns and wraps her arms around the boy in the passenger seat, who looks as confused as I do. Some things I wish I didn't lip-read. Just because I'm brown doesn't mean I'm Muslim. Not that it should even matter. Take your hate to Chik-fil-A. My only god is Banksy. I hold my hands up and back away. Donovan offers his fry-upgrade tranquilizer and it seems to work. We head over to my station. Out of sight, he pretends to refill the bags. He motions for me to add the extra fries.
I reach for one of the colder bags—as if she deserves anything for free; why should she get the good stuff?—Donovan holds the take-out bag open, and I drop the fries in, but the Band-Aid from my X-Acto slash slides in with them. Sweat and the heat, I guess. We both look up at each other, stunned.
"Screw her," he says, and takes the bag to the window.
I can't believe we served that lady a no. 4 with a side of scabs. I guess she was too rushed to notice, because she never came back. I try to forget her, and what she said. I'm thankful I don't have to hear her voice ringing in my head. All those kids, though. I feel bad for them, I wish they didn't have to hear that shit. No one should.
After my shift I get both backpacks from my locker; the black bag needs a resupply. This means it will spend a night under the driver's seat and some time hanging in my school locker. It'll be a tricky week.
Donovan comes into the locker room, already halfway through unbuttoning his shirt. _Don't look. Don't look._ He straddles the bench next to my locker, practically defining the word _swag._ Pretending to ignore him, I zip up my hoodie. He waves up at me. Okay, fine, you have my attention.
"Sorry——-that woman." He points over his shoulder.
"It's nothing." I wave the thought out of the air. "She was psycho." He understands because I use the classic pointer finger making circles near my temple motion.
"Ha-ha! Yeah," he laughs. I win. "So, you and Jordyn are tight, right?"
"Yeah..." Moment killed. Jordyn finds a way to ruin everything.
"I———take her————-cool, you know, like a date?" Can he tell how hard I'm scowling on the inside? I act like I don't know what he's asking. I wish I didn't.
"Oh, sorry." He looks at the ground. He's mad that I can't understand him. At least he doesn't realize I'm faking it. There's just no way I'm helping set them up.
I grab my black bag, fling it over my shoulder, and bang my locker shut. I wave on my way out, but he stops me. He points to my bag, and I feel like I have to get my eyes checked. He can't possibly be gesturing the words _spray paint_. There's no way.
"What?" I keep pretending that I don't understand. He points to my bag, and once again crooks his index finger and moves it back and forth in front of me, spraying invisible paint in the air.
"No." I lift my shoulders, shake my head. _Who, me?_ Of course not. Donovan tilts his head and looks at me and my black bag with more skepticism than Casey did when I told her I had friends.
"But—" He points out all the paint on my hoodie. Grabbing at the cuffs, he runs his thumbs along the stained ribbing. I yank my hands away, as much as it pains me to. Crap, I've got to get rid of it.
"See you later!" I sign with a smile, and hurry past him. I never thought I'd be running from Donovan, but even he can't know about this.
'Black bag stashed underneath the passenger seat, I pull into Urban Café's parking lot. It's closer to school, far from my home base. They love me in Urban Café. There, I'm this polite little deaf girl who comes in for treats and to write her school papers. In my head I call myself Melissa, bat my eyelashes, and wear my old hearing aids (turned off) for extra it's-rude-to-stare points.
It's not a cozy place. The lights are too bright and the floor is white tile. There isn't a little section with couches or armchairs, no fake library on the wall. All they have is tables, chrome chairs, and the counter. I imagine it was a frozen-yogurt place that folded, and the new owners couldn't be bothered to change the vibe. Anyway, the hot chocolate comes with homemade marshmallows, and that's what keeps me coming back.
I take my drink to my seat and fire up my laptop. I check the Stencil Bomb forums to see if someone was foolish enough to post a picture of their lame diss on my tag. There's a section for Long Island, but I get bored after combing through page after page of weak tags and lazy work. Amateurs posting hurried, dripping hate-paints. I can't stand them.
I log in. I have to post something. This is so unacceptable.
> H3R3: This shit gives us bad names. Why you think everyone want to lock up writers and artists? Because of people like you. What you adding to the world by doing this?? Nothing. Keep your hate out of our game. ELEVATE. Get on our level, or get out.
I log into my Hush _mail_ account and there's nothing new. This is the sort of thing you have to do to avoid getting busted by someone worse than your principal. Free Wi-Fi, incognito mode, an encrypted email address to make paint orders from, and a place—
_Crap._ I used to ship paint to the mailroom at Kingston. There was this work-study college guy that stuffed envelopes who I made friends with. Okay, I made out with him. A few times. Maybe more than a few times. He would set aside my boxes, and I would come grab them at the end of the day when the halls were chaotic. It was Jordyn's idea. She would hang out in the hall and get a kick out of my attempts to flirt. Once, after I picked up an abnormally large order, she told me I looked like a horrible kisser.
I think he knew what was in them, but he never said anything about it. So maybe I'm not such a bad kisser after all. Or he really liked me. Maybe both. It doesn't matter, because I can't show my face _there_ anymore.
I have to order paint online because I'd get carded in person. Before the whole mailroom scheme, I would just send it to my house. My moms wouldn't even think to ask what I was ordering, and if they did, I'd show them. "Working on a project," I'd say, and they wouldn't even question it.
I decided it was a bad idea to keep sending it to my house when I read on one of the forums that a kid got pinched that way. I was lucky that Mail Boy turned out to be pretty cute.
_Now_ what the hell am I gonna do? I can't mail paint to McDonald's. I'm not going to start racking; one charge on my rap sheet is enough, thank you very much. I consider if I'm old enough to open a P.O. box.
This is why my grades are tanking. After all the effort I put into my career, there isn't much energy left over for all this reproductive-system-of-a-frog junk. Who even needs to know about that?
What a bust. I have only three cans left and the paint levels are dangerously low.
> YP: I'm so bored.
>
> JULIA: Whats up?
>
> YP: Nothing to do on sundays not even good TV
>
> JULIA: Im @ Urban Cafe
>
> YP: Be there in 10
I don't know why I invited her. It just happened.
—
There's a tap on my shoulder and YP practically skips into the seat next to me. Her nose is pink from the cold, and it only adds to the cheeriness of her face. She grins and looks at her watch.
"Eight minutes --- fifty——-seconds! A—-record!"
"You ran here?" I can see she's panting a bit.
"Yep! Me, go, O R D E R," she signs, fingerspelling, and bounces off to the counter. This can't be her first coffee of the day. I take a minute to log out of everything, erase my history, clear my cache, and close my laptop.
Yoga Pants—today's pair are a dark shade of blue—heads back to our table and carefully puts her mug down. I was expecting her to get some sort of soy-caramel-extra-whip-extra-foam-type drink but no, it's plain black coffee. Before she starts fumbling through her signs again, I wave my phone, indicating we should text.
"Oh! Totally." She pulls out her cell and all her bubbles burst at once. Staring at the screen, unblinking, her eyes start to glaze over before little pools of tears well up in the corners.
"You OK?" I fingerspell for her.
"Yeah, 'scuse me for a sec." She puts her phone facedown on the table and heads to the bathroom.
What the hell happened? Her phone taunts me from her side of the table. I can't imagine anything that would dull her sparkle that fast, unless they were gonna close down Claire's, or Taylor Swift decided to quit music, or someone was kidnapped and texting for ransom, or someone, you know, died. Yikes. I reach for the phone. _No, no, I can't. I'm no snoop, I'm not going to just—_
> KATIE: Your not bad at cheer.
>
> just gross in the uniform now.
>
> No 1 wants 2 see that.
I try to unlock her phone to get to the rest of the thread, but I don't know her passcode. It's not 1234. That one text says enough, though. I know what it's like to get a text like that. I get them in real time, real life, every day at Finley. I want to get that bitch's number and text back, tell her to fuck off, shove it, and die in a fire for insulting someone like that. But I don't. YP shuts the bathroom door, and I slip her phone more or less back where it was.
"Sorry." YP rubs her fist on her chest. I flip her phone faceup and she looks deep into her mug.
"D E L E T E," I spell, and point to the phone. She looks up at me, angry. She could rightfully blow her top over my snooping, but she doesn't. She opens her newly glossed lips to say something, but reconsiders and shakes her head. I fingerspell again, "D E L E T E."
"Why? Damage————-done." She pushes the phone away from her. I pull out mine and start typing. Her phone vibrates on the table.
> JULIA: Words are only words.
>
> JULIA: People speak words about me all the time. You kno what??
>
> JULIA: I can't hear them so I let them say whatever.
>
> JULIA: The words dont make you.
>
> YOU make you.
She rolls her eyes, but I notice the slightest upturn in the corner of her lips. She points at my laptop, changing the subject.
> YP: What were you working on?
>
> JULIA: Dumb class project
>
> YP: Mr. Katzs artist report?
>
> JULIA: No! I wish. not in his class.
>
> YP: Why not!!?!? Youre so good!
>
> JULIA: How do you know?
>
> YP: Your car!
>
> JULIA: Oh! No room in his class for me
>
> YP: Thats super unfair.
>
> JULIA: RIGHT? Its the only class I really wanna take
>
> YP: jerks. they could like totally pull up an extra dang chair.
>
> JULIA: Why cant i get one good part to my day, you know?
>
> JULIA: Makes me crazy!
>
>
>
> YP: Not your fault you came to school late.
>
> JULIA: Well...
Her hands cover her mouth, eyes wide. I assume she's apologizing but I can't tell. You can't read lips when they're hidden. I pull her hands away from her face.
"Sorry!" she signs. "I———-forgot—-—-E X P E L L E D."
"Doesn't matter," I sign. It's cute how flustered she is, like I'm embarrassed about getting expelled. I'm not, but I'm sure she would be mortified if it happened to her.
I bet we look so perfectly normal, two cute girls drinking their hot beverages, texting away, not even looking at each other. We could be anyone, not some deaf chick and the girl who took pity on her. Just BFFs.
But we aren't, and I can't go spilling the beans to anyone. Yes, everyone at Kingston knew why I got kicked out, and no one was surprised. Jordyn wasn't the only person who knew about my penchant for spray paint, but she was the only person who snitched. Everyone knew I was the girl with the spray paint. The only clue about that part of my life now is Lee, and I guess my hoodie, but I'm taking care of that tonight.
> JULIA: Feel better?
I text her as we start bussing our table. She's bouncing again.
> YP: Little bit. Gonna jog back, thatll help
>
> JULIA: I can drive.
>
> YP: No way...I need it.
She pulls her sweater down around her hips. I don't get it, the nasty text, the jogging. YP isn't skinny, but she's not gross or ugly or whatever that text said. She's downright pretty. Shiny hair, blue eyes...So she's not skinny. So what?
> JULIA: nah, youre a beauty.
>
>
>
> YP: Nothing big is beautiful
After YP jogs off, I drive around for a while in my paint-stained hoodie. One last night together, gotta make it count.
I considered tagging another street sign, drawing a hoodie on Mr. Silhouette in tribute. It seems too small now. Too small, and too easy for someone to retaliate. I head over to the park where the Little Leaguers must play in the spring.
I've been eyeing the backs of the scoreboards for a month. There's one in particular that's low to the ground, so I wouldn't need to climb up anything. Definitely not a heaven piece, but I don't feel the need to put my life on the line. Not yet anyway. The field is empty at dusk, and it helps that it's starting to get really cold outside. I probably won't have to worry about some kid walking up on me.
Dusk isn't the prime hour for writing. The farther away I park, the longer it will take me to get back to the field. By then it will be darker. I take my ID out of my wallet and put it in my front pocket, leaving my phone and the wallet in the glove compartment. My car key and my house key go under the inserts in my boots. I know how noisy keys can be. _[Keys jingling]_ is in, like, every closed-captioned movie ever.
My black bag is not your standard L.L. Bean or whatever kind of backpack. It _looks_ like one, nothing fancy, just a plain black backpack. But on the inside, I sewed in a false back with a zipper you can only access once it's open, and even then, it's hard to spot.
On the _real_ inside there are holsters for six loaded cannons. Though after what happened at Kingston, I never carry more than three cans on me. The elastic holsters ensure the cans don't jostle around if I have to run. It's hard enough running with a backpack, let alone one with a bunch of junk bouncing around inside.
I have a speckled notebook in the front compartment, and a few school flyers. Nothing heavy. That's all I bring. I've debated wearing my old hearing aids so I could better sense someone approaching, but I refuse. I get through every day without them; I'm not going to "cheat" and put them on when it's convenient.
The sky darkens and the chill in the air intensifies. I'm not freezing, but I'm hardly comfortable. I wish I had a heavier coat instead of just the hoodie, but the cold air will force me to work quickly. Get in, get out, get back to my nice warm Lee.
What am I gonna say? What goes into this throwie? My last tag was for Donovan; what should this one remind me of? I never throw up a meaningless piece. It's always attached to a memory, something I'll recall every time I see it. It isn't about seeing my art on a wall. It's about putting a feeling out into the world. It's communication, a release.
Maybe I'll paint my new tag, my new name, my new place in this hood. All done up nice and clean and big. Big. I can't stop thinking about YP. About big beautiful things.
What did she mean, "nothing big is beautiful"? I pull the cords in my hood taut. What a dumb thing to say. People will always be jerks. Always. Doesn't mean mountains aren't beautiful. Or redwood trees, or oceans, or—
I've made it up to the waist-high chain-link fence in the outfield and suddenly I'm not so cold. The kindling catches; the fire's coming.
I thought the back of the scoreboard would be a slam, too public, standing right in the middle of the field. But it casts a large shadow in my direction, and shadows are a lifesaver. Stepping into its dusky protection, I disappear. I slide my bag around to my chest and open the inner compartment. The new latex gloves don't keep my fingers warm, but I'll be sweating in no time. I grab the first can. Bruised Gray, a faded purple color. It looks handsome against the dark green of the scoreboard.
I start outlining the basic shape first. My arm works diligently. It must have had these plans for a while, filed away somewhere deep, because the shape almost draws itself.
I move as quickly as I can. I've practiced these techniques so many times, they're second nature. As soon as I finish the outline I start filling it in with the same color. It's important to be fast, but how you fill in shapes distinguishes the toys from the real writers.
It's getting even darker out, so I don't hesitate moving on to the next step. My heart races along with the passing minutes; beads of moisture form and drip down my spine. I holster the gray back in my bag. Next up: Cyanide Blue outlines. Normally, the outline stage moves the quickest for me. It's pretty much a repeat of step one, except this time I add more detail. This isn't just letters, though; it needs more volume, texture. I want to make this big thing beautiful.
Capped and holstered, I'm ready for the last step. I unsheathe Siamese Sesame from the bag and get to it. I wish I had another color for this bit. Some bright white would really make it stand out, but the dusty beige will have to work. It's all I have. I need a really tight stream of paint to do the final details. I dig through my bag in the dark, trying to find my stencil cap. My fingers wrap around the little piece of plastic when the lights around the perimeter of the field flicker on.
I slam my body into the scoreboard, grateful only for the longer shadow it's casting. _Shit shit shit._ I was so close to finishing. A car or two passes. I go from flames to frozen stiff. I should be running to my car. I should be driving home. _Legs! Why aren't you running?!_
No one's coming. I peer around the board onto the diamond: empty. Maybe the lights are on a timer, I reason. _Finish it, finish it. You're so close._
I take in a deep breath and hold it. It's up to my lungs how much time I have left. It's my own timer system. I snap the stencil cap in place. Final touches demand the finest lines. My arm flies across the board. A tan mist trails behind my hand. Each puff adds a detail, a little touch that brings the whole thing to life.
My lungs are shredding. Tension rips through my chest. I run my gloved finger through the wet paint on the last little beige circle. My work looks right back at me. It's the biggest, most beautiful piece I've ever done. My gift to Greenlawn. You're welcome, Universe.
I can't hold it anymore. I exhale, coughing and dizzy. The paint goes back into my bag. I zip the inside zipper and then the outer one. I'm an action hero, walking away from an explosion. I'm so awesome I don't even look back.
I ditch the used gloves in a garbage can and walk a few more blocks over to the Walgreens parking lot. They have these big clothing donation bins there. All I'm wearing underneath the hoodie is my ratty old Wonder Woman T-shirt. I didn't think about the weather. What a day to go braless. Doesn't matter; I'll freeze for a few more blocks. I finished my piece and it was worth it. One day I'll go to L.A. and paint huge murals with Retna and never freeze again.
Of course Donovan noticed my paint-stained hoodie, but it doesn't explain how he knew I had paint in my bag. I keep my locker locked my whole shift. How could he go snooping through my stuff? My black bag doesn't have a single drip on the outside. He'll forget about it. A few more make-out sessions with Jordyn, and Donovan won't even remember I exist. He won't get to see this hoodie again as a reminder, that's for sure.
I pull the handle on the donation bin and the hinged drawer swings opens with no resistance. I squeeze the hoodie close to me, the chemical smell of fresh paint lingering. I drop it into the drawer, and it slides into the bin like it can't wait to get away from me. I hope it keeps someone else warm and safe. The thing is full of good juju, as Mee would say. I stand by the bin for a moment and sign, "Thank you." My chest feels tight again, but I'm not out of breath. I'm not going to cry. It's a piece of clothing. No crying over clothes.
A crack of light shines from the doorway onto my bed. After a moment or two it grows wider and Mee stands there, silhouetted. She holds out a tub of coconut oil for me to take as she sits on the edge of my bed. I slide to the floor and sit at her feet. She reaches into the jar and warms up a chunk of the oil between her palms. I swivel around to face her so we can talk. Mee has this perfect smell about her. Maybe it's biology and we're all programmed to love the way our moms smell. Sandalwood and sage: she's always burning something in little trays or pots. Her acupuncture studio is full of these smells, too. They cling to her, following her from room to room. But above all, she smells like sweet coconut oil, and it's my favorite smell in the whole world.
"Wanna talk?" she signs slowly, hands glistening with oil. I stiffen and sit up. She saves this phrase for when I've messed up.
"What happened?" I throw my pointer fingers down.
"Nothing, nothing. Nothing bad." She rubs the oil into my hair, starting at the scalp and working her way to the ends. "Things calming down yet?" she asks.
"A little. It's fine." I tell her what she wants to hear.
"Come on, really?"
"Yes."
"Well, I'm—" She looks over my head at the door. "I'm proud of you. I know it hasn't been easy." Mee braids my hair once it's saturated. I can treat my own hair, but there's something about the way she does it. My hair is glossy for a whole week afterwards. When I do it myself, I just end up staining all of my clothes.
I want to wrap my arms around her. I've been waiting for her to come back, for the wall to crumble down. It's getting chipped away, but I don't know what's doing the chipping. She isn't stubborn like me or Ma. Can't hold a grudge, always ready to move on. To forgive. All I manage is a shrug.
"Having to give up your art, that must be the worst of it." _Gulp._ "We should look into something for you to work on. Is there an art club? What about sets for a play? Or—"
"Yeah, maybe." I cut her off.
"Chin up, my jewel, this is only a valley." She drapes a towel across my pillow before she goes, knowing I would forget and stain the pillowcase when I sleep. Mee tells me to get some rest, kisses her finger, and touches it to my forehead. I fall back onto my pillow as she shuts the door. Taking her smell and the light with her.
—
I can't sleep. I want to sleep, but my eyes won't stay closed. They adjust to the darkness slowly, making it twice as hard to turn off my mind. My room seems so different in the dark. Cloaked in shadow, everything appears colorless and dull. I want to run down to my car, get my black bag from under the seat, and spray over every surface with the colors from my whale piece.
I blink my eyes and I see it, swimming in space, watching over me like some sort of spray-painted spirit. I sit up. There's no way I'm getting to sleep right now. I have to draw, I have to make something, to keep going. I don't want this moment to end. And who knows when I'll get to do something this big again?
Luckily, I have plenty of sketching supplies in my room. I just have to find them. I slip out of bed and a jolt of pain shoots up my leg. The culprit gleams in the moonlight: Jordyn's selfie stick. Her stupid fucking selfie stick that she insisted we buy from Chinatown. She spent all day waving it around, taking picture after picture, as if we didn't grow up in the city. I hate looking like a tourist.
Afterwards, she came over and binge-watched a bajillion hours of _Gossip Girl_ reruns while I sketched in my B-book. I guess the novelty of the selfie stick wore off quickly, considering it's still here, lying in wait for the right moment to sneak up on me and bite. Well, this is the last time she hurts me.
I shuffle around in the dark, looking for any trace of Jordyn's presence. Anything left over from when I thought having a best friend meant something. Something big. But now I know better. It's better to keep to myself, not to trust anyone, not to care. Jordyn was only thinking of herself when she got me expelled, and now it's time for _me_ to be the selfish one.
I find a pair of her hoop earrings and the old phone case that I tagged for her before she upgraded. I take those and the selfie stick and stuff them all in my empty Doc Martens shoe box. I remember her hoodie out there in my car, and I can't stand the thought of it being with Lee one more second. The hallway is dark and my parents are asleep, so I seize the opportunity to go and get it. I don't even bother putting shoes on. I want this finished.
On the way back up the stairs I spot one last friendship artifact, a photo-booth strip we took in the lobby of a movie theater. I don't even remember what movie we saw, but we were so excited to try out those fancy new closed-caption glasses. It was incredible. The glasses made us feel like we had a superpower: finally we could go to any movie, at any time, and not have to worry about whether they were going to use captions or subtitles. We looked like futuristic sci-fi nerd-bots, but it was amazing. We just had to take some pictures with the glasses on before we returned them. I swipe the strip and head upstairs.
Everything goes into the box. I take a charcoal pencil from my dresser and mark every side with a giant _X._ All my ex–best friend junk goes into the X-box. I place the lid on top and it feels like I can finally put her out of my mind. She's just some girl now, like anyone else. It's done. We're over. No more besties, no more friends.
Finley dominates the horizon as I pull onto Taylor Street. The auditorium is unmissable. It looks a lot like a whale, huge and blue, with the right side angling upward like a tail. I tried to capture a hint of it in my mural.
All the little fishy cars pull into the parking lot for another day of sink-or-swim. I can't tell which I'm doing anymore.
I didn't have time to do my history homework last night, so I decide to nap in my car until 9:15, when second period starts. I'll tell Casey I had car trouble, hit traffic, anything. I recline the seat and catch a glimpse of YP in my rearview, crying across the parking lot. It makes me feel nervous. I don't want to see her this upset, but I'm not trying to butt in. It's not my business. I can see her shoulders heave forward. Some other girl walks right past YP, not even bothering to stop and see what's wrong. YP doesn't let up, and my stomach turns. _Fine._ I lace up my boots, grab my stuff, and head over.
She's in some sort of argument with Kyle Fucking Stokers. _Shocker._ I hang back and try to get a feel for what's going on, but they're talking too fast and I can only see their profiles. I walk around the next row of cars to get a better angle. There's a lamppost there that I "casually" stand behind.
Snow starts to fall and creates a little halo around YP's blond crown. Her legs must be freezing in her signature yoga pants, but her feet are probably toasty in those sheepskin-lined boots. How could anyone fight with someone crying so cutely? But there he is, face red and scowling.
"...not my problem," I catch him saying. She stares up at him with glossy cheeks, and he doesn't even blink.
"You said you——-———-——I got back." Snow and spit obscure her lips. She tries to say something more, but KFS cuts her off.
"What——-—————about it?"
"Everyone will—-" She covers her face with her hands and sobs.
"Jesus, stop crying...." He looks around, embarrassed. His plea doesn't help: YP heaves even harder into her mittens. _Enough's enough._
I walk quickly. The falling snowflakes prickle at my nose, and I blink furiously to keep the flurries out of my eyes. I aim straight for his back and "accidentally" slam into him. He hits the ground with what I imagine is a gratifying thud.
"Oh! Sorry! I didn't see you!" I sign to him, neither caring that he doesn't understand me nor offering to help him off the ground. I take YP's arm and I lead her into the mouth of the whale, leaving Kyle Fucking Stokers for the sharks.
—
"T H A N K S," she fingerspells. I show her how to sign it.
"What was that?" I point back to the entrance of the school as we make our way to history class.
"Long story," she mimes. I'm getting tired of this answer.
"Well?" I prod.
"Not now." She looks down at the floor when she signs the words. She doesn't turn down the hall with me for class.
"Come on!" I wave.
"-——-my locker," she mumbles. Making excuses to leave me behind. "Thanks——-Julia, it——-a lot." She looks into my eyes, hoping I'll get it and leave her alone. She doesn't actually need me. "Go, go! I'll be right there," she says as the bell rings. Of course she wasn't.
—
"You haven't seen her? She wasn't in gym, either," I sign one-handed, munching on a deli pickle.
"No, maybe she went home early." Casey unwraps her sandwich and folds up the foil into a perfect square.
"Before first period is way early."
"It is." Casey tries not to smile. Seeing me get along with someone else has her almost giddy. I don't know what she's over the moon about; she had nothing to do with it. Obviously, YP and I aren't close enough for her to tell me what's _really_ going on with her. Considering she just vanished without a good-bye after I got her away from KFS this morning.
"Whatever. She can do what she wants." It's not like we're real friends or anything. I ball up my own foil and napkin and stuff them into the brown paper bag.
"I'd tell you to bring her her history homework, but I'm not sure she'd receive it," she says.
"What's that supposed to mean?"
"You've been kind of a slacker lately, haven't you?" she says in that joking-but-not-really-joking way. I give her the side-eye on my way to the trash bin. _Get your own life._
When I'm back at my seat, Mr. Katz is there, talking to Casey.
"Good news!" she signs.
"Hi," he waves. "I...uh..." He looks over to Casey and back at me.
"You're fine," I sign to him. "What's up?" Casey might be interpreting behind him but he looks at me when he talks. I love this.
"A spot opened up in my class."
It's the middle of November and I can't believe how the snow is piling up. I love when it flurries; there's something magical about it, like little stars falling all around you, getting caught in everything. Anyone who says otherwise is a bore. The only thing I hate about the snow is the cold. I'm amazed I didn't freeze to death out at the baseball diamond last night. I almost put on _two_ bras this morning to make up for it.
Despite the day's drama, it ended perfectly, so I drive the long way home to get a celebratory sighting of my whale piece. I wonder how many people have driven past it today; if I keep this going, I'll be getting up in Greenlawn in no time. My rep will skyrocket. My hands get sweaty when I turn onto Cobblestone Avenue with the field up ahead. The snow fades the grass a minty green. I drive slowly: I want a nice long look at my piece. But there's something off about it. I only used what I had left: gray, blue, tan. So what is that big pink—
_What. The. Hell._
I squint. It must be the snow playing tricks. I need glasses, right? No, it's there, almost winking at me as I drive past. This couldn't be the same poser from before, that scribble skeleton. This? This took planning. Those bones could have been cut from a textbook and pasted up, they're so precision-perfect.
Whoever it is must be freakin' fast as lightning. They had time to spot it, grab their gear, and bomb my piece all in the same damn night. I never would have pegged Greenlawn as having throw-down streets. It's all white-picket, seasonal-flag, Prius territory.
Haven't seen a lot of graffiti, at least nothing special. Yet as soon as I do some real writing, try to get up with my new name, someone's gonna just slash it like that? I didn't diss anyone. I'm not painting over anyone's work this time. I can't imagine where around Finley is considered "turf." I haven't heard of beef springing up like this since I was tagging in Queens.
Fine. If that's how they want to play it, get ready for war.
How am I going to buy paint? All that's left is a few colored puffs of air hanging tentatively in my school locker. Starting a war on Empty is a shit position to be in. Looked into the P.O. box. No one will ship paint to one. I'm not old enough to open one anyway.
Those damn bones. If they looked crappy, I could rest easy; I'd have the upper hand. But they were above and beyond—not someone I'd want to take on, normally. I don't know what pisses me off more, that they ruined my whale or that they might have made it look cooler. Dick.
The Internet hasn't offered much help in my paint-acquisition research. I'm about to call it quits for the night when my inbox goes: (1). It's a video hangout invite from YP. I throw my quilt over all the papers on my bed and take a quick glance in the mirror across the room. My hair has tangled itself into a nest on my head where a bun used to be. I yank out the hair elastic and shake out the tangles.
I've made it worse. I spot my red beanie poking out from under the bed. _Perfect._ Hair tucked neatly underneath, I hit Accept.
—
"Hi!" she waves. I wave back.
"I sign now okay?" She moves her hands slowly, but she gets it all.
"It's okay," I type into the box.
"No!" She pinches the air. "I want sign you one T H I N G." She's determined.
"OKOK."
"Sorry I go home today. Today not good day." I can tell she's rehearsed. "I thank you for today, and you very nice to me, and—" I'm relieved when she catches her super-serious face on the screen, and we both start laughing.
> jjjjjulia: Aaaaaaaaanywayyyyy........
>
> HeadsNHearts: Did i miss anything good?
>
> jjjjjulia: nah. Mr. Clarke needs trim his damn nose hairs already.
>
> HeadsNHearts: lol
>
> jjjjjulia: so...what happen????
>
> HeadsNHearts: Ok well. so me and Kyle went out like, all last year
>
> jjjjjulia: rough.
>
> HeadsNHearts: it wasnt. we were so good together. real cute and stuff.
>
> jjjjjulia: im sure you were the cute one.
YP rolls her eyes in the video window, before returning back to the keyboard, determined.
> HeadsNHearts: i got him, i knew him. like, the REAL him. we were so REAL.
>
> jjjjjulia: Kyle???!!
>
> HeadsNHearts: I thought he was...ugh im such an idiot.
>
> jjjjjulia: >:(
>
> HeadsNHearts: okok long story short, he broke up with me when I got fat.
I take my hands off the keys and look at her. Really look at her. I've never been a person to use _fat_ as an ugly word. It's something you either are or aren't. Deaf people aren't really shy about truly describing a person. It's not an insult; it's the way something is. I imagine Ma asking me about her, how would I describe YP? I'd probably use the words _shiny_ and _blond._ She's big and beautiful. So what the hell?
> jjjjjulia: what a dickhole.
>
> HeadsNHearts: He isnt!
>
> jjjjjulia:
>
> HeadsNHearts: I used to be really skinny and it turns out that was super important to him.
She's getting upset, her eyes tinting that wet shade of pink right before a tear squeaks its way out. Time to change the subject. I can't handle watching her cry over KFS twice in one day. I wave at the camera to get her attention before going back to my keyboard.
> jjjjjulia: Some good news today
>
> HeadsNHearts: Really?!
YP looks relieved to not have to go on.
> jjjjjulia: got in Mr. Katz class!
>
> HeadsNHearts: you deserve it!!! i wanna see your stuff
>
> jjjjjulia: why?
>
> HeadsNHearts: i think itd be good. Like how blind people are good at music!
>
> jjjjjulia: um. i gotta go.
The still life in Room 105 has changed since my last visit. If my heart could sink any lower into my body, it would. All drapery, all the time. Couldn't I start off with something easier? I look over to the podium and my pear drawing is still up on the wall beside it. My nerves settle a bit.
Some students are already setting up their supplies, intensely staring down the grand array of cloth. Mr. Katz has suspended some of it from the ceiling with crystal-clear fishing line, purple and red ghosts drifting over the table. Underneath is a scarf draped over a sheet draped over a chair. The scarf lies inconspicuously among all the other flashy fabric, plain gray wool, nestled in like a little cat taking a nap.
That's where I'm starting.
Mr. Katz appears in the doorway, tote bag over his shoulder, wavy black hair smoothed into place. There's a one-inch pin pinned to his (blue?!) breast pocket, but he's too far away for me to see what's on it. He pulls a record out of his tote bag and puts it down next to the record player.
"Start when you're ready, guys." He looks out over his class. Some kids, sketchbooks closed, continue on with their conversations. What are they doing in here if they're just gonna goof off? Mr. Katz taps me on the shoulder and puts a sheet in front of me. My name is typed on top in nice bold letters:
> JULIA
>
> We're starting a new still life this week, take your time with it. I suggest you start off with quick gestural drawings to get yourself familiar with the setup and build up from there. This week we will work in pencils and charcoal and next week we will move on to color.
He stays to make sure I understand; I nod and smile. His pin has George Harrison's face on it. Casey butts in and starts talking.
"That was so thoughtful of you." She points to the handout. "I'm Casey, Julia's interpreter."
"Right, right! I forgot, I'm sorry." He extends his hand and she takes it in hers. "Andy. It's nice to meet you." They shake hands a little too long. I have to think of something to say so I can get Casey's hands away from him.
"Where's the sharpener?" I ask, and she obediently interprets for me. Ugh, I'm letting her show off. He does that thing again that none of the other teachers have grasped yet: he doesn't tell Casey where the sharpener is. Instead he looks right at me and points over to the wall.
When Casey is around, people don't bother looking at me when they talk—they look at her. When I sign, people watch her and wait for her to tell them what I'm saying. It's all I can do not to wave my arms and direct people to _look at me, I'm HERE._ I thank him and take my 5H pencils over to the sharpener.
He goes back to the record player and puts the needle down on the record he took out of his tote bag. I don't know or care what it is all that much. I zero in on the little scarf-cat nestled in the folds. This is going to be tough. Again, I look over at my pear still life, and I notice for the first time how poor the drapery looks.
Did he pick this still life on purpose? _Not everything is about you, Julia._ I decide to start with a blind contour of the scarf, sheet, and chair. I put my pencil down on the bottom left-hand corner and lock eyes with the still life, trying not to peek at the drawing, not allowing my pencil to break contact with the paper. It's one of my favorite ways to warm up, really get my eyes working.
I feel the rough texture of the paper underneath through my superhard pencil. I try to keep my arm motions as fluid as the cloth looks. I'm in the zone, that place between here and my head. Tuning everything out except every fold and crease. Every wrinkle dangling in front of me and...
It's awful. I look over to the girl sitting next to me. She's shading in a nice deep shadow, perfectly capturing the drape hanging from the ceiling. Ugh, she's good. I squint over at the kids who were chatting earlier and even their drawings are in better shape than my mess. I start over.
_It's okay._ That was only my first try. I'll do one more blind contour and I'll be nice and warmed up. Pencil goes back down to the bottom left of the page, eyes focus once again on the mountain of folds in front of me.
_Garbage._ What was I thinking, begging to be in this class? I look out the window. The snow is still falling steadily: why aren't we drawing that?
"Need help?" Mr. Katz points to my book while Casey stands next to him. I timidly nod.
"Folds are all about gravity." He points to the highest drape and lets his finger trace along the outline in the air. "Starting at the bottom will only make things more difficult for you." Casey is mesmerized. So am I. "Try sketching out the whole shape first. Fill in the details later. Don't be so hard on yourself—" He stops and asks, "How do you sign your name?"
He doesn't get it right the first time; he has trouble with the hand shape. I take his fingers in mine. His hands are rough and dry. Mine are much warmer than his—it's either the coldness of his hands or the contact alone that sends a chill up my arm. I show him how to sign my name. Then he signs it himself, and this time, it's effortless.
"I love the name Julia. It's the title of my second favorite Beatles song." He smiles.
"You like the Beatles?" Casey interjects. "My favorite is 'P.S. I Love You.' "
"Oh, that's -——- -—-!" Mr. Katz says, turning away from me. They're off, chatting about the Beatles. I'm left in the dark because Casey has stopped signing.
The Beatles ruin everything. Everyone goes on and on and on about them: how revolutionary they were, how they'll never go out of style. People have asked me how I've been able to live without knowing what "Blackbird" sounds like. They give me that pity-face and say, "I don't know what I would do if I could never hear a Beatles song." I can tell you what you would do: you would get on with your life. I don't see what all the fuss is about. It's a band. Paul McCartney is old as hell. People put so much emphasis on music when I tell them I'm Deaf—like, without it, my life must not be as rich or full as theirs. Or they tell me to watch videos of people signing "Imagine." Barf. It's the same sort of shitty comment that YP let fly last night. It's either pity or fascination.
I give Mr. Katz a pass, though, because my third attempt at the still life was worlds better than my first two. That, and I didn't know my name was in a Beatles song.
Back at McDonald's, I transfer the black bag into my work locker. All of the secrecy feels pointless when my bag's empty. I'm still pissed I can't put a message out there to whoever is bombing my art.
I've decided to switch out the combo lock for one that locks with a key. Donovan asking about the paint nags at me. I don't think he would go through my stuff, but I don't actually know him all that well. I take out my visor, Velcro it under my ponytail, and lock everything else away. The key gets added to my key ring. I hook it to the same loop as my broken Eiffel Tower key chain. One day I'll go and bomb the alleys of Montmartre with C215.
From the locker room, through the kitchen, and up to my station, it gets hotter and hotter. Jordyn is standing in the drive-thru window, helping Donovan with something. He leans over and whispers into her ear.
I've never really wanted to be hearing. If I had the choice, I would choose to be Deaf. There's this sense of community. People care more, do more—at least that's what I thought when I was at Kingston. Now, I can see why Jordyn got her implant. I've never had that, that whisper thing, that let-me-lean-in-and-tell-you-a-secret-in-your-ear thing. Lots of people make a big deal out of first kisses and hookups. But I want _that._ I don't even need to know what the secret is, I just want to be whispered to, I want to feel breath on my ears. They aren't dead. They can still feel.
I don't care what he's saying to her; the fact that he gets to say anything to her is enough to make me upset. She probably doesn't understand him, either: she wasn't implanted as an infant, and it's not the magic fix everyone thinks it is. I _do_ know that she's loving the attention.
Jordyn thinks I should get CIs, too, but it's...So. Much. Money. Even if we could afford it, I wouldn't even want it, or the hours and hours of aural and speech therapy that go with it. I just want someone who will whisper secrets to me and not care if I hear them.
The manager must have said something, because Jordyn leaves Donovan's side, giggling. She pats my shoulder as she passes me, as if I'm somehow part of her shenanigans. All through the evening rush, I catch them staring at each other, smiling. I do my best to ignore them while I funnel the salt sticks into their boxes.
At the end of my shift, I serve myself an enormous fountain Coke, a sugary prize for making it through their PDA-fest. I grab an equally huge box of fries after I see them making out in the locker room when I leave.
—
_What's wrong with me?_ I don't give a shit about who Jordyn makes out with. Donovan and I wouldn't work. He chose her because she's easier—easier to get to know, easier all around. I'm not easy. Never have been. Donovan taking the easy way out says more about him than me. So why should I care?
I don't need anyone but Lee. Tonight should be the night. I should get my revenge up on a wall, make my move already. I work my best magic when I'm angry.
Lee has really warmed up—she's almost hot. Which is how I prefer the temperature. If I could, I'd keep the heat at home cranked to eighty-five degrees all the time. I'm tired of the lectures and arguments over the dial, so I layer up the sweaters. The sky threatens snow again, but it feels like Miami in here. Not that I've ever been to Miami. One day I'll do a huge piece on the Wynwood Walls with Kazilla.
I pull over in front of YP's house, bright white with a green door and little Japanese maple trees under the windows. The lights are on; it's not too late. I haven't made plans with her, but I decide to get out and knock anyway.
_Wait._ What if someone else opens the door? Do they know YP has a deaf friend? Do I ask if YP is home? Oh, God. They won't even know who YP is. Why didn't I take more speech therapy? Oh, right...hated it. I reach for my phone so I can write out some sort of greeting, but I've left it in the passenger seat. _Craaaap._
"Good evening." One of the biggest men I have ever seen cracks the door open. Imagine if the Brawny paper-towel guy and that Bunyan guy had a baby that grew up, grew a mustache, and moved to Greenlawn. I wave hello, point to my ear, then to my mouth. I form the word _deaf_ with my lips.
"Oh!——-Julia! Come——!" He opens the front door wide and firmly ushers me into the living room with a slap on the back. He turns and calls for YP, I assume, having a brief conversation with the ceiling. The whole house smells like cinnamon and apples.
"You——-—-———from—-——-slice rye?" He raises his eyebrows. I have no idea what he asked. Mustaches are the bane of my lip-reading existence. I give him one of my "Huh?" faces and he tries again. This time he gestures a bit. He makes a slicing motion across his palm and then points to his mouth.
Oh! He's offering me pie. If it's as good as it smells, I'll eat the whole thing. I nod my head enthusiastically. His smile's so wide it's a little frightening. I follow him into the kitchen.
He serves me up an enormous slice and produces a can of whipped cream from the fridge. He holds it aloft and raises his eyebrows. _Yes yes yes_. I nod my head again, and he piles it on.
This pie. This pie...this pie is no ordinary apple pie. With apple pie, I usually expect something tart, with nutmeg, sugar, and cinnamon. Basic stuff, still delicious. But _this_ pie, this pie tastes like vanilla, and instead of nutmeg, it's laced with oozing, gooey caramel. The top of the crust is sprinkled with sugar and...salt, of all things. It is art.
YP bounds into the kitchen in these knitted slippers with little pom-poms dangling from the top.
"You——more pie?" She playfully punches Mr. Brawny on the arm.
I sign to her, "Best best best pie!" Then I realize she asked _him_ about it. I look ten feet up at him and mime: "You made this?" I thought it would be YP's mom, maybe even YP herself. This lumberjack of a man making beautiful delicate pies never would have crossed my mind. Ever. He smiles and gives YP her own slice. When he smiles, I see the resemblance, he looks just like YP. An unmissable personality.
"New recipe,—-gotta—-—least a bite." He pushes the plate in front of her before pulling down three glasses from a cabinet.
He fills each glass with milk, ruffles YP's hair, and takes off down the hall. I can feel each footfall.
"He's always——————-——new flavors." YP dips her finger in some of what's dripped onto her plate and licks it off.
"Is it his J O B?"
"—started——summer. So what's up?"
Good question. Her dad and his dessert melted down my fury like ice cream under the heat lamp at work. I almost didn't care why I drove here in the first place.
"Are you OKOK?" she asks, after waiting nearly a minute for an answer. She tilts her head with genuine concern.
I down my glass of milk. It's thick and creamy, so much better than the skim that Mee makes us buy.
"Go out with me for a minute?" I point to her, then me, then the door. I pinch the air, point to my wrist. And hope she understands.
"Lemme ask." She skips off in the same direction as her dad, her blond bun dancing behind her. I rinse my dishes in the sink. She comes back holding her brown sheepskin boots and trades her slippers for them.
"He———back in thirty. That work?"
"Perfect."
—
"You told me a S E C R E T. Can I tell you one?" She's nervous about me signing (more like pointing) while driving. I don't watch my hands when I sign, and I keep one hand on the wheel. _Don't worry so much, YP. You're all right._
"What happened? What is it?————about———art class, or—" When she's excited, she talks so damn fast. I shake my head, and wave my arms to cut her off.
"You'll see," I mime.
We make the turn for Cobblestone Avenue, and the scoreboard comes into view. I pull over. My headlights illuminate my work _and_ the work of my rival. I point. First to the painting, then to myself.
"What?" She looks confused.
"I (spray-paint motion) that."
"That—that's yours?" She covers her mouth, although it looks more like her eyes might pop out of her head than her tongue. I nod. She studies the scoreboard, her eyes tracking back and forth and back again. She looks at me timidly. "Is——why—-—- expelled?"
I nod and pull out my phone.
> JULIA: I made a huge one on the back of the gym at my old school.
>
> YP: They caught you?
>
> JULIA: Yeah duh.
>
> YP: So you still do it?
>
> JULIA: Obv
She types into her phone but she keeps deleting and starting over. She looks at the whale again before finally sending.
> YP: its good. I like the different styles.
I can tell she's reaching for the compliment. I'm sure she's never so much as ended up in detention, let alone broken a law. It's kind of fun to feel like I might be corrupting her.
> JULIA: The styles is my problem.
>
> YP: Doesnt look like a problem.
Hah! Yeah freaking right.
> JULIA: I not do it.
>
> YP: I thought you said you did do it.
>
> JULIA: I not do the bones part.
>
> YP: Who did
>
> JULIA: Not 100% sure on that one yet.
>
> YP: You dont know???
>
> JULIA: No its like some sort of turf war thing.
>
> YP: War?
>
> JULIA: They did this to my work. Now I need answer back somewhere
>
> YP: So you take turns.
>
> JULIA: And its my turn.
>
> YP: But...
>
> JULIA: ?????
She doesn't look at me, only at her screen, or the field. She doesn't seem excited by it. Maybe it's not her thing, but it's mine, and for some reason I want her to at least try to understand.
> YP: Why does it have to be a war?
The drapes are looking very drapey in Advanced Art Studio. They loom overhead, taunting me with their highlights and shadows. The folds seem to have multiplied, thousands of little wrinkles and creases mock me: " _real_ artists draw drapes." Today we start color versions of the same still life. Mr. Katz puts on a record at the beginning of class again, but this time he props up the sleeve so the class can see. It doesn't have the name of the musician on the front.
The cover is a painting, really rough brush strokes—I'm going to guess in oil paint by the way the colors mix up. It's a man's face. I'm not sure if it's the performer or not, but it's not painted in a very realistic way. Thick gray outlines, a big flat light blue nose.
There's a debate going on about it. I look over to Casey, but she's preoccupied with participating in the argument, rather than clueing me in. Katz keeps laughing and smiling at her. From what I can piece together, the guy across from me with the long brown hair and holey black T-shirt has a bad opinion about whatever is playing. A lot of the other kids are chiming in, too.
Casey tells him to get to his artwork...and something about the history of music? Mr. Katz finally steps in and lifts the needle, puts it back down on the same record, and says, "——listen——-next track."
The drama stops; everyone takes a moment to consider the song change. So much fuss over a song...aren't we in _art_ class? Shouldn't you all be drawing? The next song seems to placate the room. Black Shirt closes his eyes, and nods along to the beat.
Mr. Katz comes over to me and kneels down so we're on the same level. He speaks at the perfect speed for lip-reading. Everything about him radiates a Zenlike calmness. "Did I show you where the paints are?"
I shake my head "no" and follow him over to one of the large wooden cabinets lining the back wall. He swings the door open, revealing an assortment of supplies. A box labeled "Oils," a box for acrylics. I pull out boxes one by one; dinged-up, dirty tubes fill each to the brim. Some of them must be empty, maybe only one squeeze of pigment left inside. Most are so caked with dry paint, you wouldn't even know what color you were picking out.
Underneath the paint boxes, there's a shelf full of pastels and colored pencils, followed by a shelf of watercolors. Dozens of little clear plastic trays stacked up and crammed into place. Under the watercolors there's one last shelf, and on it is an old plastic milk crate packed full...
Of spray paint.
I grab a tray of watercolors and turn away as fast as I can. I won't give it a second glance, won't risk being caught scoping the merchandise. At the sink I pick up a jam jar and fill it with water, take three brushes from the coffee can, unroll some paper towels, and head back to my seat.
Mr. Katz closes the closet. It doesn't even lock. Casey is standing in my line of sight for both the closet and the still life. I shoo her aside. _Don't even think about it._ I'm above stealing. _Where else am I going to get it?_
I dip my brush into the water, swoosh it around a bit, and dab the excess water on the paper towel. I rub the brush into the red first, getting the pan nice and wet. I'll start out with light colors since I don't really know much about watercolor painting. I grabbed the paint without thinking.
Another jam jar gets placed next to mine, and Mr. Katz calls Casey over to interpret.
"Two glasses keep the colors from getting muddy," he instructs. "One for cleaning your brush, one for adding to washes." Casey has obviously been practicing her art vocabulary. She doesn't skip a beat.
"Thanks." I smile. He moves on to the girl across from me; she's attempting to thin out acrylic paints with turpentine.
My eyes dart back to the closet again. _Stop looking!_ But it's like I have X-ray vision and I can see right through the door. The cans sit there, unused. Collecting dust. Begging. I pull my eyes away from the closet and back to the cloth. I can't steal from the one teacher who's reached out to me. That's just bad karma.
I'm sitting cross-legged in my chair. The bottom foot hums with pins and needles—I miss my big armchair. I've started by painting general shapes again. Watercolor paint is trickier to work with; you can have too much water and get gloppy textures, or not enough water and end up with scratchy strokes. Not like spray: beautifully opaque, spray performs exactly as you'd expect all on its own. No water or thinners required. With the right caps, you can achieve a lot of range, too. I'm thinking about doing a stencil piece as my next move. Stencils can be stunning. Elevated. Next level.
I realize I'm staring at the paint cans in the closet. _Wait. When did I come over here? I don't even remember getting out of my seat._ I grab a few colored pencils from the top shelf and book it like mad back to my spot.
Casey waves: the bell rang. Class is over. Mr. Katz shows me where I can put my work to dry, a big rack next to the record player. I pick up the album and flip it over.
Bob Dylan— _Self Portrait._
"It's a good—" He crosses his arms as he tells me, "I'll make you a copy."
I laugh and roll my eyes. Casey taps her foot in the hall. I close the door behind me on my way out. _Another door between me and that stash._
Clicking on my paper lamp after school, I decide to figure out the paint later. Gotta plan out the piece first: I'm finally going to tag the underpass. I crouch down next to my armchair and grope around underneath it. I have my B-book Velcroed under there.
If I sketch or plan anything that has to do with graff, I put it in here. Not in my piles of regular sketchbooks, not on flyaway pieces of paper. They all go in here, in my bible, in my little black book. And I have some real work to get to.
So how do I retaliate? This will be _my_ art. I consider doing one lone pear, but I scrap that quick. That's for Katz. What says ME? What says JULIA? What says HERE? It has to be all those things. It has to be _me_ on that wall.
I flip through some magazines, hoping lightning will strike, but there's not a cloud in the sky. No one in the pages of _Nylon_ is like me. I'm a fingerprint, an anomaly, a snowflake. Indian, Deaf, girl, two moms. You couldn't make this shit fit in the pages of those glossy mags. I think about the curtains, the paint, the smell of Room 105. I think about the Zen being that is Katz, and his hypnotic red plaid. And I remember "Julia."
I crack open my laptop and search for the lyrics to the Beatles tune I was unaware of. I wrote off the Beatles long ago, mostly because they're all over the map. I can't figure out what sort of music they play. A bunch of songs are all about holding hands and loving and _I love her_ and _does she love me_ and _she loves you,_ blah blah blah. And then you have songs that are, like, some sort of country song about a bar fight with some raccoon guy, and another song about another guy who goes around bashing people with hammers? One of their songs goes on and on about doing it in the road and that's it. That's the whole song.
Someone explain the appeal.
Anyway, I figure if my name's in a song, maybe I can use it. Might as well see. I find a link and pull up the lyrics. I scroll through the lyrics pretty quickly, and I have to scoff. _Seashell eyes? What?_ For some reason I read them again. And again. And again. The lyrics repeat in my head, and I try to imagine how they would play out loud. The words are too dreamy to be a rock song. I repeat them even slower. I pick up my pencil.
The words don't leave my head as I draw. My brain is preoccupied with the words and my hand takes over. _Am I hypnotized?_ Line after line, the drawing starts to come together. Three colors. _Where's my X-Acto? Where's my poster board?_ The song in my head plays on as I start tracing out the first image.
I know it doesn't look like much yet, but it will. I cut away all the pencil, making lace out of paper. I love this part just as much as all the rest of it. The methodical, unrushed part. The quiet creation before the spray-paint storm. I have the words memorized. _Is this what it's like to get a song stuck in your head?_ I start working on the second board when the overhead fluorescent lights flash. I shove everything under my armchair before running over to the stairs. It's Ma again.
"Everything okay?" She's checking in, but she doesn't invade my space. It's the first time since Kingston she hasn't just walked downstairs without a heads-up. It's a relief and I mean it when I tell her, "Perfect."
YP slams down her lunch tray and a few stray chips fly onto the table. Casey is thrilled to get to interpret our lunchtime conversations. Sometimes she tries to drag other kids into it, but YP is the only one who sticks around.
"What's wrong?" Casey beats me to asking her. YP ignores Casey and stares at her lunch tray.
"Everything all right?" Casey pushes. I hope Pants comes up with some sort of answer, because Casey could go on like this forever.
"Well, no. Officially lost my chance at Cheer this year."
"You actually like that stuff?" I pull my hands away from my chest, touching my middle fingers to my thumbs. Casey interprets for us.
"I liked having lots of friends." She doesn't make eye contact with either of us. Her eyebrows are hidden under her bangs. I can't tell if she's upset or angry. I'd be neither; those girls weren't her friends, not real ones anyway. What kind of friend texts you insults, abandons you?
"Can't you at least try out?" Casey urges.
"No." YP shoots me this look, one I recognize the meaning behind. She doesn't want to talk about it. I try changing the subject.
"Casey, you know that Beatles song, 'Julia'?"
"Of course! John Lennon wrote it for his mother." She cleans her round glasses on her scarf, shining proudly. She loves when I ask her questions. Especially if they involve her opinion on something.
"His mom?" I put my thumb in my chin and my fingers stick up, making the sign for _mom._ "But it's a love song!"
"You can love your mom!" Casey objects. YP and I both crack up.
"People don't write _love_ songs for their moms," YP chimes in.
"Why not?" Casey asks.
"Weird," I sign. "It's weird."
_"Tsk."_ Casey puts her glasses back on, rolling her eyes. She makes the sign for _bathroom_ and leaves. She'll meet up with me next period, no matter how much I wish she would get lost in the halls.
I text YP.
> JULIA: Why cant you try out?
>
> YP: Everyone hates me now.
>
> JULIA: For for??
>
> YP: Same reason Kyle dumped me.
>
> JULIA:
>
> Screw them try out anyway.
>
> YP: Why? So I can be humiliated?
>
> JULIA: You care too much
YP takes her tray, untouched, and dumps it in the trash on her way out. I know how she feels. This time I don't let her get away.
"Hey!" I use my voice to get her to turn around. The hall is empty; my voice is just for her.
"Sorry," she signs.
"It's OKOK," I sign back. "I understand."
The whites of her eyes are pink again: this must be really important to her. That, or she cries a lot.
"Come on." I walk in front of her down the hall. After lunch I have English, which I hate with every fiber of my being. And, shit, I've been trying. Trying to keep my texts to YP clear and understandable. Not falling back on the old texting habits I had with Jordyn. That's something I do miss, not having to watch my words when I type. Using whatever grammar I damn well please and not being called stupid or slow. I hate English. Hate. I'd have no problems ditching, but I'm the only junior with a babysitter. Sure, my teacher could _find out_ I cut, but he wouldn't be able to leave class and _search the school_ like Casey can when I don't turn up.
YP wipes her nose on her sleeve. I don't care, I'll deal with Casey later. _This is more important._ We turn down the art wing and make it to Room 105. I peek inside the long thin window and the room is empty, exactly like the first time I came here.
I open the door and motion for YP to follow. She looks around the art room and her shoulders drop. She exhales and starts bawling. I consider hugging her, but it feels weird. I never know what to do when someone is crying. Pat her on the head? I stare at the wall. It's covered in drawings of perfectly rendered cloth; they put my little pear to shame. YP slumps down into a chair next to me.
"I lost everything, all———, I thought——-—-—-be better. I———-I was better." Her lips are puffy and moist, but I get the gist.
"What happened? For real," I sign. She seems to understand.
YP covers her eyes so she doesn't have to look at me when she mouths very deliberately, "I—-sick—-getting better——-gaining weight."
Oh. She looks over her hands to gauge my reaction. I pull out the seat next to her and join her at the table.
"I was alone. Before. I didn't——anyone. I—-skinny and suddenly I—-whole entourage." She sighs, a big breathy exhale from the bottom of her stomach. What good is an entourage if they don't really know you? If they don't really see you for who you are?
"So, I didn't want to, you know, get better. —————was great—-——. And Kyle..." It's better to let her talk. I don't think she wants to hear my opinions about Kyle Fucking Stokers, or her fake-ass friends at the moment.
"He made me feel like it didn't matter. All——work—-—-counting, and pills, and walking, and everything wasn't important. ——I was more important."
"You are more important." Kyle and I actually agree on something.
"I went and got help and he——-text me and—-———————it wasn't———allowed. I got better, and I came back. And everything was so messed up. ———worse than before. Everything I loved is over."
The way she hunches over in her chair, the drapes soaring to the ceiling behind her, she looks like she has wings.
"I draw you?" I ask without asking. I pull out my sketchbook and flip to a new page. Moving quickly, I take a box of markers from Mr. Katz's podium. I wish they were paint pens. We could use the fumes.
We sit in the empty room while I outline her figure. I thought YP was one of those girls who never shut up, the ones who blather on and on just for the sake of talking, who don't listen to what you're saying. It's always about them. Turns out YP likes it quiet, too.
I swivel the book around on the table to face her.
"This, this is, perfect." She's getting better at ASL every time she signs. She must be practicing. She's right, it looks great. I'm working in my real style: fast, loose, markers, colors. It's all me, and all her. I rip it out and give it to her. YP puts the drawing down in front of her.
"What about you?" she signs.
"What about me?"
"Tell me something. I want to know more about you." She waits for me to respond, but I don't know what else to tell her. I told her about my street art, what else is there? "What about boys?" YP asks with a small smile.
_"Worst!"_ I sign with the angriest eyebrows I can muster.
"S P I L L." Her eyes widen and she leans in.
And I tell her. I tell her about Donovan, which leads to Jordyn, and all her hearie boyfriends that last a month or two at a time. How it never bothered me before, but of course it bothers me now. About how it makes more sense for him to choose Jordyn anyway. I don't think she can understand half of the words, but she nods in all the right places.
"You were friends with her?" YP signs the word _friends_ a few times with a scrunched-up forehead.
"I thought so."
"I don't know if—-—-friends with _you,_ " she says, assuming. I didn't lecture YP about her phony friends.
"But she was." I nod and sign. "She was my best friend." I think back to when Jordyn would sleep over and we would spend all night watching Bollywood movies, with Ma and Mee dancing around the living room. Jordyn would play along, but then make fun of me for days afterwards. I know Bollywood movies are silly, but they're mine. A part of me.
And then it sinks in: Is that what Jordyn is? Just a big faker? YP straightens up with wide eyes. She makes her hands into fists, and uses them to sign "shoes," then points to the door. We look around for a place to hide. I know all the closets are packed; there's no way we would fit in any of them.
"Quick!" She hops up on the still-life table and offers me her hand. I take it by instinct and she pulls me up to meet her. YP lifts up one of the hanging curtains and wraps it around us both. It reaches all the way down to the table, where it cascades onto the floor.
She looks over her shoulder, listening for the door to open. She holds a finger up to her lips, forgetting I'm not really one to make a peep. I wonder if the sound of my heart pounding in my rib cage is as loud to her as it is to me. Her eyes shift from the door to the other side of the room and back, her neck craned over her shoulder, for one of the longest minutes of my life.
"C L E A R?" I ask. Signing has its advantages.
"I think," she exhales, and starts laughing. We scramble out of the room as fast as we can.
In the hall, YP checks the time on her phone before rushing off to her AP calculus class. I should be walking to my next class with her, but I lag behind. After she turns the corner, I go back inside the art room.
_I can't just steal the paint, right?_ I've never stolen supplies in my life. It goes against everything I'm about. My graffiti is more about the art, less about the vandalism. Stealing to make it happen? That's not art.
_But no one is using it!_ It's sitting there in a cabinet untouched and unloved. How would we even use spray paint in class? Indoors? Maybe it was donated or something and Katz doesn't know what to do with it. If I don't take it, it's only going to keep collecting dust. And whoever is ruining my work gets to win our war by default.
I take three cans.
"You can't cut class and not tell me." Casey is furious.
"I was helping YP! It's not like there wasn't a reason." Some things are more important than proper sentence structure. Why don't teachers get that? Ninety percent of the time, fine, I'm there in class. Sometimes shit hits the fan, though, and we're supposed to ignore it?
"You're not understanding me—" she goes on. Damn, Casey, you're not even that old. Don't you remember what this was like? I tell her exactly what I think she wants to hear, but I can't help but roll my eyes as I do it.
"No, I understand. I have to go to class and try to be better and be brave, and if I don't start getting better at English I'm gonna make Deafies look stupid and—"
"What? No! I know your friend was upset. I saw her at lunch, too, and if you had asked me, I would have let you."
_What?_
"You make yourself look worse when I show up to your class and you don't."
_Is this real life?_
"Get it?" Her pointer finger flicks up at the ceiling. I'm so stunned I can barely nod.
"I told them you weren't feeling well, so let's go to the nurse and make it look right."
_Casey, who are you?_
"Liar," Donovan signs. "You very liar." He grins. Jordyn must be teaching him more signs. I don't know why she would bother.
"I've never lied in my whole life," I lie as fast as I can, and when he doesn't reply I puff my chest out and roll my eyes.
"You paint. You paint! I know." He signs some more then speaks. "Jordyn——me all——-it." If I wasn't sweating after my extra-long shift, I was now.
"I quit all that," I sign, dusting imaginary crumbs off my palms. "Finished with it."
My shoulders ache under my black bag. It feels like it's full of bricks, not paint cans.
"Liar," he signs again. He zips up his hoodie. The zipper looks like a spine, with a rib cage printed on either side of it.
"I gotta go." I point to the door and wave, fingers clenched around the straps of my bag. I can't get out to my car fast enough.
What's Jordyn running her mouth for? She knows the kind of trouble I could get into. As far as she knows, I've moved on from all that, anyway. So why would she go and talk to him about it?
The wipers clear the snow off my windshield. There's something stuck to one. Trash? It's yellow, like the burger wrappers inside Mickey D's. I stop the wipers and struggle, reaching out of the window, to grab it.
I wait until I'm home to inspect the note again. The snow has blurred the letters, but there it is, clear as day. This shouldn't be possible. I know it's my move, thank you very much. What I don't know is how the hell you know who I am. I shove the note in my pocket and start pacing on the sidewalk in front of our house. My footprints in the snow diagram the steps of some long-lost stress-induced dance called the Panic.
I feel sick, violated. I'm going to pass out; I need to lie down. This isn't just a major diss, it's a threat. It's _scary._ I can feel the sweat beading on my forehead. I'm hot, so damn hot. I lie on our little patch of lawn and let the snow fall on my face.
_This isn't okay. This isn't happening._ Out in the open, up on a wall, anonymous is one thing. This, though? On my car, _my_ car. I'm aware of every flake that touches down and melts on my face. They know who I am. They know I'm Julia; they know I'm HERE.
I run through that night again and again as the snow collects in my hair and around my ears. I parked at least a mile away from the field that night. After I dumped my hoodie and walked to my car, I checked over and over again to see if anyone was following me. I took random streets on the way back, not the quick or easy route.
_Who is it?_ Who could possibly be this offended by my work? Be so up on their game that they notice my graff right after it goes up. I decide to head to my basement command center and work through the fear. I have one last stencil to finish.
—
I slice through the poster board with a little more care this time. Ma and Mee are on one of their date nights, so I don't have to worry about them sneaking up on me. I can work and think. Every now and then my hand finds its way into my pocket and I wrap my fingers around the slowly disintegrating note. YOUR MOVE. _Oh, I'm sorry, am I not moving this along fast enough for you?_
The little cuts in the board shine like stars when I hold it up to the light to check my progress. I've been playing it so safe, and still I get caught. I guess I should be thankful it's a rival and not the cops.
How did Donovan know I had paint that day?
He saw my hoodie.
Is that enough?
He knows my car.
He has that hoodie. That skeleton hoodie!
Jordyn told him I _used_ to spray.
What exactly does that prove?
Why did he bring it up tonight?
Where did he go on his break?
Does he have Post-its in his locker?
_How does he know I'm a liar?_
_No. No way._ I refuse to believe it. He's not clever enough—he's all looks. This sort of work takes brains _and_ talent. He's with Jordyn, so what's he messing with me for? _Obsess much?_ I thought I was the obsessed one.
Since when are you an artist, Donovan? You never said anything before. I guess you never said anything, period. Do you _like_ me or something? Is that what this is about? Trying to play the same game as me? What else did Jordyn tell you about me? Did she whisper it in your stupid ear and you thought, OH, HEY, THAT'S COOL, LET ME GO AND MESS WITH HER ART, TOO, SINCE I CAN ONLY MESS WITH HER AT WORK. Goddamn it.
I roll up my stencils and venture out into the cold again. _My move._
I'm on edge, and I don't like it. I should be buzzing, I should be driving down to the overpass high on sweet guerrilla endorphins. I'm ready. I have everything in place, my plan committed to memory, but I'm on edge.
How does he know I'm a liar?
The wind blasts through the tunnel, stinging my face. I thought to wear more layers this time, but I'm still freezing. I decide to put my piece up near the far exit of the overpass. The one-way street will give me some time to dash if I have to. I've learned some new tricks since my last bomb.
I slide the first stencil out of my bag's straps. Each stencil is cut from black poster board: nice dark camouflage, no giant white beacon in headlights. I use Sticky Tack to get it up on the wall. I do the tack at home so each stencil already has it in place. I peel the roll apart and stick it right on up. It always ruins the stencils but I don't keep them after finishing anyway. Rip 'em and ditch 'em.
Black can goes first. I shake it up and spray across the poster board. I have to get paint in every cut. The black on black on black makes it harder to see what I'm doing, and I waste paint going over the same spots more than I need to. The wall looks like a paint-eater anyway, so I keep the spray flowing. Doesn't matter if I run out, as long as it looks perfect.
The tunnel brightens up. _Car's coming._ I grab my bag, leave the stencil on the wall, and haul ass through the nearest exit. I toss my bag, and it lands behind a bush on the hill outside of the tunnel. I leap down in the frozen dirt, next to the bag.
_No time to waste._ On the ground, I prepare for Stencil Number Two, uncap the blue and holster the black. I lie back, and my breath forms little clouds above my face. Cars are going by, I can feel it but I can't tell if they're on the overpass or under it.
The sky is blanketed with heavy gray clouds. Every now and then a star is bright enough to peek through. Maybe it'll snow again soon, but I hope not tonight. On the other hand, if it snows, maybe people will stay home and I'll have enough time to finish. _Have I waited long enough?_ My neck is freezing down here on the ground. I wait another full minute before I decide to dash back to the tunnel.
I rip down Stencil Number One. I try my best to get the cutouts on the second board to register in the right place, but it's okay if they're slightly off. I think it looks cooler that way sometimes. The Sticky Tack doesn't hold up too well in the cold, so I have to really press it into the concrete. I have blue ready to go. This time the lighter paint color makes it much easier to see if I've sprayed in the right spots. I double-check over both shoulders while I spray. The coast is still clear and I throw up the final stencil, the first two in shreds at my feet.
The final layer is the magical one for me. It's what makes the whole thing come alive. The highlights, the special moments, all happen on that last round. My arm and fingers shake. My right hand is frozen into that spray-claw shape so badly I worry that I won't be able to pry the purple can out of it. Everything aches, even though I haven't been at it all that long. Stencils don't care if I'm shaking, so no need to spray in a straight line.
I rip down the last stencil and ball all three up. I'll toss them in random trash cans on the way back to Lee. I take my Screaming Silver paint pen out of the front pouch and sign the piece. _HERE._
The still life is down in art class this week. Nothing is set up in the center of the room. Casey is already in the middle of a conversation with Mr. Katz (green flannel today) near the record player. When he sees me take my seat, he holds up a finger and grabs something from inside his tote bag.
"Here, I made—--copy." He slides a big used yellow envelope on my desk. I slip out the stack of papers inside and on top is a color printout of the Bob Dylan cover. I flip through the stack: there's a new page for the lyrics to each song. Katz has drawn little doodles in the margins: fish swimming, horses, hands reaching for things.
"I thought that—"
I cut him off with a wave, he doesn't need to say anything. I put my hands over my chest, one on top of the other, trying to keep in the warm feeling that's started to radiate there. I don't want Casey to interpret. I want Mr. Katz to understand me like I understand what he just did for me. It's hard to look him in the eyes; I felt something in me unlock and I don't want to share it. But if I want him to understand how grateful I am without words, I have to.
When our eyes meet, we both must look so worried, so serious, it's funny. I burst out into a laugh, and I'm relieved when he starts laughing, too.
"Thank you." I slide the pages back into the envelope and tuck it away into my bag. _This is mine._ It's all for me. Then the guilt worms its way into my chest. Why did I take that paint? I immediately start thinking of ways to repay him. What would be an equivalent gesture? I can't really make him a mixtape, considering I don't know all that much about the art of it. Maybe my next piece, something with red, something—
Black Shirt hands out paper and Sharpies to everyone in the class. Casey takes her position as Mr. Katz starts explaining our next unit.
"For the next few weeks, we're going to be talking about street art." My ears are hot. There's no way he can know. I glance around the room, scanning faces to see if anyone else is flipping out. No, it's just me.
"I wasn't planning on doing this lesson until the spring, but I was inspired on my drive to school this morning."
_Gulp. Katz saw the mural. He must have. He can't know it was me, I've been so careful. How many cars drove by when I was painting? It couldn't have been that many. It's just a coincidence, at least it inspired him. Stop blushing, you're only making it worse._
"Before we start, though, I need you all to promise me that you will practice your work only here in class."
"What's the point, then?" a girl with freckles asks.
"The point is to learn about a different art form, add more styles to our toolbox. Who knows? Maybe you'll be hired to paint a mural one day."
Aw, what a cute notion. Styles in our toolbox, and getting hired for murals? As if that's the same thing as real street art. Mr. Katz, I love you, but sometimes you're so corny.
"We're going to start out using our real names for our work here. No nicknames yet. So, first, I would like you to start experimenting with the letter forms of your name. Feel it out. Try not to rely on your pencils and only use the pen. You wouldn't get to erase in the real world."
True, that.
Tagging with my own name feels wrong. Even on a piece of non-incriminating paper, it feels off. _This isn't how it works._ You don't sit down with a Sharpie and write your name over and over. Well, I guess you sort of do. But definitely not in a room full of other kids all doing the same thing.
Ugh. My fingers get dappled in Sharpie. Dead giveaway. I'll have to try and explain this to my parents. _I swear, it's for class_ probably won't cut it. Maybe Casey will back me up on this one.
When I tag, it's not about slapping my name on a wall. It's more than that. Right now, though? I'm not jamming out. I'm not going into that amazing, humming, buzzing trance that happens when I'm dreaming up new work. It's just my name, just paper. The _J_ is ugly and the stupid _I_ looks like another _L._
Mr. Katz comes over and looks at my paper.
"Hmm. I thought --- ----- be good at this." Wait. Did he—or am I misreading lips again? How would he know? Some gossiping secretary who knows why I was expelled? Teachers want to act like they're above rumors and gossip, but I know shit spreads faster in the main office than in the cafeteria. He laughs to himself as he moves on to the girl sitting next to me.
BOOM! _I win._ I drive down Spring Road on my way to work. My fingers, still smudged with Sharpie, wrap around the steering wheel. My piece is still burning under the overpass, and, more important, untouched. Now this, _this_ is real street art. Not some Sharpie doodled on poser-printer paper in art class.
This takes ovaries.
I wonder if Donovan has seen it yet. Maybe he's planning his next move, or maybe he thinks _I_ haven't made a move yet. Whatever. Move made. _BOOM!_
I strut into work like a boss. Like a queen. Like a CEO. Like nothin's gonna bring me down. Check my swag, D. It puts yours to shame. Jordyn is already getting changed when I strut into the back room. I'm glowing and nothing she says is going to kill my buzz.
"I wanted to talk to you," she signs. I pull down my visor to avoid looking at her.
"Doesn't matter." My hands swipe back and forth.
"I think I'm getting serious with Don—"
"And?" I cut her off.
"Well, I need you to back off."
"Back off what?" I snap. I'm losing my patience with Jordyn and her demands. My conversation with YP replays in my head. Jordyn was a real friend, I didn't make that up. Right? I was always there for her—all she had to do was text. I remember bringing her chocolate shakes after a particularly dramatic breakup. We ate them with spoons on her fire escape, and she cried and I made jabs at her ex. That's not fake.
"Him. I know you like him. And he keeps talking about you."
"Why?" Seriously, I want to know. He shouldn't be talking about me to anyone. Did he tell you about our little war?
"I don't know, I don't really get it." Real nice, Jordyn. I wouldn't expect you to get it. "So, would you mind backing off? I'm with him now."
"Doesn't matter, okay? I'm over it, and I'm over you. You lost all your clout when you sold me out."
"I had to! They thought it was me!" she signs. Pathetic. She's never even taken an art class.
"No, they didn't. I stood up for you! I painted that wall for you! You were my best friend. What happened?"
"I wouldn't say _best_ friend." She shrugs her shoulders like it's nothing. Like what she just said wouldn't slice through my heart.
"You're serious?"
"I mean, it's not like we hang out all that much, not unless we're working. Or at school. Don't make this weird, okay?"
"What was I to you?" My hands can barely sign the words, they're shaking with anger, with exasperation.
"Look, none of this has to do with Don—"
"Go bang every hearie in the world, for all I care." I cut her off furiously, my hands a blur, and I'm out the door. I might be burning bridges, but they're my bridges to burn.
—
The heat radiating off the fryer is welcome for a change; it's been so damn cold out. I should be standing here steaming mad after being passed over by Donovan, only to have him come back and invade my space after all. Does he like me or not? Is this about art or something else?
But all my anger is reserved for Jordyn. She stole my school and then she stole Donovan. They aren't worth the trouble anymore. I don't want her to think I'm on her thieving level. I'll back off, but I can't force him to do the same. The breakup chat we couldn't actually have plays out in my head.
—
_We're both standing under the overpass by my latest. He looks so cute when he's defeated._
_"What's wrong?" I ask._
_"I can't do it, can't add to this. I wouldn't want to ruin it." He might be holding back a tear or two._
_"That's all right. You tried." Pat, pat on the head._
_"You're so much more talented than I am." Donovan looks at his feet._
_"Maybe one day." Poor little toy._
_"I'm sorry I ditched you for Jordyn. I thought she'd be easier to talk to, you know?"_
_"Was she?"_
_"Sure, but all she ever does is talk. She never listens."_
_"Shame that implant goes to waste, then."_
_"Hey, I was wondering—"_
_"Lemme stop you there, D. I'm sorry it's not going great with Jordyn. That's too bad. I'm also sorry your graff game is so damn weak. But I can't help you with that. You have to earn it. This could have been great. But I'm out."_
Drop the mic, etc.
—
Too bad I have to keep staring at the back of his head for hours. I wonder if I should leave a note on his locker: _Your move now._ Nah, let him find the underpass on his own.
I line up some fries under the lights, and start folding up little boxes for the next batch. I hate folding these things; they have such a freaking weird shape. No other box is shaped like a fry box. It's a singular thing, and it's annoying as hell.
Donovan keeps looking over his shoulder, except this time I swear he's looking at me, not Jordyn. She keeps filling soda after soda for him, back and forth, getting between us the whole shift. Every time she passes me she glares, or bumps into me a little too obviously while squeezing by. I'm surprised he's not in a diabetic coma by the time we leave.
—
Jordyn leans against my car, obviously pissed off, all toe-tapping and folded arms and a scowl visible from space. I can still smell the fries in the air, and the glow of the arches tints her skin a sickly yellow color. What happened? How did we ever end up like this?
"What?" I ask her before I even reach the car.
"You didn't let me finish," Jordan snaps.
"Finish what?" I reach past her and unlock Lee. She pushes against the door. I fight every urge to yank it open and send her flying.
"Knock it off." She touches my shoulders, forcing me to face her.
"You knock it off! I told you—"
"Shut up, Julia!" she signs and shouts. I back off. Let her say whatever she wants and after, I'll add her to my X-box and shove it in the back of my closet to rot.
"Do you like him?"
"I don't want to talk to you about boys."
"Please stop playing games, I need to know." She looks so hurt, and it kills me. Why doesn't she care about me the way she cares about him? I thought our friendship was worth more than some dude who works the drive-thru. I want to know why she sold me out, why she totaled our friendship but acts like it's still drivable.
"No. I don't like him." I lean next to her. We both stare off at some unknown point in the universe.
"Would you tell me if you did?" she asks without looking at me.
No.
"Yes."
"It's just that..." She starts pacing. "I don't know. I really like him. But he's dated a lot of girls at work, you know?" she asks, but it doesn't feel like she's actually asking me, or even talking to me. It's like she's trying to figure it out for herself while I'm on standby. It makes me miss YP and our talk in the art room. How she sat and listened, even if she had no idea what I was saying—she knew it was important. The more Jordyn rambles, the more upset I get.
"I guess it feels like, maybe he doesn't like me." She finally pauses and looks over to me, expectant. Waiting for me to comfort her, but I can't. I won't.
"Not everything is about you."
"Don't be so jealous," she says, laughing. But none of this is a joke to me, not even remotely. I swing open the car door, but before I can peel out in a fiery rage, someone zooms into the parking lot on a bicycle. It's YP.
"Are you OKOK?" I sign over the car to her.
"Fine, fine," she signs back, leaving one hand on the handlebars. She leans her bike up against Lee, and they look like they belong together. She's panting, hard.
"You rode all the way here?" I ask. It must have taken her hours.
"No, no. Bike, L I R R, and bike again," she signs, smiling and proud.
"What happened?"
"Not. Good." She takes her time, signing with purpose. Her eyebrows angled down, she shoots Jordyn some shade.
"Doesn't matter. What happened?" I ask again.
"What's going on?" Jordyn tries to get our attention.
"O V E R P A S S," she spells. "Not good."
I tried to shake Jordyn, but she wasn't having it. If only YP could read my mind so we could talk without Jordyn butting in. Pants is getting good enough at sign; I just wish telepathy were the next step.
"Just tell me what happened," I sign to her. Jordyn must be having a conversation, out loud, with YP at the same time; YP keeps talking over her shoulder.
"Hello?" I wave.
"You want to tell her?" YP tries to sign out of Jordyn's sight. My need for info is trumping my beef with Jordyn. It's not like she can get me expelled from Finley. YP looks disappointed.
"Don't tell me you're painting again!" Jordyn squeezes between the seats. She's crammed in the back next to YP's bike wheel. The rest of the bike just barely fits in the trunk.
"She knows?" YP scowls from the passenger seat. Jordyn replies, but I have no idea what she says. I'm trying not to crash the car. It looks like they're arguing, from the few glimpses I catch.
I make the last turn onto Spring Road and there's a cop parked behind a pillar under the overpass. Lights off, he thinks he's being clever. He's not fooling anyone. Least of all me.
"Should we just drive through real quick, like normal?" I ask Pants.
"Wait!" She holds her arm out like she's protecting me from stopping short. Probably the first time a passenger has tried to protect the driver. A second cop walks across the underpass and gets into the cruiser. "We can't—-——see your car." Good catch, YP. I turn onto a side street, trying to make it look natural.
"D A I R Y B A R N?" I sign.
"Fine." YP's still upset.
Carefully avoiding Spring Road, Cobblestone Avenue, and Broadway, we snake through side streets, getting closer to our quasi-hideout. It's a long, winding drive, the silence punctuated by Jordyn and YP sizing each other up in the rearview mirror. I just want to see what happened on the underpass. Having my work painted over is hard enough to deal with; I can't be concerned with the possible hurt feelings of an ex-friend. I'll sort YP out later, she'll at least understand.
Dairy Barn stands tall and proud at the end of the road. I'm dying to know what those cops were up to. They didn't have a paint roller or anything, and cops don't usually do patch jobs themselves. We pull under the carport and YP leans over me to order a large iced tea. I pay with some quarters from the center console and we park.
"When did you start all this again?" Jordyn asks, shocked.
"I never stopped."
YP sits with her arms crossed, head leaning on the window. I nudge her shoulder and offer her the first sip of iced tea.
"Please! Tell me how it looks!" I beg.
"I think your move now." YP signs as best she can.
"What? No. That's not possible," I sign.
"Why not?"
"Because the guy who was doing it was at work with me all afternoon."
_"Donovan?!"_ YP obviously says.
"Donovan did what now?" Jordyn butts in.
"Nothing," I tell her.
"Someone's been bombing her work," YP explains to Jordyn.
"Bombing?"
YP and I share a smile.
"It wasn't him," I say to myself more than anyone else.
"Then who?!" YP demands.
We all sit for a moment without saying anything. I really can't think of who it could be. Who knows me well enough? But the _who_ isn't what's most important right now.
"Describe it," I plead.
"It's not— Don't— Don't get mad."
"What."
"It's not bad."
"I have to see it."
"Take my bike." She points to the backseat and pedals her arms. "We—-wait here——-——car, right?" YP asks Jordyn, who only nods, dumbfounded by the whole situation.
I don't hesitate.
—
I pedal as hard as I can. Icy air burns my throat and stings my lungs. Puffs trail from my mouth like I'm a speeding steam engine. I don't bother with back roads. I'm just a girl on a bike, right?
I don't know what's racing faster, my mind or my legs. It's _not_ Donovan. I was so damn sure. This means they still have a leg up. They know me and I don't know shit about them. My thighs burn as I push up the hill on Cobblestone. I haven't biked since I was a child, and I'm out of shape and practice. It doesn't help that YP is much taller than I am and my feet don't reach the ground. I wobble every time I have to stop.
The overpass is straight ahead, and thankfully, the pigs have moved on to haunt someone else. _For how long, though?_ I take mental inventory. I don't have any paint on me, the Sharpie ink has faded a bit, but that's not proof of anything. I decide to risk it and bike through. That's what I came here for.
I slow down as I get to the entrance. My knees shake from biking, and from nerves. I try to swallow, but my throat is dry. Every breath stings. My chest heaves up and down. That dickwad. That asshat. That—
Genius. Goddamn it. YP was right. It's not bad. It's not even not bad; she was being kind for my sake. It's good. It's really fucking good. They elevated it. Brought it up to this whole new level of, well, art, I guess.
It was great before, but with two people working on different shifts, and more hours devoted to the piece, the more detailed and beautiful it became. Is it my move now? If it were up to me I would call this one finished. Clearly, they know their stuff.
They haven't stopped calling me out. It's obvious now that they really know me.
There should be a picture of this. But I know I can't take one, especially when the cops are snooping. Now I keep my phone and feed free of evidence. I stall another moment to look it over and really memorize it. I want this stuck in my head. Every shape, every line. The bones, the colors, the hearts, the hand...Wait.
The skeleton hand doesn't register up with the one I stenciled. It's pointing. Up. I look at the underside of the overpass. Nothing there. I walk the bike out of the tunnel, eyes locked on the pointing finger. I follow the bony index finger up again, and I see it. Screw you, Universe.
Maybe they _don't_ know who I am. I smash my face into my hands, alone in the yellow glow of my paper lamp in the basement. There's no way in hell I can pull off a heaven piece like that. And for that matter, how did they? Climbing up a water tower? _Are you kidding me?_
Nothing makes sense anymore. I want to go back to Kingston, to my little school where no one challenged me, where I didn't have to worry about anyone butting in, where I was alone and happy with my work. Never mind that no one really noticed it there.
Back then, I'd have these fantasies of the cops trying to track me down and all the ways I'd elude them. People would notice my work popping up everywhere, would wonder if it's graffiti or art. Or if graffiti _is_ art. I'd get up. I'd be a queen. All that good stuff. Now they're only snooping on my work because someone else had to come along and show off. Do your own work. Why drag me into it? _Am I not good enough on my own?_
I rub my eyes. This isn't worth crying over. _Don't be so weak._ I need more time, more supplies, more planning. There's no way my opponent is backing down, retaliating so fast and now one-upping me. It's not fair, I have school and parents and a job. I can't just drop everything to plan and paint in a day. I thought they knew me.
How do they know me? How am I going to get up to that water tower? _How? How? How?_ I sink into my chair. I have to stop thinking about it or I'm going to lose it. My legit backpack nags at me from the floor. I guess I could do some homework for once in my life.
I pull out my illustrated Bob Dylan album and flip through the pages. The pages smell like firewood. I try to imagine Katz sitting there in his house, fire going, drawing in the margins. All that work, just so I could be included. I'm not sure I even deserve it.
Little fish swim all around one page, some realistic-looking with long flowing tails, some no more than goldfish-cracker-looking doodles. I laugh. I get up and grab one of my little staple-bound sketchbooks.
There was a sale on them last year and I stocked up. I have about five or six left, blank, ready and waiting for whenever I need them. I pick one with the kraft paper cover. I'll make him a book full of _my_ language, poems in little drawings of hands and hand shapes. First, I'll draw his name sign.
I came up with it almost instantly. I can just picture the dark shade of green Casey will turn when she finds out I named Katz before her, too. But his was just too easy to come up with. I draw it spanning the inside cover and the first page. I don't like to waste any surfaces.
I have one of those cubby shelves stuffed with all sorts of art materials. Mostly dead markers, art sets from when I was a little kid, that sort of thing. I can't bring myself to get rid of any of it. I grab a shoe box full of gel pens/brush pens/Crayola markers and color in the lines. I leave parts uncolored. Too much color makes it heavy, or I get carried away and end up ruining it. I'm learning to hold back. That's something else writing out there in the world has taught me. Being efficient means being minimal. Beauty can be found in only three colors.
I like this. I flip to the next page and the stress over my rival is only a slight hum in the back of my mind. The next sign I draw is the sign for _art_. It's a good opener. I do another full-page spread.
And another.
—
I keep sketching out words until the smell of pizza drifts downstairs.
We have a thing for comfort food in our house. It's sort of a family tradition. We'll order your favorite to celebrate for you, or if you've had a bad day. Twice, if you've had a bad week. Mee and I have the same favorite: Indian takeout from Rajdhani's. Ma is very into organic food and all that, but even she has a weakness. Hint: it's pizza. So when the smell of cheese wafts down to the basement, I know something's up with Ma.
I follow the scent upstairs and into the kitchen. Two boxes are stacked on the table next to a bottle of red wine and a tallboy of Arizona Iced Tea. Mee pulls down three glasses from the cabinet and slides them next to the pizzas.
"What's the occasion?"
"We both had a rough day at work. Ma'll be home soon."
Damn, bad-day pizza means I can't fling open the box and chow down like with good-day pizza. It's their bad day, so they get to choose who should get the first slice. I decide to test the waters with Mee, get the art-class news out there.
"We're doing a street-art lesson in my new class." I pull out some paper napkins and plates.
"No, you're not."
"Well, not on the actual streets or anything."
"You're not joking?" Mee does not look excited.
"No, you know, there's a lot of cool history and stuff—"
"Of course there is. I know that. Don't you understand why that worries me?"
"You don't need to worry."
"It's like an alcoholic going to meetings at a bar." Screw this, now _I'm_ having a bad day, too. I open the topmost pizza box.
"Hey! We're waiting!"
"I can't believe you compared me to an alcoholic."
"I didn't mean it in a bad way." Right. How else am I supposed to take that comment? As if taking a class on street art would force me to do it. I don't need a class for that. I'm not addicted: it's my _life,_ not a bad habit that needs to be broken.
"Look, I think it's great you're learning about the history of graffiti and whatnot. I just want you to be careful." We both feel the door slam. Ma's home. She rushes into the kitchen, coat still on, and collapses into the chair across from me. She swings the pizza box around to face her and pulls out a slice.
"My day was hell," she signs while chewing. Another bonus of knowing sign: you can talk and eat at the same time without being totally gross. Mee takes Ma's coat off and rubs her shoulders before grabbing her own slice and joining us. The plates I brought out remain unused. We just scarf it up.
"First, these parents wouldn't let up about having a gluten-free classroom." Ma teaches the kindergartners at Kingston.
"No pizza for the kiddies, then." I shove another bite into my mouth.
"I mean, it's their choice, that's fine. I can keep gluten-free snacks around, but I can't impose them on the whole class."
"Separation of bread and state," Mee quips.
"And then"—Ma grabs another slice—"these parents tell me I should act more 'reasonably' because I've chosen an alternative lifestyle for myself." Ma wiggles her left ring finger, and Mee almost chokes on a piece of pineapple. (Hawaiian pizza is the king of pizzas.)
"Fuck 'em," I say. It's my mantra.
"Language!" Ma still manages to smile.
"What's wrong in the shop? People demanding gluten-free acupuncture?" I ask Mee.
"No, it's all this paperwork. I'm drowning in it."
"Paperwork for what?" Ma asks, wiping some grease from her cheek.
"Oh,...um...it's for this thing, for the shop. Trying to...ah...do some renovations." The pizza acts as a conversational sedative, and we all calm down between bites of ham, pineapple, and heaps of delicious gluten.
YP waits for me at my locker before class this morning. She texted last night, asking how I was doing, but I couldn't come up with an answer. I just didn't know.
"OKOK?" she asks, picking up on the way I sign the phrase. I waggle my head noncommittally, not really sure if I am OKOK. I wave for her to walk with me to class. I don't actually need anything in my locker anyway.
"You think painting was really bad?" YP signs most of the words.
"You know it _wasn't bad._ "
"So why you—" She looks over her shoulder and stops walking. Kyle Stokers walks by us, arm slinked around some girl, hand crammed into her butt pocket. _Gross._ They're laughing. Arm Candy's covering her mouth, so I can't see what she's saying. I have a feeling it's not anything nice, judging by the look on YP's face.
"Uch. Let's go." She loops her arm through mine and we start down the hall again. It's strange. I thought the sight of KFS and Co. would bring on YP's waterworks. But she looks pissed. And it looks good on her.
She speeds up, walking faster in order to overtake him. Once we do, she turns to me and signs while walking.
"D O U C H E B A G."
"I couldn't agree more!" I sign.
"How sign 'asshole'?" she asks with a smile. I go right ahead and show her.
—
"Maybe YP can tutor you after—" Casey is droning on, as the three of us leave history class together.
"I'm not sure that's, like, the best idea." YP thankfully cuts her off.
"C'mon, Casey, I'm not that bad."
"What's going on?" Casey signs. A crowd blocks off the hall in front of the main office. YP and Casey both jump as someone slams the door to the office from the inside, keeping out the onlookers. Casey seems more curious than either of us and picks up the pace.
Shutting the door didn't do much to stop the crowd. The office is lined with windows facing the hallway. YP and Casey try to pick up on the students' whispers, but I'm still in the dark. Instead of looking into the office, I scan people's faces. Lots of concerned brows, some slack-jawed staring. Everyone seems to be asking the same thing: _Who?_
YP is the first of us to get a glimpse inside the office and makes a quick 180 to face me.
_"Go. Class. Now,"_ she signs, and runs off without another word. Casey and I elbow our way to the window just in time to see a fully uniformed police officer sternly shut the blinds.
Everyone in Room 105 sits quietly, waiting for Mr. Katz to show up. Casey asked, but none of the students know where he is.
"What happened? Anyone know?" She's asking about the cop. _No one's going to talk to you, Casey. No one's going to spill it to a teacher._
"I heard————graffiti." Freckles proves me wrong.
"Where, here at school?" Casey asks, eyes bugging out behind her glasses.
"Nah,—-——think——kid here did it," Black Shirt adds. He mumbles, his face is deadpan, I can't read it. I wave to Casey and ask her to start doing her job so I don't have to lip-read. She blushes and quickly starts interpreting for Freckles.
"That's so stupid. No one here is _that_ good. Have you seen it?" I try to take her comment as a compliment, but she's so snotty about it, I can't tell if she's being sarcastic.
"Yeah," Black Shirt answers, "that skeleton is sick!"
_The skeleton!_ What about the rest of it?! My ears burn hot again. All of our sketchbooks are ready, but there's nothing to draw. If the cops are here about the overpass, I doubt we're going to continue on with the street-art unit. _I can only hope._ The door swings open and Mr. Katz scrambles in, his hair falling wildly across his eyes.
"So sorry, class! Let's talk." He pulls a stool over to his podium and takes a seat. Everyone in the class watches him expectantly. He taps his foot and opens his mouth, but says nothing. He has a few frustrated false starts before he finally begins.
"Who can...ah...tell me the difference between graffiti and vandalism?" I'm not going to be the first with my hand up.
"Is there a difference?" This question gets more responses. Everyone nodding, saying yes, there is a difference.
"So, what is it?"
"It's when you spray-paint something that's public property," the toy to my left answers.
"Is that vandalism or graffiti?" Katz raises his eyebrows.
"Oh, um..." She doesn't know.
"Graffiti elevates," I sign. Casey interprets, and all eyes are on me. _Shit._
"Meaning?" Katz asks.
"I guess I'm saying that vandalism doesn't add anything to the world. It's all bravado."
"You don't think there's ego in street art?"
"That's not what I mean." I'm having a conversation. With a teacher. In a mainstream class. No one is laughing. My pits start to prickle with sweat. "Street art—graffiti—adds something to the world, something that makes you think, that makes you stop and notice something you might not have noticed before. Vandals tag for the sake of putting their name on a wall. Their intentions are crappy." Casey even says the word _crappy_ verbatim. I can tell by the expression on Black Shirt's face.
"Do you think street art should be legal?" Katz breaks eye contact, directing the question at the whole class.
"Sure, if you get permission," Freckles chimes in.
"Yeah, there should be some sort of system in place," Black Shirt says. Funny that our token teen anarchist is calling for a system.
"Why?" I ask him. He hesitates, looking back and forth between me and Casey. He addresses her instead of me.
"The art can be in, like, specific places. Not randomly in your face or wherever."
"But there are billboards and advertisements everywhere in my face. No one asked for _my_ permission." He looks confused. I press on.
"It's corporate vandalism, if you ask me. If they get to do it, why can't—" _Shit shit shit shit shit shit._ "Why can't...um...artists do it?"
—
The blinds are still drawn when YP and I pass by the main office again. I'm starting to realize what a problem my custom-painted car might be. Walking through the parking lot, I remember painting Lee. How hot it was, Jordyn hanging out in her two-piece, winking at the guys who walked by as I worked. When I finished, I knew. I knew there was no going back. I had found my place. This was my art.
> YP: Youll raise red flags if you paint over it.
>
> JULIA: You think?
>
> YP: People will wonder why
>
> JULIA: True
YP gets in shotgun. Her dad baked up a new pie recipe, and I begged to be a taste tester. She wants to talk to me about my next move, but it feels weird, talking about something that used to belong to me, and me alone.
> YP: No talking plans round my dad okok?
>
> JULIA: ofc.
She opens the front door to her house, and again the smell is heavenly. This time it's toffee, butter, something bitter. I love my parents, I really do. But this house makes me dream of what it would be like if Ma could bake something that doesn't come out of a tube.
"Julia!" I can feel her dad's deep voice in the air. I wave hi.
"I——good news!" he exclaims, pulling down plates. "Diane—-—--you can do Cheer tryouts—-—-problem." A slice of pecan pie lands heavily in front of YP.
"You did what?!" YP's hands fly into the air. She crosses the kitchen, shouting and gesturing, facing away. I can only see the hurt look in her dad's eyes as she stomps off to her room.
"Sorry,—-————," he mumbles to the ground before excusing himself. I'm sure the last thing YP wants right now is pie, but I bring two slices on my search for her bedroom. I follow her upstairs.
I can feel the floor vibrating as I get closer to her door; she's blasting some sort of music. Her door wrapped in that "POLICE LINE DO NOT CROSS" tape. I don't bother knocking. Police lines never stopped me before.
WHAT I EXPECTED YP'S ROOM TO LOOK LIKE:
Pink. Pink everywhere. Posters of dazzling blond pop singers. Pom-poms hanging from the footpost of a white bed with swirling ironwork, adorned by frilly pillows and a marshmallowy down duvet. A little vanity table cluttered with makeup and earrings dangling from a stand. Topped off with a layer of stuffed animals as far as the eye can see.
WHAT YP'S ROOM LOOKS LIKE:
White. White everywhere. The walls are white, the bed is white, her curtains are white. She doesn't have a single poster, of a pop singer or otherwise. The only picture hanging on her wall is the one I drew of her the other day. Not even a family photo sitting on her white dresser. Even the speakers she's blasting music from are white. The only bit of color is that yellow tape on her door. And her, sitting on the edge of her bed, chewing her fingernail.
—
"What are you doing?" she asks without fingerspelling.
"Bringing you pie?"
"No, what you going do? What next?"
"What was that all—" She cuts me off, waving the situation out of the air.
"Not matters, we have work to do," she tells me.
_"We?"_
"You——think I'm letting you——-next—-alone,—you?"
## GRAFF OATH
# FUCKING BINDING AND NON-NEGOTIABLE
> I HEREBY PROMISE TO KEEP MY LIPS LOCKED. THIS MEANS:
>
> NO TELLING KIDS AT SCHOOL
>
> NO TWEETING
>
> NO FB
>
> NO INSTAGRAM
>
> NO BRAGGING
>
> NO PARENTS
>
> NO COPS
>
> IF SOMETHING GOES DOWN I WILL RUN AS FAST AS I CAN TO THE MEETUP SPOT. I WILL NOT RUN TO THE SPOT:
>
> IF THE COPS ARE TAILING ME
>
> IF SOMEONE IS WATCHING ME
>
> IF THE RUNNING DOESN'T WORK AND I GET CAUGHT:
>
> I GOT MY LIPS LOCKED UP. NO NAMES
>
> MY MOUTH STAYS CLOSED
>
> NO NAMES
>
> COPS CAN'T CHARGE YOU WITH ANYTHING UNLESS THEY CATCH YOU IN THE ACT. ADMIT NOTHING.
>
> IF I GET CAUGHT I REALIZE THAT IT'S PART OF THE GAME, AND I WILL TRY NOT TO TAKE IT OUT ON JULIA.
>
> JULIA SWEARS TO DO EVERYTHING IN HER POWER TO MAKE SURE WE GET IN AND OUT SAFE, AND HOME IN TIME FOR PIE.
>
> SIGNED
>
> * * *
She signs the paper without hesitation, not a question about one single bullet point. She doesn't make a fuss about what could happen if she gets caught. That's how I know she's really up for it. We burn it in her fireplace before clinking mugs of cocoa that her dad made by way of apology.
I feel better. Coming clean to YP about my whole deal means I don't always have to have my guard up. I know I should be more careful, but right now it feels like all my bases are covered.
Making my next move seems slightly less terrifying, knowing at least I have someone to keep my 6. Someone I can _actually_ trust. Now all I need is more paint. And, you know, a game plan.
I drive right past the big blue whale-shaped school the next morning, checking up on the underpass. I wonder if the vandal squad has painted over it yet. Heh, the vandal squad. I wonder if they even _have_ a vandal squad in the suburbs. Hard to believe they would need one, before now.
The cops aren't there this morning, but someone else is. I slow down a bit. Someone is taking pictures of it, with an old-school Polaroid camera, of all things. The photographer turns to leave the tunnel right as I drive past. Four big brown eyes lock onto each other for what seems like the longest second of my life. I blink and I'm under the sky again, leaving Mr. Katz, pockets full of pictures, dumbfounded in the tunnel.
Static. That's all my brain is capable of processing right now. After seeing Katz in the wild, I let my body switch to autopilot, and here I am in history class, unable to think about anything. My brain buzzes with fuzzy, scrambled images. Every thought I have is on some sort of weird delay, like when pixels on TV can't keep up with the video feed. Broken. _My brain is broken._
Buzz through history, buzz through the halls, buzz through getting changed for gym. YP must know something is up; she's keeping her distance today. I can feel her watching me. _Do I tell her?_ My brain won't allow me to think about what I saw for too long, yet it seems like it's the only thing I can think about. It makes so much sense, and no sense at all. My brain is pins and needles.
I've almost made it to winter break. One more day of classes until we are off for the holidays, and I'm spending it in a haze. Ms. Ricker has decided that our last gym class should be something "fun." She hands each of us a little device from a plastic basket. It looks like a cheap digital watch.
"Everyone clip the——-——–-your———-okay?" she bellows, holding up one of the devices and clipping it to her shorts. I follow along, still on autopilot.
"We'll have a contest!———-——-—-most steps wins! Start running ------- go!" I was too busy staring at the ceiling to notice when she announced for us to start, and YP tugs me along on her second lap of the gym.
The little display counts up by one with every step I take. I like watching how fast the numbers climb. Ten, twenty-two, thirty. I jog around the gym with everyone else, letting the static wash over my thoughts until there's nothing but the numbers and the pounding of my heart.
YP is fast, much faster than me. I should be keeping in shape; I should be as fast as she is. How am I supposed to outrun the law at this pace? Maybe I should buy a bike. She laps me again, determined. She doesn't look down at her screen, she only looks ahead, staring at some imaginary finish line.
Kyle Fucking Stokers runs up beside her and says something that only makes her run even harder. He stops running altogether. I pass him and smirk over my shoulder. _Ha! Yeah, that's right, leave her alone._ The look he shoots back at me is almost enough to trip me and cause a six-kid pileup.
I'm excellent at reading facial expressions; they're an important part of my language. YP only signs with her hands. She doesn't have a grasp on her face yet. I understand her without expressions to read, but it's like she's speaking with an accent. It's like watching someone dance but they don't move their arms. Awkward.
KFS's expression isn't one I've seen before. It's not pity, not the look you'd give to a wounded animal, a look I'm used to getting daily. It's an odd mixture of pure hatred and hopelessness. It asks: _What are you trying to do?_ What _am_ I trying to do? Why is _he_ asking me?
—
"What's up with you today?" YP signs back in the locker room. She's flushed after winning the mini-competition.
"I'm worried, I guess." I pull my black sweater over my vintage, thrift-shop Keith Haring tee.
"About—" She looks over her shoulder, forgetting no one else in here can understand us. "About W A T E R T O W E R?"
"DUH," I sign, and stick out my tongue.
"I have plan. Urban Café Sunday?"
"Not now?"
"No, no, no, I have-—-——- I need to get———-." YP opens her prize for winning, a chocolate protein bar, with her teeth and takes an enormous bite. She smiles at me, mouth full, on her way out of the locker room.
—
Mr. Katz doesn't show up for his class.
On my way to work, my focus sharpens. Everything that was covered in static becomes clearer the farther I drive from school. The street-art unit, the paint in the supply cabinet. He's seen my car, he's heard the song. He knows it's me, and I _know_ it's him.
Taking Polaroids, showing up late, not showing up at all. He's revealing his hand. And after this morning, it might as well be painted red. Maybe that's why he didn't come to class—he couldn't face the fact that his rival is onto him.
Part of me wants to text YP right now, tell her what was really going on this morning. I was fine blowing up Donovan's spot when I thought it was him. But Katz? He could get in real trouble. I guess we both could. So I'll keep his secret as long as he keeps mine.
I want to take it as a compliment, that a teacher I look up to decided my art was worthy of the conversation, but I can't get my gut to agree with my brain. Just because he's a teacher doesn't make him better, doesn't give him the right to my art, my walls. He can't insert himself into my conversations like that. I didn't give him permission. Permission. _Fuck._ That whole street-art talk, all that crap about permission. Was he trying to absolve himself?
_See, Julia?_ Katz asks in my head. _No one needs anyone's permission, you said so yourself. Your art's open season. Let me show you how it's done._
I change into my fry-girl uniform and lock away all my stuff. I haven't talked to Jordyn since that night in my car, so when she sashays into the locker room and sees me standing there, she does a little double-take.
"Coming or going?" she asks.
"Coming."
"So, you _are_ back at it. Why didn't you tell me?"
"Are _you really_ asking _me_ that?"
"But you told that girl." I don't respond. She won't understand, I've been anti-hearie, anti-implant, anti-friends, anti-everything for so long, so how do I explain that this random, bubbly cheerleader happens to _get_ me? I hardly understand it myself.
"She's fat. And weird." Jordyn wiggles her fingers under her nose, grimacing.
"Shut the fuck up. You don't know her."
"There's something not right there."
"I'm done talking to you."
"She's hiding something." She puts her visor on and leaves without letting me have my say.
I know YP. We've talked more than enough for me to piece it together. The pies, the weight, the cheerleaders, Kyle. It's not exactly an uncommon story. She hasn't explicitly said what happened, but she doesn't need to. The fact that Jordyn spent all of an hour with YP and feels entitled to judge her reminds me exactly why I didn't tell her that I'm "back at it." Why I'm never telling her anything again.
—
When I get to my station I see that Donovan is planted in the drive-thru. Great, another third-wheel shift. I throw down my first batch of fries and hope the steam is so hot I evaporate with it. Jordyn is still pretty protective of him, too, winking and blowing kisses whenever she catches his eye. Making a big show of their relationship, just in case I forget. He looks worn out, exasperated. It's a familiar expression. I've seen it on Jordyn's face dozens of times. Once again I'm thankful for my deafness. It's much easier to ignore Jordyn's giggling face than the actual giggles.
The oil is sparkling, rolling, bubbling, beautiful, yellow. That color would look sweet on the water tower, if I could only find a way to get up there. I'm actually excited that YP has thought of something. I expected to be put out, but I've never seen her so revved up before. Never seen her shove food into her mouth with a smile. Maybe she'll feel better soon, cry less. Learn to not give a fuck. Like me.
Golden, foamy, deep-fried letters standing out against the mint green of the water tower. A grease stain you can't wash out. I'm HERE, Katz. _Now what?_
—
My socks are soaked through with sweat when I peel them off after my shift. Donovan comes into the locker room and flicks the lights to get my attention. He has it. He leans against the door, nervous. His hands tremble as he reaches into his pocket and waves for me to come over. I pad over the cold tile floor in my bare feet.
I arch my eyebrows, miming, _Yes?_
"Here." He pulls out a little package wrapped in silver paper, nearly drops it on the ground before putting it in my hand.
"Merry Christmas," he mumbles.
"It's not—"
"Open it," he demands, still barricading the locker-room door with his back. I peel the tape off. It still has the little plaid Scotch tape–header piece on it. The small present is heavy, and wrapped without a box.
"Hurry up!" he motions, paddling his hand in a circle. I slide the contents into my hand.
Magnets. It's a column of plain, black, round magnets. I look up at him and he's giving me his best Donovan drive-thru megawatt smile.
"Thanks?" is all I can sign.
"—-for——paint, you liar."
"What?"
"You know," he signs haltingly, trying to remember the right hand shapes, "how I know about you?"
I shake my head. My face feels hotter than it gets over the fryer.
"——cans rattle," he says.
"No!" I cross my arms. He's wrong. He has to be. I always holster each one in a loop, to avoid that very thing.
"—-———-inside the can, you know,——you shake it?" Donovan mimes shaking a can.
_Holy shit. I never..._
He cracks up. I'm sure the look on my face is priceless.
"Put them on the bottom of the cans." He acts out the motion with an invisible can, when suddenly the door opens a crack, pushing him forward. He shoves me away and jolts to his locker before Jordyn tries to open the door again. The three of us change without so much as a sign.
I wait until Jordyn and Donovan leave before pulling my black bag out from under my car seat. Damn, here I was dreaming up some masterpiece in yellow, and all I have is the near-empty Katz cans. Black, blue, and purple.
They're freezing from sitting in the car; can't be good for the paint. I shake up the purple can. I can barely feel the little ball sliding back and forth in there. It's that noisy? Noisy enough for him to hear it through my bag. _Shit._ I shake it up again, trying to feel the sound, putting it up to my cheek. It's barely there, and I might be imagining it. I've always seen _[keys jingle],_ but _[paint rattles]_? Never.
The magnets are nestled in my coat pocket back in their silver paper wrapping. If this had happened a few weeks ago, it would have made sense: another tipoff to the role I thought he was playing in our game. Now, I'm confused as hell. How would he even know about this little trick?
I can't help but think it's pretty cute. The silver paper, the tape. If Jordyn was jealous before, I can't imagine how she feels now. I wonder if he got _her_ a Christmas gift. I'm supposed to be backing off. I already broke up with him in my head. It seems impossible to break it off with him now.
It's getting late. The magnets snap to the bottoms of the cans. I shake them one by one, hoping that's enough to guide the metal balls to the magnets. It's brilliant. Whatever sound they made before, they certainly won't make again. _Stupid sound._
Ma is asleep on the couch when I get home. I hate bringing my stash anywhere near the house, but I can't leave it in school unattended, especially with the cops snooping around. Mee waves from the top of the stairs.
"Come come come on!"
"OKOK." I kick off my shoes before joining her.
She leads me into my room and sits on my bed next to a yellow folder. She's practically bouncing with excitement.
"I got you a present!" she finally announces. _Two presents in a day?_
"For what?" I furrow my brow.
"Just, because." She can tell I don't buy it, so she continues. "Because you made it through the semester. I know it wasn't easy." My stomach churns. She has no idea. It wasn't easy, not because of classes, but because of stencils and rivals and stolen paint. And now she's rewarding me for—what? For going behind her back, after I promised them both it was over? I wish she would stop telling me how good I'm being.
"Here." She hands me the folder and claps her hands together. I don't want to open it. I don't deserve whatever it is.
"Go on," she urges.
I stick my fingers in the folder like I'm holding my place in a book. I should hand it back, tell her everything. My stupid hands work against me and open it anyway. Paperwork. I flip through the sheets, all headed with the words _QUEENS COUNTY._ The more I read, the more excited Mee gets and the worse I feel. I can't believe she did this. She got me a wall. The biggest side of her acupuncture studio. A legal wall for me to paint on whenever I want. And what have I done? Lied to her. All damn year.
I'm going to throw up.
I tell Mee a thousand times how great it is, how happy I am, it's so perfect, thanks so much! I'm praying she can't tell as my stomach somersaults. I hug her out of my bedroom before heading to the only room in the house with a lock, and hurl up a small fry and Coke.
My forehead rests on the toilet seat. I'm not getting up until I'm sure I'm finished. I feel my phone buzz in my pocket; it's YP.
> YP: omgcheck yr email.
I close out the message and open up my mail. She's sent me a link with no subject line or any other info. It leads to an article:
TEACHER TRYING TO SAVE STREET ART
In the small Long Island town of Greenlawn, one art teacher doesn't want to clean up his neighborhood's graffiti—he wants to preserve it. Andy Katz, 31, a teacher at Finley High School, has been in talks with City Council and the Greenlawn Police Department for the past week, defending the graffiti as works of art and attempting to bar the city's typical cleanup procedures.
The first recipient of Katz's preservationist attention was a mural on the back of a scoreboard at Tri-Village Field, depicting a whale and its skeleton. After meeting with Mr. Katz, the owner of the park, Cliff Ferguson, has decided to preserve the "art."
"I think it's interesting," Ferguson said about the mural. Because the graffiti was created on his property, if Ferguson chooses not to press charges, law enforcement must drop the incident.
A second work of graffiti that local police say was painted by the same individual appeared recently on the Spring Road underpass. It depicts a woman with seashells for eyes, similarly overlaid with a skeleton. It's been painted on public property managed by the Suffolk County Department of Public Works. Local government, however, is adhering to policy in responding to the incident.
"You vandalize city property, you're going to have a bad time," Watch Commander Cox quipped in a phone interview. He was also quick to dismiss the efforts of Mr. Katz to preserve these works. "I get it. Artists like to stick up for their own. But when it comes down to it, it's not real art, is it? It's the defacing of public property that costs the city money to clean up."
Besides presenting his impassioned view in a City Council public hearing last Thursday, Katz has filed a petition with Suffolk County to prevent cleanup of the Spring Road mural. He is also "investigating [his] options" for obtaining an injunction against the DPW. According to Katz, the fight is not over. Whether it is an eyesore or an artwork, at this moment, the Spring Road mural is still there.
I hurl one more time for good measure.
—
Underneath my quilt, the light shines through the squares like stained glass. I'm supposed to want this. Public reaction, someone saying they want to actually preserve a piece. This should be the happiest night of my life.
What an ego, that Katz. Let your graff go. Don't babysit your writing, don't take pictures, don't talk to the fucking press! Oh, and how about, don't bring your girlfriend into it? And while I'm at it, don't date your student's interpreter! You're a great artist, but your toy is showing.
City property. This is the problem with wars. Considering the article and the cops, I wouldn't be caught dead bombing public property twice in a row. Hit up the side of a shop or something.
It's too hot. I can't breathe. It stinks under here. I need a shower. I need a plan.
I need YP.
> JULIA: Come over?
The hall lights flash off and on; she's here. I step into my slippers and go downstairs. Mee is already standing in the doorway, YP slowly signing introductions. Mee asks YP to take off her shoes. I must have looked exactly like that when her dad answered the door for me. Scared out of my mind, totally out of my element. Mee turns around to get my attention but I'm already standing behind her.
"You didn't tell me you made a new friend!" Mee smiles uncontrollably. YP and I smile, too.
"It's late; is everything okay?" she asks, addressing both of us. YP picks up on the _okay_ bit and signs that she's fine.
"We're having a sleepover," I explain as I pull YP past Mee and up to my room. My mom stomps her foot when we reach the top of the stairs.
"Love you."
"Your mom deaf, too?" YP sits down on the floor in front of my bed.
"Yep. Both of them are," I sign.
"Wait, both? Both what?"
"Both moms."
"And they are deaf, and you are deaf?"
I nod.
"How?"
This is the problem with me and YP. It's nothing I wouldn't tell her—I mean, she already knows the real secret stuff. I want to be able to tell her, to talk to her like we both speak the same language, not to have to reach for a phone or paper. Every big conversation has to be a struggle.
She waves me back from space.
"Sorry," I sign. "It's C O M P L I C A T E D."
"Here." She grabs my laptop off of the bed. I roll my eyes. Always with the typing.
"Don't be stupid." She hits her forehead, her fingers in a V shape. Pants opens up the laptop on the floor and lies down in front of it. She waves me down to join her. I open up a Word file and let my fingers fly, English grammar be damned. YP won't care.
> My moms wanted baby right? So they decided they would get a donor and have one that way. Ma was working for tenure so Mee say that she will have the baby. So they had this friend who's Deaf give them...the stuff right? And blah blah I was born.
>
> SO like, if you're both deaf you'll have a deaf baby?
>
> No, its a chance. They not know for sure that I would be Deaf. They say even if i was hearing they wouldnt care, they just wanted a baby. But I turned out to be Deaf.
>
> Are you mad?
>
> for what?
>
> Maybe you wouldn't be deaf if...
>
> I like being Deaf.
>
> Oh.
>
> Im big D deaf btw.
>
> What?
>
> use the big D Deaf not little d deaf.
>
> Theres a difference?
>
> yeah bc being Deaf is part of who i am. Im proud of it, I have a community. Im part of it.
It's so hard to explain, I don't know how to make YP understand the ins and outs of Deaf culture in a way that would make sense to her, or any hearing person. Ma is better at this stuff. The whole "it's not hearing-loss, it's Deaf-gain" thing. I'm not a spokesperson for us, but I do love our community. I know how to explain it in sign, but not so well in English.
> Thats really cool!!! I had no idea :)
>
> you seem like youre feeling happier.
>
> I feel like...like I'm me again. But better.
>
> even without cheerleading or whatever?
>
> Cheer wasn't me.
>
>
>
> SERIOUSLY!!! I liked it because it meant I had friends.
>
> but they're so mean!
>
> I know, but I didn't care! I didn't have to be alone.
>
> Alone not so bad.
>
> It gets old.
>
> so you're done with them? all the cheer girls??
>
> Yes. I feel like, I'm back...I'm here.
I pick at the carpet. I think YP senses something is off. Her eyes dart around my room like she's looking for something new to talk about. I rip out a thread when she finds the Jordyn box.
"Ooh, ex-boyfriend stuff?" she signs, and shakes the box like a kid with a present.
"Ex-friend stuff." I snatch the box away from her.
"—-I see?" YP scoots next to me. We lean against the edge of my bed.
"Sure, I guess." I give the box back to her and let her open it. As soon as she does I can't help but laugh. It's unexpected and I'm caught off guard, but it all seems so silly now. I was moping over what, a selfie stick?
"Oh, it's just stuff." YP reads my mind again. "I thought———-be more, I don't know." We lay it all out on the floor and finally have a real sense of what my friendship with Jordyn was about. Convenience. Outcasts banding together so we wouldn't be so alone. And for a while that worked, but we're just too different. And not in that good, "opposites attract" way. Now I see that she sort of glommed on because I didn't judge or shame her. I'm sure that was comforting for her, and I don't think she's done anything to be ashamed of, but what exactly did she add to my life? A shoe box full of random stuff. It's hard to believe I was so heartbroken over it.
"I'm happy I met you," I sign to YP, and she practically glows.
"Same." She uses the right sign, sliding her hand in the Y shape back and forth between us. "Oh, I———forgot! Check——out!" YP stands up and unzips her quilted duffel bag and pulls out two yellow vests. On the backs, the words _SUFFOLK COUNTY_ are stenciled in black.
"I have jumpsuits, too!" She reveals a pant leg from the bag.
"Isn't this going to make us, you know, stand out?" I pulse my fingers like flashing lights.
"Nuh-uh." She shakes her head, her bangs swoosh around. "————-look like the city hired——fix it."
"That's brilliant!" I sign, and flop onto the bed. "But—" I sign to the ceiling. She's really thought this through; I'm seriously impressed. The effort that went into the outfits...
"What——?" She flops down next to me.
"I don't know if I can do it."
—
Someone is shaking me along with my alarm. I open my eyes and a wide-eyed YP points to the bed over and over again.
"Where is it?!" She blushes and covers her mouth, waiting for my response.
"What? Where's what?" I sign.
"Your... _you know._ " She shakes her fingers, then starts spelling, "V I B R A–"
"No!" I cut her off, pinching my fingers. "Are you crazy?"
"It must've switched on—————-," she says, lifting a corner of my quilt.
"I don't have a—" _Oh._ I'm finally awake enough to realize what's going on. I slap the button on my alarm clock and the bed stops vibrating. My bed alarm is intense. I had to get one called the Ultra Shaker. It was the only alarm annoying enough to actually get me out of bed.
"My alarm," I tell her. She doesn't catch the sign, so I spell it out.
"Jeez,——- I thought——————an earthquake or————-. Then I thought...um...well. Yeah."
She _is_ crazy. I comb out my hair with my fingers before tucking it under my beanie.
"So? You ready?" She picks up the duffel bag.
"I told you, I don't think I'm ready. I don't think I can—"
"Uch, don't be all like that. Get dressed; we're going out." She skips into the hall with her toothbrush.
Since when is she in charge? I know I _can_ do it; that's not the point. This whole mess hasn't given me a second to think. They tag back too fast. Stealing paint, hitting city property, and telling people—it doesn't feel right. I like to plan, take my time. Be thoughtful, not reckless. It feels messy. YP doesn't care though, she wants me to see it through. I really don't know why she cares so much. Not just about this, about everything.
"Where are we going?" I sign with one hand and swipe my MetroCard with the other.
"I read——-——place on Google." I want to tell her you don't read about anything on Google itself, but whatever.
It's freezing this afternoon up on the elevated platform for the 7 train. It hasn't snowed in a few days, but gray drifts clump to the bases of trash cans and benches. It's disgusting. The snow over by Finley stays beautiful and clean for weeks. Not this nasty, gritty sludge that coats the entire borough of Queens.
"You drive long to go F I N L E Y every day, huh?" YP signs.
"I guess. Didn't have a choice."
"You go to the city a lot?" she asks.
I want to lie and say, "Of course, I'm always there. Definitely not spending all my free time at McDonald's and in my basement." The train pulls up and spares me from answering her. YP sits on her knees facing the window, her finger running across someone's scratch tag.
"You like?" she asks.
"No." I shake my head, and sit forward.
"Y?" she signs, using shorthand of her own invention. Sort of like texting, she'll sign only one letter instead of the word. It's really wrong, but it's pretty cute.
"Trashy. Ugly. Doesn't _mean_ anything," I tell her, and she smiles.
"Come on, it's our stop." YP walks to the door before the train stops, like a true New Yorker.
"Already?" I stand up to follow her, and as soon as I glance out the windows, I know where we're going.
—
It's 5 Pointz. The Institute of Higher Burnin'. Anyone who's ever dreamed of bombing a wall in New York knows about 5 Pointz, the giant yellow graffiti Mecca. I've never done any writing on those sacred walls. My work is better than ever, but I know I have a ways to go before I can throw up anything worthy there.
We walk up Jackson Avenue, the building towering over us, colors spilling down every surface. Painted, bombed, muraled, tagged, and painted over and over again. Paradise.
"I lied." YP stops walking and looks down at her feet. "I not Google it."
"So?"
"Mr. Katz took a few of us here last year."
"You went?" I ask. She nods yes, frowning. "I don't understand."
She takes out her phone and starts texting, even though we've been doing great without it today.
> YP: Dont be mad.
>
> JULIA: ok what
>
> YP: He took some kids from his class
>
> JULIA: ok so and?
>
> YP: I was in his class.
>
> JULIA: art class?
>
> YP: yea.
>
> JULIA: you do art?
>
> YP: used to. for fun or whatever.
>
> JULIA: why you think I'm mad?
>
> YP: It's your thing.
>
> JULIA: I dont own all art. katz class is fun.
>
> YP: I know.
>
> JULIA: too bad we not in class together.
>
> YP: yeah. anyway. I thought you should see this place. get inspired
>
> JULIA: it might be working ;)
YP puts her phone away and points to a chain-link fence blocking the entrance to the building. We both grip the fence, our fingers curling around the links, and read the official sign posted in front of us.
# NO TRESPASSING:
# VIOLATORS WILL BE PROSECUTED
# COMING SOON:
# THE BALLSTON
# LUXURY LIVING IN THE HEART OF QUEENS.
"No way," YP looks back up at the building. "They can't do that, can they?"
"Fuck no, they can't."
We run along the fence looking for a break in the chain link, anywhere we can squeeze through. Turning the first corner, we see the fence is secure. It extends around the entire perimeter, down to the ground and at least six feet up, and is topped with barbed wire. I slow down and walk along it, my fingers brushing the links.
"Maybe———keep—-——, like artsy condos?" YP suggests, and I give her the most skeptical face I can muster.
The fence rattles against my hand. We both look down the length of it. There's a man trying to get over the barbed wire from inside the fence. He's put his black blazer over the barbs to protect his hands, but his tie snags on the way down. He wildly swipes at his caught tie with one hand, hanging on with the other. I grab YP's arm as we both run to meet him. I throw her my bag and climb up alongside the man.
"Go,—-, get—-—-ere!" He swats at me, trying to keep me from helping him.
"Stop struggling!" I use my voice, and he looks at me, startled. I reach up and pull his suit jacket from the barbs. It comes free cleanly and I toss it down to YP. I climb a little higher to get within reach of his tie. I think he starts talking again, but I don't look back at him. My gaze is fixed in front of me—past the tie, past the fence, to the wall straight ahead.
It's the most beautiful throwie I've ever seen. A spray can turns into a tower on fire that turns into a blossoming tree. Little pink flowers bloom from charcoal branches.
YP shakes the fence, snapping me back to reality. She signs for me to hurry up and I focus on the stranger's tie, the end of which is splattered with droplets of pink paint. As soon as I unhook him, he jumps down to the pavement and starts running. YP takes off after him, and I follow behind.
The dude is fast, but YP might be faster, even while carrying his jacket and both of our bags. She's gaining on him, and I'm struggling to keep up. _Run faster!_ My Doc Martens dig into my ankles and my run turns into more of a skip, which turns into an all-out _splat_ as I turn the corner. _Damn ice._
I can't see YP or the Suit anymore, but she ditched my bag in the alley. I guess it was weighing her down? The alley branches off onto at least three streets, all bustling with people. I try to read expressions, see if anyone looks shocked or is watching the runners, but everyone is preoccupied with their phones, or work, or whatever.
I'm not rolling the dice on this one. The odds are stacked against me. YP will head back to 5 Pointz eventually, right? That's what I would do, I wouldn't just abandon her. My socks have slipped down from my ankles and only cover my toes; I unlace the boots and hike them up into place before heading back the way I came.
No point in running. I take my time. I can't get the image of the stranger in the suit out of my head. The look on his face...He was so _angry._ I was only trying to help. His dark skin and wild hair, the crinkles around his eyes. Not really old, but older than I would have pegged a bomber brave enough to jump barbed wire, and in the middle of the day. Who does that?
I know who. I just can't believe it.
What's he doing in Queens? If that's him, everyone's got it wrong. I always held out hope that Banksy was a woman. But I'll take a dude with an Afro in a business suit over an old pasty guy any day. No one would believe me! I barely believe it myself. In fact, I wouldn't at all if I hadn't seen his graffiti. There's imitators of his work, sure, but there's no faking his style.
I check my phone. No texts from Pants, no missed calls. I go to open my mail and there's a tap on my shoulder. I jolt and the phone slides from my fingers and smashes into the ground.
"Fuck!" I yell, pick up my phone, and turn around.
Don't. Don't move. Can you move? Can you breathe? What are you doing? Say something! Hello? Say something! Lift your hands and tell him you know. Don't just stand there. Are you dreaming? There he is, right there. So close you could...what? Touch him? _Do not touch him._ When did the sky get so pink? How long have you been standing there? Say something! He smells like aerosol and dynamite. His eyes are black; the grin he cracks is even blacker, devilish and dark.
_What? What? What?_ I can't understand him, it's like he's speaking another language. The mouth shapes aren't making sense. He's talking too fast. My stomach turns over, I'm missing it. _He's talking to me and I'm missing it._
He smiles again and leans in. He places a hand on my shoulder and whispers in my ear. I can feel the hum of his voice, his warm breath snakes in and tickles a spot in me I didn't know was there. He steps back and all I can do is shake my head. I don't break eye contact. YP turns a corner, a block behind him, and stops in her tracks.
The Suit's smile vanishes. He points to his ear and raises one perfect eyebrow. I nod my head, raise my arms. _Oh, fucking well._ He reaches out and puts his thumb in the dimple on my chin. Then, he stands straight up and he signs:
That's it. He straightens his tie and walks off.
—
"What—-—say?!" YP huffs, on our way back to the 7 train.
"I don't know," I sign. "I couldn't understand him." She looks disappointed. She has only her bag now; she must've ditched his coat in the chase. "It's not my fault! He was talking too fast or something."
"He had an A C C E N T," she clarifies, and bites one of her polished pink nails.
"Shit."
"Yeah. Shit," YP signs, and starts walking in front of me.
"You're mad at me?!" I turn her to face me.
"You let him go!" she yells, hands in her pockets.
"What do you care?"
"You not know, who that?!" Her signs are sharp and angry.
"Yes!" I shake my fist. "I didn't _want_ to let him—"
"You could have stopped—"
"Wait." I cut her off. "You _know_ him?" I see her searching for the words, her fingers curl and uncurl by her sides.
"Whole world knows him." She boards the train.
I ignore her by sketching a few more signs in the sketchbook I'm making for Katz. But somewhere between Bliss Street and Jackson Heights we both calm down and exchange apologies.
"Why paint there now?" she asks. "It's not gonna last."
"Maybe it will. Maybe someone will want to save it."
"Like Mr. Katz?" She elbows my side and giggles.
" _He's_ trying to save his own ass. Banksy wants to save us all." We get off the train and wander back to where I parked Lee. YP puts her bag in the backseat next to mine before sitting backward in the passenger seat.
"What you mean, save ass?" YP asks, leaning against the glove box. When we sit like this, I don't have to keep looking over while I drive.
"You mean with Katz?" I sign one-handed. "He's trying to save his own graff. It's so..." I bring my hands up to my head and pull them away. "Big-headed, you know?"
"He's saving _your_ work," she says.
"And _his_!"
"You mean, you think?"
"Totally."
"Wow!" She shakes her hand. "That's pretty cool, right?" Her signs keep getting better. If I drove her somewhere a month ago, we'd look out the window and not say a word. Now we're hanging out on the weekends and chasing down Banksy.
"Not really," I tell her. "He should do his own work, he doesn't have to school me on the street."
"But and...um...don't take———-wrong way—don't you think it looks good?"
"Psh! Yeah. I do. That's the problem. He isn't some sort of co-conspirator." (I spell it out for her.) "He's competition."
"But why?" She deflates. She always wants to be friends with everyone. Even if it means being sick to keep them. Not everyone has to be your friend, some people are just enemies. I sigh, and put both hands on the wheel. How do I explain the rules of a game that has no rules? He could have asked first, could have let me know some other way. He didn't have to goad me on with personal Post-its and one-ups. He obviously knows it's me and doesn't care. _It's insulting._
"I don't know," I tell her as we drive up to her house. She reaches over the seat and grabs her stuff before getting out of the car.
"When?" she signs through the window.
"When I get paint."
"Paint?" she signs, shocked. "No no no, W H E A T P A S T E."
Wheatpaste has always seemed like cheating to me. It takes a lot of the fun out of writing, since there's so much less risk involved. Yeah, it's still putting art up on a wall for the world to see, but it's too easy.
All I would have to do is mix up some flour and water, easy enough to sneak past my parents, and I'd have some pretty impenetrable glue. I could spend hours on the art itself, spread out on papers across the basement floor, and just tell my folks it's for class. Pack it up if they do catch on, but I could always pull it out and work on it some more. Then, I'd take the papers, and the glue, and a big-ass brush down to Greenlawn. It'd be up on that water tower in what—a minute?
_Shit._ She's right. Again. Wheatpaste might seem like cheating under normal circumstances, but these circumstances are far from normal. Everything else is a huge risk, so why not make it easier on myself? I don't even have to get my hands on spray paint. I can do it all with markers.
—
The light above my workstation blinks and I shake the excess oil off of the finished fries. School's out, but Mickey D's never closes. Come on down and get some fries at 4:30 a.m. You can be sure someone will be here to shake, salt, and sack 'em up for you.
Thankfully, I never have to work those weird night-owl shifts. It's only 10:00 p.m., and I'll be out soon enough. I put in the next frozen batch and the oil bubbles up as they're submerged. Maybe this isn't what I want to say on my tower piece. I thought I wanted to write about this place, this bit of my life. But that was before. Before 5 Pointz, before the barbed wire, before Banksy. Now I'm in the after.
Two days A.B. and fryer oil–styled letters leave a bitter aftertaste in my mouth. It's not big enough. It has to be huge. Enormous. Fat. Grande. Plus-sized. Supersized. The lights blink again, and I shuffle more fries into their boxes and bags.
Donovan turns around and wipes his forehead with his arm, exhaling deeply. I expect him to wink and flash one of his smiles, but instead, he walks to the locker room without a glance in my direction. Any swag he ever had has been sapped away. It wasn't even a busy shift. Who broke his crayons? Jordyn isn't on the schedule today, so neither of us has to work under a microscope. It's refreshing. I don't need any more lectures from her about my choice of boys or friends or "pastimes." I let him sulk in the drive-thru and go back to my fries.
It's nice having a monotonous job on days when my brain is full. I could handle the station one-handed—heck, one-legged. Sometimes I think my manager invented the position for me so he can say he hired a disabled person. Clearly he's obeying whatever laws he needs to obey. Invented position or not, I still get paid for doing hardly more than sweating ten gallons a shift. Not too bad.
Two days A.B. and I know it's time to aim higher, cut deeper. _Make a fucking statement._ I feel my phone buzz in my pocket. I want to check it, but I've been busted checking it before. I wonder how many warnings I get before they fire me over it. I check my 6 and it's clear. The place is dead; only one lonely old man sits in a booth toward the front. He comes in a lot. He likes to nibble at a no. 4 while people-watching for hours. I risk it and slide my cell out of my pocket just enough to read the text.
> 555-8920: Can we talk?
Should I know this number? I've never plugged it in. Then again, I don't really take the time to add contacts when I meet people. I have about six people on my phone. Before I can ask who it is, he texts again.
> 555-8920: Ill wait.
The last hour of my shift trudges on. The chick with the glasses clocks in, and there's no way I'm texting back with her around. Rumor has it she's a tattle, and she loves to stare whenever Jordyn and I sign. Tonight, she has nothing better to do than watch me like I'm some sort of unicorn, because we only get four more customers. I can't distract myself with my blinking alarm or bagging fries, because no one's buying.
My phone burns a hole in my pocket. I keep wrapping my fingers around it, running my fingertips along the cracks in the screen. I'll never have the cash to fix it. It's not on my priority list right now. I want him to text again. Gimme a little help here. I pace back and forth in front of my station, waiting for another message or customer or grease fire to break out, anything. Screw it. It's only five minutes early. I pull the baskets out of the oil and head through the kitchen. Glasses waves her arms at me.
"What?" I express with a sharp gesture rather than words. She sternly points to the clock, eyebrows arched so high they may fly off her head. I grandly present to her, with a sweep of my arm and bend of my knees, one empty chain restaurant. I wiggle my fingers as I wave good-bye.
—
Donovan sits hunched over on the bench in the locker room, head in his hands. I've never seen him like this, defeated. Did he not hear me come in? I stomp my foot and he jolts around to face me.
"You?" I hold up my phone.
"Me." He points to himself. I text him back.
> JULIA: how you have my number?
>
> 555-8920: stole it
>
> JULIA: whats wrong?
>
> 555-8920: wat u think?
>
> JULIA: she break up with you?
>
> 555-8920: no
>
> JULIA: then what?
>
> 555-8920: i need 2 break up w u
What is he talking about? I look up from my phone. He's facing the wall again. I wish he wouldn't; I need to see his face. Expressions don't come across in texts, though I suppose I can guess.
> JULIA:
>
> we are not together.
>
> 555-8920: no shit
"So, what the fuck?" I stomp my foot again, and gesture sharply. He turns and scowls before going back to his phone.
> 555-8920: dont play stupid u know she doesnt want us to talk.
>
> JULIA: so? dont talk to me then.
>
> 555-8920: but wat if i want to talk 2 u
I don't want to blush, but it creeps over my cheeks anyway. Donovan notices and smiles, that stupid perfect smile, and I try my best not to melt away. I'm cold as ice, I don't care about him or the magnets or his perfect arm hairs.
> JULIA: u want to talk break up with her.
>
> 555-8920: well thats no fun.
Gross. I brush past him to get to my locker. Get me out of here. I have bigger things to worry about. Jordyn's love life and Donovan's gross games do not register on the list. Donovan places a hand on my shoulder and turns me around to face him.
"I can't," he signs slowly. "I like you. And I know you like me."
"Not anymore."
Everyone in my dreams has telepathy. It's rare for anyone to sign or talk. We're all psychically linked. But tonight, there's no one to connect with. Someone must have poured sand in my boots because my feet are heavy; they drag along the pavement. Every step I take is more labored than the last. I have to go faster than this. Why are my shoes making me so slow?
I use my arms to lift my legs, step after step, pound after pound. I'm not getting any closer. _I have to go faster than this._ I should take the boots off. But I won't. I need them.
_STOMP. STOMP. STOMP._ The ground shakes. I look up. The water tower looms over my head, water dripping and spraying from the rivets. I need to go, I need to run. Why can't I run? I look down at my yellow shoes. _Just take them off, Julia!_
My feet start to fuse with the asphalt. Everything gets hot. Too hot. I can feel the water spraying my neck, soaking my hair. Where is everyone? Someone help me! I can't stay here. The heat rises up through my feet, past my ankles, burns my calves. It's unbearable. The supports start to buckle and shake.
_"Fine!"_ I yell up to the tower. I have to take off my boots, but as soon as I touch the laces, everything flashes white. It's unbearable. I can't do it, I can't move, I can't take them off. I'm on fire. I'm drowning. I'm burning. I need help. Someone...
_STOMP. STOMP. STOMP._
_STOMP. STOMP. STOMP._
_STOMP. STOMP. STOMP._
—
My bed shakes me awake.
Christmas comes and goes, and the few days after fly by. YP and I finally set a date: December 31, 11:59 p.m. Tonight we tag the water tower. My parents are going to their annual Deaf karaoke New Year's extravaganza, and with any luck, the cops will be overwhelmed with all sorts of non-graffiti-related shenanigans.
The basement is cold and I don't have to meet YP for at least an hour. I plug in my space heater and settle into my armchair. I flip open the Katz sketchbook and draw the signs for the words _queen_ and _win,_ even though I haven't actually won yet. I try to think of a few more words but they don't come. My pencil's ready, but my arm isn't. Maybe I don't want to draw more. Maybe, right now, nothing is better than something.
I miss being invisible and impossible to understand. Everyone is onto me, getting into my business. Donovan, Katz, Casey. Even Banksy signing at me. _Family?_ Really? You don't know me. No one actually knows me. Except YP. She's allowed to stick around. Everyone else is a poser.
She's been texting me all day. Not saying anything important, but obviously very enthusiastic. I don't know why I'm not. Mee gave me some paint rollers for Christmas, and wants to know why she can't see the plans I've been working very hard on down in my lair. I don't know what to say to her anymore. Every word I sign is a lie, and it's exhausting.
—
The clock counts down to go time, and I throw in the towel with this whole sketching thing. It's obviously not coming to me tonight. This can't be what Katz wanted. He was probably trying to be playful, encouraging even. It's not like he made my art worse, not like he actually defaced it. What we make _is_ bigger than both of us. But he took it too far. Expected too much of me. I wish I could talk to him.
I wear my Warhol flowers T-shirt underneath the coveralls YP gave me. My red beanie calls to me from the floor, but no municipal worker would ever wear one. I opt for one of Ma's baseball caps. We have a full-length mirror by the front door. Ma insisted on it: she likes to do a last look before heading out. My last look is solid. I really do look the part. Except for the whole sixteen-year-old-girl thing. And the striped socks thing.
Everything's ready. I go to put on my boots and—my boots. My breath catches in my throat, and I'm hit with this feeling. Not anger or disgust. Not fear, not nerves. _Doom._
The doom hangs heavy in the front hall and keeps my eyes fixed on my boots, sitting on the bottom stair. The boots Mee gave me. Even after the expulsion, after all her disappointment, she never quit. She gave me a wall. _Who does that?_ And what do I do to thank her? Lie. Over and over. The doom is suffocating me. The doom helps me decide. I can't do it. Not tonight. Not to Mee.
I leave my boots, bag, all my supplies, and head out to meet YP.
I park near Dairy Barn and start walking toward the water tower. She'll understand, right? She's the only person who does. YP has to understand. I can't do it. We can do something else. We don't have to respond.
I keep chanting it over and over in my head. _She'll understand, right? She's the only person who does._ I pass the whale. _She'll understand, right? She's the only person who does._ I pass the school. _She'll understand, right? She's the only person who does._ Walk through the overpass. _She'll understand, right? She's the only person—_
Her arms wrap around me and squeeze so tightly I might pass out from air loss. She pulls back, hands on my shoulders, and smiles. I don't want to break her heart. But I'm sure she will understand. _She's the only person who does._
—
YP takes my hand and leads me under the tower. I look up and imagine it buckling, crashing down on top of us both. Doom. We make it to the ladder and she offers her hands for a boost. This is the only time I've seen her look like a cheerleader since that day she decided to talk to me for the first time.
"Come on! We can't waste any time!" She bounces her hands, fingers interlocked forming a step up to the ladder overhead, demanding I place my foot there.
"Wait a minute. Please."
"Hey, you OKOK?"
"No, I'm not." I slink down to the ground and she kneels beside me.
"Hey," she waves. "It's scary-looking, but I know you can do it."
"How?" I twist my fists together.
"Are you kidding? You shouldn't————a pep talk! Look——-—-——-you've done so far. This is your life, your art! Remember?" She brings her thumbs together.
"But this?!" I point up above us. "It's too much." She should recognize that. I didn't come here to argue with her. I didn't think I would have to.
"Not for you."
"Yes, for me! I'll do something else. Somewhere else. Another time. I just can't." My skin feels tight and itchy, it's not just the cold. I need her to be okay with leaving. What happened to Little Miss Rule Follower?
"Take a deep breath, okay? You can't quit."
"Why not? You quit Cheer."
"I never gave a shit about Cheer. This isn't about me." Her forehead crinkles as she curses. She looks furious. She shouldn't be mad at me; she should be mad at my rival writer. She's supposed to be on my side.
"I know! It's about _me._ How am I supposed to do this? What's next? The Empire State Building?" Her anger is contagious. I wrinkle my forehead to match hers.
"You're signing too fast!" she says as I steamroll ahead.
"Try to understand, it's only going to escalate until someone gets caught. I can't have it be me! Please—"
"Stop saying _can't._ You can!" she signs.
"How? How can I get up there, do this? All the pressure is on me! Not you! Back off, okay?"
"Back off?"
"I mean—"
"Me help you! I know how to get it done!"
"How? How can you possibly know that?"
"Because I did it."
"You?!" I yell, stabbing my finger at her. "The whole time?" YP stands still and exhales. She nods.
"Of all the idiot people in the whole fucking world, _you_ did this to me?" It doesn't matter if she can't understand what I'm saying. My fear and anxiety twist into a deep dark rage. I can't believe it's happening again. Another knife lodged into my back, right next to the one Jordyn left.
"I didn't know it was you." YP arches her eyebrows, pleading for me to believe her.
"Bullshit. Maybe that first time, yeah. Sure. But you can't say that for the others. I told you! I told you it was me!"
"Too fast!" she signs again, tearing up.
"How could you do this to me?" I ask again.
"I didn't want a war. I thought the art looked great together. They fit together." She twists the knife.
"You think I'm talking about the stupid graffiti? Are you seriously as dumb as you look? You lied to me. Over and over. To my face."
"What sign this?" YP asks and repeats the sign for _liar._
"L I A R. Liar, liar, liar. Every day since I met you has been one big L I E."
"Not true!"
"Oh, really? How many times did we talk about who the other tagger was? How many times could you have just come out with it?"
"I not want you to hate me!" she signs through pitiful tears. I don't feel one speck of guilt over them.
"How can I hate you when I don't even know you? Who the fuck _are_ you?"
"You do know me! You know me better better better," she signs. "Don't be like this."
"Don't act like you care all of a sudden," I tell her. YP stiffens, her expression shifts from grief to bitterness.
"I care! I dropped out of art class last quarter so you could have my spot." She stands there, arms crossed, proud of herself. I'm disgusted. It takes me a second to respond to this newest low. I had her so wrong. Not only does she disrespect my art, my friendship, she pities me. I won't have it.
"How. _Dare._ You."
"What?!"
"Poor deaf girl can't get into art class! I know, I'll be the better person and help her out and take a knee so she can go use her magical deaf art powers." I use my voice, make her listen to how stupid she must sound when she talks. "Like...um...like blind people and music."
"I did it 'cause you're my _friend._ " Of course YP starts sobbing. For once I don't feel bad for her. She brought this on herself.
"You're not my friend. Never were."
—
She knows better than to follow me right now. At least I hope she does. She was right about one thing. I do hate her. Hate, hate, hate her. I flick my middle fingers out from my thumbs again and again. _Hate hate hate hate._
This whole time, patronizing me with her fake ignorance. _Oh, wow, graffiti, street art, so cool._ Signing that stupid-ass oath. Of course she knew who Banksy was, of course she suggested wheatpaste. She told me about the underpass tag—what, _moments_ after _she_ tagged it? She wasn't afraid of my reaction—she was playing me. _Hate hate hate hate hate._ I break into a run past Finley.
It wasn't Mr. Katz. He just really likes my art, that's all. Maybe he didn't even know it was me. I never gave him that sketchbook, I got so wrapped up in all this. Maybe I should give it to him after all. _UGH. None of this matters right now, Julia._ I head toward Lee, fists balled up, knuckles white.
_This. Is. What. You. Get!_ I slam my fists onto the steering wheel over and over. What you get when you give a fuck about anything. I'm sobbing so hard I expect puddles of tears to rise up to my ankles. I should have known better, especially after Jordyn. How long has it been since she fucked me over, a semester? I should have learned my lesson, not gone confiding in some lame, cheery-cheeked cheerleader. Snot starts dripping from my nose, and I do my best to sniff it back up. Crying's disgusting. I'm a mess. Who cries over graffiti, anyway? My neck is cold and wet; I wipe away what I can with my sleeves. I want to drive home but I'm too upset. If I get pulled over and have to open the car door, a wall of tears will flood out like in those drunk-driving ads. And on New Year's Eve I doubt the cops will believe it's tears.
Fuck it. Fuck them all. I get out of the car. A boiling, snotting, dripping mess, I head to the Little League field.
—
There it is, the big whale and bones. The sight of it makes me want to scream again. This never would have happened if Casey didn't try to force me into friendships. I don't _do_ friends, I don't do _friendly._ I don't play nice, because I get played. _That bitch._
The whale glares at me, taunting me. Showing off. I trudge through the snow on the field, my feet freezing in my old, worn-out sneakers. Fuck footprints. I don't give a shit anymore. Come and find me. I'm responsible. With each step, my nostrils burn from anger mixed with the icy air.
_The whole time._ She knew the whole time, every conversation laced with lies. What did she think was going to happen? High fives? Hugs? Was she _ever_ going to tell me? Oh, my God, what if she never said anything?
The paint pen is in my hand before my brain realizes it's there. My arm starts writing before I even know what I want to write. You can write a lot faster when you don't care how it turns out—I'm back on the road in under thirty seconds, paint dripping and drying in the darkness.
Lee obediently waits for me, alone in the parking lot. Good. I'm finally ready to get as far away from here as I can. A car drives past, headlights sweeping across the pedestrian-crossing sign. Something in my memory sparks. The scoreboard _wasn't_ the first time! That day, when we walked past drinking iced tea. The skeleton on my tag, the X-ray of the crossing man. I knew _then_ that war was brewing. She pulled me away without saying a word.
Donovan tastes like Mountain Dew and sweat. It's gross, but good: I don't want to enjoy it. Not too much, anyway. He moves his hands up my shirt, and I hit my head on the roof of his car.
"Your hands are _freezing,_ " I mime, and slide his hands under my ass for warmth. I can feel him laugh through the kisses. I knock off his visor and toss it into the backseat before I run my fingers through his blue-black hair. He tilts his head back and I kiss his neck. He's moaning; I feel his throat buzzing.
His car is a dump. I can count at least six soft-drink cups from where I'm sitting. Not to mention all the grease-stained bags, some clearly containing half-eaten no. 3s. It's more than a turn-off. I grind into his lap; he holds on to my hips and slips his salty-sweet tongue back into my mouth.
I close my eyes and try to imagine that it's months ago. That Jordyn never went out with him. That she let me have him after she got me expelled. That he approached me first. Back when I was crushing hard, when Donovan could do no wrong. Why couldn't he have given me the magnets then? Why did he only start liking me when he was already attached? I thought I could have been the exception. What a fool I was. What a fool he is.
I lift up my shirt and let my hair fall down over my shoulders. His eyes light up and he greedily reaches for my breasts, arm hairs all smooth and perfect. I kiss him again and he closes his eyes. _Good._ I keep one hand on his chest while I use the other to wiggle my phone out of my back pocket. While he's busy kissing my neck, I hold the phone out behind him just high enough to fit us both in the frame.
—
"Thanks," I sign to him before reaching for my shirt.
"Wait, don't go. Not yet." He hangs on to my hips, his eyes search my face and my body. He's hoping for more.
"Sorry," I sign, getting out of the car. "Got what I needed."
"What?" He looks so stupid sitting there in the passenger seat. All worked up, no idea what's about to happen. I'd pity him if it weren't his own damn fault.
"I might be a liar," I sign with one hand, drafting a text with the other. "But at least I'm not a cheater."
> Message _sent._
The snow is melting; traffic lights reflect in the road on the way back to Finley, Red Bull nestled between my thighs. It's like my first day all over again. I won't fuck it up this time, though, that's for sure. Late last night I considered getting myself expelled again, but where would I go? Getting expelled again would destroy Mee. So, fresh start at Finley it is.
Bundled up, hands tucked under her armpits, Casey waits for me at the entrance to the school. _Crap._ I forgot to ask for a new terp. Maybe I can talk her into quitting. And by "talk her into" I mean "torture her into," obviously.
"Julia!" she signs, and waves. I keep walking. "How was your break? Did you do—" I stop her hands.
"Are you interpreting for anyone right now?" I ask.
"Um...no."
"Then you don't need to be talking to me." I blow past her toward my locker. Why do the slow fade when you can do the torch-and-burn? My locker springs open and I hang my coat up. My breath catches. I only notice it when I go to shut the door. She was here.
I want to laugh, not because YP's cheered me up or brought me around, but because she doesn't know how pitiful the gesture is. _Give it up._ The pink letters catch the light as I start to swing the door closed. I can't believe she thinks this is the way to handle the situation: breaking into my locker and tagging over my work yet again. Showing me up.
My new schedule commands me to head over to the math hall for intermediate algebra. Math's not too bad; numbers I can handle. The rules are the same no matter what language you speak. And bonus, Casey hates math. Lots of fingerspelling, lots of numbers. Not a lot of fun for someone who became an interpreter because _sign language is so beautiful._ Few people could make solving for _x_ beautiful in any language. Definitely not Casey.
"You didn't see it yet, did you?" Casey asks, taking her spot next to the chalkboard.
"See what?" I reply with the stink-eye instead of words.
"Your English class," she signs without making eye contact. The schedule peeks out from the pages of my textbook. I slip it out, and lo and behold, I've been moved to ESL. _Shit._ I would say I tried, but I didn't, really. Here's the part where I would normally fly off the handle, rage against the English machine and all that. But after everything that's happened, _who cares_?
—
The first half of the day I spend zoning, fuming, working, and figuring out what the hell I'm going to do next. I point to a soft pretzel in the lunch line and fill a little paper cup with mustard. It's still bitter and cold outside, but I haven't run into _her_ yet, and I don't plan on it. The concrete table in the courtyard is my new lunch hangout. I sit with my back to the windows. They can see me, but as far as I'm concerned, I'm eating my snack in Siberia. Alone.
I want to go out, I want to bomb every wall, every sign, every lamppost. I don't care if she was here first. She doesn't get to win this war. She won't be able to walk a single block without a reminder of her betrayal, her crimes against whatever friendship I thought we had. Friendship. Friend _shit._
I know it's Casey tapping my shoulder; I don't have to turn around to know it's her, but I oblige. Her nose is already running from the cold. _Toughen up._
"Come inside." She motions to the door.
"Are you interpreting for someone right—"
"Stop it with that; look, I know ESL sounds bad, but—" We take turns cutting each other off.
"I don't care about ESL."
"Oh, then why..." Casey looks me over, searching for the right words.
"Unless you're out here, interpreting for a teacher or something, I don't see how it matters to you."
"Of course it matters—"
"Let me put it this way: it's none of your business. I'll see you in class." I turn back to my pretzel and she takes the more-than-obvious hint to leave. I wish I was a smoker. I feel stupid sitting outside with nothing to do but look at my busted-ass phone. Jordyn has texted, but I haven't replied yet. Not much to say to her anymore. Soon she'll wise up and leave me alone and I'll finally be invisible at work, too. No way Donovan's ever looking at me again. They deserve each other.
My butt starts feeling numb from the cold concrete bench. I look over my shoulder and notice the cafeteria has emptied. _Damn it!_ Now would have been more helpful to butt in, Casey. I have Mr. Katz next, and I'm sure she wants to be there even more than I do.
—
The halls are clear as I rush to the art wing. This is bad—how late am I? It's the first day of the new quarter, so hopefully I'm not the only one. I pick up the pace. The door to Room 105 is closed. _Weird._ He never closes the door during class. I turn the knob and the whole class stares as I creep through the doorway.
"Sorrysorrysorry," I sign. Mr. Katz looks less than pleased at my late arrival. He points to an empty chair. Casey leans against the display wall and glares at me. It's okay. I made it. Everyone gets to be late on the first day of new schedules, right?
"I think it sucks," good ole Black Shirt says. He's back for more art this quarter. I survey the room. Black Shirt, Freckles, Pigtails, and YP. Wait. _What?_ Our eyes meet and I immediately look away.
"What sucks about it?" Mr. Katz asks him.
"Why ruin a good thing? It was cool before, now it's all messed up," Casey interprets for Black Shirt.
"Is it any different than what the first artist did?" Katz continues his line of questioning.
"Duh." Are they talking about what I think they're talking about? Black Shirt continues. "It's like what she said." He points at me and I don't appreciate the second round of stares from the whole room. "What they did didn't add anything to the art. They had crappy intentions."
Ugh, using my own words against me like that. Gross.
"How do you know?" I sign at him. He looks at me when he answers this time.
"C'mon, did you see it? It's totally ruined." He actually looks at me when he speaks.
"Who says it's ruined? Who are you to judge?"
"You're joking, right? I can't tell with your, like, lady over there. You can't say it's better now than it was before."
"I think it was better when it was just a whale. Everything after that ruined it." I can feel YP staring at me from the other side of the room. My eyes want to look, too, but my willpower holds out. Casey starts interpreting again.
"You're wrong." I can guess who said that. I keep my eyes fixed on Casey.
"What makes you say that?" Katz asks.
"It was good," Casey continues interpreting. "The whale. And the skeleton? That was good, too. But together? They're awesome. Brought up to another level."
"Who are you to say?!" I turn and sign to YP. "Why are we talking about this at all? It's art class!"
"It's street _art,_ " Pants emphasizes.
"It's not meant for some stupid roundtable discussion about what counts as art, or what's good and what's bad. Let it be what it is!"
"And what's that, Julia?" Mr. Katz tries to call me back down to earth with his question, but I'm too far gone.
"It's over. I'm done with this class." I grab my bag and slam the door behind me on my way out.
I don't care about still lifes and draperies, so what do I need advanced art class for? Nothing. I have everything I need right here. The basement is covered in papers and pencil shavings, left over from what was once the Big Plan. I sit cross-legged in my armchair and look over the sketch wasteland. When I snap to, Ma is staring at the floor alongside me. I don't shove the papers away. It's too late, she's already seen it all. At this point, I hardly care. This piece is never going up anywhere. It'll only ever live on paper. Nothing to see here.
"You're home?!"
"Obviously," she says, eyes fixed on the floor.
"What's wrong?"
"I should be asking you that." She sits on the arm of the chair next to me.
"Why?"
"I heard you quit art class."
"Casey needs to stop calling you, it's not fair—"
"Why would you quit art class?" She stares off into the room, not waiting for my answer. "I kept asking myself that. You really wanted to be there. You _needed_ it after—after everything." I know better than to interrupt her. "So, why would you quit? Either something happened, or..." She sighs and motions to the floor. "You're back at it again."
"This? No, this isn't that." It was, but it's destined for the trash now.
"Don't lie to me, Julia." Ma picks up one of the loose pages. "If you quit class, what's all this for?"
"I'm allowed to draw on paper, aren't I?"
"Don't get defensive. Can't you tell me why you dropped out?" She releases the paper and it floats back to the ground. I want to tell her it's none of her business, that once again she's overreacting to information fed to her by an overreaching terp. I can't tell her about YP and me; she'll tell me to deal with it, stay in class, keep my commitment, don't let teenage drama keep me from my education, blah blah blah.
"Julia, I know these aren't for Mee's store wall." She indicates the piles of papers and pens. I've never wanted to be alone so badly in my life. I change tactics.
"Didn't Casey tell you? I got moved to ESL. No time for art if I want to pass."
_"Really?"_ Ma digs her fists into her hips. This is her thinking-with-purpose stance. She stares off at some distant spot on the wall. "It's not something else?" She looks at the papers again, and the corners of her mouth turn down. I can tell she doesn't know if she should believe me, but I don't feel panicked. I feel empty. It's not like I'll ever use the plans, nothing here is worth the interrogation.
"I honestly thought that you—"
"Really. I'll take art next year or something. Whatever."
"Are you sure? Because I'm not so sure you need ESL. I could call the school and—"
"Ma! I'm fine, everything is fine, okay?" I flick the _K_ hand shape at her. I thought I was done getting the third degree.
"Okay, Miss Attitude. Dinner soon."
Once I'm sure she isn't coming back, I pull out the very last of what was once my stash. After I got caught at Kingston, I got rid of nearly everything I had at home and opted for the new routine, the shell game with my black bag in different lockers. There's only one small box left. You wouldn't know it's contraband by the looks of it, which is why I kept it around.
I'm sure if Mee or Ma ever found the box, they'd assume it's just a bunch of empty glue pens. They don't scream _graffiti material,_ not when they're empty. I can feel Ma walking around above me in the kitchen. I didn't tag the water tower, but I didn't exactly stop. I haven't stopped lying, I've stopped caring. Am I any better than YP? Maybe everyone on earth is a liar.
I'm not the cops, I'm not her dad. I'm her friend. I _was_ her friend. That is what makes it unforgivable. She had no reason at all to lie to me, and I have every reason in the world to lie to my moms. I dump the contents of the box into my backpack and gather the papers off the floor.
Screw planning, screw big thoughtful pieces. I don't need planning. I don't need respect. I need revenge.
The bass blasts so loud in my car, the seats vibrate with every beat. I haven't turned on the radio in a year, maybe more. Tonight I need the distraction. _THM THM THM,_ the steering wheel hums under my fingers. I don't want to think my way out of this one. I turn the knob up a little more.
Plans are for pussies. For toys who are afraid of getting caught. I don't need a disguise or an alibi. All I need is some paint and a wall. Everything else is a distraction from the real deal.
I roll through Dairy Barn. The cashier, frowning at the music, hands me a huge Styrofoam cup of iced tea.
"It's late!————down!" he yells, turning an imaginary knob. I wave and pull away into the parking lot across the street. I don't want that guy to think I'm lowering the volume for him, but I have markers to fill, so I shut off my car and the bass thumping ceases. The best thing about mops is you can fill them with craft-store acrylics. The watery stuff that grammies use for stenciling birdies onto flowerpots and bathroom walls works best.
No one gives you a second look when you buy the paint for this stuff. Even so, I gave up on mops a while back; they drip more and don't mix well. Generally, they're harder to control and don't make anything nearly as beautiful or perfect as spray. But I'm not looking for perfection anymore. Perfect is the enemy of getting it done. Right now.
First, I unscrew the caps and squeeze the paint from their tubes into the pens. A little bit of the Martha Stewart's Pursed Red Lips color dribbles onto the passenger seat. I try to rub it off with my sleeve but that only scrubs it deeper into the fibers. And now it's on my shirt. _Whatever._ I put the mops in my coat pockets and lock up Lee.
I bet I could walk from Dairy Barn to the underpass backward with my eyes closed at this point. How many times have I made this trip? I have to start switching it up, I can't keep tagging in the same places. Gotta expand, move on.
How did she do it? How did YP have me fooled for so long? Not just her innocent act—the logistics of the whole situation baffle me. She knew where I bombed, when I bombed. She retaliated so swiftly, but her art looked like it took weeks to plan. It doesn't seem possible that she could live all those lives at once.
_I'm_ supposed to be the one you'd never suspect. I should be able to write at lightning speed, with no pauses for planning. No time for second guesses. How did a bouncy blond babe beat me at my own game? My fingers wrap around one of the mops as I walk under the overpass, each one filled with a different shade of gory red.
YP can suck it. Jordyn can suck it. Donovan, Casey, they can suck it. The red paint drips down the wall as I work over the mural. _Huh._ Two hearts, she put two hearts in the skeleton. She wasn't toying with me enough. Had to leave a little hint in there. I mop on a deep-burnt-red broken heart over one of the originals. I leave the other one untouched.
I use the brightest red for crossing over the eyes and let every drip run its course to the ground. Does she really expect me to forgive her? I scrawl the last few letters up and run the mop across the length of the wall as I leave.
"I don't understand why, though!" Jordyn signs, during our break.
"Does it matter?" I need to quit this job. I'm exhausted by all the drama. I thought what I did would finally put an end to it. That they would both hate me so much, they would finally, finally leave me alone.
"Yes! It does!" She blocks me from leaving the room.
"You were worried he would mess around, so..."
"No, I wasn't!"
"Why did you ask me to stay away from him?" That shuts her up for a second. "What did you see in Donovan, anyway?" I don't know what I saw in him, either. The only thing I see now is that they are perfect for each other. Users. People who love you when you're new and shiny or when they need you for help, but the minute you need them—they vanish. They should get married, and divorced, and married again.
"I liked him!"
"You like everyone! _I_ liked him, and you got bored and decided to swoop in."
"That's not—" Jordyn stops herself, she looks up to the ceiling and exhales. I can't tell if she finally understands me, or if she's plotting some new way to get my sympathy.
"Look, you wanted to know if he was a cheater, and now you know. I did you a favor."
"Jesus, Julia. You're so fucked up. I'm tired of pretending to be friends with you." She blows past me to the kitchen.
"So don't," I sign to the closed door.
I'm fucked up _because_ people like her stab me in the back all the time. If everyone left me alone, I'd have nothing to be fucked up over. I straighten my black polyester collar and catch my reflection in the mirror on the wall.
I look _wrecked._ Like I need five hundred hours of sleep to make up for the past two weeks. My eyes are sunken in and darker than usual. I'd feel crappy about it except they remind me of someone's. They have that same tired look as _his._ That day when I pulled him off the fence, he looked ragged, but like a pro. I stand a little taller, thinking he and I might have more in common than I thought.
I flip off Donovan every chance I get. Granted, it's only in my head so I don't get canned, but I like to think he can feel it in the air. I'm sure I'm getting the same treatment from him, what with the death stares and clenched fists. Without Jordyn or Donovan to worry about, I'm on top of my game tonight. I got fries lined up for days. Evening rush? Bring it on.
"Your shift's over." The manager taps me on the shoulder and motions for me to leave. "Nice work tonight."
Wasn't expecting that. I raise my head high and smile on my way into the locker room. I don't need anyone to tell me I've done a good job, but I can't say I mind.
I'm all changed out of my grease-coated uniform when Donovan bursts into the room.
"You think you're funny?"
"No." I duck my head and click my lock shut.
"You realize—-——fucked over the only two————-liked you———place."
"You fucked yourself over," I sign.
"I can't understand you,——talk, damn it!"
"Ha!" I cough out for him to hear. I grab one of the mops still in my coat pocket and go up to his locker.
"You done being pissed yet?" YP signs to me, this time in the locker room. She's in three of my classes _and_ has the same lunch period. I swear, it would be so hard to avoid her if I was a hearie. Thankfully...I turn my back and it's like she's not even there.
"——on, give it up———-." She moves in front of me. I close my eyes as I pass her on my way out. No, I'm not giving up that easily. The fact that what she did to me still doesn't strike her as a big deal only strengthens my resolve. I'm not the kind to kiss and make up; she should know at least _that_ about me.
The gym is set up for indoor volleyball. I'm relieved to find that I'm not placed on a team with YP. Forty minutes of hitting a ball back and forth over a net and I'm out of here. I get to zone out in ESL, then eat lunch in my igloo. The feeling of relief doesn't last long. I feel it slip away as the ball rushes toward my face.
Stars flash and everything goes black. I feel a pop in the bridge of my nose that sends pins and needles shooting across my cheeks, as if the ball was slammed into the funny bone of my face. It's a shooting pain followed by numbness.
Am I bleeding? My hand rushes up to my nose and comes back dry. I'm fine, but damn, it hurt enough to be bleeding. I look up and Kyle Fucking Stokers glares at me. A chill runs over my arms. I signal that I'm okay to Ms. Ricker and the game continues. I'd expect KFS to high-five a bro, giggle over hitting the deaf girl in the face again. This was different. He wasn't joking, he wanted to hurt me.
I spend the rest of class avoiding the spike zone and generally trying to be out of the way. My nose is still throbbing—one more hit like that and I'm sure it'll break. _What's his problem?_ Up until now, KFS has escaped my wrath entirely. I kind of forgot he existed, to be honest. I suppose it's possible he's truly a psycho and wants to screw with me because he can. I wouldn't put it past him.
YP must have gotten dressed fast, because there's no sign of her when the bell rings. _Good._ I get changed in peace, take my time, and drag my yellow boots to ESL. I assume Casey is already there and waiting, no more hallway walk-and-talks. Nice to know at least she's catching on. I pass the art room and Katz's red flannel catches my eye through the window. He looks so serious, talking to YP, who stares at the floor and picks at her nails. I try to read her lips but they're in profile and obscured by her hair. Mr. Katz must have felt me watching, because he looks up and frowns. I book it down the hall.
—
What a weird day. It'll be over soon. Everyone will stare and scowl and frown for now. But soon they'll stop caring, just like me, and I can go back to life under the radar. The ESL teacher gave me a sheet to evaluate where I'm at. Essay-type questions, so Casey is looking bored. The first question is about my influences. I know I can't write about Banksy or Swampy or Miss Van. Not with Katz talking about street art down the hall and my paint still drying on the underpass. What do they _want_ to hear? I'll write about Mee and Ma. That should count for something.
Who are the most influential people in your life? How have they contributed to your life?
I think my moms are the most influential people in my life right now. They are good role model because even when they don't get along they still love together. They had to overcome a lot because one, they are Deaf, and two they are together.
While I'm writing, the teacher stands next to my desk and looks over my shoulder. Casey stands by, hands at the ready. The teacher addresses her.
"Can you tell her to stop? I'm going to help her."
Casey tells me to stop, then explains that Teach can talk directly to me. I'm not sure he gets it.
"So"—he takes out a red pen from his pocket and starts marking up my unfinished answer—"this isn't too bad. Here."
_I think my_ moms _Mom_ are _is the most influential_ people _person_
"Hey!" I wave for him to stop. "I have two moms."
"Oh. Uh. Really?" He raises his bushy eyebrows at Casey.
"Really," we both sign/say at the same time. Casey smiles at me. I'm a stone wall. The bell rings and Mr. T practically shouts at me. There's no way I could understand his distorted mouth shapes without Casey interpreting.
"Sorry about that! Take the paper home with you, bring it back next class. O-kay?!"
Shout all you want, Mister. I ain't gonna hear you.
—
This has to stop.
The parking lot is full of kids getting in their cars, leaving. All the little fishies swimming home for the night. How in the ever-loving world did YP pull this off during school hours? I noticed it before I even stepped off the sidewalk. I'm stuck pacing back and forth from the curb to the flagpole, hoping each time the heart will be gone. It never is.
_Of course._ I pick at one of the edges of the heart: wheatpaste. I was so oblivious. She was dropping hints left and right. She should have told me. I can't relent now. I expected this from nearly everyone else in my life. But not her. She needs to understand that. I look down at the heart. Doesn't she get it? She broke mine.
—
Everything's playing out pretty well. Casey stopped bugging me; Jordyn and Donovan kissed and made up—both of them too bored or lazy to move on to someone new. I thought Jordyn would have stuck up for herself, but they deserve each other. At least they've stopped talking to me. I'm about 90 percent transparent, 5 percent visible to my parents, and for some reason, Katz is holding on to that last 5 percent.
I see him watching me watch everyone at lunch. I still sit outside every day, but now I face the windows, so I know when to head back to class. It's not that cold anymore. Either that or I'm growing thicker skin. I like watching everyone eat, mill around the cafeteria, through the glass. All the fish hanging out in the whale's guts. Throwing a party, oblivious that they were swallowed whole.
YP dumps her tray into the trash. I try not to notice it's full of food. I try to forget this is the fourth day in a row. Her tags have stopped. No more hearts, no more quotes. Nothing. _Weak._ If she really loved graff so much, she would be out there. She wouldn't let me stop her.
My tags are everywhere. Dripping red marks on the slide at the park, on the backs of stop signs. Some days it feels like I'm running out of places for it. It hasn't been easy. I avoid driving anywhere near my tags. I don't want people putting two and two together. I don't need to look at them anymore.
The fish stir in the whale's guts. Throwing away trash, hiking up their bags, and heading off to class. YP and Katz go to the art wing together. She's stopped checking over her shoulder for me when she leaves for class. She's stopped looking at me altogether.
Good.
"Why haven't you started yet? You told me you would start in February, and it's already March," Mee says. She takes her spot on the edge of my bed.
"I don't know what to paint," I tell her, and for once, it's the truth. I know the wall is there, free, legal, and all mine. But I can't bring myself to plan it out. It's too much work. Scrawling my new tag over every surface in sight is easier.
"No sketches? Nothing?" She looks around my room for any relevant scrap of art.
"Nope." I hang my head. I've disappointed her. Again.
"Ma said you had some out on the floor?"
"For something else."
"Something...illegal?"
"No! She already asked me that."
"I don't understand, Julia. Is it only fun if you're not allowed? Why shouldn't my wall count?" She swallows hard, and braces for my response. When Mee is worried, she takes deep breaths through her mouth. She's asked me to do the same thing when I'm upset. It tricks your brain into calming down. I've never gotten it to work for me.
"It's not like that."
"You aren't yourself lately."
"I'm always like this."
"Not true. Moody? Sure. Angry? Sometimes. But you don't smile anymore. You're a dark cloud."
"I'm fine." I'm so beaten down, telling her would just make it too real.
"Come and paint the wall, it will cheer you up."
"I told you. I'm fine." Mee wants to keep pushing the issue, but enough one-word answers force her to give in. I hate upsetting her, but I _am_ always like this. I flop facedown on my bed. My nose collides with the mattress and I wince, the pain from the ball zings back into my sinuses. I let myself wallow for five more minutes. Life handed me a shitty year, and I want to roll over and sleep out the rest of it. But it won't help. Maybe I _should_ paint Mee's wall; at the very least it would make her happy. And I can show YP just how fine I am without her.
I open my laptop. Maybe I'll find some inspiration online. My Hush _mail_ has a bunch of junk I need to clear out. The forums spam my inbox a bunch, and I haven't been active on them in a long time. I log in and read. It's too risky to post on the thread I find.
> SIBERxREBIS: wtf is up wit ths toy taggin evrywher? ne of u seen ths?
>
> GNOMES: thaats fukked up. that first piece was tits. fuk toys earn ur stripes on stikies or somethin god.
>
> T.HUB: ive seen that shit!
>
> KORE: You've seen it? Who did it?
>
> SIBERxREBIS: Kore gtfo we all kno ur a cop.
>
> KORE: That is not true.
>
> T.HUB: cop
>
> GNOMES: cop
>
> KEZTECK: cop
>
> SOPROOAKS: cop
_Fuck and double fuck._ I've never been called out online before. Never posted pictures for this exact reason. Every now and then some cop gets onto the forums and tries to squeeze info out of toys. I am _not_ a toy. These punks don't know who I am. They look up to Neckface, but diss my new stuff? We're not any different.
I read over the comments a few more times. They liked it. Well, they liked it before my latest addition. It's better now. They don't know what they're talking about. I don't regret it. Not at all, so stop asking.
Everything's been painted over. The scoreboard, the underpass. It's all gone. Most of the little tags I've thrown up from Greenlawn to Queens have been covered or worn away. Craft paint doesn't really hold up to the elements. My mops are still there in my coat pocket, refilled and ready, but I haven't found a decent spot since reading the forums.
"Hey! Pay attention!" Donovan shoves me and jabs a finger at my timer before pulling the fries out of the oil himself. "Clock out before you burn the place down." His promotion is going to his head. I don't stay to argue, though—I'd love to get out of here early.
My phone buzzes after I get changed.
> MEE: Will you be home for dinner?
Of course it's Mee, she's the only person who texts me now. My phone has become this weird paperweight in my pocket, my personal mom-communication device. I should toss it. Be the only teen on earth without a phone. I let her know I ate as I get into my car.
Lee's too obvious. She's probably attracting way too much attention. The cops are stupid, but they aren't blind. I should have painted her back to solid when I first saw them snooping around the tunnel instead of tempting fate. _Let's go, Lee, time for a trip to the hardware store. You're getting a makeover._
Buying spray paint might be off-limits, but I don't need it for what I'm doing. Lee is ancient. I love her, but her paint job doesn't need to be glamorous. I put some rollers—I'm saving the gifted ones from Mee for her wall—and a tray in my orange shopping basket and head for the paint aisle.
I'm trying to decide whether to paint Lee white or black when I see her. Quickly, I take two steps back and pretend to look at light-switch plates. I don't think she saw me. I peer around the corner, leaning far enough to see YP put a spray can in her own orange basket.
How does she plan on buying that? She's only two months older than me, and I know her birthday isn't until June. We had this whole birthday thing planned. _Whatever._ I watch her pick out two, three, four more cans before she's done. _What a haul._ I follow far behind her to the checkout counter.
_Oh, look at these power-drill things, so reasonably priced. Just checking out tools over here, nothing to see. I'm certainly not stalking anyone._ YP swoops her hair over one shoulder and saunters up to the cashier. He looks _really_ happy to see her.
She puts her basket on the counter and leans over it, giggling, pushing her boobs together. The checkout guy checks out more than her paint. He turns red as she giggles again and bites her lower lip. I know this game. I used to play it with Mail Boy at Kingston. No, we aren't alike. We're nothing alike.
He doesn't ask to see her ID, doesn't even hesitate ringing her up. Puts the cans into a paper bag, and she's out of the sliding glass doors with a wave and a wink. A lady in an orange smock taps me on my shoulder.
"Can I help you——?" She gestures to the power tools.
"I'm good," I say out loud, and head back to the paints. I choose two big cans of dark gray enamel paint and use the self-checkout.
—
Everyone's asleep by the time I get home. The gate to our driveway is closed. I hop out of the car, leaving her running while I open the little chain-link fence.
We don't have any outdoor lights on the side of our house, but there's a street lamp that provides me with enough light to see what I'm doing. I'd rather paint her now, no one walking by, no one asking questions.
I pry open the first can with one of my keys. I forgot to buy paint stirrers, so I mix up the paint with a pencil before pouring some into a tray. The roller sops up some paint and I squeeze out the excess. Can't let it drip all over the driveway. Ma would kill me.
The paint rolls on in a thin coat; I'll probably have to go over her twice. I start with the trunk. I thought YP quit writing. I haven't seen anything around that looks like her work. There have been a few new tags popping up around school, but none of them are good enough to be hers. Where has she been painting?
I wrap around to the right side of the car, letting the roller do the work, rolling paint in W shapes so it doesn't streak too much. Maybe she's been planning this the whole time. I know what she's capable of, so I can't imagine what she'd pull off with months of downtime to plan. She did buy _a lot_ of paint.
I roll back around over the trunk and to the left side of the car. She can do whatever she wants. She can go paint a huge piece and this time _I'll_ tag over it. Then we'll be even. Except for all the lies. Can't forget that. I climb up on the hood to reach the roof. I keep climbing and take a break on the top of my car.
There aren't any stars here. When I go out writing in Greenlawn, they're always up. But not in Queens. The lights on Citi Tower and in Manhattan were my stars. Now, they aren't enough. The sky is a hazy, dark gray color; it'll be black soon. I dunk the roller into the paint well in the tray and finish up the roof. All that's left is the hood. I paint right over YP's heart in three big strokes.
—
Gray was a nice choice. Lee looks nearly invisible sitting in the driveway. Invisible car for an invisible girl. Perfect.
"We were going to pick up some pizza last night," Mee informs me as she fills a glass with grapefruit juice. "You must have gotten home late."
"Yeah, sorry. My phone's acting weird since I dropped it." It's sort of true.
"Where were you?" she asks as casually as she can.
"Hardware store. I gave Lee a makeover."
"You did?!" Mee rushes to the window and pulls open the little half-curtain. "Oh. It's..."
"Gray."
"So plain. I thought you might have done something more colorful, I guess."
"Nothing wrong with gray." I down my glass of grapefruit juice before grabbing my bag.
"I suppose not." She looks solemnly over the new paint job from the window. "Do you need some money for supplies? Is that why you haven't started on my wall? You know I'll get you whatever you need to start."
"I don't need money."
"What do you need?" she begs. There was a time when I might have taken advantage of this, but I can't take anything else from her.
"Nothing."
"I'm not sure I understand." Mr. T crosses his arms in front of his chest. "I don't think she needs to be in ESL...more like remedial English."
"Why not?" Casey asks him for me.
"She knows English! With her hands or something." He waves his arms around. I don't know who rolled their eyes first, me or Casey. She turns to me and signs, "Please."
"Fine." I give in. "Listen, English _is_ my second language. I speak American Sign Language. It's not English. It's not charades, not miming. It's a language. How did you get to be a language teacher, anyway? I'm not so sure I can learn English from you." The smirk on Casey's face grows wider as she interprets.
"Excuse me?" Mr. T backs away.
"I don't think I need to be here either, honestly. But it's supposed to be your job to teach me, not to kick me out because you assume I already speak English. Would you throw out Philippe because he already knows English?" Philippe is the only other person in my ESL class. He's a tiny freshman with a bowl cut and a little shadow of a mustache on his upper lip. It's so ugly it's cute.
"Philippe doesn't know English!" Mr. T's face starts turning red. Philippe's eyes dart from my hands to Casey's to Mr. T's face. Clueless. "How am I supposed to know you're learning? You don't talk!" Mr. T argues.
"I don't need to—"
"Yes! In order to speak English, you have to SPEAK!" He must have really shouted. Casey looks stunned, and poor Phil looks scared out of his mind.
"I was saying—" I start in on a new rant, but Casey cuts me off. She stands in front of me; I can't see her face. Her gestures have nothing to do with sign language, that's for sure. Phil hooks his finger into his collar and pulls it away from his neck, as if to say, _Jeeeeez._ We both start cracking up. Both the adults turn back to us with furious faces. Casey tells me to get my belongings. We're leaving. I leave class first, with a little nod to Philippe. Casey ushers me in front of her before flipping off Mr. T through the window in his door. Her face is still curled into a snarl when she storms away. It's the most badass thing I've ever seen within the walls of a school.
"Where are we going?" I ask her as we hurry away.
"Don't know yet. Never done that before." She pulls on her scarf anxiously between sentences.
"Leave a class?"
"Curse someone out." She pivots back the way we came. "I should apologize."
"Are you kidding?" I keep pace with her so she can see what I'm saying.
"You're right." She swivels back. I think she's sweating. "He made me so mad! He's supposed to be a professional." She starts biting at the cuticle around her thumb.
"I get it."
"This was a mistake," she signs with her left hand, still gnawing at her right. "I wasn't ready for all of...this." She gestures at me.
"You're fine, Casey."
"Um...no, Julia, I'm not."
"He's an idiot! You were right!"
"Not right to say what I said. How I said it! Oh, God." She starts pacing again.
"Just relax!"
"Relax?! I'm going to get fired—for what? For you? You don't even like me!" We face each other in the hall. Casey stares me down, panic-stricken, waiting for me to say something. But I can't. I don't know what to say. She's right, isn't she?
She turns her back to me and changes directions twice more before deciding to go into the main office. She shuts the door before I can follow her inside.
—
Students flood the hall between periods. Casey still hasn't left the office. I wonder how long I should wait for her. I wish I knew what she said to Mr. T; it must have been pretty harsh. Damn, that would have been fun to know. I work my way to my locker, slowly, giving Casey more time. If she doesn't come back soon, I don't know how I'll make it through the rest of my classes. I'm not voluntarily going into that office, though. I'll wait.
I unlock my locker but the door jams and gets caught on something. I pull harder and it jerks open. A paper bag falls to the floor. The folded top must have been crammed between the hinges of the locker. "Hey, HERE" is written on the front.
How does YP keep getting into my locker? I don't want to open the bag in the hall in front of everyone, but I'm growing more invisible by the day. No one's going to notice anything.
The contents of the bag:
Three spray tops: one stencil cap (fine lines), one pink dot cap (super-fat spray), and one gold dot cap (a happy medium).
A disposable respirator.
Black vinyl gloves.
A note written on a "Hello, my name is" sticker:
I put everything back in the paper bag and push it down into my black bag, which I've just started using as my normal, boring backpack. My folks have given up on the at-home inspections.
Why do I even bother locking this thing anymore? I slam the door shut before I realize I'm in the hallway by myself. The door to the office opens and the vice principal, I think, steps out. He searches around, catches my eye, and waves for me to come over.
"Your——is—-—-so—say——to—tha—--ifthatsokaywithyou." He talks so fast and jittery I'm surprised I caught even that last bit. I take my phone out of my coat pocket. The screen has a big red splotch of paint on it. I try scratching it away with my nail. Seeing it sets something off in him. He pinches the bridge of his nose and points to the phone, annoyed.
"No phones—-a the——s—." He points to my cell one more time to drive his point home. I leave the paint alone for now and type into the notes app.
> I do not understand you. i am sorry.
He takes the phone from me and his eyebrows arch up, his mouth makes an O shape, and he starts feverishly typing.
> Your translator has quit. I asked her to finish out the day, but she refusd.
>
> What i do now
>
> Go to your classes.
>
> How will I understand?
>
> try fora bit someone else is on th way.
He hands the phone back to me and shoos me off to class in the wrong direction.
—
Going to class without a terp would be the biggest waste of everyone's time. I can't believe she actually left. That was the plan, get her to quit, but I didn't think she would really give in. She seemed tougher than that. I don't think I can even take the credit for what set her off anyway.
With her gone, nearly everything is going how I pictured it, with the exception of YP breaking into my locker. I wonder who can see me now. Anyone?
I stand as still as I can in the entrance of the school. Main doors in front of me, gym directly behind. Cafeteria to my left, office to my right. How long will it take for someone to notice me standing here, doing nothing?
I start pacing along the front doors, all the way around and back. I switch and go in the opposite direction. Still nothing. Everybody's in class. Pacing gets boring. I sit under the pay-phone bank. I wonder why we still have them. I've never seen a kid pick one up and use it.
I take out the paper bag again. There's no way I'm going. She can paint all on her own. I might go after, to see what she does, but I don't have to be there for it, or participate. Plus, I can't exactly copy her tactic for buying paint. Not so easy to flirt it up when you don't speak the same language.
_Which doesn't matter, because I'm not going._ When class lets out, I pace around and around again. Waiting for someone to bump into me or call me out for cutting. To yell at me to get back to class, to throw something at my head. Anything.
—
Nothing.
I did it. I'm actually invisible.
No one cares.
Perfect.
I take out one of my mops and tag both of the pay-phone receivers.
No one notices. I sit underneath them and sketch out more signs in the dictionary I'm sketching for Katz. I add the signs for _fire, liar,_ and _hurt._ Being invisible is boring.
—
You know what, I'm not giving YP the chance. Or the satisfaction. I'm retaliating _right now._ Not waiting for the end of the week and the wee hours of the morning, to show up and be shown up. _This. Ends. Now._ I zip up my bag and stride toward the art room, invisible and unstoppable. I'm Julia. I'm on a mission. I'm HERE.
Unsure if there's a class in session, I approach Room 105 very slowly. The art gods, once again, shine down on me. The room is empty and dark. Most important, it's unlocked. This will only take a minute.
The door swings open and I beeline for the supply cabinets. I don't waste any of my attention on the latest art projects hanging on the walls. I'm here for only one thing. Paint.
Last time I took the first three cans I could reach. But this time, I'm putting an end to our war. It needs to be better than anything we've done together. I pull out a few more cans and inspect my color options.
I'm instantly drawn to a can of yellow. Old habits die hard. I shake it up. No good: I can feel it's almost empty. I'm going to need a lot. I'm not planning this one out. No more plans. I have to take the fluorescent orange, that's for sure. I shake it up and put it in my bag.
Shake-test a can of red, a can of teal. Take 'em. Shake up a can of white, take it. Shake up a—
The lights flick on and off, and the can of purple I was holding falls to the ground. Can I make it to the window? I don't want to turn around, can't bear to see the look on his face. The lights flash again. I'm frozen, breathing deeply, trying not to have a panic attack.
"Julia." Mr. Katz comes over to me and signs. His eyes, they stab me in the heart. I can't look at him.
"Julia, please..." He points to himself. I look up, my lip trembles. I'm mortified. I can see he is having a hard time figuring out how to talk to me. It's the longest minute of my life. Silence is the loudest sound.
"Where C A S E Y?" he finally asks.
"Q U I T," I tell him, head hanging down toward the floor. When I look up, he's no longer disappointed. He's pissed.
"Let's go." He takes my bag and gestures for me to follow him out of the classroom.
Mr. Katz didn't plead for leniency on my behalf, but he evidently didn't bring up the paint, and I'm sure as hell not about to. I could have sworn he told the principal that he would call my parents to address the situation, but that never happened. I would take a whole week of in-school suspension in exchange for sparing me from that phone call. I was only sentenced to a day. God bless you, Mr. Katz.
The only good thing about being stuck in ISS is that I don't feel the need to look for an interpreter until I'm released. I spent an hour here yesterday, sitting in utter silence with the temp terp they must have called in, phone obscuring his face all afternoon like Magritte's apple. Nobody's on my case to find a new terp until I wait out my sentence and I'm allowed to talk again.
I sit here in the tiny cell of a room, door open, across from the office. Alone. I was told to wait for a pile of work from my classes. So, I'm waiting. And waiting. They gave me a sheet with the rules so I don't get "confused."
# IN-SCHOOL SUSPENSION RULES AND REGULATIONS
_1._ _No talking at any time._
_2._ _You are permitted to do only the work provided for you by your teachers. If you do not complete the provided work, you will receive a zero grade for the day._
_3._ _No reading._
_4._ _No drawing._
_5._ _No cell-phone use._
_6._ _You will sit in your assigned seat with both feet in front of you, facing your desk._
_7._ _You will keep your area neat and clean. Trash can be disposed of only on a break or at the end of the day._
_8._ _You will be allowed to purchase a lunch. You must eat your lunch in the ISS room._
_9._ _No other food, gum, candy, etc., will be permitted._
_10._ _Bathroom and water breaks will be provided. If you must use the bathroom, speak to Mrs. Gomez. You are permitted to be in only the ISS room and the office._
_11._ _Sleeping is not permitted._
ISS is all about waiting and no gratification. I put my head down to wait for my assignments and Mrs. Gomez comes in and starts yelling at me, pointing at the intercom over and over. I take out a notebook and write out: "I AM DEAF. SORRY." She crosses her arms over her giant boobs and taps her foot.
"You—-—-can pretend——but I know——that never——ever—" I cut her off by waving the paper again. This time I speak, so she'll actually believe me.
"I'm really Deaf. I'm sorry." She turns three shades of red, and I sit a little taller having put my jailer in her place.
"Oh!" She raises her thick arms to make a desk and puts her head down on it. Then she wags her finger back and forth.
"No, no, no, okay?"
"Okay," I laugh as she bustles out the door.
Twenty minutes later, still no assignments. I count the tiles on the floor. I count how many people pass by in green shirts. Blue shirts. Red flannel. Mr. Katz walks by ISS swiftly, glancing back over his shoulder as he disappears out of view. Making sure I'm there. I look up at the ceiling and count those tiles.
Mrs. Gomez is back, arms crossed, at the door. She looks down the hall and crooks her finger in the air as if to say, "Come here." She punctuates it by pointing at the floor, sharp and stern: "NOW." She holds out her hand, and Kyle Fucking Stokers reluctantly hooks his backpack over it. Mrs. Gomez tells him where to sit, shuffles in behind him, and hands him a copy of the same welcome-to-hell sheet. She opens his backpack and hands him one notebook and a pen from inside. The rest she zips back into the bag, which she brings into the office with her. I wonder what happened to my black bag.
KFS slams his notebook down on the desk and pushes the chair against the wall. He yells something at the door, spit flying from his mouth. A vein in his neck is raised and purple. It's intimidating. I try not to stare. Either Mrs. Gomez is using the intercom, or he's having a conversation with the ceiling, or God. Whoever it is, KFS is pissed.
"What're _you_ looking at?" I guess I am staring. I sign the word for _nothing_ and look back down at my desk.
"Listen..." He gets in my face, pointing.
Before he can let it rip, Gomez is back in the room, scolding him. I can see why they put her in charge of the ISS kids. She doesn't stand for any shit. Except instead of yelling, she sort of scolds you, like a mom. A very strict, no-nonsense mom.
"You're———it worse, Kyle. I don' wanna——ackere——orrow, kay?" I bet she's got an accent.
"Whatever." KFS slumps in his chair.
"Good boy." Mrs. Gomez smiles and turns on her little kitten heel.
—
Eventually, someone drops off a pile of papers for me and another for him. I start on my math sheet. The rest of the papers look like busywork; they have nothing to do with what we've been working on in class. KFS sits staring at the door, not even looking at the stack of papers on his desk.
Isn't he bored out of his mind? Sure, I don't want to be sitting around filling out worksheets, but the alternative is staring into space for six hours. I wish I had that kind of resolve.
"You need the bathroom?" Mrs. Gomez asks from the doorway. Kyle practically bum-rushes her on his way out. I don't have to go, but a change of scenery would be nice, even if it's only toilets and sinks. Mrs. Gomez waits between the bathroom doors for us to finish before ushering us back to our cell.
My head hurts. I need caffeine. I slide my head down onto my history paper and close my eyes. This only lasts a moment before I'm jolted back into reality. KFS kicks my chair. "————head up, re-mem-ber?" I give him a thumbs-up.
"Shut up with that," he snarls.
"What?" I act out, raising my shoulders, arching an eyebrow.
"You know," he says.
"No, I don't." I shake my head.
He turns to face me and starts in on a rant, talking so fast I don't even try to lip-read. Instead I focus on his expressions, but he really exhibits only one. _Contempt._ I didn't get him sent here. I don't know why he's here with me. What's he so pissed at me for? I raise my shoulders again, trying to get him to stop. I hold up a finger and write out in my notebook:
> I cant understand you
He rips the notebook from my hand and starts scrawling his response.
> _stop acting like your so fucking cool. your not. all you are is a bitch._
>
> and youre a dick. what do you care anyway
>
> _i dont care about you at all. but if she gets sick again. thats on YOU._
Mrs. Gomez shuffles back into the room: it's time for lunch.
—
I thought having Casey sit with me at lunch was bad. This is so much worse. Everyone knows, and everyone gossips. We aren't allowed to go to the cafeteria and come back alone. Mrs. Gomez waits in the lunch line with us. Our ankles might as well be shackled together.
We both point to our selections. Even though KFS can speak, he doesn't. Mrs. Gomez chats happily with the lunch ladies and other kids in line. Everyone loves her. Everyone who's not in ISS, anyway. It feels like all eyes are on us when we leave the kitchen. I miss my little table outside, looking in.
I spot YP: floral-print yoga pants, white top, suit jacket. Her hoop earrings are so huge you could use them to hula. I hate that my first thought is: _Shit, she looks cool._ She doesn't have a lunch tray. She's texting or Tumblring, sucked into her phone. Frowning.
"What—-—-do, man?" a guy in a jersey asks KFS. He glares back, and the guy takes the hint. What _did_ you do?
Mrs. Gomez chaperones us through lunch. Once we're back in our cell, I can see she's lecturing KFS, but not in a condescending way. She obviously cares: her gestures are gentle and expressions are soft. Every now and then I get a sympathetic glance, but nothing more.
Once she leaves, I flip open my notebook.
> Shes not my responsibility
>
> See you are a bitch. i knew you werent really friends with her
>
> you dont know what you're saying. i was a really good friend for her.
>
> then you ditched her or some stupid shit right?
>
> how you know any of this
>
> you got all friendly and fucking ditched her, and all im sayn is if she gets sick again its your fault.
>
> MY FAULT? YOU DITCHED HER all because she got FAT. YOU BROKE HER HEART.
>
> shut up
>
> YOU broke up with her when she got FAT
>
> fuck you
I throw the book at his chest. He stands up, his chair crashing into the desk behind him. He glares down at me. I'm not afraid, I stand straight up and meet him face to face. _Try me._
"You," he spits, "you don't..." Kyle slumps back into his chair and looks up at me. "You don't—————. It can't -- -- me anymore." His eyes water, he turns away to hide them. Why wouldn't it be on him? He's the one who broke her.
I pick my notebook up off the floor and hand it to him.
> _She never had friends, she always hung out in the art wing doing her art thing whatever. No one gave a shit about her cause she was fat i guess? thats what she said to me. i took a class with her. i was like the only dude to ever talk to her. so she went and got herself all skinny and pretty and made cheer and friends. and she said she was happy, and i liked her and yeah we went out and shit. but i noticed that she never ate, and was always sick and she said she was happy but she wasnt._
He wipes his nose with his sleeve before going back to the note.
> _and one day she fainted at my house and it was just too much for me to handle. okay??? i have my OWN shit to worry about too you know? im not her dad im just a guy. so i told her dad and he sent her to get help. but after that i was just done. she's a chill person or whatever but i can't handle that. I didn't break up with her cause shes fat. i dont give a shit if she weighs a fuckton. i got her better and thats all i could do. and NOW you're fucking it all up._
The rest of our ISS sentence flies by. It's 2:45 and we're reunited with our backpacks and cell phones.
"Be good—-," Mrs. Gomez urges as she locks the door behind us. Free to go. Thankfully, there aren't many students hanging around to stare, so we brave the halls on the way to the parking lot. I break away for my car, but Kyle stops me with a hand on my shoulder.
"What now?" I use my voice; I've had enough of him for a lifetime. Especially after today.
"Don't fuck it up." He thrusts his finger into my chest. I swat him away, ticked.
"Don't touch me, okay? Stop that. You understand me?" I'm talking to him out loud, my throat feels dry and scratchy.
"Yes." He backs off. "It's just—"
"I know."
"Do you?" He speaks slowly, wanting me to understand him. "Because I really did try to help her and—"
"I get it, but listen—" I start. "You listening?" Kyle nods but doesn't hesitate to roll his eyes. "I know you did what you had to do. I understand why. But that doesn't make you some kind of saint." I feel my voice catching in my throat on certain words, but he looks annoyed, which means he must understand what I'm saying. "You were horrible to me. All year. Awful. You don't get a free pass."
"Like you———a brat, too? Sulking———acting——you're better——everyone else?"
"I didn't deserve—" I try to explain that he's partially to blame for my attitude but he cuts me off with a wave. He lifts his hand and slowly and awkwardly spells out:
T
R
U
C
E
He laughs, because my jaw must be scraping the pavement. Who in the whole damn universe taught Kyle Fucking Stokers how to fingerspell?
"Doesn't——-kid learn——like, kindergarten?" he explains, reading my confused expression.
"Fine. Truce, for now."
—
I sit in my car, reading and rereading his note. Thinking about YP, friendless in the art wing. What's so wrong with that? Being alone isn't so bad. She's always been so sensitive. I think about her being a cheerleader, surrounded by buzzing girls, boys crushing on her. Dating Kyle, of all people. Of course she quit Cheer. She only wanted the friends, and those friends sucked. I feel a small comfort in knowing I outranked them in her mind.
It's not fair. Why should I be forced to give in because she's delicate? She really did lie to me, I'm not making that up. I didn't ditch her for no reason. I've been alone since then and I've been—
No. I haven't been okay. _Shit._ We're both messed up.
After today, I think I've earned an iced tea. I pass three of my butcHEREd tags on the way to Dairy Barn. No wonder I've been avoiding these streets. They look like ass. What was I thinking? I have to go to 5 Pointz. Revenge or no revenge, I need to make up for all this garbage writing.
When I get my phone to type out my order, there's a text notification on the screen. The attendant is already waving impatiently at me. I flash my order on the screen, pay, and drive on through.
> YP: this cant wait til friday, 5ptz NOW. something happened.
I hit every red light on my way to the highway. I flip off each one until it turns green. She must be okay; she couldn't text if she was hurt. Or caught. Doesn't matter. I press down on the accelerator. I need to be there, _now._ I should be driving to work. I consider texting Donovan, telling him I'm sick, but I never want to text him again. Consider this my resignation. _Sayonara,_ Mickey D's. Find someone else to deep-fry fat sticks. I'm out.
The drive is a blur: suburbs, highway, city. I park as soon as I see a spot. I won't find one anywhere close, so I lock Lee up, leave everything behind, and run.
There she is, flower print, suit jacket, and all. Blond hair draped over a silver backpack, a Nordstrom shopping bag at her feet. She's reading the "coming soon" condo announcement again. The building's been painted since we were here. No murals, nothing but white primer covering every surface, windows included. All the graffiti ghosted underneath the thin layer of white. _Fuckers._
"Hey." She turns and waves sheepishly. She must've heard me coming.
"This sucks." I point up to the building.
"You didn't see it yet, did you?" she signs with complete confidence. She must have been practicing this whole time.
"No, I haven't been back. When did they paint it over?"
"Not this!" YP points over her shoulder. "This isn't why I texted you."
"What, then?"
She starts chewing at her nails. "I really am sorry, you _have_ to know that," she explains.
"What you did, it broke my heart," I tell her.
"I know, mine was broken, too. I didn't mean—"
"I know." I cut her off. She looks so defeated. Tired. I really hope she ate lunch. Her shoulders droop forward, her jacket rumples, the sleeves dappled with pink paint. "Hey, is that... _his_ jacket?" I tug at the sleeve.
"Oh,...uh..." She turns red. "Yeah."
"Ha! It looks good on you!"
"Listen, small or big, choose."
"What do you mean?" I ask.
"I have two things to show you. Small or big first?"
"Small, I guess. I'm not ready for anything too big."
YP swings her bag around front and unzips it. I catch a glimpse of school papers and books, but not much else. She pulls out a small flat package, rectangular and wrapped in pink wrapping paper, and hands it to me.
"It was in the coat," she says as I unwrap the paper. It's a Moleskine sketchbook—shiny, soft, and black. "I think you should have it." I flip through the pages. Oh, my God. It's _his._ It's his B-book.
"This is insane! You don't want it?"
"Well, I look so good in the jacket!" she signs, and twirls around. I can't help it. I swoop in and wrap my arms around her.
She gives me her death squeeze, but I don't care. It's not tight enough. I want us to get out of here. Hop in my car, head to her house, split ten pies and talk about nothing for a month straight. She pulls away too soon and sees that I'm crying.
"I'm sorry, too. Really," I tell her.
"You don't have to be."
"You know that's not true." I hug her again, and I can feel her laughing. I pull away and sign, "Thank you." We flip through the pages some more.
"This dude is magic," I tell her, and she hugs me again.
"I'm really glad you came. I didn't feel right stashing it in your locker. I wanted to give it to you."
"Oh! My locker! What did you want to paint? What happened?"
"That's the big thing. You ready yet?"
"Yeah, I'm ready."
This time, YP leads the way around the fence. I can just barely make out the old graffiti beneath the primer. Ghosts of writers past. It's depressing, and sort of spooky, thinking about how in a few years, no one will even know it was here. A whole gallery of graffiti, gone. When we reach the spot where we rescued the Suit, YP points up to the building, but I'm already looking. It's unmissable.
He came back.
"Did you...?" I turn to YP, picking my jaw up from the concrete.
"You kidding? I can't do _that_!"
"How does he know we..." I spray invisible paint in the air.
"I have, like, literally no idea," she laughs. (She signs the word _like._ It's hysterical.) YP's hair glows orange in the setting sun. We both hang on the fence, saying nothing, smiling and carefully looking over every detail of his piece. He got us. I don't know how, but he did.
"Let's put him in his place." YP throws the jacket on top of the barbs and climbs up and over as if she's done it every day of her life. Piece of pie. "Hurry up!" She checks in both directions as I climb over the fence. She tosses me a can of Jet Black and vinyl gloves from her Nordstrom bag and we get to work.
—
YP is smart. No supplies in her backpack, everything in the shopping bag under a thick layer of tissue paper. She pushes a pink dot cap into place on a can of yellow and shakes it up.
"Hey!" I wave and get her attention. "You have M A G N E T S?" I point to the bottom of the can.
"What T O Y sold you on that trick?" she laughs.
"It doesn't work?"
"Most cans," she signs, "have G L A S S balls in them." Face, meet my palm.
Embarrassed, I uncap and walk to the wall. I reach my arm out, but YP stops me. She pulls a paper respirator over my nose and mouth.
"You need. Is important," she signs, before pulling hers down. Only YP would be worried about fumes when the clock is ticking. Never saw the point of a face mask, still don't. I hang back and watch her work first.
Her left hand clenches into a fist and relaxes a few times. She's deciding where to start. I can literally see her _Aha!_ moment in her body language. She rushes the wall and sprays two giant yellow circles over our likenesses' faces. _Oh, my God. I get it._
I grab a stencil cap and hook it onto the collar of the black can. We can't have our faces up on a wall, a giant picture of us spraying. YP is smart. I add some smiles. We fall into a rhythm, each taking turns, watching the other, then adding our own touches. Once we've been at it for a while, we have to move faster. The politeness ceases and we go for it, moving quickly, trading places, swapping colors. We're a blur of color, painting until YP hears footsteps and we run like hell.
—
Everything gets put into the Nordstrom bag and tossed into the first Dumpster we pass. She doesn't save her paint. I want to keep running, but YP's practically window-shopping.
"You work so C L E A N!" I tell her.
"You work fast!" She snaps her fingers. I see her bike chained to a post near the 7 train entrance.
"You want a home?" I ask as she unlocks the bike. Her face goes white. She swallows hard. "What? What is it?"
"S T A Y C A L M," she spells quickly before turning around, face to face with two police officers. She smiles at them. I want to run. Everything in my body screams for me to get the fuck out of there. This time, my head knows better than my body. Running would only guarantee I'd get dragged in. I stand shoulder-to-shoulder with YP.
"What—-—-two girls——----- evening?" the bigger of the two cops asks. I can't make out what YP says. She's talking fast, grinning wide.
"Is——so?" the lady cop asks me. I slowly point to my ears and shake my head. YP steps in and tells them I'm deaf. The cops exchange a skeptical look.
"It's true," YP signs and speaks.
"We—-call————-girls————your description————-—vandalizing———-property."
"Sorry," YP signs/speaks. "We're coming from—"
"Hold—-——hands," Big Cop demands. YP starts to interpret but Big Cop cuts her off again. "I——, hold out——hands!" He mimes for me and I comply. Fem Cop flips our hands over, inspecting every crease and cuticle. Mine are sweating so much she has to wipe her own hands on her pants when she's finished. Our hands are clean, but she isn't satisfied.
"Open your bags," Fem Cop orders.
"I don't———-to——search," YP says, holding her bag.
"Good for you," Big Cop huffs, and snatches the bag from her. "So———you're here——-train station,—free—search you."
_Oh, fuck._ Is there paint in my bag? Mr. Katz _had to_ have taken it back, right? I hand my bag over to Fem Cop. He wouldn't have left it in there. I feel a bead of sweat roll down my back. No, he _would_ have. That's exactly the sort of thing Katz would do. He _likes_ the art, but hated my behavior. Fem Cop unzips the front of my bag. I'm light-headed, I feel like I'm going to pass out. I take a breath, but it feels like I can't get any air in. Please, Katz, be a hard-ass. Just this once.
Fem Cop pulls out my notebook and busywork from earlier in the day and throws them on the ground. No paint. _Katz, I could kiss you._ I risk a glance at YP, who looks more annoyed than terrified. Her books and papers are also on the ground at her feet. I have no idea how she's staying so calm.
"Arms out." I do as Fem Cop says. YP does the same, but Big Cop takes a step back. Fem Cop runs her hands up my right sleeve and back down again. I'm going to throw up. I don't know how I'm still standing. I've dreamed of running from the cops, fooling them at every turn, with Pum Pum, with Creepy, with Wurstbande. In Buenos Aires, in Australia, in Berlin. But I can't handle it as she fem-handles my left arm. In this moment I don't pretend to be tough. I cry. The tears fall. They run and drip off my chin and onto my shirt. Fem Cop notices and takes a step back.
"What's—-got——-about?" she demands of YP. I've never been so scared and confused in my life.
"She doesn't know what's going on," YP signs, and explains, pointing to her ears. I sob. Big Cop laughs. His face is smug, punchable. He motions for Fem Cop to get on with it. She does. And I'm an idiot.
I'm an idiot for blowing up like I did, for tagging like a toy. For telling myself planning is for pussies. For lying to my moms over and over. I'm an idiot and I deserve everything that's about to happen when Fem Cop pulls out the blood-red mop from my coat pocket.
"What's this?" Fem Cop holds the mop up in my face. All the color drains from YP's. She's so smart, she has all her bases covered. Here I am, the toy with paint in my pocket. I'm sinking us both. I start to explain, but Big Cop tells me to keep my arms at my side.
"Pen and paper," I say aloud. "I—I—I n-need a pen and paper."
"Finish first," Big Cop tells Fem Cop. She hands him the mop marker and pats down my legs before moving on to YP. YP is clean like Greenlawn snow. I'm as dirty as Queens sludge. She never has to apologize to me again. I'm the one who fucked us.
"So, what's this?" Fem Cop repeats herself.
"Pen and paper," I remind her. Instead of getting me a pad to write on, she turns to YP. So against the law, but the officer doesn't seem to care, or know. Fem Cop talks fast, her lips are thin and coated in gloppy gloss. I have no idea what she's saying. More tears cling to my chin, then drip to the ground. I don't dare lift my arm to wipe them away.
"You can't prove—-—that!" YP doesn't appear to be yelling, but she's nowhere near as calm as she was before. Big Cop doesn't care. He opens the back door of his cruiser, and ducks her head inside.
"Pen and paper! Pen and paper! Pen and paper!" I shout over and over. They have to give it to me. It's the law. I can't believe I thought cops who cut corners were only in movies and bad TV. Fem Cop ducks me into the seat next to YP. They take our school IDs and leave us in the backseat. We can see them through the windshield, talking into their radios.
"OKOK?" YP signs.
"No!" I shout. I put my face in my hands. So thankful they didn't bother cuffing us.
"Hey." She pulls my hands away. "It'll be okay. OKOK." She looks deep into my eyes with certainty.
"How can you say that? How can you be so calm?" I demand.
YP signs low in the seat so she doesn't draw attention to our conversation. "Before Kyle, before Cheer, before you, I was B U S T E D."
"No shit?"
"And not by my dad, or principal. Real, real B U S T E D. They————my room, took———everything."
"That's why your room is so empty!" She must have been doing some really serious art to get the cops to raid her room. If the thought of a raid at the moment wasn't so terrifying, I would be impressed.
"Never—————replace any of it."
"Did they...?" I pantomime handcuffs, grab invisible bars in front of my face.
"Nah, in the end they not P R O V E it. And they not P R O V E this, either." She reaches out and squeezes my hand.
Fem Cop is holding up the mop marker, examining it from different angles, shoulder crunched against her radio, probably relaying a description. Big Cop stands to the side, looking more punchable than ever, copying info from our IDs onto yellow slips of paper.
"I tried. I tried telling you at the water tower," I start. How do I explain this to her? She's calm, I'm a mess. She's not going to want to stop. Why should she?
"What?" she asks.
"I can't keep doing this." I point around the car. "I can't keep lying to Mee and Ma; I can't risk getting kicked out of another school. Especially not now."
"There—-other schools."
"None with you in them."
We both cry, holding each other's hands. We almost laugh, until we remember where we are.
The cops, each taking a side, open the doors of the cruiser and order us out. They lead us around to the front of the vehicle. "Tilt your heads back," Big Cop barks, and motions for my benefit. We comply. Fem Cop takes a flashlight off her belt. _What are they doing?_ They wouldn't give me paper, but they wouldn't...beat us or anything, would they? Fem Cop clicks on the flashlight and shines it up my nose. _What the fuck?_
She does the same to YP and shakes her head no to Big Cop. His shoulders drop in disappointment. He hands back our IDs, along with the yellow slips he was writing on earlier. "We took——all——information so—————you—-get away—-———," he says to YP. Fem Cop makes a show of keeping my mop. _Great._ She can have it.
"—-—free to go?" YP asks. Big Cop dejectedly tells her yes. YP grabs me by the arm and leads me away. Thank you, Universe.
—
We bike around Sunnyside, me standing on her back pegs. She pedals steadily. We're taking the longest and windingest path back to Lee. Don't want the cops knowing my car, my plates, or where I live. We take our time.
"Want to eat?" she asks at a red light. My stomach has barely settled from our near-arrest, but if YP wants to eat, I'm not going to say no. We lock her bike outside of a place called Tofu and Noodles. We sit at the table in the window so YP can keep an eye on her wheels. The waitress pours us waters and sets out five tiny bowls, each with a different type of kimchi.
"Kimchi smells like S O C K S," I whine.
"It's good!" She fills up a chopstickful and hurries it into her mouth. We pore through the pictures on the menu. I'm only halfway present. The other half of me is still in the back of that cruiser, headed for a holding cell.
"What you want?" YP flips through the menu.
"Split this with me?" I point, and her eyes light up.
"My favorite!" YP smiles, tapping her chin. She tells the waitress our order and eats more mouthfuls of kimchi cucumbers and kimchi classic. I still don't have an appetite.
"Why aren't you freaking out?" I ask her. I can't stop bobbing my knee up and down. I tap the table with the white plastic chopsticks. YP reaches over and stops my hand.
"It's over, we're fine," she signs with one hand. It's amazing how far her sign language has come. We haven't had our phones out since 5 Pointz, and we haven't missed a beat.
"Well, I'm freaking out." I start bouncing both my knees.
"I see that." YP takes her hand back so she can eat some more kimchi.
"What are we going to do, though?" I can't stop looking back over my shoulder, at her bike, still locked to a street sign, but no one is coming. We should be fine, but I don't feel it yet.
"First, we eat. Then, bike to L E E. Then—"
"No, about us. About..." I mime spraying paint across the air.
"Oh, I don't know," she says, forgetting to sign, mouth full. I'll never eat kimchi, never.
"Stop, right? Shouldn't we stop?" My right hand chops my left palm. "Why didn't you stop when you got caught?"
"I did. For a long time. I was done. I was out."
"What happened?" I'm rapt, still. On the edge of my seat, and she practically spits out her soda, laughing at me. "What?" I shake my hands furiously.
"I saw your whale and I couldn't help myself." She almost looks bashful, breaking eye contact and looking down at the table. Our waitress brings over a huge plate, steam rising off of the whole grilled octopus that we ordered. She takes out a pair of scissors from her apron and cuts it up for us before leaving us to it. It smells and looks delicious, and suddenly, finally, I'm starving.
We pop the front wheel off YP's bike and angle the body into the back of my car. I'll take her home eventually, but I'm not ready to let her go yet. She takes up her usual position, sitting back against the glove box, cross-legged in the passenger seat. I turn on Lee and crank the heat.
"Can I ask you a T O Y question?" I ask before pulling out.
"You're not T O Y."
"Why did that cop do that? With the flashlight?" I mime shining one.
"To see up your nose."
"For what? D R U G S?" This gets a big laugh. "Told you I'm a toy."
"That's why you wear the mask," she motions. "The paint—-stain——nose hairs."
"You're kidding, right?" I ask, and she crosses her heart with her pointer finger. I turn the key. YP reaches out and touches my arm.
"Hey, can I ask you a question?" We should probably stop asking if we can ask and just ask.
"Just ask from now on, OKOK?" I tell her, and she smiles.
"Why you call me YP?"
"Uh...Don't be mad, OKOK?"
"Never."
"Your pants."
"My pants?!"
"You always wear Y O G A pants, Yoga Pants, YP." I squint and look away without really looking away.
"Ha! I like it," she says. "I hate J E A N S. _Hate._ "
"What'd they ever do to you?"
"Make me feel huge. Yoga pants don't really have numbered sizes."
"Do you want a new name?"
"Nope, never."
I start to drive, but somewhere along Queens Boulevard I decide we aren't going back to her house. Not yet.
We pull into the small parking lot next to the acupuncture shop as the streetlights flicker to life against a navy sky. I should text Mee.
> JULIA: be home soon-ish. At the wall with YP.
>
> MEE: :) :O :) YAY. See u soon my love.
"So," I ask YP, "what do you think?"
"Of what?" YP looks around the small parking lot, confused.
"The wall!" I spread my arms out, presenting it through the windshield.
"I think you were right."
"I want to, and I don't." I bite my lip, it's not fair. I don't want to force her to give up her game. That's her choice. I'm worried my wall is a poor substitute for the real thing.
"Same." YP's hand moves back and forth.
"You and me, _together,_ it's next-level art."
"T R U T H," she spells with a pursed smile.
"Why do you love G R A F F so much?" I ask her. YP takes her time before answering. She looks up and down Mee's wall.
"I love making art you can't escape."
"Big." I grin.
"Yeah, big."
"Big like a whale. Big and beautiful."
"Yeah, big," she repeats.
"Big is best." I elbow her.
"Okay!" She lets out a laugh and rolls her eyes to the roof. We take some time to consider the wall and our future in front of us. I can still smell the paint in the air from our 5 Pointz piece, but I'm sure it's only in my head. It was magic, working alongside her instead of against her. The closest thing to telepathy I've experienced outside of my dreams. Sitting in the back of that cruiser, though, I never want to experience that again. Mee will be thrilled if we paint her wall, but I'm sure YP will think it's too easy.
"Would you still love it, if it was legal?" I break the silence.
"Maybe. I might miss the R I S K. But I wouldn't stop."
"If it's legal, are we posers?" I ask.
"You'll never be a poser. That underpass was the best thing ever."
"Then we can't quit," I say, determined.
"But what about—"
"Cops can't stop us from writing on our own wall."
—
By the time I drop YP off at her house, my stomach has unknotted itself. It's almost relief. _We're okay._ Yes, we're safe and free and clear. But we're okay. _Us._ We're going to be OKOK. And we get to keep painting. Bless Mee. It's not about the risk. It's about the art. I know YP and I could make something impressive without the pressure of cops breathing down our necks. I can only imagine.
I notice my backpack in the rearview and realize there are still a few kinks in my gut to work out. Katz. And Casey. Casey wasn't the most professional interpreter I've ever met, but is professional really what I want? At least she cares. It could be a little much, but that's nothing time couldn't fix. I should have cut her some slack.
I feel the worst about Katz. I burned him pretty bad. All he ever did was help me, include me, even when he couldn't. That illustrated album. _Jesus, Julia._ You were such a bitch. I have to fix this. I have to apologize.
—
Greenlawn Drugs is still open. I'm here for one thing, I just hope they have it. As I look up and down the aisles I consider the letter I'll write Casey. I'll grovel if I have to. She should get to come back. That is, if she wants to. If I were her, I'd be through with me.
I zero in on a big bucket of sidewalk chalk. Perfect. I toss it on the passenger seat and rifle through all the papers on the car floor. All I need now is the yellow envelope that had my Bob Dylan album in it. The lyrics are all here, but I hope the envelope didn't get dumped out of my bag back in Queens.
My phone buzzes in my pocket: Ma checking in. My new rule is always answer their texts. I've put them through enough, and they only know the half of it. I'd like to keep it that way. I let Ma know where I'm at and my ETA, and she seems satisfied, even a little cheery. I crouch down and feel around under the seat, find a crumpled take-out bag. I reach in again and I'm in luck, there it is.
I bet Katz didn't realize his address was on the front when he gave it to me, reusing the envelope from some other package. Just what I needed. I tear off down the street in Lee, chalk safe beside me.
Mr. Katz lives in a big farmhouse. Smoke rises up from his chimney; he must be home. There are lights on, but they're all on the opposite side of the house. It's only chalk, right? I cut the headlights and pull over behind a big fir tree. Lee's gray paint job camouflages her well. After grabbing the bucket of chalk and the sketchbook full of signs, I close the door as quietly as possible.
His driveway is long, his house set far off the main road. At the end of the driveway, in front of the house, are two cars: his and Casey's. Ooh-la-la. I think fast, changing up my plans now that I know she's here. Nothing drastic, though. Still going for it.
I carefully pick out colors of chalk by the light of my phone. Despite the size of the bucket, the shades are limited. They're all washed-out pastel blue, green, pink, or orange. I'll make it work.
This'll be my first—and last—chalk piece ever. _Why can't you be more like paint?_ I try my usual outlining method, and it works all right. But only all right. I'm going for _epic apology_ here, and _all right–looking_ pisses me off.
I push through, filling in the shapes, getting chalk fingerprints all over everything, mostly my clothes. Strange that the legal stuff is twice as messy. Takes twice as long, too. I really want to cover the driveway, but if I did, it would take until first period tomorrow.
The last orange stick wears down to a pebble and I pack it in.
I take out Katz's sketchbook, the one I illustrated all the signs in, and go to leave it in his mailbox on my way out. Unfortunately, his mailbox is attached to his house, right next to the door. But I just bombed 5 Pointz. I can sneak a book in a mailbox.
I creep up the steps slowly, hoping that the old wood isn't groaning under me. When I reach the top, the porch lights flick on. I dash to the mailbox as quickly as I can and try to fit the sketchbook in the slot. My hands are trembling and I accidentally drop the book. As I'm picking it up, I feel the door swing open.
"What are you doing here?!" Casey signs from the doorway. She wraps a blanket around her shoulders, covering up her matching pajama set. She's obviously pissed, but she looks so dorky I can't help but smile. "You've caused me enough trouble. Go home." I stop her from swinging the screen door shut. I angle myself so she's looking at me and not my chalk masterpiece in the driveway. Let her see that in the morning.
"Wait! No. I—I wanted to say I'm sorry," I tell her. Casey looks over her shoulder into the house.
"Well, he's sleeping, so tell him at school."
"Not to him, to you." She's surprised to hear me say it and I feel so guilty for it. She never expected an apology from me, and not because she doesn't deserve one. She really, really does. I look down at the sketchbook I made for Katz and realize it shouldn't be for him. Of course it should be hers. "Hang on," I sign, and she wraps the blanket tighter, covering her arms while she doesn't have to speak.
I take out a pen from my pocket and uncap it with my teeth. I open up the book to two blank pages, and on the first I scribble out the sign for the word _sorry._ Then I draw her name sign, the one she's wanted since the day we met. I hope she understands. I hope she likes what I've chosen. I think it suits her.
"Here. For you." I hand her the sketchbook. She flips through every page, the blanket resting on her shoulders like a cape. When she gets to the last two pages, she looks up and signs,
"Forgiven."
The kitchen light is on when I get home. I debate getting back into Lee and driving around the block until it goes dark. I don't really have a curfew, more like an unspoken rule that I should be home before midnight. I have three minutes. I park in the driveway. It's not like Ma will know what I've been up to.
But Ma isn't in the kitchen having her usual late-night Sleepytime tea fix. It's Mee, surrounded by tinfoil take-out containers. The whole room smells of onion, garlic, and butter. With the smallest hint of garam masala.
"Want to share?" she asks, without looking up.
"Sure." I sit across from her and scoop some chicken makhani onto a slice of almost-stale naan. Mee is an expert at reheating food. Every time I try to reheat leftovers, they come out scalding, and I burn my tongue so badly I can barely taste the rest of the meal. Ma is useless at it too, but she has the opposite problem. Cold in the middle. Mee's technique, whatever it is, is foolproof. Every bite warm and perfect.
"Makhani is my favorite," I sign, and chew.
"I don't doubt it, you're practically made of the stuff." She smiles.
"What do you mean?"
"When I was pregnant, it's all I wanted to eat." Mee takes the tin of shrimp pakoras and dumps the remaining two onto my plate. She folds up the tin and tosses it into the recycling bin. "Your poor mother. I made her go out and get it for me all the time. Almost every day. Once, I made her go twice in one day. I swear Rajdhani's is still in business because of you. It's certainly not because of that guy who works there, the rude one?"
"Avi," I remind her.
"Right. I'm lucky I met your mom; she doesn't put up with any shit. Like you." She takes my empty plate away and I help her clean up the table.
"Mee." I stop her before she begins washing our dishes. "I'm going to start the mural. But I'm going to paint it with my friend from school."
Mee hugs me and lifts my feet an inch off the ground. She bounces in place without letting go. I try to sign to her, tell her to stop, but my arms are pinned to my sides, so I just give in. I feel her laughing, and everything about it comforts me.
"I'm sorry about this year," I tell her when she finally lets me go.
"You're sixteen. It's to be expected." She brushes my hair behind my shoulders. "When I was sixteen, I tried to take a train across the country to see Nirvana play in Seattle."
"You did what? How? When? Wait...what?" I follow her around the kitchen as she puts everything away.
"That's a story for another time." Mee flicks the lights off and then kisses her thumb and presses it to my forehead. "I love you."
"I love you, too."
I would never have gone back. I would have left my job at Mickey D's without a second thought. But I left my black bag in my locker and I want it. I want to use the last of my overpass paint for our new piece. I want all of our histories, YP's and mine, tangled up together on the wall, our wall.
"Thank God, you're here." Jordyn rushes over to me as soon as I enter the locker room, her face slick with tears.
"What happened to you?"
"Donovan," she signs, sobbing. "He fucking dumped me, can you believe it?"
"Yes."
"Come on, Julia. I need you."
"Of course you do," I tell her with calm confidence. "This is what always happens. You only ever want me around when you need something."
"Don't be so dramatic." She wipes her face dry.
"You're the one crying over that loser with Mountain Dew breath," I joke, and she chuckles.
"What am I gonna do?" she asks, eyes fixed on the floor.
"I don't know."
"You're supposed to make me feel better."
"Why should I?" I actually want her to tell me. I wonder if she even realizes how hypocritical she's being.
"Because we were friends."
"Were we?"
"Sometimes." She almost looks embarrassed. Almost.
"That's not good enough anymore. I don't want sometimes friends."
"Right, you don't want any friends because you're too cool or whatever. You're better than friends."
"I'm not better than friends, I want _better friends._ I want friends who are all in, all the time. It can't just be all on your terms. You have to care, care about more than just yourself."
"This isn't about you! It's about Donovan breaking up with me for that über-bitch with the glasses."
"Look, you'll get over it." I wrap my arm around her shoulder. "You always do. This isn't the first time this has happened, and it won't be the last." I open my locker and grab my black bag. I stuff whatever is left behind in the front pocket. Jordyn calms down, she pulls her hair up, ready to start her shift.
"You're right," she says.
"And when it does happen again"—I stop and look directly into her eyes; I need her to understand me—"I don't want to hear about it." I fling my bag over my shoulder and head out the back door. She stomps her foot over and over, and I turn around.
"Wait! Where are you going?" she pleads, eyebrows arched.
"I quit."
It will be the biggest piece we've ever done. Bigger than the underpass, bigger than 5 Pointz. We're going to have to work our asses off. YP hauls paint from my trunk while I lay out some tarps. It feels strange to be setting up properly, in the light of day. I've seen two cop cars pass us, and each time I swear I froze in place. They didn't give us a second glance.
YP's dad brought over two ladders this morning and now he's at breakfast with my parents. They seem to be getting along, even though he's probably the worst texter I've ever met in my life. I finish with the tarps and back up, trying to take it all in. A blue mote of plastic underneath our White Dove wall.
"Do you think we got enough purple?" YP asks, signs springing from her fingertips.
"Are you kidding? We probably got too much." We did go overboard buying the paint. Our parents pitched in, and it was so much easier without all the covert ops that we ended up buying enough paint for two murals.
I wrench open the small can of Clear Skies, a light blue color we're going to use to outline the mural with. It's light enough that it won't show through the finished piece. YP brings over some brushes.
"You have the paper, right?" I ask her.
"Paper? For what?" Her eyebrows angle together. Even her facial expressions are getting better. She's losing her hearie accent.
"Our plans! The plans for the wall?" YP pats her pants out of instinct, I guess, because I don't know what pair of yoga pants has pockets. Her hands shoot up to her mouth, she's panicked.
"Oh, my God, I can't believe I forgot it. We can start tomorrow, I'm so sorry. Julia. Please, you have to—"
I hold my hands up to stop her. "Don't worry about it." I try to calm her down.
"Are you sure? What do you want to do? Should we wait, you want to drive to Greenlawn?"
"Whoa, whoa. Okay, deep breath." I take one for myself and mime for her to do the same. She does, but she still looks uneasy. "We don't need the plans, we've gone over it a dozen times."
"You want to start anyway?"
"Why the hell not? Since when do either of us carry around cheat sheets?"
"True."
"So chill. We got this." I dip both of our brushes into the paint and swirl them around. I look up at YP; a smile creeps over her face as she looks at our wall. She's ready. I stick out one of the brushes, a bead of Clear Skies drips over both our shoes.
"It's only the beginning."
# ACKNOWLEDGMENTS
I have more people to thank than stars in the sky, and every last one of them makes my universe brighter.
Huge thanks to my team at Knopf. Thanks to Marisa DiNovis for the tour and her contagious enthusiasm for books. To Ray Shappell for helping me design the heck out of my color-bombed cover; Stephanie Moss, who laid out the pages with great finesse and care; and, of course, my brilliant editor, Stephen Brown, who knows just how to make my stories shine. This book is beautiful because of all of you.
Thanks to everyone at my wonderful agency, Triada US. To Uwe Stender for his constant encouragement, and my dear agent, Brent Taylor, whose unfailing optimism and fierceness never cease to amaze me. Thank you for being my champion, agent, and friend.
To all of my d/Deaf, Hard of Hearing, and interpreter sensitivity readers, who helped make Julia's life and experiences as truthful as possible. Kelsey Young, Brook Wayne, Kimberly Bull, and Kate Boyd—thank you for your insight, honesty, and understanding. Very special thanks to Kalen Feeney, my wonderful ASL tutor and mentor in Deaf culture. I love chatting with you until my fingers hurt. If Julia ever makes it to the big screen, you're writing the script.
Nisha Sharma for your notes and enthusiasm for Julia's heritage and love of makhani.
Thanks to Jen at _Pop! Goes the Reader_ for my gorgeous cover reveal.
My first-ever critique partners, Kiersi Burkhart and Cynthia McGean. Thank you for pushing me to get words on the page week after week.
To everyone who keeps me going through countless chats, DMs, and texts: Jon & Vicky, Blair, Brooks, Lygia, Nita, Summer, and Kaitlyn. To Taryn, who had milk shakes with me on the day we sold the book. And especially Fred, who sent me the nicest email I've ever received.
To Mr. Nick Maravell, greatest art teacher of all time. Thank you for getting me through high school and inspiring me even now.
To my two perfect pugs, especially Gouda, who doesn't know what a book is or how to read, but has had such an impact on my life, I must include his fuzzy face.
Grandma Pat, who always encouraged me to sing as loud as possible, and Grandma Alice, who wanted me to be quiet and read more books. You both shaped me in very deep and obvious ways.
Mom, thank you for letting me paint all over my bedroom walls and encouraging me to keep painting everything around us. Aunt Linny, thank you for inspiring me to create beauty out of the ordinary. You're the strongest women I know.
I consider Best Friend to be a tier, not a person, and I'm so lucky to have three incredible best friends.
Arielle Gardner, my sister and my first-ever BFF. There are pieces of you in everything I've written. Some more obvious than others. Thank you for sharing all of your stories with me, and keeping thousands of my secrets. I love you.
Cara Hallowell. For all the Fridays in comic shops and Collage. For knitting and puzzle games and podcast recaps. For baking obscene amounts of cookies in your perfect tiny kitchen. Now that we've met, I can't imagine Portland without you. Thank you.
Brie Spangler. You've been my sister-in-arms through our publishing journey. I don't think I would be writing novels if it wasn't for your encouragement to break out of comic panels. Meeting you was one of the best things about moving to Portland. One of the best things, period. Hand in hand into the void. Thank you for believing.
This book is for you, Roger. But you know that. Thank you for investing in my dreams, in me, in our little life together. I promise it's only going to keep getting better. I love you more.
Thank you, Universe.
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1. Cover
2. Title Page
3. Copyright
4. Contents
5. Dedication
6. Chapter 1
7. Chapter 2
8. Chapter 3
9. Chapter 4
10. Chapter 5
11. Chapter 6
12. Chapter 7
13. Chapter 8
14. Chapter 9
15. Chapter 10
16. Chapter 11
17. Chapter 12
18. Chapter 13
19. Chapter 14
20. Chapter 15
21. Chapter 16
22. Chapter 17
23. Chapter 18
24. Chapter 19
25. Chapter 20
26. Chapter 21
27. Chapter 22
28. Chapter 23
29. Chapter 24
30. Chapter 25
31. Chapter 26
32. Chapter 27
33. Chapter 28
34. Chapter 29
35. Chapter 30
36. Chapter 31
37. Chapter 32
38. Chapter 33
39. Chapter 34
40. Chapter 35
41. Chapter 36
42. Chapter 37
43. Chapter 38
44. Chapter 39
45. Chapter 40
46. Chapter 41
47. Chapter 42
48. Chapter 43
49. Chapter 44
50. Chapter 45
51. Chapter 46
52. Chapter 47
53. Chapter 48
54. Chapter 49
55. Chapter 50
56. Chapter 51
57. Chapter 52
58. Chapter 53
59. Chapter 54
60. Chapter 55
61. Chapter 56
62. Chapter 57
63. Chapter 58
64. Chapter 59
65. Acknowledgments
1. Cover
2. Cover
3. Title Page
4. Contents
5. Start
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| {
"redpajama_set_name": "RedPajamaBook"
} | 3,286 |
\section{Introduction}
Anomalous rare-earth and sctinide compounds are studied extensively starting
from the middle of 1980s \cite{Stewart,Brandt}. They include so-called Kondo
lattices (with moderately enhanced electronic specific heat) and
heavy-fermion systems demonstrating a huge linear specific heat. Main role
in the physics of the Kondo lattices \cite{Brandt,Zwicknagl,Col2}
belongs to the interplay of the on-site
Kondo screening and intersite exchange interactions. Following to Doniach
criterion \cite{Don}, it was believed in early works \cite{Brandt}
that the total suppression of
either magnetic moments or the Kondo anomalies takes place. However, later
experimental data and theoretical investigations made clear that the
Kondo lattices as a rule demonstrate magnetic ordering or are close to this.
This concept was consistently formulated and justified in a series of the
papers \cite{IKFTT,IKZ1,Von2,kondo} treating the mutual
renormalization of two characteristic energy scales: the Kondo temperature
T_{K}$ and spin-fluctuation frequency $\overline{\omega }$. A simple scaling
consideration of this renormalization process in the $s-f$ exchange model
\cite{kondo,kondo1} yields, depending on the values of bare
parameters, both the \textquotedblleft usual\textquotedblright\ states (a
non-magnetic Kondo lattice or a magnetic state with weak Kondo corrections) and
the peculiar magnetic Kondo-lattice state. In the latter regime, small
variations of parameters result in strong changes of the ground-state
moment. Thereby high sensitivity of the ground-state moment to external factors
like pressure and doping by a small amount of impurities (a characteristic
feature of heavy fermion magnets) is naturally explained.
During 1990s, a number of anomalous $f$-systems (U$_{x}$Y$_{1-x}$Pd$_{3}$,
UPt$_{3-x}$Pd$_{x}$, UCu$_{5-x}$Pd$_{x}$, CeCu$_{6-x}$Au$_{x}$, U$_{x}$Th
_{1-x}$Be$_{13},$CeCu$_{2}$Si$_{2},$CeNi$_{2}$Ge$_{2},$ Ce$_{7}$Ni$_{3}$
etc.) demonstrating the non-Fermi-liquid (NFL) behavior have become a
subject of great interest (see, e.g., the reviews \cite{Maple,Stewart1}).
These systems possess unusual logarithmic or power-law temperature
dependences of electronic and magnetic properties.
Various mechanisms were proposed to describe the NFL behavior \cite{Proc},
including two-channel Kondo scattering \cite{Tsv,Col}, ``Griffiths
singularities'' in disordered magnets \cite{Castr}, strong spin
fluctuations near a quantum magnetic phase transition \cite{Col1,Vojta2}.
It is important that experimentally the NFL behavior (as well as heavy-fermion
behavior) is
typical for systems lying on the boundary of magnetic ordering and
demonstrating strong spin fluctuations \cite{Maple,Col2}.
The NFL behavior close to the quantum phase transition was theoretically
studied in a number of works \cite{Kim,Si,Vojta,Vojta2}.
In particular, renormalization group investigations near the quantum phase transitions
connected with the topology of the Fermi surface were performed \cite{Si}.
A scaling consideration of the Kondo lattices with account of
singularities in the spin excitation spectral function (which are owing to
Van Hove singularities in the magnon spectrum) yields the NFL behavior in an
extremely narrow interval of bare parameters only \cite{kondo}. As
demonstrated in Ref. \cite{nfl}, when taking into account renormalization of
spin-excitation damping, the region can become considerably broader.
The systems under consideration demonstrate both local-moment and itinerant
features.
Moreover, large linear specific heat and NFL behavior is observed
also in some $d$-systems including layered ruthenates Sr$_{2}$RuO$_{4}$ \cit
{ruthenates} and Sr$_{3}$Ru$_{2}$O$_{7}$ \cite{ruthenates1}.
It is well known that magnetism of itinerant systems is intimately related
to the presence of Van Hove singularities (VHS) near the Fermi level.
Therefore, it is instructive to treat the Kondo effect in systems with a
singular electron spectrum. This is the aim of the present paper.
It is evident that the Kondo effect in such systems has a number of peculiar
features. In particular, for the logarithmically divergent density of state
\begin{equation*}
\rho (E)=A\ln \frac{D}{B|E|}
\end{equation*
(the energy is referred to the Fermi level, $D$ is the half-bandwidth, the
constants $A$ and $B$ are determined by the band spectrum) the Kondo
singularities at $E_{F}$ become double-logarithmic. Perturbation expansion
yields a non-standard expression for the one-centre Kondo temperature,
\begin{equation}
T_{K}\varpropto D\exp \left[ -1/(AI)^{1/2}\right] , \label{perturb1}
\end{equation
instead of the result for the smooth density of states, $T_{K}\varpropto
D\exp [1/2I\rho (0)]$, $I$ being the $s-f$ exchange parameter. The
logarithmic divergence in $\rho (E)$ is typical for the two-dimensional case (in
particular, for the layered ruthenates). However, similar strong
Van Hove singularities can occur also in some three-dimensional systems like
Pd alloys and weak itinerant ferromagnets ZrZn$_{2}$ and TiBr$_{2}$ \cit
{VKT,pickett}.
In the present work we consider the Kondo problem and the NFL behavior with
the singular electron density of states for the lattice of $d(f)$-spins where a
competition with spin dynamics takes place.
In Sect. 2 the renormalization group equations in the presence of VHS are
presented. In Sect.3 the scaling behavior for a paramagnet and
for magnetic phases with account of spin-excitation damping is considered.
In Sect. 4
we treat the scaling behavior for the magnetic phases with account of the
incoherent contribution to spin spectral function. In Conclusions,
various electron properties and general physical picture of
magnetism are discussed.
\section{The scaling equations in the presence of Van Hove singularities}
We use the $s-d(f)$ exchange model of a Kondo lattice
\begin{equation}
H=\sum_{\mathbf{k}\sigma }t_{\mathbf{k}}c_{\mathbf{k}\sigma }^{\dagger }c_
\mathbf{k}\sigma }^{{}}-I\sum_{i\alpha \beta }\mathbf{S}_{i
\mbox {\boldmath
$\sigma $}_{\alpha \beta }c_{i\alpha }^{\dagger }c_{i\beta }^{{}}+\sum_
\mathbf{q}}J_{\mathbf{q}}\mathbf{S}_{\mathbf{-q}}\mathbf{S}_{\mathbf{q}}
\end{equation
where $t_{\mathbf{k}}$ is the band energy, $\mathbf{S}_{i}$ and $\mathbf{S}_
\mathbf{q}}$ are spin-density operators and their Fourier transforms, $J_
\mathbf{q}}$ are the intersite exchange parameters, $\sigma $ are the Pauli
matrices.
The density of states corresponding to the spectrum $t_{\mathbf{k}}$ is
supposed to contain a Van Hove singularity near the Fermi level. In
particular, for the square lattice with the spectrum
\begin{equation*}
t_{\mathbf{k}}=2t(\cos k_{x}+\cos k_{y})+4t^{\prime }(\cos k_{x}\cos k_{y}+1)
\end{equation*
we have the density of state
\begin{equation}
\rho (E)=\frac{1}{2\pi ^{2}\sqrt{t^{2}+Et^{\prime }-4t^{\prime 2}}}K\left(
\sqrt{\frac{t^{2}-(E-8t^{\prime })^{2}/16}{t^{2}+Et^{\prime }-4t^{\prime 2}}
\right) \simeq \frac{1}{2\pi ^{2}\sqrt{t^{2}-4t^{\prime 2}}}\ln \frac{1
\sqrt{t^{2}-4t^{\prime 2}}}{|E|}
\end{equation
where $K(E)$ is the complete elliptic integral of the first kind, the
bandwidth is determined by $|E-8t^{\prime }|<4|t|$. For $t^{\prime }=0$ we
deriv
\begin{equation}
\rho (E)=\frac{2}{\pi ^{2}D}K\left( \sqrt{1-\frac{E^{2}}{D^{2}}}\right)
\simeq \frac{2}{\pi ^{2}D}\ln \frac{4D}{|E|},~|E|<D=4|t|
\end{equation
so that, according to (\ref{perturb1}),
\begin{equation}
T_{K}\varpropto D\exp \left[ -\left( \frac{\pi ^{2}D}{2I}\right) ^{1/2
\right] . \label{perturb}
\end{equation
Note that the expression for the Kondo temperature in the parquet
approximation has a different form \cite{gogolin}.
\begin{equation}
T_{K}\varpropto D\exp \left[ -\frac{1}{(AI/2)^{1/2}}\right] =D\exp \left[
-\left( \frac{\pi ^{2}D}{I}\right) ^{1/2}\right] \label{gogolin}
\end{equation
However, the expression (\ref{perturb}) agrees with the numerical Wilson
renormalization group calculation for the square lattice \cite{zhuravlev},
unlike the result (\ref{gogolin}); the corresponding problems of the parquet
approximation in the Hubbard model are discussed in the works \cite{parquet}.
In Refs.\cite{kondo,kondo1,nfl}, the interplay of the Kondo effect and
intersite interactions was investigated by the renormalization group method.
This starts from the second-order perturbation theory with the use of the
equation-of-motion method (within the diagram technique for pseudofermions,
such an approximation corresponds to the one-loop scaling).
We apply the \textquotedblleft poor man scaling\textquotedblright\ approach
\cite{And}. This considers the dependence of effective (renormalized) model
parameters on the cutoff parameter $C$ which occurs at picking out the
singular contributions from the Kondo corrections to the effective coupling
and spin-fluctuation frequencies.
Using the results of Refs.\cite{kondo,nfl} we can write down
the system of scaling equations in the case of the Kondo lattice for various
magnetic phases.
In the calculations below we use the density of states for a square lattice
with $t^{\prime }=0$ both in the two-dimensional (2D) and three-dimensional
(3D) cases as a phenomenological one, so tha
\begin{equation}
\varrho (E)=\varrho (-D)F(E),\varrho (-D)=\frac{2\ln 4}{\pi ^{2}D},\
F(E)=\ln \frac{D}{|E|}+1
\end{equation}
We adopt the definition of the effective (renormalized) and bare $s-f$
coupling constan
\begin{equation}
g_{ef}(C)=-2\varrho I_{ef}(C),\ g=-2I\varrho ,~\rho =\varrho (-D)
\end{equation
where $C\rightarrow -0$ is a flow cutoff parameter. Other relevant variables
are the characteristic spin-fluctuation energy $\overline{\omega }_{ef}(C)$ and
magnetic moment $\overline{S}_{ef}(C)$.
To find the equation for $I_{ef}(C)$ we have to treat the electron
self-energy. For a ferromagnet (the case of an antiferromagnet is considered
in a similar way, see Ref.\cite{kondo}) the second-order Kondo contribution
reads
\begin{equation}
\Sigma _{\mathbf{k\pm }}^{(2)}(E)=\pm 2I^{2}\overline{S}\sum_{\mathbf{q}
\frac{n_{\mathbf{k-q}}}{E-t_{\mathbf{k-q}}\pm \omega _{\mathbf{q}}}.
\end{equation
where $n_{\mathbf{k}}=f(t_{\mathbf{k}})$ is the Fermi function. Then we have
\begin{equation}
\delta I_{ef}=[\Sigma _{\mathbf{k\downarrow }}^{(2)}(E)-\Sigma _{\mathbf
k\uparrow }}^{(2)}(E)]/(2S)
\end{equation}
Picking out
in the sums the contribution of intermediate electron states near the Fermi
level with $C<t_{\mathbf{k+q}}<C+\delta C$ we obtain
\begin{equation}
\delta I_{ef}(C)=2\rho F(C)I^{2}\eta (-\frac{\overline{\omega }}{C})\delta
C/C \label{ief}
\end{equation
where $\eta (x)$ is a scaling function which satisfies the condition $\eta
(0)=1$ which guarantees the correct one-impurity limit.
In the magnetically ordered
phase, $\overline{\omega }$ is the magnon frequency $\omega _{\mathbf{q
},$ which is averaged over the wavevectors $\mathbf{q=}2\mathbf{k}$
where $\mathbf{k}$ runs over the Fermi surface (for simplicity we use a
spherical Fermi surface). In the paramagnetic phase (the problem of
localized moment screening) $\overline{\omega }$ is determined from the
second moment of the spin spectral density.
Now we treat the singular correction to $\overline{\omega }_{ef}$ and the
effective magnetic moment $\overline{S}_{ef}$. We have within the spin-wave
picture
\begin{equation}
\delta \bar{S}=-\sum_{\mathbf{q}}\delta \langle b_{\mathbf{q}}^{\dagger }b_
\mathbf{q}}\rangle \label{Ssf}
\end{equation}
The singular contribution to magnon occupation numbers occurs owing to the
electron-magnon interaction. Calculation for a ferromagnet from the
corresponding magnon Green's function yields \cite{kondo}
\begin{equation}
\delta \langle b_{\mathbf{q}}^{\dagger }b_{\mathbf{q}}\rangle =I^{2}S\sum_
\mathbf{k}}\frac{n_{\mathbf{k}}(1-n_{\mathbf{k-q}})}{(t_{\mathbf{k}}-t_
\mathbf{k-q}}-\omega _{\mathbf{q}})^{2}} \label{bbf}
\end{equation}
We see that, when considering characteristics of localized-spin subsystem,
the lowest-order Kondo corrections originate from double integrals over both
electron and hole states. Then we have to introduce two cutoff parameters
C_{e}$ and $C_{h}$ with $C_{e}+C_{h}=C$ ($C$ is the cutoff parameter for the
electron-hole excitations), $\delta C_{e}=-\delta C_{h}~$to obtain
\begin{equation}
\delta \overline{S}_{ef}(C)/S=2\rho ^{2}I^{2}F(C/2)F(-C/2)\eta (-\frac
\overline{\omega }}{C})\delta C/C \label{sef}
\end{equation
The renormalization of spin-wave frequency owing to magnon-magnon scattering
is given b
\begin{equation}
\delta \omega _{\mathbf{q}}/\omega _{\mathbf{q}}=-a_{\mathbf{q}}\delta \langle b_{\mathbf
q}}^{\dagger }b_{\mathbf{q}}\rangle /S
\end{equation
Further on we pass to the magnon frequency averaged over the Fermi surface.
Then we have ($a_{\mathbf{q}}\rightarrow a$
\begin{equation}
\delta \overline{\omega }_{ef}(C)/\overline{\omega }=a\delta \overline{S
_{ef}(C)/S=2a\rho ^{2}I^{2}F(C/2)F(-C/2)\eta (-\frac{\overline{\omega }}{C
)\delta C/C \label{wef}
\end{equation
The latter result holds for all magnetic phases with $a=1-\alpha $ for the
paramagnetic (PM) phase, $a=1-\alpha ^{\prime }$ for the antiferromagnetic
(AFM) phase, $a=2(1-\alpha ^{\prime \prime })$ for the ferromagnetic (FM)
phase. Here $\alpha ,\alpha ^{\prime },\alpha ^{\prime \prime }$ are some
averages over the Fermi surface (see Ref.\cite{kondo}), $\alpha ^{\prime }=0$
in the nearest-neighbor approximation.
This approximation enables us to use a single renormalization parameter,
rather than the whole function of ${\bf q}$.
For simplicity, we put in numerical
calculations below $a=1$ (although the deviation $1-a$ just determines
critical exponents for physical properties, see Ref.\cite{nfl} and
Conclusions).
The scaling picture (which determines the NFL behavior)
is influenced by not only real, but also by imaginary part
of the spin-fluctuation energy.
The latter is even dominating in the paramagnetic phase (e.g., in the Heisenberg model
a spin-diffusion picture can be adopted at high temperatures).
In the magnetically ordered phases, the damping comes from paramagnon-like excitations.
They can be taken into account starting from the magnon picture of
the localized-spin excitation spectrum.
In the $s-f$ exchange model the damping is proportional
to $I^{2}$ and to the magnon frequency (for the details, see Ref.\cite{nfl}).
The dependence of the damping on the magnetic moment $\overline{S}$
(which is strongly renormalized) is crucial for the size of the NFL region.
The calculation of the damping in the second
order in $I$ yields the contributions of the order of both $I^{2}\overline{S}$
and $I^{2}$ \cite{Aus,afm} (formally, they correspond to the first and
second order in the quasiclassical parameter $1/2S$). The corresponding
problems from a semiphemenological point of view are discussed in Ref.\cit
{sokol}. Similar to Ref.\cite{nfl}, here we do not
introduce the additional factor of $\overline{S}$ to obtain in terms of
renormalized quantitie
\begin{equation}
\overline{\gamma }_{ef}(C)=kF(C/2)F(-C/2)g_{ef}^{2}(C)\overline{\omega
_{ef}(C),
\end{equation
the factor $k$ being determined by the bandstructure and magnetic ordering;
we put in numerical calculations $k=0.5$.
When taking into account spin-wave damping $\overline{\gamma }$ we have
\begin{equation*}
\eta \left( \frac{\overline{\omega }_{ef}(C)}{|C|}\right) \rightarrow \eta
\left( \frac{\overline{\omega }_{ef}(C)}{|C|},\frac{\overline{\gamma
_{ef}(C)}{|C|}\right)
\end{equation*
Replacing in the Kondo corrections $g\rightarrow g_{ef}(C),\,\overline
\omega }\rightarrow \overline{\omega }_{ef}(C)$ we derive the set of scaling
equations with account of the energy dependence of the electron density of
states:
\begin{eqnarray}
\partial g_{ef}(C)/\partial C &=&F(C)\Lambda , \label{gl} \\
\partial \ln \overline{\omega }_{ef}(C)/\partial C &=&-aF(C/2)F(-C/2)\Lambda
/2, \label{sl} \\
\partial \ln \overline{S}_{ef}(C)/\partial C &=&-F(C/2)F(-C/2)\Lambda /2
\label{ssl}
\end{eqnarray
with
\begin{equation*}
\Lambda =\Lambda (C,\overline{\omega }_{ef}(C),\overline{\gamma }_{ef}(C))
\frac{g_{ef}^{2}(C)}{|C|}\eta \left( \frac{\overline{\omega }_{ef}(C)}{|C|}
\frac{\overline{\gamma }_{ef}(C)}{|C|}\right) .
\end{equation*
Similar equations can be obtained for the general $SU(N)$ Coqblin-Schrieffer
model \cite{kondo} where $a/2\rightarrow a/N$ (there are some peculiarities
for FM case owing to asymmetry of spin-up and spin-down states).
One can see that the renormalizations of the spin-fluctuation energy
\overline{\omega }_{ex}(C)$ and the damping are more strong than that of
g_{ef}(C)$\ owing to the factors of $F(\pm C/2).$ We obtain from (\ref{sl}),
\ref{ssl})
\begin{equation}
\frac{\overline{S}_{ef}(C)}{S}=\left( \frac{\overline{\omega }_{ef}(C)}
\overline{\omega }}\right) ^{1/a} \label{sscc}
\end{equation
However, the simple expression for $\rho =~$const,
\begin{equation}
\overline{\omega }_{ef}(C)=\overline{\omega }\exp (-a[g_{ef}(C)-g]/2),
\label{linsmooth}
\end{equation
does not hold for the logarithmic density of states; no simple relation with
the quantity $\widetilde{g}_{ef}(C)=F(C)g_{ef}(C)$ is obtained either.
Expanding in $1/\ln |D/C|,$ we deriv
\begin{equation}
\overline{\omega }_{ef}(C)\simeq \overline{\omega }\exp \left( -\frac{a}{2
\int_{-D}^{C}\frac{dC^{\prime }}{C^{\prime }}[g_{ef}(C)-g_{ef}(C^{\prime
})]-a\ln 2[g_{ef}(C)-g]\right) \label{compl}
\end{equation}
Now we treat the scaling functions $\eta $. In the paramagnetic case we use
the spin-diffusion approximation (dissipative spin dynamics) to obtain (cf.
Ref.\cite{kondo}
\begin{equation}
\eta ^{PM}(\frac{\overline{\omega }}{C})=\left\langle \frac{1}{1+\mathcal{D}
\mathbf{k-k}^{\prime })^{2}/C^{2}}\right\rangle _{t_{k}=t_{k^{\prime }}=0},
\overline{\omega }=4\mathcal{D}k_{F}^{2}
\end{equation
where $\mathcal{D}$ is the spin diffusion constant, the averages go over the
Fermi surface. Integration yield
\begin{equation*}
\eta ^{PM}(x)=\left\{
\begin{array}{cc}
\arctan x/x & d=3 \\
\{\frac{1}{2}[1+(1+x^{2})^{1/2}]/(1+x^{2})\}^{1/2} & d=
\end{array
\right.
\end{equation*
In the FM and AFM phases for simple magnetic structures we have
\begin{equation}
\eta \left( \overline{\omega }_{ef}/|C|,\overline{\gamma }_{ef}/|C|\right)
{\rm Re}\left\langle \left( 1-(\omega _{\mathbf{k-k}^{\prime
}}^{{}}+i\gamma _{\mathbf{k-k}^{\prime }}^{{}})^{2}/C^{2}\right)
^{-1}\right\rangle _{t_{k}=t_{k^{\prime }}=0} \label{etafm}
\end{equation
For an isotropic 3D ferromagnet integration in (\re
{etafm}) for $\gamma =$ const and quadratic spin-wave spectrum $\omega _
\mathbf{q}}\propto q^{2}$ yields
\begin{equation}
\eta ^{FM}(x,z)=\frac{1}{4x}\ln \frac{(1+x)^{2}+z^{2}}{(1-x)^{2}+z^{2}}
\label{intfm}
\end{equation
where $x=\overline{\omega }_{ef}/|C|,z=$ $\overline{\gamma }_{ef}/|C|.$
Although details of the spin-wave spectrum are not reproduced in such an approach,
the renormalization of spin-wave frequency (which is an cutoff for the Kondo divergences)
is adequately reproduced.
In the 2D case we obtain in the same approximation
\begin{equation}
\eta ^{FM}(x,z)=\frac{1}{2}{\rm Re
\{[(1+iz)(1+iz-x)]^{-1/2}+[(1-iz)(1-iz+x)]^{-1/2}\}
\end{equation
For an antiferromagnet integration in (\ref{etafm}) with the linear spin-wave
spectrum $\omega _{\mathbf{q}}\propto cq$ gives
\begin{equation}
\eta ^{AFM}(x,z)=\left\{
\begin{array}{cc}
-\frac{1}{2x^{2}}\text{ }\{\ln [(1+z^{2}+x^{2})^{2}-4x^{2}]-2\ln (1+z^{2})\}
& d=3 \\
2{\rm Im}(x^{2}-1-2iz+z^{2})^{-1/2} & d=
\end{array
\right. \label{intafm}
\end{equation
which modifies somewhat the results of Ref.\cite{kondo}.
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig1a.eps}
\includegraphics[width=0.45\columnwidth]{fig1b.eps}
\caption{
The scaling functions $\eta (x)$ for a ferromagnet (solid line),
antiferromagnet (long-dashed line) and paramagnet (short-dashed line) in the
2D (a) and 3D (b) cases
}
\label{fig:1}
\end{figure}
The plot of the functions $\eta (x)=$ $\eta (x,0)$ for different magnetic
phases is shown in Fig.1. Note that for a 2D antiferromagnet $\eta (x)$
vanishes discontinuosly at $x>1$. However, a smooth non-zero contribution
can occur for more realistic models of magnon spectrum.
\section{The scaling behavior in paramagnetic and magnetic phases}
We start the discussion of scaling behavior
from the simple case of the Coqblin-Schrieffer model in the limit
N\rightarrow \infty .$ Then the renormalization of the magnon frequency is
absent and the scaling behavior can be investigated analytically, similar to
the \cite{kondo}. We have
\begin{equation}
1/g_{ef}(C)-1/g=G(C)=-\int_{-D}^{C}\frac{dC^{\prime }}{C^{\prime }
F(C^{\prime })\eta (-\frac{\overline{\omega }}{C^{\prime }}) \label{intG}
\end{equation
The equation (\ref{intG}) can be used even for $N=2$ provided that $g$ is
considerably smaller than the critical value $g_{c}$. The effective coupling
$g_{ef}(C)$ begins to deviate strongly from its one-impurity
behavior
\begin{equation}
1/g_{ef}(C) \simeq 1/g-\frac{1}{2}\ln ^{2}|D/C|
\end{equation
at $|C|\sim \overline{\omega }.$ The boundary of the strong coupling
region (the renormalized Kondo temperature) is determined by
G(C=-T_{K}^{\ast })=-1/g.$ Of course, $T_{K}^{\ast }$ means here only some
characteristic energy scale extrapolated from high temperatures, and the
detailed description of the ground state requires a more detailed
consideration. In the PM, FM and 2D AFM phases spin dynamics suppresses
T_{K}^{\ast }.$ To leading order in $\ln (D/\overline{\omega })$ we hav
\begin{equation*}
T_{K}^{\ast }\simeq (T_{K}^{2}-\overline{\omega }^{2})^{1/2}
\end{equation*
with $T_{K}$ given by (\ref{perturb}). [However, owing to the minimum of the
scaling function (\ref{intafm}) (Fig. 1b), in the 3D AFM case spin dynamics
at not large $\overline{\omega }$ results in and increase of $T_{K}^{\ast }
.]
Provided that the strong coupling regime does not occur, i.e. $g$ is smaller
than the critical value $g_{c}$, $g_{ef}(C\rightarrow 0)$ tends to a finite
value $g^{\ast }.$ To leading order in $\ln (D/\overline{\omega })$ we have
\begin{equation}
1/g_{c}=\frac{1}{2}\ln ^{2}\frac{D}{\overline{\omega }}=\frac{1}{2}\lambda
^{2}
\end{equation
(which yields also a rough estimate of $g_{c}$ for $N=2$). An account of
next-order terms results in an appreciable dependence on the type of
magnetic ordering and space dimensionality. For PM, FM and 2D AFM phases the
critical value $g_{c}$ is given by $1/g_{c}=-G(0)$, and in the 3D AFM case
g_{c}$ is determined by the minimum of the function $G(C)$.
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig2a.eps}
\includegraphics[width=0.45\columnwidth]{fig2b.eps}
\caption{
The scaling trajectories $g_{ef}(\xi =\ln |D/C|)/g$ (a) and the
corresponding dependences $\ln (S/\overline{S}(\xi ))=\ln (\overline{\omega
/\overline{\omega }_{ef}(\xi ))$ (b) for 2D paramagnets with the logarithmic
density of states (solid lines, $g=0.029<g_{c},g=0.030>g_{c}$) and constant
density of states (dashed lines, $g=0.14<g_{c},g=0.145>g_{c}$),
$a=1,\lambda =\ln (D/\overline{\omega })=5$
}
\label{fig:2}
\end{figure}
The numerical calculations for $N=2$ were performed for $\lambda =\ln (D
\overline{\omega })=5.$ The plots are presented in Figs.2-5
for both smooth and singular bare
densities of states. We compare these cases in the
same relative interval of the coupling constant $|g-g_{c}|/g_{c}$. The
scaling process for finite $N$ in the former case is
described in Ref. \cite{kondo}. It turns out that the qualitative
picture near the
critical value of the coupling constant is rather universal for the same
interval $|g-g_{c}|/g_{c}$ depending mainly on the scaling function and the
damping, but not on the details of the bare electron density of states (even
on its singularities). The shift of the Van Hove singularity below the Fermi
level does not influence strongly the results too, although we cross the
singularity during the scaling process (the singularity is in fact
integrable).
An important quantitative difference in the presence of VHS is that the
renormalized coupling constant is considerably smaller. Moreover, relative
renormalization of the coupling constant is also smaller (Fig.2-4). This
makes using perturbation theory and lowest-order scaling analysis
more physically
reliable than for the smooth density of states.
At the same time, in the presence of VHS the renormalization of
spin-fluctuation frequency (and of magnetic moment) becomes larger (a
\textquotedblleft soft mode\textquotedblright\ situation, which favors a NFL
behavior). Formally, this is due to that the derivative $\partial \ln
\overline{\omega }_{ef}(C)/\partial C$ is proportional to $F^{2}(C/2)$, and
\partial g_{ef}(C)/\partial C$ to $F(C)$ only [see Eqs. (\ref{sl})-(\re
{ssl})]. Such a situation is similar to the scaling in the large-$l$ limit (
l$ is the number of scattering channels for conduction electrons) where the
effective coupling is not renormalized, $(2l+1)g^{2}/2=\widetilde{g}^{2}=~
const, since $a\rightarrow a(2l+1)$ \cite{kondo}.
For a paramagnet with pure dissipative dynamics, the one-impurity behavior
1/g_{ef}(\xi =\ln |D/C|)=1/g-\xi $ is changed at $\xi \simeq \lambda $ by a
NFL-like (smeared quasi-linear) region where
\begin{equation}
~\ln [\overline{\omega }/\overline{\omega }_{ef}(\xi )]=a\ln [S/\overline{S
(\xi )]\simeq (aA/2)\xi \label{lina}
\end{equation
with $A<2/a$. Such a behavior takes place both for $g<g_{c}$ and $g>g_{c}$
in a wide region of $\xi ,$ i.e. up to rather low temperatures (Fig. 2b, for
a discussion of physical properties see Conclusions).
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig3a.eps}
\includegraphics[width=0.45\columnwidth]{fig3b.eps}
\caption{
The scaling trajectories $g_{ef}(\xi )/g$ (a) and the corresponding
dependences $\ln (S/\overline{S}(\xi ))=\ln (\overline{\omega }/\overline
\omega }_{ef}(\xi ))$ (b) for 2D ferromagnets with the logarithmic
(solid lines, $g=0.052<g_{c},g=0.053>g_{c}$) and constant
density of states (dashed lines, $g=0.186<g_{c},g=0.189>g_{c}$), $k=0.5$,
other parameters being the same as in Fig.2
}
\label{fig:3}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig4a.eps}
\includegraphics[width=0.45\columnwidth]{fig4b.eps}
\caption{
The scaling trajectories $g_{ef}(\xi )/g$ (a) and
the corresponding dependences $\ln (S/\overline{S
(\xi ))=\ln (\overline{\omega }/\overline{\omega }_{ef}(\xi ))$ (b)
for 2D antiferromagnets
with the logarithmic (solid lines,
g=0.0361<g_{c},g=0.0365>g_{c}$) and constant density of states (dashed
lines, $g=0.153<g_{c},g=0.155>g_{c}$)
}
\label{fig:4}
\end{figure}
For magnetic phases, the singularities of the scaling function $\eta
(x\rightarrow 1)$ play the crucial role. A rather distinct NFL behavior
takes place in a more narrow region where the argument of the function $\eta
$ is fixed at the singularity during the scaling process, so that
\begin{equation}
\overline{\omega }_{ef}(C)\simeq |C|~,~\ln [\overline{\omega }/\overline
\omega }_{ef}(\xi )]\simeq \xi . \label{w=c}
\end{equation
Then for a smooth density of states we obtain from (\ref{linsmooth})
\begin{equation}
g_{ef}(\xi )-g\simeq 2(\xi -\lambda )/a \label{linnn}
\end{equation
However, in the presence of VHS the relation (\ref{compl}) between
\overline{\omega }_{ef}(C)$ and $g_{ef}(C)$ is more complicated.
In the case of a constant magnon damping considered in Ref. \cite{kondo},
the region (\ref{w=c}) is not too narrow only provided that the bare
coupling constant $g$ is very close to the critical value $g_{c}$ for the
magnetic instability ($|g-g_{c}|/g_{c}\sim 10^{-4}\div 10^{-6}$). However,
when taking into account the magnon damping renormalization, this region is
considerably wider, although being smeared \cite{nfl}, and the influence of
the energy dependence $\rho (E)$ becomes stronger.
One can see that VHS lead to a considerable increase of the effective moment
$S^{\ast }=\overline{S}(\xi \rightarrow \infty ).$ For the chosen deviation
|g-g_{c}|/g_{c}\sim 1\%$ the moment renormalization $S/S^{\ast }$ can make
up one-two orders of magnitude; even for $|g-g_{c}|/g_{c}\sim 10-20\%,~$the
moment can decrease by several times.
Since in 2D antiferromagnet $\eta (x>1)=0,$ a sharp transition to the
saturation plateau occurs, unlike the FM case (cf. Figs.3 and 4).
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig5a.eps}
\includegraphics[width=0.45\columnwidth]{fig5b.eps}
\caption{
The scaling trajectories $g_{ef}(\xi )/g$ (a) and
the corresponding dependences $\ln (S/\overline{S}(\xi
))=\ln (\overline{\omega }/\overline{\omega }_{ef}(\xi ))$ (b)
for 3D antiferromagnets
with the logarithmic (solid lines, $g=0.072<g_{c},g=0.073>g_{c}$)
and constant density of states (dashed lines,
$g=0.228<g_{c},g=0.231>g_{c}$)
}
\label{fig:5}
\end{figure}
The situation in 3D systems with a logarithmic density of states can be
considered in a similar way. Surprisingly, for 3D ferromagnets to high
accuracy the scaling trajectories (except for a narrow critical region near
g_{c}$) and the critical values $g_{c}$ turn out to be very close to 2D case
(for both smooth density of states and with VHS). Therefore we do not show
the corresponding plot. On the other hand, for 3D antiferromagnets VHS does
not lead to suppression of magnon frequencies (Fig.5). This is due to the
influence of the minimum in the scaling function (Fig.1b).
\section{The scaling behavior with account of incoherent contributions}
The magnon approximation used in the previous Section is not quite valid since
this underestimates the role of the damping. In fact, the spin spectral
function should have an intermediate form between large-damping and spin-wave
pictures, containing both coherent (magnon-like) and incoherent
contributions A simple attempt to construct the corresponding scaling
function as a linear combination was performed in Ref. \cite{kondo}. In
particular, in the case of a ferromagnet we have near the magnon pole
\begin{equation}
\langle \langle S_{\mathbf{q}}^{+}|S_{-\mathbf{q}}^{-}\rangle \rangle
_{\omega }=\frac{2\overline{S}\mathcal{Z}_{\mathbf{q}}}{\omega -\omega _
\mathbf{q}}^{ef}}+\langle \langle S_{\mathbf{q}}^{+}|S_{-\mathbf{q
}^{-}\rangle \rangle _{\omega }^{incoh} \label{gz}
\end{equation
where the inverse residue at the pole is determined by
\begin{equation}
1/\mathcal{Z}_{\mathbf{q}}=1-\left( \frac{\partial \Pi _{\mathbf{q}}(\omega
}{\partial \omega }\right) _{\omega =\omega _{\mathbf{q}}}
\end{equation
$\Pi _{\mathbf{q}}(\omega )$ being the polarization operator of the magnon
Green's function. Besides that, there exists the singular contribution which
comes from the incoherent (non-pole) part of the spin spectral density. Then
we get
\begin{eqnarray}
\partial g_{ef}(C)/\partial C &=&F(C)\Lambda \label{gll} \\
\partial \ln \overline{\omega }_{ef}(C)/\partial C &=&-aF(C/2)F(-C/2)\Lambda
/2 \label{wll} \\
\partial (1\emph{/}\mathcal{Z})/\partial C &=&\partial \ln \overline{S
_{ef}(C)/\partial C=-F(C/2)F(-C/2)\Lambda /2 \label{zl}
\end{eqnarray
wher
\begin{equation}
\Lambda =[g_{ef}^{2}(C)/|C|][\mathcal{Z}\eta _{coh}(\overline{\omega
_{ef}(C)/|C|)+(1-\mathcal{Z})\eta _{incoh}(\overline{\omega }_{ef}(C)/|C|)]
\label{l1}
\end{equation
with $\eta _{coh}=\eta ^{FM}$. The choice of $\eta _{incoh}$ is a more
difficult problem; here we put simply $\eta _{incoh}=\eta ^{PM}.~$
According to (\ref{zl}) we have
\begin{equation}
\frac{1}{\mathcal{Z}(\xi )}=1+\ln \frac{S}{\overline{S}(\xi )} \label{1/ZS}
\end{equation
Consequently, the increase of magnetic moment owing to the Kondo screening
leads to a considerable logarithmic suppression of magnon contributions to
the spectral density.
The role of the incoherent contribution becomes important only provided that
$S/\overline{S}(\xi )$ and $\mathcal{Z}$ deviate appreciably from unity.
However, such a moment suppression just occurs at passing the region of
singularity in $\eta _{coh}(x=1)$, the further scaling process being
determined by the incoherent contribution.
\begin{figure}[tbp]
\includegraphics[width=0.45\columnwidth]{fig6a.eps}
\includegraphics[width=0.45\columnwidth]{fig6b.eps}
\caption{
The scaling trajectories $g_{ef}(\xi )/g$ taking into account
incoherent contributions (a) and the corresponding
dependences $\ln (S/\overline{S}(\xi ))=\ln (\overline{\omega }/\overline
\omega }_{ef}(\xi ))$ (b) for 2D ferromagnets with the logarithmic
(solid lines, $g=0.0330<g_{c},g=0.0335>g_{c}$) and constant density
of states (dashed lines, $g=0.143<g_{c},g=0.145>g_{c}$),
$k=0.5,a=1,\lambda =\ln (D/\overline{\omega })=5$
}
\label{fig:6}
\end{figure}
The corresponding scaling trajectories are shown in Fig.6. Now we have a
two-stage renormalization. One can see that the well-linear
\textquotedblleft coherent\textquotedblright\ behavior region (which is
rather narrow in Fig.6) is changed by a PM-like \textquotedblleft
quasi-linear\textquotedblright\ behavior (\ref{lina}) with increasing $\xi.$
This crossover occurs when the function $\eta _{coh}$ reaches its maximum
value at the singularity (cf. Ref.\cite{kondo}).
The \textquotedblleft quasi-linear\textquotedblright\ behavior, although
being somewhat smeared, is pronounced in a considerable region of $\xi $
even for not too small $|g-g_{c}|$. The difference with the PM case is in
that $g_{ef}(\xi )$ increases considerably at the first stage of
renormalization owing to the singularity of the function $\eta _{coh}$.
A similar consideration can be performed for the antiferromagnetic phase
(cf. Ref.\cite{kondo}).
The account of incoherent contribution results in a smearing
of the non-monotonous behavior of $g_{ef}(\xi )$ in the 3D AFM case, so that
at small $|g-g_{c}|$ the maximum in the dependences $g_{ef}(\xi )$ and $\ln
\overline{\omega }/\overline{\omega }_{ef}(\xi )]$ vanishes completely.
\section{Discussion and conclusions}
In the general problem of metallic magnetism, the peaks in the bare density
of states (which are usually connected with VHS) near the Fermi level play a
crucial role. Here we have investigated their influence starting from the
Kondo lattice ($s-d(f)$ exchange) model.
For $g\rightarrow g_{c}$ we obtain the magnetic state with small effective
moment $S^{\ast }$ and a NFL-type behavior. The corresponding dependences
\overline{S}(T)=\overline{S}_{ef}(|C|\rightarrow T)$ describe an analogue of
the \textquotedblleft temperature-induced magnetism\textquotedblright\ \cit
{Mor}. Such a picture is based on the many-electron renormalization
(compensation) of localized magnetic moments and differs outwardly from the
ordinary mechanism for weak itinerant ferromagnets with small $\overline{S},$
which are assumed to correspond to the immediate vicinity of the Stoner
instability.
However, the physical difference is not radical. In fact, a continuous
transition exists between the highly-correlated Kondo lattices and the
\textquotedblleft usual\textquotedblright\ itinerant-electron systems. In
particular, one may view Pauli paramagnets as systems with high $T_{K}$ of
order of the Fermi energy; for enhanced Pauli paramagnets like Pd, Pt, UAl$_2$,
where the Curie-Weiss holds at high temperatures, one introduces
instead of the Kondo temperature the so-called spin-fluctuation temperature.
A combined description of the Kondo
lattice state and weak itinerant magnetism has been considered recently by
Ohkawa \cite{Ohkawa}. Remember that the Kondo systems with VHS near the
Fermi level under consideration just possess high values of $T_{K}$ (see the
Introduction).
In this context, it would be instructive to describe weak itinerant magnets
not from the \textquotedblleft band\textquotedblright\ point of view, but
from the side of local magnetic moments which are nearly compensated. Since
a number of cerium NFL systems demonstrate
itinerant-electron behavior \cite{Proc}
and it is customary now to treat UPt$_{3}$, CeSi$_{x}$ and CeRh$_{3}$B$_{2}$
as weak itinerant magnets, the second approach appears already by far less
natural than the first (see Refs.\cite{613,IKZ1}).
From the formal point of view,
perturbation calculations in the Hubbard model, which describes
itinerant-electron systems, are similar to those in the $s-d(f)$ model,
provided that one postulates the existence of local moments. Besides that,
for two-dimensional itinerant systems with strong spin fluctuations the
semiphenomenological spin-fermion model can be used which separates electron
and spin degrees of freedom and is somewhat similar to $s-d$ exchange model
\cite{spin-ferm1,ruthenates2}.
Further on, the question arises about the role which many-electron effects
play in the \textquotedblleft classical\textquotedblright\ weak itinerant
3D magnets like ZrZn${}_{2}$ and TiBe$_{2}$. Indeed, one can hardly believe
that the extremal smallness of $\overline{S}$ in these systems is due to
accidental bare values of $N(E_{F})$ and Stoner parameter. Moreover, the
Stoner criterion is not valid even qualitatively (in particular, due to
spin fluctuations the critical coupling $U_{c}$ in the Hubbard model is
finite when the Fermi level tends to VHS \cite{ruthenates2}). Thus a scaling
consideration in the presence of VHS would be of interest, especially
with account of chemical potential renormalizations (cf. Ref.\cite{VKT}).
In particular, owing
to VHS the chemical potential can depend weakly on electron concentration (the
pinning phenomenon \cite{pinning}). In the Kondo systems such a treatment
may lead to a renormalization of the scale $|g-g_{c}|$ itself.
Now we discuss some real layered systems. Specific heat is considerably
enhanced in ruthenates Sr$_{2}$RuO$_{4}.$ A gradual enhancement
of the electronic specific heat and a more drastic increase of the
static magnetic susceptibility were observed in Sr$_{2-y}$La$_{y}$RuO$_{4}$
with increasing $y$. Furthermore, the quasi-2D Fermi-liquid
behavior observed in pure Sr$_{2}$RuO$_{4}$ breaks down near the critical
value $y=0.2$. The enhancement of the density of states can be ascribed to
the elevation of the Fermi energy toward a Van Hove singularity of the
thermodynamically dominant Fermi-surface sheet. The NFL
behavior is attributed to two-dimensional FM fluctuations with
short-range correlations at VHS \cite{ruthenates}.
The bilayered ruthenate system Sr$_{3}$Ru$_{2}$O$_{7}$ in the ground state
is a paramagnetic Fermi liquid with strongly enhanced quasiparticle
masses. The Fermi-liquid region of the phase diagram extends up to 10-15 K
in zero field and is continuously suppressed towards zero temperature upon
approaching the critical field of $B=8$T. In the vicinity of the putative
quantum critical end point, NFL behavior has been observed in
various macroscopic quantities including specific heat, resistivity and
thermal expansion and has been described on the basis of phenomenological
models \cite{ruthenates1}.
We can mention also some layered $f$-systems. The layered Kondo lattice
model was proposed for quantum critical beta-YbAlB$_{4}$ where
two-dimensional boron layers are Kondo coupled via interlayer Yb moments
\cite{YbAlB4}. CeRuPO seems to be one of the rare examples of a ferromagnetic
Kondo lattice where LSDA+U calculations evidence a quasi-2D
electronic band structure, reflecting a strong covalent bonding within the
CeO and RuP layers and a weak ioniclike bonding between the layers \cit
{CeRuPO}.
To describe layered antiferromagnetic cuprates, the 2D $t-t'$ Hubbard model
is often used which also describes Fermi-liquid and NFL regimes.
Despite the density-of-state logarithmic singularity, the staggered spin susceptibility
in this model does not diverge within the Fermi-liquid approach,
the reason being the appearance of the logarithmic singularity
in the quasiparticle mass \cite{Hlubina}.
The effective mass renormalization is beyond our lowest-order (one-loop) scaling
consideration, but may play a role at an accurate treatment.
The two-loop considerations of the flat-Fermi-surface and $t-t'$ Hubbard models
\cite{Freire,katanin2} yield an (generally speaking, anisotropic)
suppression of the quasiparticle weight (inverse effective mass) along the Fermi surface,
the staggered spin susceptibility remaining divergent, although the divergence is
considerable weakened.
Note that the picture in the Hubbard model and $s-d(f)$ model (where ``direct''
exchange interaction is present) can be considerably different.
Recently, the $\epsilon$-expansion has been used for a scaling consideration
of the 2D antiferromagnetic Kondo lattice with the use of non-linear sigma model
\cite{2Drg}.
In the case of a smooth electron density of states, various physical
properties of NFL systems are discussed within our approach
in Ref. \cite{nfl}. The temperature behavior of
magnetic characteristics $\overline{S}$ and $\overline{\omega }$, which
depend exponentially on the coupling constant, is decisive for the NFL
picture under consideration.
At the same time, the presence of VHS near the Fermi level
influences strongly all the electronic, magnetic and transport properties in
the scaling approach, as well as in the one-electron theory. The replacement
$\rho ^{2}g_{ef}^{2}(T)\rightarrow \rho ^{2}(T)g_{ef}^{2}(T)$ with $\rho
(T)$ being considerably temperature dependent owing to VHS may modify
somewhat the behavior of observable quantities.
Consider the temperature dependence of the magnetic susceptibility $\chi
\propto \overline{S}/\overline{\omega }$. Using the scaling arguments we can
replace $\overline{\omega }\rightarrow \overline{\omega }_{ef}(C),$
\overline{S}\rightarrow \overline{S}_{ef}(C)$ with $|C|\sim T,$ which yields
$\chi (T)\propto T^{-\zeta }$. The non-universal exponent $\zeta $ is
determined by details of magnetic structure (the difference $a-1$ can be
used as a perturbations, see Ref. \cite{nfl}). Besides that, a number of
crossovers are characteristic for NFL behavior under consideration. In the
coherent \textquotedblleft magnon\textquotedblright\ regime we have $\zeta
=(a-1)/a,$ and in the \textquotedblleft quasilinear\textquotedblright\
(incoherent) region $\zeta =(a-1)A/2$.
The temperature dependence of electronic specific heat can be estimated from
the second-order perturbation theory, $C_{el}(T)/T\propto 1/Z(T)$ where
Z(T) $ is the residue of the one-electron Green's function at the distance $T$
from the Fermi level. Then we have
\begin{equation}
C_{el}(T)/T\propto \rho ^{2}(T)g_{ef}^{2}(T)\overline{S}_{ef}(T)/\overline
\omega }_{ef}(T)\propto \rho ^{2}(T)g_{ef}^{2}(T)\chi (T)
\end{equation
which results in a non-trivial behavior of the Wilson ratio.
Following to Ref.\cite{nfl}, a simple estimation of the transport
relaxation rate (which determines the temperature dependence of the resistivity
owing to scattering by spin fluctuations in AFM phase) yields
\begin{equation}
\ \frac{1}{\tau }\propto T^{2}\rho ^{2}(T)g_{ef}^{2}(T)\overline{S}_{ef}(T)
\overline{\omega }_{ex}(T)\propto T^{2}C_{el}(T)/T \label{taul}
\end{equation}
However, a more refined treatment in spirit of Ref.\cite{Hlubina}
would be useful in some cases.
In Ref.\cite{nfl}, a mechanism of NFL behavior owing to peculiar behavior of
spin spectral function was proposed. Here we treated a similar, but somewhat
more simple and natural mechanism which is connected with the singularities in the bare
electron spectrum.
Of course, a more accurate treatment of magnetic fluctuations near the quantum phase
transition is required.
Therefore detailed investigations of the NFL behavior for a
realistic Fermi surface and spin spectral function are of interest. An
accurate investigation of the situation where VHS is shifted from $E_{F}$
or two peaks are present below and above $E_{F}$ \cite{VKT} would be also
instructive.
The research described was supported in part by the Program
\textquotedblleft Quantum Physics of Condensed Matter\textquotedblright\
from Presidium of Russian Academy of Sciences.
The author is grateful to M.I. Katsnelson and A.A. Katanin for discussions of
the problem.
| {
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Viktor Arkadyevich Bryzhin (en anglais), souvent Bryzgin comme en russe, en , Viktor Arkadiïovytch Bryzhine ; en , Viktor Arkadievitch Bryzguine, né le à Vorochilovgrad, est un athlète ukrainien ayant concouru pour l'Union soviétique dans l'épreuve du 100 mètres.
Carrière
Viktor Bryzhin fait ses débuts sur la scène internationale durant la saison 1983 à l'occasion des premiers championnats du monde en plein air tenus à Helsinki. Éliminé en quart de finale du 100 m, il remporte en fin de compétition la médaille de bronze du relais 4 × 100 m aux côtés de ses coéquipiers soviétiques Andreï Prokofiev, Nikolay Sidorov et Vladimir Muravyov.
Aux Championnats d'Europe d'athlétisme 1986, il prend la huitième et dernière place de la finale du 100 m en 10 s 38 et s'adjuge le titre continental du relais 4 × 100 m avec le temps de 38 s 29. L'année suivante, Bryzhin termine en sixième position de l'épreuve-reine des Championnats du monde de Rome mais sera reclassé cinquième (meilleur placement d'un Soviétique sur 100 m) à la suite de la disqualification pour dopage du Canadien Ben Johnson. Dans l'épreuve collective, l'équipe d'URSS composée de Aleksandr Yevgenyev, Viktor Bryzhin, Vladimir Muravyov et Vladimir Krylov remporte la médaille d'argent du 4 × 100 m en établissant un nouveau record d'Europe de la discipline en 38 s 02.
Bryzhin obtient la consécration internationale en 1988 en enlevant le titre olympique du relais 4 × 100 m des Jeux de Séoul, associé à Vladimir Krylov, Vladimir Muravyov et Vitaliy Savin. L'URSS devance avec le temps de 38 s 19 le Royaume-Uni et la France, Viktor Bryzhin prenant le départ de la course.
Viktor Bryzgin est l'époux d'Olha Bryzhina, coureuse de 400 mètres titrée à plusieurs reprises lors des Jeux olympiques et des Championnats du monde et le père d'Elizaveta Bryzhina et d'Anastasiya Bryzhina.
Palmarès
Jeux olympiques
Jeux olympiques d'été de 1988 à Séoul :
Médaille d'or du relais 4 × 100 m
Championnats du monde
Championnats du monde d'athlétisme 1983 à Helsinki :
Médaille de bronze du relais 4 × 100 m
Championnats du monde d'athlétisme 1987 à Rome :
Médaille d'argent du relais 4 × 100 m
Championnats d'Europe
Championnats d'Europe d'athlétisme 1986 à Stuttgart :
Médaille d'or du relais 4 × 100 m
Records
100 m : 10 s 23 (1987)
Notes et références
Liens externes
Athlète soviétique
Sprinteur ukrainien
Champion olympique soviétique
Champion olympique d'athlétisme
Champion d'Europe d'athlétisme
Coureur de 100 mètres
Relayeur (athlétisme)
Détenteur d'un record d'Europe d'athlétisme
Athlète (homme) aux Jeux olympiques d'été de 1988
Naissance en août 1962 | {
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Q: ¿Cómo puedo actualizar un jtable que obtiene los datos de la BD en tiempo real sin necesidad de oprimir un botón? Lo que intento hacer es que un jtable se actualice automáticamente sin la necesidad de presionar un botón para mostrar que los datos han cambiado en tiempo real, solo quiero que la tabla lea la BD y cambie si es que se han agregado o eliminado datos en tiempo real.
Soy novato en programación por lo que no se mucho del tema.
void mostrardatos (String valor) {
conectar cc=new conectar();
Connection cn=cc.conexion();
DefaultTableModel modelo=new DefaultTableModel();
modelo.addColumn("ID");
modelo.addColumn("Nombre");
modelo.addColumn("Apellido");
jTable1.setModel(modelo);
String sql="";
if (valor.equals("")){
sql="SELECT * FROM idconexion";
}
else{
sql="SELECT * FROM idconexion WHERE (id='"+valor+"' OR nombre='"+valor+"' OR apellido='"+valor+"')";
}
String []datos=new String [3];
try{
Statement st=cn.createStatement();
ResultSet rs=st.executeQuery(sql);
while(rs.next()){
modelo.addRow(new String[]{
rs.getString("id"),
rs.getString("nombre"),
rs.getString("apellido")
});
}
jTable1.setModel(modelo);
}catch(SQLException ex){
Logger.getLogger(datos.class.getName()).log(Level.SEVERE,null,ex);
}
}
Esta función es la que hace que cuando presione El botón "mostrar" la tabla se refresque y muestre los datos en ella.
A: Podrias colocar un TIMER para realizar esa acción.
import java.awt.event.*;
import java.util.*;
import javax.swing.Timer;
public class temporizador {
public static void main(String[] args) {
// Creamos una instancia de la clase mostrarHora que implementa
// la interfaz "ActionListener"
ActionListener objHora=new mostrarHora();
// Creamos una instancia de la clase Timer indicando que cada
// 5 segundos ejecute el metodo "actionPerformed" de la clase
// mostrarHora que implementa la interfaz "ActionListener"
Timer objTimer=new Timer(5000,objHora);
// Iniciamos el timer
objTimer.start();
// Esperamos la pulsación de la tecla enter para finalizar el timer
Scanner entrada=new Scanner(System.in);
System.out.println("Pulsa la tecla ENTER para finalizar...");
String valor=entrada.next();
entrada.close();
// finaliza el timer
objTimer.stop();
}
}
// Clase mostrarHora que implementa la interfaz ActionListener que nos
// obliga a implementar el metodo "actionPerformed"
class mostrarHora implements ActionListener {
// Definimos el metodo "actionPerformed" para que nos muestre la hora
// cada vez que sea llamado por el Timer en la linea 16.
// Este metodo es de la interfaz "ActionListener", por lo que estamos
// obligados a implementarlo.
public void actionPerformed(ActionEvent e) {
// Mostramos en consola la fecha actual
Date fechaActual=new Date();
System.out.println(fechaActual);
}
}
FUENTE: https://www.lawebdelprogramador.com/codigo/Java/3634-Ejemplo-de-la-utilizacion-de-la-interfaz-ActionListener-para-mostrar-la-hora-cada-5-segundos-con-el-objeto-Timer.html
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Court orders Canada's elections chief to reconsider changing date of federal vote
Moira Warburton
Steve Scherer
Published: Jul 24, 2019 at 12:05 p.m.
By Steve Scherer and Moira Warburton
OTTAWA/TORONTO (Reuters) - A judge said on Tuesday that Canada's chief electoral officer (CEO) must reconsider the Oct. 21 date for the federal election because it coincides with a Jewish holiday, according to a copy of the ruling.
Stephane Perrault has only a week to appeal the Federal Court ruling in Toronto or change his recommendation for the date of the vote before an Aug. 1 deadline.
"Elections Canada will act in a timely manner in accordance with the directions provided by the Court," Perrault said in a statement. "I will make public my final decision as soon as possible."
Canadian Prime Minister Justin Trudeau and his Liberal Party face a tight race for re-election against Andrew Scheer's Conservatives.
In June, an Orthodox Jewish Conservative Party candidate and a Canadian Jewish activist filed a lawsuit arguing that the current date should be changed because it falls on the Jewish holiday of Shemini Atzeret.
On Shemini Atzeret, voting and campaigning are forbidden for Orthodox Jews, who number some 75,000 in Canada, the Jewish advocacy group B'nai Brith Canada said in a statement. B'nai Brith was granted "intervener" status in the case. It called for the date to be moved to Oct. 28.
"We certainly hope the decision (the CEO) makes is the right decision ... to change the election date," said Colin Feasby, a lawyer for B'nai Brith.
Instead of changing the date, Elections Canada had said it would seek to accommodate Jewish voters with a range of early voting opportunities and argued that moving it back by a week would cause significant logistical problems.
Judge Ann Marie McDonald urged Perrault to reconsider the date, saying there was not enough evidence showing the CEO had weighed everyone's constitutional rights when deciding to keep the election on Oct. 21.
"Parliament has granted the CEO discretion to make a recommendation for a change to the election date up until August 1," MacDonald wrote. "Although the August deadline is fast approaching, legal counsel for the CEO indicated that he is prepared to take whatever action is necessary as a result of the Court's decision."
The CEO has the power to recommend the election date, but it is up to the prime minister and his Cabinet to determine when Canadians will go to the polls.
(Reporting by Steve Scherer in Ottawa and Moira Warburton in Toronto; Editing by Peter Cooney)
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<?php
namespace Gerfaut\Yelp\Exception;
use \Exception as BaseException;
/**
* Class DeserializeException
* @package Gerfaut\Yelp\Exception
*/
class DeserializeException extends BaseException
{
}
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Eniola Aluko - Biography and Images
Home • Biography & Images • Eniola Aluko
Eniola "Eni" Aluko – born 21 February 1987 – is an English footballer who plays as a forward for Serie A club Juventus.
Aluko made 102 appearances for the England national team from 2004–2016 and competed at the 2007 FIFA Women's World Cup in China, 2009 UEFA Women's Euro, 2011 FIFA Women's World Cup in Germany, 2013 UEFA Women's Euro, and 2015 FIFA Women's World Cup in Canada.
At the 2012 Summer Olympics in London, she represented Great Britain.
Aluko previously played for Birmingham City, Charlton Athletic, and Chelsea in England's FA Women's Premier League.
She played for Saint Louis Athletica, Atlanta Beat, and Sky Blue FC in the American Women's Professional Soccer (WPS) from 2009–2011.
After a short stint with Birmingham City in England's new top-division league, FA WSL, she signed with Chelsea where she played from 2012 to 2018.
She served as a pundit on ITV's World Cup coverage in 2018.
Posted in: World Cup 2018 ITV Team
Roy Keane - Biography and Images
Roy Keane – born 10 August 1971 – is an Irish football assistant manager and former professional football player. He is the joint-most successful Irish footballer of all time, having won 19 major trophies, 17 of which came at Manchester… Read more
Patrice Evra - Biography and Images
Patrice Evra – born 15 May 1981 – is a professional footballer who played for Manchester United and West Ham United. Originally an attacker, he primarily plays as a left back. Evra, whom Sir Alex Ferguson praised for his leadership,… Read more
Henrik Larsson - Biography and Images
Henrik Edward Larsson – born 20 September 1971 – is a Swedish professional football manager and former player. He was known as a striker, whose main attributes were his goal scoring prowess and on-field intelligence. In June 1996, Larsson married… Read more
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Terkotka (ang. noisemaker, Cog Rattle lub Ratchet; niem. Knarre hiszp. matraca) – instrument muzyczny (samodźwięczny z grupy idiofonów), wykonany z drewna. Działa na zasadzie mechanizmu zębatkowo-zapadkowego. Koła zębate terkotki wraz ze sztywną płytą są zamocowane na uchwycie, który obraca się swobodnie. Podczas ruchu płyta uderza w koła zębate, drgając za każdym razem gdy przechodzi przez ząb, powodując hałas podobny do klikania i grzechotania. Nazwa instrumentu pochodzi od przymiotnika "terkoczący", a więc "wydający suchy, jednostajnie przerywany dźwięk". Brzmienie terkotki można usłyszeć w zdigitalizowanych zbiorach Polskiego Muzeum Instrumentów Ludowych.
Budowa i rodzaje terkotek
Instrumenty mają różne rozmiary. Mała terkotka ma około 15 cm długości i jest zbudowana z elastycznej listewki wydającej trzaski poprzez przesuwanie się po zębatym walcu, w wyniku obrotowych ruchów ramki (walca), do której jest przymocowana. Ruch ten wywoływany jest przez trzymającą instrument rękę. Dynamika dźwięku jest kontrolowana przez szybkość wykonywanego ruchu. Całość trzyma się na ramie.
Duże, drewniane terkotki są zbudowane w kształcie taczki (stąd nazwa: terkotki taczkowe) i również wydają serie trzasków w następstwie ślizgania się po zębatym kółku lub wałku sprężystej listewki, umocowanej w ramce obracającej się wokół osi wałka, osadzonego na uchwycie instrumentu. Instrument pcha się tak jak taczkę.
Zazwyczaj cały instrument jest wykonany z drewna, jednak we współczesnej praktyce ludowej różne części terkotek (szczególnie listewki) są wykonane z tworzywa sztucznego lub blachy. Elementy wykonane z drewna są niekiedy zdobione przy użyciu dłuta.
Funkcja obrzędowa
Tradycja rzymskokatolicka
Katolicka tradycja używania terkotek wywodzi się ze średniowiecza, kiedy to wprowadzono nieformalny zakaz używania dzwonów, począwszy od Mszy Wieczerzy Pańskiej aż do Wigilii Paschalnej z powodu uroczystej żałoby spowodowanej ukrzyżowaniem Jezusa i szacunku dla czasu jego śmierci trwającej do Zmartwychwstania. Aby zastąpić dzwony w czasie Triduum Paschalnego stosowano terkotki i inne pokutnie brzmiące instrumenty perkusyjne. We Włoszech, Francji, Anglii i Niemczech popularne były duże, stołowe terkotki z pudłami rezonansowymi, napędzane korbą, które z jednej strony zwiększały ich głośność, a z drugiej nadawały im głębsze brzmienie. Z kolei w Hiszpanii i Portugalii stosowano duże terkotki bez pudeł rezonansowych, w których dźwięk wydobywa się obracając bębnem umocowanym na specjalnym stelażu. Hiszpańska wersja terkotki to matraca. Słowo to pochodzi od arabskiego mitraqa (młot) i taraq (uderzenie). Dwumetrowa matraca była używana w Hiszpanii i Portugalii w celu wezwania wiernych do kościoła.
W Polsce
Dawniej za pomocą terkotek tworzono rytualną wrzawę podczas ludowych obrzędów kolędowania. Głównie instrument był używany w okresie Półpościa i Wielkiego Tygodnia poprzedzającego święta wielkanocne. Przykładowo, na Kujawach terkotki zastępowały dzwony kościelne, które na czas Wielkiego Piątku milkły w celu upamiętnienia śmierci Chrystusa: Biegano wtedy z nimi, hałasując, a dla wzmocnienia efektu konstruowano terkotki taczkowe, zwane "taradajką" lub "świnią", i objeżdżano nią całą wieś. Z kolei na ziemi chełmińskiej w Wielki Czwartek chłopcy wędrowali po wsiach i przy użyciu dźwięku terkotek wyganiali złe moce.
We Włoszech
Raganella to włoska nazwa terkotki. Nazwa instrumentu pochodzi od płaza z gatunku rzekotki drzewnej (łac. Hyla arborea). Raganella jest nadal używana w niektórych regionach Włoch: Veneto i Padanii, w Kalabrii, Apulii i Sardynii podczas uroczystości religijnych i festiwali. Jest nadal stosowany w Molise, w kościele podczas Wielkiego Tygodnia zamiast dzwonu. Instrument we Włoszech ma swoje regionalne nazwy. W Wenecji jest to ràcola, w Salento ṭrènula lub ṭròzzula, a w Tarent w ròzzələ lub ruèzzələ.
W Czechach i na Morawach
Terkotka (cz. klapotkami, rapace, Tragacete, grzechotka i hrkači) jest w Czechach do dziś ważnym elementem obrzędowości Wielkanocnej. Legenda głosi, że w Wielki Czwartek wszystkie dzwony kościelne lecą na pielgrzymkę do Rzymu. Ich miejsce zastępują grupy chłopcy z terkotkami (zwani klapotáři, rapačáři, trakačníci). Przewodniczy im "kapitan", z którym aż do Wielkiej Soboty obchodzą okoliczne gospodarstwa domowe, robiąc przy tym za pomocą klapotek dużo hałasu. Pierwszy obchód (cz. obchůzka) zaczyna się w Wielki Czwartek i powtarza się w godzinach porannych, w południe i w nocy. Za to w Wielki Piątek klapotáři chłopcy obchodzą własne wsie aż czterokrotnie. Pierwszy raz o 6 rano, drugi w południe, trzeci o godzinie trzeciej po południu i czwarty raz w godzinie ukrzyżowania Chrystusa. Mówi się, że dzieci w ten sposób ścigają Judasza. Podczas ostatniego obchodu wsi w Wielką Sobotę grzechoczący chłopcy idą dookoła wioski z koszami, do których właściciele odwiedzanych domów zazwyczaj wrzucają im jakąś nagrodę. Kiedyś były to owoce, pieczywa i jaja. Jednak dziś coraz częściej są to słodycze lub pieniądze.
Pokaz tradycyjnego obrzędu Klapotáři można obejrzeć co roku podczas Wielkanocnego festiwalu folklorystycznego odbywającego się w Skansenie wołoskim Valašské muzeum v přírodě w miejscowości Rožnov pod Radhoštěm.
Tradycja w innych kręgach religijnych
Tradycja żydowska
Mała terkotka w tradycji żydowskiej jest znana w języku jidysz jako gragger lub grogger (oryginalna pisownia: גערַגרא) lub ra'ashan (w języku hebrajskim: רעשן). Instrument używa się podczas święta Purim każdorazowo, gdy imię Haman (prześladowcy żydów) wspomina się podczas czytania Księgi Estery. Wytwarzany przez gragger hałas ma symbolicznie zagłuszyć imię żydowskiego prześladowcy każdorazowo, gdy pojawia się ono podczas czytania tekstu. Istnieją współczesne przypowieści o rosyjskich Żydach obchodzących Purim, którym nieskutecznie próbowano zabronić używania instrumentu podczas święta z uwagi na jego głośność.
Tradycja ukraińska
Ukraińską wersją instrumentu jest derkacz lub derkach (ukr. Деркач). Był on znany od czasów biblijnych i używany na Bliskim Wschodzie w celach rozrywkowych w grach ludowych. Na Ukrainie grzechotka znana od czasów starożytnych, pełniła funkcję alarmu. Hałas wywoływany przez derkacz informował o pożarze. Był również używany podczas polowań na wilki lub króliki. Na Wołyniu derkacz stosowano przy kolędowaniu. Czasem pojawia się przy okazji występów zespołów folklorystycznych. Derkacz składa się z czterech części: uchwytu, zębatego walca, ramy i płytki.
Pozostałe funkcje
Miejski alarm
Już w średniowieczu straż miejska używała terkotek jako sygnału alarmowego. W czasie kolejnych fali epidemii dżumy w XV, XVII i XVIII w. terkotka stała się typowym instrumentem ostrzegawczym stosowanym przez grabarzy zwłok w całej Europie, którzy odstraszali przy jej pomocy przypadkowych gapiów i złodziei.
W XVIII i XIX w. w Wielkiej Brytanii i Ameryce Północnej terkotka stała się instrumentem alarmowym milicji miejskich, z których wyewoluowała współczesna policja. Terkotka policjantów (ang. policeman's rattle) służyła w miastach do przywoływania pomocy. W roku 1658 Peter Stuyvesant, ówczesny dyrektor generalny Nowych Niderlandów zorganizował nocny patrol, który stanowił pierwszą w mieście policję (podwaliny przyszłej Policji Miasta Nowy Jork). Stuyvesant nazwał grupę stróżów Rattle Watch, od nazwy terkoczącego urządzenia, który służył do wznoszenia alarmu informującego o niebezpieczeństwach. Grupa Rattle Watch składała się z czterech mężczyzn, wybranych przez zarządcę, którzy spacerowali nocą po ulicach od zachodu do wschodu słońca. Mieli oni obserwować co się dzieje i wypatrywać niebezpieczeństwa. Nosili mundury w kolorze holenderskiej flagi, latarnie koloru zielonego i drewniane terkotki. Jeśli podczas obchodu zauważali cokolwiek niepokojącego, trzęśli terkotkami w celu zaalarmowania śpiących mieszkańców przed niebezpieczeństwem. Niebezpieczeństwo oznaczało cokolwiek, od pijackiej awantury na ulicy, po ogień, który mógłby rozprzestrzenić się i zniszczyć bardzo szybko sporą część miasta. Jeśli problemem był pożar, strażnicy dodatkowo krzyczeli "ogień", aby zmobilizować ludzi z wiadrami wody. Również w Bostonie – 20 lat wcześniej niż w Nowych Niderlandach – zorganizowano podobną grupę, zwaną również rattle watch.
Przez kolejne 200 lat terkotki były regularnie używane w Ameryce, jak i Wielkiej Brytanii, zarówno przez policję, jak i oddziały straży pożarnej. Jako narzędzia wznoszące alarm miały trzy podstawowe zalety: były tanie w produkcji, łatwo przenośne i bardzo głośne.
Alarmy chemiczne w czasie I wojny światowej
Podczas I i II wojny światowej terkotki używano jako urządzenia do ostrzegania przed obecnością bojowych środków trujących w powietrzu. W momencie, gdy siły brytyjskie i amerykańskie zdały sobie sprawę z tego, że Niemcy wykorzystują broń chemiczną, terkotka okazała się wygodnym sposobem ostrzegania żołnierzy przed zbliżającym się zagrożeniem. Nie można było używać gwizdków, bowiem te służyły do ogłaszania rozpoczęcia własnych ataków i podrywały żołnierzy z okopów. Siły brytyjskie i amerykańskie wytwarzały terkotki oznaczone napisem "tylko alarm gazowy" (ang. gas alarm only) dla żołnierzy z pierwszej linii. Istnieją nagrania wideo pokazujące wykorzystanie terkotki podczas I wojny światowej.
Inne zastosowania
Niegdyś terkotki stosowano przy polowaniach z naganką do płoszenia zwierzyny. Terkotka jest używana na spotkaniach sportowych. Umożliwia ona bez znacznego wysiłku wytworzenie głośnego dźwięku. Na rynku można kupić produkty przeznaczone zarówno dla dzieci i kibiców. Jest też sprzedawana jako zabawka dla dzieci.
Muzyka
Terkotki były wykorzystywane przez różnych kompozytorów. Lista przykładowych utworów:
Ludwig van Beethoven Zwycięstwo Wellingtona, czyli bitwa pod Vittorią (tzw. Symfonia Bitewna) (ang. Wellington Victory), 1813. Instrument jest użyty do zaprezentowania ognia karabinowego.
Joseph Haydn (Leopold Mozart) – Toy Symphony, 1820.
Maurice Ravel – Dziecko i czary(fr. L'Enfant et les sortilèges), 1925.
Modest Musorgski – Obrazki z wystawy (ros. Картинки с выставки), 1874.
Richard Strauss – Don Kichot (ang. Don Quixote), 1897.
Richard Strauss – Till Eulenspiegel's Merry Pranks, 1895.
Ottorino Respighi – Pinie rzymskie (ang. Pines of Rome), 1924.
Josef Strauss - polka Plappermäulchen op. 245 (1868)
Zobacz też
Przypisy
Bibliografia
Linki zewnętrzne
Etnografia
Instrumenty ludowe
Idiofony | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,733 |
Various demos/benchmarks.
### Example Name [ODE solver library used]
- Van Der Pol [scipy]
- Bouncing ball [pydstool]
- Bouncing ball + sinusoidally oscillating platform [Assimulo]
- Helicopter [scipy]
### Dependencies
- scipy
- pydstool: `pip install pydstool`
- Assimulo: [Install](http://www.jmodelica.org/assimulo_home/installation.html)
Assimulo can be installed from PyPI.
Before installing Assimulo, make sure that [sundials](https://computation.llnl.gov/casc/sundials/download/download.php) built with -fPIC is installed in the system.
Download sundials and read the docs to install it. Version 2.6.2 can be installed as below.
```
tar -xvf sundials-2.6.2.tar.gz
cd sundials-2.6.2
mkdir build
cd build
cmake ../ -DCMAKE_C_FLAGS=-fPIC
make
make install
```
Now install Assimulo using `pip install assimulo`.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,337 |
Q: Paraview wind velocity does not match cloud movement I was visualising my wind velocity using glyphs and cloud water content at the same time. However, I notice that the direction where the clouds move do not match the direction the glyphs pointing.
Below are the steps how I create the output:
The data is a netcdf file with wind variable array "ua" (eastward_wind_speed), "va" (northward_wind_speed), and "wa" (wind_vertical_velocity).
*
*I used a cell_data_to_point_data filter to convert them into point data.
*Then I combined these 3 arrays using a Paraview calculator with the equation iHatua + jHatva + kHat*wa.
*Then do a glyph filter to visualise the wind velocity.
The problem is, the clouds are moving to the left(east), which does not match where the glyphs are pointing at (south).
What would be the possible reason for this error?
TIA
Update:
For anyone that might have the same problem:
Just solved the problem and the glyphs make much more sense now.
*
*Switched off the spherical coordinate
*Transform filter to scaled down the vertical components
Then do the contour filter and glyph filter as usual.
A: There are two things to consider.
*
*Some weather agencies use a convection of wind direction from which is blowing. However there are agencies that have a wind direction to which is blowing.
*Probably you are not using the wind direction at the same height as the clouds.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,536 |
\section{Introduction}
During the last two decades, there has been great progress in the design of quantum devices based on waveguide structures. The latter is comprised of quantum emitters of natural or artificial atoms (qubits) that are coupled with the one--dimensional optical channel. Numerous applications in quantum simulation, quantum information processing, and communication have already been discussed in the literature\cite{qed1, qed2, qed3, qed4, QIT1, QIT2, QIT3, QIT4, QIT5}. Despite all efforts, the level of control and preservation of quantum coherence achieved in these media are still beyond the requirements for a practical realization of operable quantum information devices. This difficulty could be overcome by a better understanding of the nature of the interaction between matter and radiation. Accordingly, intensive studies of light--matter interaction in waveguide structures are necessary.
Superconducting metamaterials (SCQMMs) are man-made material units that are both interesting and quite promising for the ultimate fabrication of quantum devices \cite{zag, qmm1, qmm2, qmm3, Rakhmanov2008, Shvetsov2013, Asai2015, Asai2018}. These engineered media, comprised of periodically arranged artificial atoms that form superconducting quantum bits (SCQ) while interacting with EM fields inside one--dimensional transmission lines. Owing to spatial confinement, tunability of the SCQ parameters, and the ability to tailor photon dispersion relation by their particular setup, SCQMMs may be conveniently engineered to provide tunable ``atom'' -- field interaction that can reach regimes ranging from weak to ultra strong coupling. This is of particular interest in the case of qubit interaction with quantized radiation fields when strong qubit--photon coupling leads to effective photon--photon and qubit--qubit correlations. The latter allows for the emergence of novel interference effects with possible practical applications. For example, photons may exhibit a nontrivial dispersion relation such as band edges and band gaps. In this way, QMM may be viewed as a photonic crystal \cite{qmm0}. This enriches their potential for practical applications and provides novel means for devising comprehensive studies of practical and fundamental aspects of the artificial atom--field interaction.
Investigations of the emergence of an atom (or other emitter, qubit in particular)--photon bound states \cite{aphbs1, aphbs2, aphbs3, aphbs4, aphbs5, aphbs6, aphbs7, aphbs0, prohib, aphbs8, aphbs9, aphbs10,aphbs11} are of particular importance due to their consequences for radiation propagation \cite{prohib,ilja, prohib2, prohib3}, preservation of quantum coherence and entanglement \cite{ilja,ent1,aphbs11}. For example, the prohibition of the free propagation of radiation could be attributed to the formation of these bound states. Their creation within the continuum potentially can be used for the storage of quantum information \cite{prohib,prohib2,prohib3} and construction of photon memory devices \cite{aphbs3}.
On the other hand, the recent discovery of topological excitations in SCQMMs implies that, by the engineering of topologically nontrivial QMM, it would be possible to tackle unavoidable structural irregularities in SCQMMs. This is since the creation of photon bound states provides the preservation of quantum coherence for times large enough to perform quantum information processing \cite{ilja}. A further important possible application is the exploitation of qubit--photon bound states as a mechanism of the entanglement preservation in quantum information processing \cite{ent1,aphbs11}.
In this paper, we study the qubit--photon bound states emerging through the interaction of an EM field propagating through SCQMM, consisting of massive, two stripe-superconducting resonator filled with a large number ($\mathcal{N}\gg 1$) of Cooper pair box (CPB) or charge qubits. Such, essentially three--dimensional (3D) structure, substantially differs from the most common realization of SCQ based waveguiding structures \cite{qed1} and SCQMM setups \cite{qmm1} in which point--like SCQs are built-in \textit{coplanar} resonators. In such two--dimensional (2D) architectures, qubit--photon interaction is described within Jaynes (Tavis)--Cummings \cite{qed1}, Dicke model Hamiltonians \cite{aphbs8,aphbs9}.
A realistic theoretical model for the proposed setup is derived in the next section in terms of classical variables is, while its quantization is performed in the third section. It substantially differs from those used in the studies of the consequences of ``atom''--light interaction in the engineered media so far \cite{aphbs8, aphbs9}, i.e. modified Dicke Hamiltonian. In particular, while the pure photon part is practically identical to those encountered so far, the qubit -- photon interaction is quite different and contains two terms that may be attributed to attractive and repulsive interaction, whose competition determines the character and existence qubit--photon bound states.
The paper is organized as follows: Description of the model and classical Hamiltonian are introduced in the second section. The quantization procedure is given in the third section. Two--particle Schr\"odinger equation and its solutions are discussed in Section 4. Results and conclusions are summarized in the fifth section. Details of the mathematical derivation are given in the Appendices.
\section{Experimental setup: the proposal}
We investigate the non--classical properties of electromagnetic radiation propagating along the SCQMM in a construction visualized in Fig. (\ref{fig1}). It is made of an infinite ($\mathcal{N}\rightarrow \infty$), one--dimensional (1D) periodic SCQ array, with period $\ell$, placed in a transmission line (TL) consisting of two infinite bulk superconductors separated by a distance $d$; we consider $d$ being of the same order of magnitude as $\ell$ \cite{Rakhmanov2008, Shvetsov2013, Asai2015, Asai2018} (Figs. 1a and b). For simplicity, we took that the thickness of superconducting strips is $\ell$. Each SCQ is a tiny superconducting island connected to each bank of the TL through a Josephson junction (JJ). The control circuitry for each SCQ (Fig. 1c), consists of a gate voltage source $V_g$ coupled to it through a gate capacitor $C_g$ and allows for local control of the SCQMM by altering independently the state of each SCQ \cite{zag}. The SCQs exploit the nonlinearity of the Josephson effect and the large charging energy resulting from nanofabrication to create artificial mesoscopic two-level systems.
\begin{figure}[h]
\begin{center}
\includegraphics[height=7cm]{fig1}
\vskip -0.5 cm
\caption{Illustration of the proposed setup of SCQMM: (a) A chain of Cooper pair box qubits inside the two--stripe transmission line. Each unit cell contains a tiny superconducting island connected with TL banks through two Josephson junctions, for the regions of the dielectric layers (blue). The propagating electromagnetic vector potential pulse is also shown schematically out of scale.\\
(b) The side view of the SCQMM. The magnetic field penetrates through free space between the islands.\\
(c) A unit cell of the SCQMM showing the control circuitry of the charge qubit, consisting of a gate potential $V_g$ applied to it through the gate capacitor $C_g$.}
\label{fig1}
\end{center}
\end{figure}
\subsection{Classical model Hamiltonian}
In order to set up the problem, we first derive the classical model and subsequently perform its quantization. We assume that an electromagnetic (EM) wave with vector potential $\vec{A}=A_z (x,t) \hat{z}$ propagates along with the superconducting TL. The direction of propagation is parallel to the superconducting electrodes and while $\vec{A}$ is perpendicular to the direction of the EM wave propagation.
Let us now derive the model Hamiltonian of the system under the consideration. First, we recall, that each CPB qubit is comprised of \textit{double barrier} JJ (DBJJ), i.e. two JJs connected in a series. Similar DBJJ has been widely studied and used in different contexts \cite{zag, tinkh, JJrow, JJrow1, zag1}.
Hamiltonian of DBJJ may be obtained employing the straightforward extension of the Feynman semi--classical approach \cite{feynm} in which the dynamics of the single JJ is described within the simple two--level model: wave function of the CP condensate (Ginzburg--Landau -- GL order parameter) on each side of JJ is represented as $\varPsi_{p=1,2}=\sqrt{n_p}e^{i \phi_p}$, while the Cooper pair tunneling is accounted through the phenomenological coupling term. In such a way, single JJ dynamics may be described employing the nonlinear model Hamiltonian of the single variable: \textit{Josephson phase} representing the difference between the phases of GL order parameter in particular superconductors.
\begin{figure}[h!]
\begin{center}
\hspace{-1cm}
\includegraphics[width=0.55\textwidth]{twoJJj}
\vspace{-1.5cm}
\caption{Simplified illustration of the CPB qubit
consisting of two Josephson junctions connected in series.}
\label{fig2}
\end{center}
\end{figure}
In the case of DBJJ, three superconducting segments separated by two JJs (\ref{fig2}), wave function in particular segment, upper, middle and lower, may be written as: $\Psi_p(t)=\sqrt{n_p}e^{i\phi_p(t)},\;\; p=u,m,l$, while the tunneling between them now is simply: $$H_t=-V\left(\Psi^*_u\Psi_m+\Psi^*_m\Psi_l+c.c\right)$$
where $V$ represents phenomenological parameter so called Josephson constant \cite{feynm}.
Substitution of the CP wave function as given above and, in analogy with \cite{feynm}, assuming that CP numbers in each segment are almost the same and equal to $n_0$, we found that the Hamiltonian of double JJ system is the sum of Hamiltonians of two independent JJs:
\begin{eqnarray}\label{single1}
H= -E_c\sum_{i=l,u}\frac{\partial^2}{\partial \varphi_{i}^2}-E_{J}\sum_{i=l,u}\cos{\varphi_{i}},
\end{eqnarray}
\noindent Here $\varphi_{u,(l)}$ denote the Josephson phase differences at lower (upper) junction. In our model we restrict ourselves to \textit{zero voltage} case \cite{Asai2015} when $\phi_l=\phi_u\equiv 0$ so that these phases read: $\varphi_{u}=\phi_u-\phi_m\equiv -\phi_m$ and $\varphi_{l}=\phi_l-\phi_m\equiv -\phi_m$.
The energy parameters $E_c=\frac{2e^2}{C_J}$ and $E_J=\frac{\Phi_0 I_C}{2\pi c}$, $\Phi_0=\frac{h c}{2 e}$, $I_c$, $C_J$, and $c$ are the junction charging energy, so called Josephson energy, flux quantum, critical current, junction capacity, and speed of light, respectively.
In the presence of EM field Josephson phase difference $\varphi_{i}$ acquires the gauge term and reads:
\begin{equation}\label{gauge} \varphi_u(t)-\varphi_l(t)=-\phi_m\pm \frac{4\pi}{\Phi_0}\int_{1}^{2}\vec A(\vec r)\cdot d\vec l. \end{equation}
Generalizing (\ref{single1}) to whole qubit lattice and accounting for the energy of EM field inside the SCQMM ($H_{em}=
\frac{1}{8\pi}\int\left( E^2_n(\vec r)
+B^2_n(\vec r) \right) d^3r$) we derived a total model Hamiltonian:
\begin{eqnarray}
\label{tot}
\nonumber
H=&&\sum_n\bigg[\frac{2\hbar^2}{E_c}\dot{\phi}^2_n -2E_J\cos\phi_n\cos\alpha_n+\\
&&\frac{2\hbar^2}{E_c}\dot\alpha^2_n+ \dot\alpha^2_n +E_{em}(\alpha_{n+1}-\alpha_n)^2\bigg].
\end{eqnarray}
Here we have introduced the dimensionless amplitude of the vector potential $\alpha_n=\frac{2\pi d}{\Phi_0}A_n$.
To facilitate practical calculation the charging term has been redefined using $\frac{d \phi_n(t)}{d t}\equiv \dot{\phi_n}=-i\frac{2e^2}{\hbar C_J}\frac{\partial}{\partial \phi_n}$, while the gauge term in Josephson phase difference has been approximated as $\int_{1}^{2}\vec A(\vec r) \cdot d\vec l\equiv \frac{2\pi d}{\Phi_0}A_n$. The identity holds in the present setup where it was assumed that the separation $d$ between the superconducting stripes and the period $\ell$, i.e. the center--to--center distance between qubits, are of the same order of magnitude and much smaller than the wavelength of EM radiation; this fact enables us to neglect the variation of the vector potential within each cell. As a result, the integration in Eq. (\ref{gauge}) is trivial (see for example \cite{Rakhmanov2008, Shvetsov2013, Asai2015}).
Finally, in evaluating EM energy integration is taken over the entire unit cell. Thus, in accordance with the approximation adopted above, we neglect spatial variation of electric and magnetic field within the unit cell, so that the energy of EM field in particular unit cell was approximated as:
\begin{eqnarray}\label{emfapp}
\nonumber && H_{em}\approx \frac{V}{8\pi}\left( E^2_n+B^2_n\right), \\
&& V=\ell^2d, \; \; - \mathrm{volyme\;\; of\; the\;\; unit\; cell. }
\end{eqnarray}
For simplicity, we took that the width of superconducting stripes is equal to inter--qubit distance: $\ell$.
Following \cite{Rakhmanov2008} and \cite{Asai2015} we have neglected the contribution of the electric field, while the fraction that originates from the magnetic field was accounted for through the discretization procedure introduced in \cite{Rakhmanov2008, Shvetsov2013, Asai2015}:
\begin{equation}\label{mf}
B(x,t)=\frac{\partial A(x,t)}{\partial x}\rightarrow \frac{A^z_{n+1}-A^z_n}{\ell}.
\end{equation}
Here $E_{em}=\frac{1}{8\pi \ell d}\bigg(\frac{\Phi_0}{2\pi}\bigg)^2$, is the so called \textit{electromagnetic energy} introduced in \cite{Rakhmanov2008}, determining the speed of "light" in the qubit chain, which, in dimensionless units, reads $\beta=\sqrt{E_{em}/E_J}$. It, together with the ratio $\gamma=\frac{E_C}{E_J}$ represents the main quantitative characteristic of CPB qubits, their derivatives, transmon for example, and networks made of them.
\section{Quantization and two--level approximation}
The quantum--mechanical versus a (semi)classical description of the qubit--EM field coupled systems still has certain controversies \cite{contro}. Nevertheless, at low temperatures, a fully quantum treatment is justified, while the dissipation is negligible. Under these conditions, the quantum state of an island is determined by the number of extra Cooper pairs on them. In addition, EM radiation exhibits quantum features for weak (small amplitude) EM fields when their modes are populated with just a few photons, one or two, per wavelength \cite{fist2}.
At this stage, we must note that the tunneling of the single CP between the banks and island does not affect the state of the former which contains a large number of CPs so that the deficiency or the excess of the single CPs has no particular significance.
Formally we quantize our model by introducing the photon creation and annihilation operators in real (direct) space and Josephson phase and Cooper pair number operator in Cooper pair number basis. In such a way, through a few intermediate steps, described in Appendix 1, the classical Hamiltonian Eq. (\ref{tot}) can be approximated by the quantum one describing the interaction of a collection of the two--level systems and the quantized multimode electromagnetic field.
\subsection{EM field}
In the quantum regime the electromagnetic field is weak, i.e., the dimensionless amplitude of its vector is small and can be treated as quantum fluctuation, i.e. $\alpha_n\rightarrow\hat \alpha_n\ll 1$. This enables us to expand $\cos\hat\alpha_n\approx 1-\frac{\hat\alpha^2_n}{2}$. Next, we quantize the EM field in two steps: first we define the generalized momentum $P_n=\frac{2\hbar^2}{E_c}\dot{\alpha_n}$ canonically conjugated to $\alpha_n$. Subsequently we treat photon variables as operators $\alpha_n\rightarrow\hat \alpha_n$, $P_n\rightarrow \hat P_n$ satisfying the commutation relation $[\hat\alpha_n,\hat P_m]=i\hbar \delta_{m,n}$. It holds for the transformation Eq. (\ref{quantiz1}) through which we introduce photon creation and annihilation operators in real (direct) space:
\begin{eqnarray}\label{quantiz1}
\hat \alpha_n=
\frac{1}{2}\sqrt{\frac{E_C}{\hbar\omega}}\left(a_n+a^{\dagger}_{n}\right),\;\;
\hat P_n={i\hbar}\sqrt{\frac{\hbar\omega}{E_c}}\left(a^{\dagger}_{n}-a_n\right).
\end{eqnarray}
\subsection{Qubit subsystem}
Similarly, in quantization of the CPB qubit subsystem we introduce the pair canonically conjugated variables (operators): the \textit{phase} $\phi\rightarrow \hat \phi$ and Cooper pair number operator $\hat N=-i\frac{\partial}{\partial \hat\phi_n}$, $[\phi_n, \hat{N_n}]=i$. Then we rewrite Eq. (\ref{tot}) in the Cooper pair number basis $|N\rangle$, using the correspondence: $\hat N=-i\frac{\partial}{\partial \phi_n}$ and noticing that $e^{\pm i\hat \phi_n }|N\rangle = |N \pm 1\rangle$.Next, in the obtained Hamiltonian we exploit the fact that in \textit{charge and transmon} regime only a few lowest levels are relevant and we may restrict ourselves to the reduced state space in which the single island can be unoccupied ($N=0$) or occupied by single Cooper pair ($N=1$). The resulting Hamiltonian is nondiagonal in reduced number basis ${|0\rangle,\; |1\rangle}$, and in the next step we diagonalize the free qubit part by means of transition to energy eigenbasis (${|e\rangle}-\mathrm{excited\; state},\; |g\rangle-\mathrm{ground\; state}$) performing the norm preserving unitary transformation Eq. (\ref{UT}). Finally, after neglecting the photon number non-preserving terms, i.e., those $\sim a^2_n$ and $\sim a^{\dagger 2}_n $, we obtain the quantized model Hamiltonian:
\begin{eqnarray}\label{qb}
\nonumber H=&&\Delta\sum_n|e\rangle_n\langle e|+\\
&&\hbar\omega\sum_n a^{\dagger}_n a_n-J\sum_n a^{\dagger}_n(a_{n+1}+a_{n-1})+\\ \nonumber &&\sum_n\bigg[B(|e\rangle_n\langle g|+|g\rangle_n\langle e|) -A|e\rangle_n\langle e|\bigg]a^{\dagger}_n a_n
\end{eqnarray}
Here the first term represents the Hamiltonian of the qubit subsystem with level splitting between the excited and ground-state $\Delta=2\epsilon$ ($\epsilon=\sqrt{E^2_J+E^2_C}$). It is represented here in terms of the operator $|e\rangle\langle e |$ to emphasize that initially system is prepared so that all qubits are excited. Such "atoms" are usually called \textit{emitters}.
In the pure photon Hamiltonian, the two terms in the second line, correspond to typical boson \textit{tight binding} model describing photon hopping between neighboring qubits. Parameters $\omega$ and $J$ stay for the photon frequency and the photon inter--qubit tunneling amplitude, respectively:
\begin{eqnarray}
\label{param1}
\hbar\omega=\sqrt{2E_{em}E_C+\frac{E_CE^2_J}{2E}},
\qquad
J=\frac{E_{em}E_C}{2\hbar\omega}.
\end{eqnarray}
Considering the noninteracting case, pure photon, and qubit system, the present model is analogous to those appearing frequently in a theoretical description of charge and energy transfer in various contexts. Recent application concerns the photonic bandgap materials where it addresses the photon hopping motion in \textit{coupled resonator(cavities) waveguides} \cite{aphbs9,aphbs10}. Quantum metamaterials built of such structures with embedded tunable quantum emitters, i.e., qubits, opened a new perspective for further development of novel, quantum, technological devices, and for studies of nonclassical features of light \cite{fist2}.
Finally, the last term is related to the qubit -- photon interaction. It possesses two components: the attractive one, measured by the parameter $A$, and repulsive $\sim B$.
\begin{eqnarray}\label{param2}
A=\frac{E^2_JE_c}{4\hbar\omega \epsilon},\;\; B=\frac{E_JE^2_c}{8\hbar\omega \epsilon}.
\end{eqnarray}
For the convenience we rewrite the interaction Hamiltonian in terms of "atomic" (pseudo--spin) operators ($\sigma^{\dagger,-,z}$):
\begin{equation}\label{atomic}
H_i=\sum_n [B(\sigma^{\dagger}_n+\sigma^{-}_n)-A\sigma^{\dagger}_n\sigma^{-}]a^{\dagger}_na_n.
\end{equation}
The operators in the attractive interaction term may be rearranged as follows: $\sigma^{\dagger}\sigma^-a^{\dagger}a\equiv \sigma^{\dagger}a\sigma^-a^{\dagger}- \sigma^{\dagger}\sigma^-$. Thus, it may be understood to originate on account the simultaneous excitation ($\sigma^{\dagger}a$) and de--excitation ($\sigma^{-}a^{\dagger}$) of the $n$--th qubit by an absorption and emission of the single photon. On the other hand, repulsive interaction comes from the photon scattering by qubits resulting in their excitation ($|e\rangle\langle g|$) and de--excitation ($|g\rangle\langle e|$).
These mechanisms differ substantially from those accounted for within the Dicke and Jaynes--Cummings models coming from the excitation qubit, atom in general, by the absorption of the single photon ($|e\rangle\langle g| a$) and vice versa: qubit deexcitation by the emission of the single photon.
In coplanar geometry setups \cite{qed1,qed2} a qubit--photon interaction substantially differs from the present one, and, in the rotating wave approximation, reads:
$$H_{JC}=g\sum_n \sigma^{\dagger}_n a_n+\sigma^{-}a^{\dagger}_n$$.
Note that here we can not distinguish whether the interaction is attractive or repulsive. This becomes possible only after deriving the exigent--value equation, counterpart equation (\ref{eigen1}) from the next paragraph, based on the sign of effective interaction parameter.
So far, the interaction resulting from the set--up proposed here was not encountered in the studies of the light interacting neither with natural nor artificial media. Nevertheless, formally very similar models may appear in solids, magnetic semiconductors \cite{sd}, when a single electron creates micro--ferromagnetic domains flipping the spins of neighboring ions, while the interaction Hamiltonian is given in terms of the $s-d(f)$ being very similar to (\ref{atomic}.)
We also point out that, in the present setup, the waveguide is the chain of unit cells (sketched at Fig. 1b) each of which contains a single qubit ("atom") and may be viewed as an optical resonator. That is, our waveguide is the set of the large number ($\mathcal{N}\gg 1$) of coupled resonators (unit cells) with one "atom" per "cavity", which imply translational invariance of the system. Nevertheless, most often, the waveguide is the set "resonators" designed independently of "atoms". In these structures "atoms" are arranged arbitrarily, depending on the particular application--research subject. Various settings are possible and a particular waveguide may be populated by a few ($\mathcal{N}$) "atoms", with one or more "atoms" per cavity \cite{aphbs3, aphbs4, aphbs5, aphbs6, aphbs7, aphbs0, prohib, aphbs8, aphbs9, aphbs10, aphbs11}.
One more distinction must be made in comparison with related systems. In that respect we refer to quantum metamaterial designed of coplanar, mostly superconducting, resonator waveguide and several embedded qubits \cite{qed1, qed2, qed3, qed4}, where the qubits are linearly\footnote[2]{The interaction is of the first order in field amplitude and contains only the terms linear in photon operators.} coupled to the resonator modes.
\section{qubit photon bound states}
\subsection{Vector of state and Schr\"odinger equation}
The wave function which diagonalizes Hamiltonian Eq. (\ref{qb}) has a form of a
single photon dressed qubit (atom) state:
\begin{eqnarray}\label{psi}
\nonumber |\Psi\rangle=
&&\sum_m u_m a^{\dagger}_m|0\rangle|g\rangle +\sum_{m,n}\Psi_{m,n}
\sigma^{\dagger}_n a^{\dagger}_m |0\rangle|g\rangle_m, \\
&&\sigma^{\dagger}_n=|e\rangle_n\langle g|,\;\; \Psi_{m,n}=\Psi_{n,m}.
\end{eqnarray}
Here the probability amplitudes satisfy the normalization condition
\begin{equation}
\label{norm}
\sum_m |u_m|^2+\sum_{m,n}|\Psi_{m,n}|^2=1.
\end{equation}
The first term in state Eq. (\ref{qb}) corresponds to the case when a single photon is excited in site $m$ with probability amplitude $u_m$, while the qubit remains in its ground state. The second term of a vector of state Eq. (\ref{qb}) corresponds to synchronized excitation of $n$--th qubit and photon at site $m$. The symmetry property $\Psi_{m,n}=\Psi_{n,m}$ reflects the translational invariance of chain: solutions must remain invariant when photon and qubit excitation exchange position simultaneous excitation of qubit at site $m$ and photon at $n$--th site.
Owing to orthogonality of $\langle g|\langle 0|a_m$ and $\langle g|\langle 0|\sigma^{-}_m a_n$ and $|\Psi\rangle$ we may project Schr\"odinger equation $H|\Psi>=E|\Psi>$ onto $\sigma^{\dagger}_m a^{\dagger}_n|g\rangle|0\rangle$ and $a^{\dagger}_m |g\rangle|0\rangle$. In this way we obtain a system of coupled equations for the amplitudes $\Psi_{m,n}$ and $u_m$:
\begin{eqnarray}\label{SE}
\nonumber &&(\mathcal{E}-\Delta)\Psi_{m,n}
+\frac{J}{2}\bigg(\Psi_{m,n+1}+\Psi_{m,n-1}
+\left\lbrace m \rightleftarrows n \right\rbrace \bigg)=\\
&&\nonumber -A\Psi_{m,n}\delta_{m,n}+B u_m\delta_{m,n},\\
&& \mathcal{E}u_m+J(u_{m+1}+u_{m-1})=B\Psi_{n,n}.
\end{eqnarray}
We will solve it by employing Fourier transform. Owing to the translational invariance we pick:
\begin{equation}
\label{ft}
\Psi_{m,n}=\frac{1}{\sqrt{\mathcal{N}}}e^{i\frac{K(m+n)}{2}\ell}\Phi_{m-n},\;\;
u_m=\frac{1}{\sqrt{\mathcal{N}}}\sum_k u_k e^{ikm\ell}.
\end{equation}
In this way, the second equation in Eq. (\ref{SE}) attains a simple form and may be readily solved for $u_m$, which then may be eliminated from the first one. In the resulting equation we employ the translational invariance and took $m-n=l$; next we perform Fourier transform $\Phi_l=\frac{1}{\mathcal{N}^{1/2}}\sum_q \Phi_q e^{iql\ell}$. This finally yields:
\begin{eqnarray}
\label{basic}
&&\nonumber [\mathcal{E}-\Delta+2J\cos(Kd/2) \cos q] \Phi_q=\\
&&\bigg[-A+\frac{B^2}{(\mathcal{E}+ 2J\cos K)} \bigg]
\left(\frac{1}{\mathcal{N}}\sum_q\Phi_q\right),
\end{eqnarray}
here $K$ and $q$ stand for center of mass and relative qubit--photon quasi--momenta, while $\mathcal{E}=E-\hbar\omega$.
On the basis of this equation it is easy to find relation for eigenvalues: we first find $\Phi_q=...$, then we multiply both sides of the last equation with $1/\mathcal{N}$ and then sum up both sides over $q$. This results in:
\begin{equation}
\label{eigen0}
1=\frac{1}{\mathcal{N}}
\sum_q\frac{1}{(\varepsilon-\delta+ \cos (K\ell/2)\cos q\ell)}\bigg[-a
+\frac{b^2}{(\varepsilon + \cos K\ell)} \bigg].
\end{equation}
Bound state solutions, if any exist, must lie outside the free state continuum appearing in the absence of qubit--photon interaction. In that case Eq. (\ref{basic}) has solution
\begin{equation}
\varepsilon(q,K)=\delta - \cos q\ell\cos \frac{K\ell}{2},
\end{equation}
so that the bound state energy must lie either below the lower energy bound $$ \delta-|\cos(K\ell)/2|,$$or above the higher one $$ \delta+|\cos(K\ell)/2|.$$
\subsection{Eigenvalue equation}
The summation over $q$ may be performed in accordance with the rule: $\frac{1}{\mathcal{N}}\sum_q<....>=\frac{1}{2\pi \ell}\int_{-\pi/\ell}^{\pi/\ell}d q<...>$. This, provided that $|\varepsilon-\delta|>1$, yields the self--consistent equations for energy eigenvalues:
\begin{eqnarray}\label{eigen1}
\nonumber 1=a'(K) \frac{sgn{(\varepsilon-\delta)}}{\sqrt{(\varepsilon-\delta)^2-\cos^2 K\ell/2}},\\
a'(K)=-a+b^2\frac{1}{\varepsilon+\cos K\ell},
\end{eqnarray}
where $a=A/2J$, $b=B/2J$, $\delta=\Delta/2J$ and $\varepsilon=\mathcal{E}/2J$, stand for the normalized coefficients. For further convenience we express Eq. (\ref{eigen1}) in terms of just two parameters, $\beta$ and $\gamma$, which fully characterize proposed system:
\begin{eqnarray}
\label{norpar}
\nonumber a=\frac{1}{4\beta^2}\frac{1}{\sqrt{1+\gamma^2}},\;
b=\frac{\gamma a}{2},\\ \frac{\hbar\omega}{2J}=2+\frac{1}{2\beta^2\sqrt{1+\gamma^2}},\\
\nonumber \delta=2\sqrt{2\frac{(1+\gamma^2)}{\gamma\beta^2}+\frac{\sqrt{1+\gamma^2}}{2\gamma\beta^4}}.
\end{eqnarray}
Eigen--equation (\ref{eigen1}) is a nonlinear (in $\varepsilon(K)$) transcendental equation and can not be solved analytically. Nevertheless, its nonlinearity implies that it may have multiple solutions. That is, qubit--photon bound states if any exist, should exhibit multi--band structure.
To facilitate practical calculations, to examine the possible appearance of multi--band structure of the qubit--photon spectra, finally, to compare present analysis with the related preceding ones \cite{bs1,bs2} we rewrite Eq. (\ref{eigen1}) in the self-consistent form
\begin{equation}\label{selfc}
\varepsilon(K) -\delta=\pm \sqrt{a'^2(K)+\cos^2 K/2},
\end{equation}
in which, on the right hand side, $\varepsilon(K)$ appears implicitly through the $a'(K)$ in accordance with Eq. (\ref{eigen1}).
This "solution" recalls much the exact one in the limit $a'(K)\rightarrow a$, appearing frequently in different contexts. Examples are numerous, and, despite different physical backgrounds, formally identical solutions, may be found in many cases such as \textit{bound states of two photons, phonons, excitons} \cite{bs1}. In addition, the problem of the bound state of an impurity atom and its vibrational or magnetic environment \cite{bs1}, within the simplest models, also reduce to this elementary solutions.
\subsection{Existence of solutions}
Solubility of Eq. (\ref{eigen1}) requires non--negativity of its right hand side, thus, for $\varepsilon - \delta<0$ ($\varepsilon - \delta >0$), eigen--energy solutions exist provided that $a'(K)<0$ ($a'(K)>0$). Accordingly, signs ($+$ or $-$) in Eq. (\ref{selfc}) stand for $\varepsilon - \delta<0$ and $\varepsilon - \delta>0$, respectively. Also, throughout the paper, we may call $a'(K)$ the \textit{effective} qubit--photon interaction strength. The term ``effective'' is used here to emphasize the self--consistency of (\ref{selfc}), and to point to its \textit{formal equivalence} with the exact ones appearing when $a'(K)\rightarrow a$.
To find $\varepsilon(K)$ we have performed the numerical calculation focusing ourselves to the case $\varepsilon - \delta<0$ when an effective qubit--photon interaction is attractive. An opposite case was not considered since our numerical calculations have shown that the solutions of the eigenvalue problem exist for unrealistic values of system parameters. For example for $\gamma \sim 100$.
\subsection{Solutions: analytical considerations}
Before presentation of the results our numerical calculations we perform some auxiliary analytic analysis evaluating explicitly eigen--energies at band edges: $\varepsilon(\pm \pi)\equiv \varepsilon(\pi)$. In that limit (\ref{selfc}) become:
\begin{equation}
\label{edge}
\varepsilon(\pi) -\delta
=\pm a\left(1-\frac{a(\frac{\gamma}{2})^2}{\varepsilon(\pi)-1}\right).
\end{equation}
The signs $(+)$ or $(-)$ correspond to $\varepsilon -\delta > 0$ and $\varepsilon -\delta < 0$, respectively. The last equation, in both cases, is the quadratic in $\varepsilon(\pi)$ implying the appearance of two bands, both for attractive and repulsive effective interaction.
\begin{figure*}[t!]
\subfigure[]{\includegraphics[width=0.35\textwidth]{b01g02ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b01g1ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b01g5ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b01g10ph}}
\caption{Graphical illustration of the energy spectrum ($\varepsilon(K)$) of system for $\beta =0.1$, and for four different $\gamma$. Green shaded area corresponds to free states. Blue solid lines correspond to qubit--photon bound states. For the comparison we gave a band of bound states in the case in the absence of repulsive interaction--red dotted lines, and pure photon dispersion curve $\varepsilon(q)$ -- green dotted lines.}
\label{fig02}
\end{figure*}
\begin{figure*}[t!]
\centering
\subfigure[]{\includegraphics[width=0.35\textwidth]{b02g02ph}}
\subfigure[]{\includegraphics[width=0.375\textwidth]{b02g1ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b02g5ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b02g10ph}}
\caption{ Energy spectrum ($\varepsilon(K)$) of system for $\beta =0.2$, and for the same values of $\gamma$ as in the preceding case.}
\label{fig03}
\end{figure*}
\begin{figure*}[t!]
\centering
\subfigure[]{\includegraphics[width=0.35\textwidth]{b05g02ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b05g1ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b05g5ph}}
\subfigure[]{\includegraphics[width=0.35\textwidth]{b05g10ph}}
\caption{ Same as in previous cases for $\beta=0.5$.}
\label{fig04}
\end{figure*}
Solutions of Eq. (\ref{edge}) are:
\begin{eqnarray}
\label{attr}
\nonumber \varepsilon_{\pm}(\pi)=&&\frac{1+\delta-a}{2}\pm\frac{1-\delta+a}{2}
\sqrt{1+\left(\frac{a\gamma}{1-\delta+a} \right)^2 }.\\
\nonumber && \mathrm{for\;\; attractive\;\; effective \;\;interaction},\\
\nonumber \varepsilon_{\pm}(\pi)&&=\frac{1+\delta+a}{2}\pm\frac{1-\delta-a}{2}
\sqrt{1-\left(\frac{a\gamma}{1-\delta-a} \right)^2 },\\
&& \mathrm{for\;\; the\; repulsive\;\;one.}
\end{eqnarray}
In the present context $\delta$ is large as compared with other system parameters. Thus, the ratios in both square roots may be regarded as small quantities. This enables us to expand both square roots in Eq. (\ref{attr}) in terms of "small parameter" $(a\gamma/(1-(\delta\pm a)))^2$ which yields the corresponding asymptotic relations:
\begin{eqnarray}
\label{asym}
\nonumber \varepsilon_{-}\approx&&\delta - a -\frac{(\frac{a\gamma}{2})^2}{1-\delta+a},\\
\nonumber \varepsilon_{+}\approx&& 1+\frac{(\frac{a\gamma}{2})^2}{1-\delta+a},\\
\nonumber&& \mathrm{For\; "attractive" \;effective\; interaction,}\\
\nonumber \varepsilon_{-}\approx&&\delta + a -\frac{(\frac{a\gamma}{2})^2}{\delta+a-1},\\
\varepsilon_{+}\approx&& 1+\frac{(\frac{a\gamma}{2})^2}{1-\delta-a},\\&&
\nonumber \mathrm{for \; "repulsive"\; effective \; interaction.}
\end{eqnarray}
Based on these equations we may estimate under which conditions particular types of solutions exist. For that purpose, we recall the existence conditions of the solutions--\textit{subsection C}. We focus on repulsive interaction for which our numerical calculations do not find meaningful solutions for reliable parameter values. According to Eq. (\ref{eigen1}) its solutions exist provided that $a'(K)>0$. Substituting the corresponding asymptotic solution from Eq. (\ref{attr}), the third equation in Eq. (\ref{asym}), into $a'(\pi)$ we obtain the following condition:
\begin{equation}
\label{conit}
a(\frac{\gamma}{2})^2>\varepsilon(\pi)-1\Leftrightarrow \delta+a<1+
a(\frac{\gamma}{2})^2 \;\;\mathrm{for\; \; \varepsilon_+}.
\end{equation}
On the other hand, solution $\varepsilon_-$, after subtracting the $\delta$ on both sides, attains the form: $$ \varepsilon -\delta=1-\delta+\frac{\frac{\gamma^2}{4}}{\delta+a-1}.$$ Note that neither of these conditions can be satisfied in the present case. Namely, the condition for the existence of solutions in the case of repulsive interaction reads $\varepsilon-\delta<0$, which cannot be satisfied in practice due to large values of $\delta$. In particular, for that purpose $\gamma \gtrsim 100$ is required.
\subsection{Solutions: numerical results}
Numerical calculations, were performed for the values of system parameters covering both \textit{charging (large $\gamma$)} and \textit{Josephson ( small $\gamma$)} regime. Note that there is no any particular restrictions on the value of dimensionless speed of light $\beta$ in QMM. In particular, in literature \cite{Rakhmanov2008,Shvetsov2013,Asai2018,scir} $\beta$ was taken to vary from few tenths up to \emph{1}. Here we restrict ourselves to $\beta \eqslantless 0.5$ since the results for its larger values do not exhibit any \emph{substantial qualitative} difference. Thus we used $\beta =0.1; 0.2,\; \mathrm{and}\; 0.5$, while, for each $\beta$, we took
four values $\gamma: 0.2,\; 1, \; 5\; \mathrm{and}\; 10$.
Our results are illustrated in Figs. (\ref{fig02}) -- (\ref{fig04}). The energy spectrum consists of the free state continuum, green shaded area, and two bands of qubit photon bound states which are observed for each set of system parameters. The higher energy band (Band 1) lies below the free state continuum and, for large values of $\beta$ ($\beta= 0.5$ as presented at Fig.(4)), is practically indistinguishable from the bound states appearing in the case of pure attractive interaction corresponding to \textit{ad hoc} choice $B=0$. For small $\gamma$ Band 1 is well separated from the continuum approaching it for larger values of $\gamma$. Band 1 features profoundly change as $\beta$ decreases. For example, for $\beta=0.1$ (Fig. 3) the magnitudes of the Band 1 bound states energies and those of the free states, for each $K$, are almost twenty times higher than for $\beta=0.5$. In addition, for small $\gamma$ ($\gamma = 0.2$), Band 1 is practically indistinguishable from bound states corresponding to pure attractive interaction. As $\gamma$ rises Band 1 and solutions for the pure attractive interaction separate and both gradually tend towards the free state continuum. Qualitatively the same behavior is observed for $\beta = 0.2$ with a somewhat different degree of changes.
As presented at the lower part of figures (\ref{fig02})--(\ref{fig04}), in parallel with Band 1 the second one (Band 2), appears. This is the novel band lying deeply below Band 1. It emerges from the competition between the attractive and repulsive interaction and lies below the free photon band. Its dependence on parameters $\beta$ and $\gamma$ exhibits similar behavior as for Band 1. That is, for large $\beta$, irrespective of the values of $\gamma$, Band 2 and free photon band are practically identical, due to complete compensation of the effective attractive and repulsive interactions. That is QMM is fully transparent, and there are no bound states. For smaller values of $\beta$ attractive interaction dominate over the repulsive and qubit--photon bound states to emerge, providing that $\gamma$ is high enough. Nevertheless, QMM is still transparent but for qubit--photon bound states.
\section{Concluding Remarks}
In this paper, we have studied the energy structure and cooperative qubit--photon excitation of a one--dimensional superconducting quantum metamaterial. The system consists of the large number ($\mathcal{N}\gg 1$) periodically arranged charge qubits placed inside the massive two-strip superconducting resonator. In such a setup each unit cell of SCQMM (sketched at Fig.1 b) can be viewed as an electromagnetic resonator, while the system as a whole, represents a coupled-resonator (cavities) waveguide with single an "atom" per cavity.
This setup, upon quantization, exhibits some novel features in comparison to those used so far in the studies of the matter--light interaction. In particular, the system is translationally invariant since the number of "cavities" and "atoms" match: each cavity contains a single qubit. So far the studies on the subject were carried under the condition that the individual "atoms" \cite{aphbs3, aphbs4, aphbs5, aphbs6, aphbs7, prohib, aphbs8, aphbs9, aphbs10, aphbs11} or their ensembles \cite{aphbs0} are placed in different resonators and where translational invariance has been rarely accounted \cite{aphbs11, cohout}.
Furthermore, the qubit--photon interaction is substantially different from that utilized in most studies on the subject \cite{aphbs3, aphbs4, aphbs5, aphbs6, aphbs7, aphbs0, aphbs8, prohib, aphbs9, aphbs10} which were carried out within the certain modifications of the celebrating Dicke model \cite{dicke}. The essential difference is that it now has two components: the attractive and the repulsive one originating on account of different mechanisms: i) simultaneous excitation ($\sigma^{\dagger}a$) and deexcitation ($\sigma^{-}a^{\dagger}$) of the $n$--th qubit by absorption and emission of the single-photon, attractive one and ii) the repulsive one from the photon scattering by qubits accompanied by their excitation and deexcitation.
The main consequence of these peculiarities is the emergence of the mixed qubit--photon bound states. In particular, the energy spectrum of the qubit--photon bound states consists of two widely separated bands. The higher energy one lies far over the photon continuum. It is very close to that observed in the simple case of pure attractive interaction and appears for large $\varepsilon$ when $a'\rightarrow a$. The results, almost identical to the preceding ones \cite{bs1,bs2}, were observed. The lower band, near the band edges, lies within the photon continuum.
Based on the recent findings \cite{aphbs8,aphbs9} we expect that these bound states may exert a considerable influence on the photon transport properties. It relies upon the possibility of radiation trapping due to the creation of these bound states \cite{aphbs8,aphbs9}. In the present case, due to translational invariance of the system, radiation trapping concerns the qubit dressing by photon cloud. The formation of bands of such complexes implies their free propagation. Band flattening with changes of the values of system parameters points to the slowing down of these mixed states. The emergence of the flat bands implies a possible stopping light which indicates that the proposed setup could be used for manipulating the opens up the novel means for realizing operable quantum devices.
The proposed setup is convenient for the practical realization of such devices with controllable parameters which could be achieved by applying a constant external magnetic field in parallel with propagating EM field. In such a way, vector potential attains an additional constant term $\alpha_n\rightarrow \alpha_n+\alpha_0$ so that interaction term in (\ref{tot}), after straightforward calculation, reads:
$$ H_i\approx-2E_J\cos\varphi_n\left[ \cos\alpha_0\left( 1-\frac{\alpha^2_n}{2}\right) -\sin\alpha_0 \;\alpha_n\right]. $$
Varying external field it would possible to change the tunneling energy and to "flip" between different regimes. A particularly interesting situation arises when $\alpha_0=\pi/2$ when the interaction term, upon quantization, attains the form identical to that encountered in coplanar arrangements.
Finally, let us comment on the generality of our results. We do not expect that the features of the propagating signal, in the proposed geometrical arrangement, should not qualitatively depend on the particular choice of the type of qubit \cite{zag}. Thus, for the simplicity and certain flexibility for the manipulation of the single qubit, we use here charge qubits, while any other type would give analogous results.
\begin{acknowledgments}
We thank D. Kapor for fruitful discussion and useful comments on the manuscript.
This work was supported by the Ministry of Education, Science, and
technological development of the Republic of Serbia. Z.I. acknowledges support by the "Vin\v ca" Institute -- special grant No. 104-1-2/2020-020, dated 11.01.2021. We also acknowledge the co-financing of this research by the European Union and Greek national funds through the Operational Program Crete 2020-2024, under the call "Partnerships of Companies with Institutions for Research and Transfer of Knowledge in the Thematic Priorities of RIS3Crete", with project title "Analyzing urban dynamics through monitoring the city magnetic environment" (project KPHP1 - 0029067) and also by the Ministry of Science and Higher Education of the Russian Federation in the framework of Increase Competitiveness Program of NUST "MISiS" (No. K2-2019-010), implemented by a governmental decree dated 16th of March 2013, N 211. N.L. acknowledges support by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) (Code No. 203).
\end{acknowledgments}
\section{Appendix 1: quantization of the model Hamiltonian}
\subsection{Quantization of the qubit subsystem}
After expansion $\cos\alpha_n\approx 1-\alpha^2_n/2$, and transition in Cooper pair basis number basis $|N\rangle$ together with the correspondence: $\hat N=-i\frac{\partial}{\partial \phi_n}$ and noticing that $e^{\pm i\hat \varphi_n }|N\rangle = |N \pm 1\rangle$ we rewrite Hamiltonian Eq. (\ref{tot}) in the charge basis as follows:
\begin{eqnarray}\label{CPN}
\nonumber H=&&\sum_n 2E_C\hat N^2_n|N\rangle_n\langle N|-\\ \nonumber && E_J\sum_n|N\rangle_n\langle N+1|+|N+1\rangle_n\langle N|+\\ \nonumber && \frac{E_J}{2}\sum_n\bigg(|N\rangle_n\langle N+1|+|N+1\rangle_n\langle N|\bigg)\alpha^2_n+ \\ &&
\sum_n\left(\frac{2\hbar^2}{E_c}\dot\alpha^2_n+ E_{em}(\alpha_{n+1}-\alpha_n)^2\right)
\end{eqnarray}
In the reduced state space, in which the single island can be unoccupied ($N=0$) or occupied by a single Cooper pair ($N=1$) we obtain reduced Hamiltonian:
\begin{eqnarray}\label{reduced} \nonumber&&
H=-E_c\mathcal{N}+ \sum_n\bigg[E_c \tau^z_n-E_J\tau^x_n\bigg]+\\
&&\sum_n\left(\frac{2\hbar^2}{E_c}\dot\alpha^2_n+ E_{em}(\alpha_{n+1}-\alpha_n)^2+\frac{E_J}{2}\tau^x_n\alpha^2_n\right).
\end{eqnarray}
where $\tau^x_n=|1\rangle_n\langle 0|+|0\rangle_n\langle 1|$ and $\tau^z_n=|1\rangle_n\langle 1|-|0\rangle_n\langle 0|$, while in deriving the above result we have used an apparent relation $\hat N_n=|1\rangle_n\langle 1|+|0\rangle_n\langle 0|\equiv 1$
Qubit component of this Hamiltonian may be diagonalized by means of the norm preserving ( $1=|e\rangle_n\langle e|+|g\rangle_n\langle g|$) transformation:
\begin{eqnarray}\label{UT}
\nonumber
\tau^x_n=\cos\eta\left( |e\rangle_n\langle g|+|g\rangle_n\langle e|\right)-\sin\eta \left( |e\rangle_n\langle e|-|g\rangle_n\langle e|\right),\\
\tau^z_n=\cos\eta\left( |e\rangle_n\langle e|-|g\rangle_n\langle g|\right)+\sin\eta \left(|e\rangle_n\langle g|+|g\rangle_n\langle e|\right),\\
\nonumber \tan\eta=\frac{E_J}{E_C},\;\sin\eta=-\frac{E_J}{\sqrt{E^2_c+E^2_J}},\; \cos\eta=\frac{E_C}{\sqrt{E^2_c+E^2_J}}
\end{eqnarray}
In such a way, up to an irrelevant constant, above Hamiltonian became
\begin{widetext}
\begin{eqnarray}\label{htl}
\nonumber H= &&\sum_n \left\{ 2\epsilon|e\rangle_n\langle e|+\left[\frac{E_JE_c}{2\epsilon}\left(|e\rangle_n\langle g|+|g\rangle_n\langle e|\right)-\frac{E^2_J}{\epsilon}|e\rangle_n\langle e|\right]\alpha^2_n \right\}+ \\ && \sum_n\left(\frac{2\hbar^2}{E_c}\dot\alpha^2_n+ E_{em}(\alpha_{n+1}-\alpha_n)^2+
\frac{E^2_J}{2\epsilon}\alpha^2_n\right).
\end{eqnarray}
\end{widetext}
Here $\epsilon=\sqrt{E^2_c+E^2_J}$, so that the $\pm \epsilon$ denote the energy eigenstates: ground (-) and excited (+) one.
\subsection{Quantization of EM field}
As usual we consider $\alpha_n \ll 1$ and expand corresponding \textit{"cosine"} term in interaction. First we define the generalized momentum $P_n=\frac{2\hbar^2}{E_c}\dot{\alpha_n}$ canonically conjugated to $\alpha_n$. Now we treat photon variables as operators $\alpha_n\rightarrow\hat \alpha_n$, $P_n\rightarrow \hat P_n$ requiring that they satisfy commutation relation: $[\alpha_n, P_m]=i\hbar\delta_{m,n}$, which holds for the following transformation:
\begin{eqnarray}
\label{quantiz2}
\nonumber \hat \alpha_n=\frac{1}{2}\sqrt{\frac{E_C}{\hbar\omega}}\left(a_n+a^{\dagger}_{n}\right),\;\;
\hat P_n={i\hbar}\sqrt{\frac{\hbar\omega}{E_c}}\left(a^{\dagger}_{n}-a_n\right),
\end{eqnarray}
Substitution of the above expressions in Eq. (\ref{htl}) yields the following model Hamiltonian:
\begin{widetext}
\begin{eqnarray}\label{hf}
\nonumber H=\nonumber\sum_n \left[2\epsilon|e\rangle_n\langle e|+\hbar\omega a^{\dagger}_n a_n-\frac{E_{em}E_C}{2\hbar\omega}a^{\dagger}_n\left(a_{n+1}+a_{n-1}\right) \right]
+\frac{E_JE_C}{8\hbar\omega \epsilon}\sum_n\left[ E_c\left(|e\rangle_n\langle g|+|g\rangle_n\langle e|\right)-2E_J|e\rangle_n\langle e|\right]\left(a^{\dagger}_n+a_n \right)^2
\end{eqnarray}
\end{widetext}
\begin{widetext}
\begin{eqnarray}\label{modH}
\nonumber H_s=\Delta\sum_n|e\rangle_n\langle e|+\hbar\omega\sum_n a^{\dagger}_n a_n-J\sum_n a^{\dagger}(a_{n+1}+a_{n-1})-\sum_n\left[A|e\rangle_n\langle e|-B(|e\rangle_n\langle g|+|g\rangle_n\langle e|) a^{\dagger}_n a_n\right]
\end{eqnarray}
\end{widetext}
\begin{equation}
\hbar\omega=\sqrt{2E_{em}E_C+\frac{E_CE^2_J}{2E}},\;\;
J=\frac{E_{em}E_C}{2\hbar\omega},\;\;
A=\frac{E^2_JE_c}{4\hbar\omega E},\;\;
B=\frac{E_JE^2_c}{8\hbar\omega E}.
\end{equation}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,918 |
(thing) by CrazyIvan Tue Apr 04 2000 at 19:50:47
The imported fire ant is a notorious plague in the southern United States, as anyone from Texas can attest. The imported fire ant, Solenopsis wagneri or Solenipsis invicta, came to the US on a boat from South America in the early 1900s. It has since spread like wildfire through all the southern states, as far west as California, and as far north as Tennessee. These evil little red bastards have a sting far more powerful than their size would indicate, and swarm over their prey like army ants. They regular kill livestock, wild animals, and occasionally, people (generally the elderly). They are thought to be responsible for the near/probable extinction of native species such as the Texas Horned Toad or Horned Frog. They are ubiquitous in Texas, picnics are impossible anywhere that has not been treated with insecticide in advance. They also have a strange affinity for electric fields, and regularly short out transformers and air conditioners. They have no natural predators in the United States, though research is being done into indroducing their natural predator from South America, the Phorid Fly. Those of you living in the north will never be afflicted with them, they cannot survive cold winters.
(thing) by kozmund Mon Sep 30 2002 at 13:10:11
Space Ghost Coast to Coast - Episode Guide
Season 6 - Episode 75 - First aired December 10th, 1999
Previous Episode - "Fire Ant" - Next Episode
Guest: Conan O'Brien
Written by: Dave Willis, Matt Maiellaro, Pete Smith, Jim Fortier, Mark Banker
The episode starts with Conan O'Brien being lowered from the ceiling in the monitor while theme music is played. Space Ghost is then lowered from a harness and wires while Conan watches. Space Ghost asks Moltar to attach a speaker to his head because "world hunger is something that affects us all." After Moltar screws the speaker on to his head it emits a variety of buzzing and static. After Space Ghost demands Moltar removes the speaker, Moltar informs Space Ghost of an imminent calamity:
Moltar: No rush, but...uh...Alpha Centauri's gonna explode in about 15 seconds.
Space Ghost: And you know of this?
Moltar: Yeah, some friends of mine are doing it.
Space Ghost: Gas up the wench, and I'll need some longer straps. So get me the catalog, my stationery and...
Something explodes in the distance
Space Ghost: There's no time! Now hoist me to Alpha Centauri!
Space Ghost begins to ascend towards the ceiling to music when one of the wires on his harness breaks. While Space Ghost is swinging back and forth on the wire he and Zorak have a brief discussion about how they fixed it the last time he was dangling from the ceiling. They agree on beating Space Ghost with sticks until he falls down, which they do in short order. After Space Ghost falls from the wire, the interview proper starts. Rather quickly, Space Ghost and Conan start bickering over Space Ghost's superhero origins.
Conan O'Brien: Space Ghost is obviously a space man who died and became a space ghost. Now, I know that you don't want the kids to know that you died, Boo hoo hoo, but you died, baby! And you got to get down with that.
Space Ghost: No!
Conan O'Brien: Face it, Space Ghost!
Conan O'Brien: You're a space man that choked on a muffin!
Space Ghost: That, sir, is impossible because I am allergic to muffins!
Space Ghost then lectures Conan on the WB's programming standards. He gets worked up enough to slam his fist into the table hard enough that his hand bursts into flames. The interview starts to seriously falter here, as Space Ghost has a hard time reading his notes as they are on fire. The flames spread over his whole body while Conan O'Brien makes fun of the sixties Space Ghost show. Space Ghost has a flash back sequence from the show featuring chanting monsters. When the flash back ends, Moltar and Zorak explain to Conan how the sixties show ruined their careers. There is some confusion about which person was Zorak and which was Moltar so Zorak claims that he's Wallace.
Space Ghost: Let's say Zorak was Wallace.
Zorak: I am.
Space Ghost: What would you do with Wild Wallace?
Conan O'Brien: I think what I'd do at first, is I'd hold out my arms like this.
Conan holds his arms out in front of him
Space Ghost: And shove him to death? Oh, good one.
A discussion of weapons ensues. Space Ghost becomes very cold towards Conan after he admits that he doesn't have a weapon. Space Ghost becomes distracted by an ant crawling near his desk. The ant bites him so he blasts it with his power bands. Conan then goes on a rant about what an awful interviewer and person Space Ghost is. Space Ghost again becomes distracted by an ant crawling under his desk. He decided that the new ant is the original ant's twin brother, there to avenge his brother's death.
Space Ghost: I'm gonna follow him home. Kill his whole family.
Conan O'Brien: Well, Space Ghost, at the end of an interview, it's traditional for the talk show host to say, "Thanks for being here, Conan. This was Conan O'Brien. Check out his show on NBC at 12:35." You didn't do that. You completely blew me off. For all these people know, my show is a cop show on, uh, Fox or something, thanks to you.
Space Ghost: Isn't it?
This is where the short version and the long version of the episode diverge. In the long version, Conan O'Brien chats briefly with Moltar and Zorak while Space Ghost follows the ant off the set. There is then a 10 minute sequence of Space Ghost following the ant. After the trek, he finally arrives at the ant's home. The door opens and a giant ant is standing there. He demands to know what the giant ant is going to do about his son biting Space Ghost. The giant ant then chases Space Ghost as Space Ghost shouts "Your son is a moron!"
In the short version, it cuts from the studio to a screen reading "The Next Day" straight to Space Ghost arriving at the ant's home. Presumably this is because having a ten minute sequence of Space Ghost slowly following an ant was a silly idea in the first place.
Watching the episode too often.
http://snard.com/sg/guide/?ep=75&fmt=0
Oh, and I stole DyRE's layout.
(thing) by GrouchyOldMan Fri Jan 10 2003 at 22:26:02
We'd only been in the Lone Star State for a few weeks, and weren't even properly moved in yet. There were still packing boxes half full of stuff everywhere, and hardly a place to sit down. Perfect time for a party! So we got a big galvanized wash tub and filled it with ice and Lone Star beers, and fired up the barbecue to toast up some hot link sausages we'd gotten from the butcher at the Piggly Wiggly market. Somebody brought their guitar, and someone else brought a big bowl of guacamole and chips. Everyone was smiling and having a great time until the talk turned to fire ants.
The red imported fire ant, Solenopsis invicta, is the critter in question, believed to have arrived in the United States sometime in the 1930 as stowaways in the soil carried aboard cargo ships as ballast. These tiny illegal immigrants probably first came ashore in Alabama and Florida, but apparently they found it to their liking and wired home to the relatives, "c'mon out." By the time the first Fire Ant census was performed by the USDA in 1953, they had spread to over 100 counties in 10 states. They are currently common along the entire southeastern border of the United States and that most definitely includes Texas.
Our new Texas Aggie friends down at Dudley's, the classic student hangout across the street from Texas A & M University1 had warned us about fire ants, but we sort of laughed it off. I mean how much trouble can a two millimeter ant cause anyway? None of us paid much attention, and the Texans just sort of pulled their hats down a little lower over their foreheads and smiled quietly.
As it turns out, they were right and we were wrong, way wrong! In 1957, the U.S. Congress allocated $2.4 million dollars for the purpose of eradicating fire ants. That was real money back then, but it didn't do the job. Since that time, they've tried poisoning the individual mounds, broadcasting fire ant bait from airplanes and a whole catalog of ingenious biological controls, but those didn't do the job either. Recently, fire ants have shown up as far west as California, and as far north as Kansas and Maryland. Coming soon to a lawn near you!2
Back at the party, after the first few rounds of Lone Star longnecks had been ceremonially gulped, and the smell of burning hotlinks wafted through the halls, the discussion turned to the wonders and delights we had all recently encountered in our new homeland. Dennis and Roy had chased an armadillo for an hour, but when they finally had the poor little monster cornered, Katie disdainfully informed them that armadillos carried leprosy, so they chickened out and let him go.
My report was the discovery of a man made lake (called a "Tank" in Texan) on the outskirts of town where we could all go windsurfing. The discovery of Bryan Utilities Lake Park (BULP) was welcome news because it had become brutally clear to our crew of displaced Californians that there was no ocean here, hence no surf, hence no surfing. It looked like it was boiling down to windsurfing or alcoholism. As history revealed, it is actually possible to blend the two of these nicely.
Anne's contribution was the report of a huge fire ant mound, right out in the back yard! The availability of this marvel close at hand won the day and after stocking up on more longnecks, we all drifted outside to have a look. The dome-shaped mound of a fire ant nest stands about a quarter of a meter high, and is about half that much around. It's made from the dirt the ants excavate when they build their home beneath. Over time, the mound is soaked by rain and dried again, so the surface of the mound develops a thin crust that protects it from the odd accidental encounter. You won't usually see many ants coming or going from the mound itself because the clever little fellows have built tunnels stretching away from the mound to handle the everyday traffic.
The mound we saw in the back yard was typical in all respects, made from the brownish-orange clay soil characteristic of the Brazos Valley. There weren't any ants in sight as we edged closer for a good look. To their credit, the girls were unanimous in their instinct about where this was headed, and after verbally abusing us for the testosterone poisoned morons that we arguably were they sensibly returned to the house, food, and music.
This was even before Bradley managed to get himself bitten. He skillfully found one of the fire ant off ramps and stuck his finger in front of it to obtain a specimen. He called us all over to display his trophy and while we were gawking, the ungrateful wretch bit him. We laughed. He laughed too, but then he went inside with his "serious," face on. We laughed again when he came back out with an ice cube sitting on the angry red blotch that had appeared on his palm. I think we were still laughing when Gus took a long pull on his beer and called Brad a pussy.
This wasn't exactly a "fightin words" challenge or anything. We all worked on research ships and pussy is sort of a favored technical term out on the high seas. Cindy, our first female marine technician had called me a pussy on the last cruise because I hesitated at plunging my upper body into a 50 gallon vat of used compressor oil to rescue our new torque wrench when it fell in. She shot me a disgusted look and plunged in herself, becoming an instant legend in the galley that evening. So having Gus call you a pussy wasn't dangerous exactly, but it was a serious charge, mostly just because it was Gus.
Gus was our leader, our boss, our role model, our hero. He was a huge guy, built like a linebacker who has let himself go a bit. He had small, fierce, pale blue eyes deeply set in a heavy viking brow that was emphasized by his flaming red moustache and an out of control beard. Picture Gimli the Tolkien Dwarf, only big. To have Gus focus his attention on you was to know the meaning of the verb "to squirm". Gus was rumored to be an ex green beret, and to have "played some football," but no one had the cojones to actually come out and ask him for the details. When he wasn't on the ship, he had a side business, demolition. He told me one time that it wasn't because he needed the money, but just "cause I like blowin shit up!" Gus was larger than life.
So when he called Brad out on his alleged pussy-ness, we knew there'd be more to follow. Brad threw away his ice cube, tried to regain his dignity and said, "Right Gus, you try it then..." Part of the mystique about Gus was that he wasn't instinctively macho. He had that natural conservatism that people who have actually been in life and death situations often seem to develop. When Brad hurled his challenge, I saw Gus' eyes dart over to the fire ant mound, then back to the angry welt on Brad's hand. He hesitated a moment, and I thought he was going to laugh it off, but instead, he walked slowly over to the mound, took another long pull of his Lone Star and plunged his thick fist right down inside the thing until only his dive watch was showing.
Fire ants are known for their synchronized stinging behavior. When they perceive a threat, they summon the troops and swarm the enemy, initially, without biting. Then, once some critical mass has been achieved, a pheromone signals the attack and everybody lets loose at once. Fire ants can bite multiple times and generally don't stop until they are physically scraped off or killed. Kamikazes of the Solenopsisfamily! Fire ant attacks have been responsible for the death of livestock and even some humans. Their stings initially cause an intense burning sensation (hence the name) followed by the development of white pustules that take weeks to completely heal and are easily infected.
To his credit, Gus didn't pull his hand right back out, an empty gesture that wouldn't have fooled this crowd for a minute. He held it in there longer even than any of us thought necessary, then longer than we could even believe. Then, longer than he should have. It was apparently just long enough for the fire ant generals to command the troops, and the captains to deploy them and the sergeants to get their platoons locked and loaded. And then, as fire ants are wont to do, they all stung at once.
A mature fire ant colony can house over a quarter of a million ants. Fire ant queens live up to seven years and can produce over 600 eggs per day for their entire life. It's not clear how many fire ants all bit Gus at once, but I think that it's safe to say that the orgy of ant squishing that erupted probably didn't affect the hive's productivity as much as it did Gus's. We could all tell it the moment they attacked because he pulled his paw out of the mound and scrutinized it with an astonished, puzzled and deeply sad look. Then he dropped his half finished beer in the dirt and started madly scraping the rusty sludge of squirming ants off. Unfortunately, this only succeeded in transferring large quantities of ants onto his other arm, clearly not a winning strategy.
Finally, in a gesture of grace and humanity that I will always remember, Brad strode over and turned on the garden hose and directed the pressure nozzle onto Gus's swollen arms. This appeared to work and, crestfallen at seeing our hero laid low, the rest of us drifted back to the food and music.
About the only good thing to be said about fire ants is that it is enormously entertaining to kill them. Despite many years of study, entomologists have yet to devise a method of controlling fire ants on a large scale. In the end, it's every man for himself with respect to fire ants, and if you don't want them around, you have to kill them off, mound by mound. The good news is that there are very effective fire ant baits expressly for this purpose. Better news yet is that the poisons are delightfully malevolent in their action. The fire ant bait3 I came to prefer consisted of cheery yellow nuggets that looked like Grape Nuts cereal only golden. This stuff was designed somehow, to trick the worker ants into thinking it was a special treat that should be delivered immediately to mommy, the Queen. You find an offending fire ant mound, sprinkle a circle of golden goodies around it and then stomp a few times to let the crew know you're there. The little suckers swarm out the doors, and dash around madly looking for something to attack. Then they discover the bait and queue up into nice, obedient little lines, each carrying a chunk of certain death home to the boss. In a couple of days, no more mound. It's not immediate and violent, but still somehow deeply satisfying.
Gus and Brad came in to the party a few minutes later looking a little sheepish. The blood had drained from Gus's face and, looking back on it, I'm pretty sure that he was experiencing a mild case of shock. We settled him next to the beer tub so he could soak his hands in the ice water and he switched from beer to Jack Daniels for the rest of the evening. As far as I know, nobody ever mentioned the event to him again.
That's the way it goes with heroes, and fire ants.
1 TAMU Fire Ant website: http://fireant.tamu.edu/
2 They're in Australia too: http://www.dpi.qld.gov.au/fireants/
3 My favorite fire ant poison: http://www.superkill.ws/
I like it! 13 C!s
Portuguese man-of-war Army Ants fruit fly elimination effort Space Ghost Coast to Coast
Deep Sea Drilling Programme My favorite things are discontinued King Dead Orchardist's Journal
Shaving my genitals Knifin' Around Slaver Ants velvet ant
Tunnel rat Acacia-Ant Mutualism Space Ghost Kentucky Nightmare
Flipmode The world's largest ant colony That which does not kill me... The little smile of recognition, peculiar to noticing a stranger reading your favorite book
You Better Watch Out, or the Insects Will Get You Insecticide Song of Ceber 14: Town Life The Ants
I reserve the right to club you and eat your bones
Take now your son, your only son, Isaac | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,382 |
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## CONTENTS
Epigraph
Chapter One: The Carpenter's Toolbox
Chapter Two: Turnscrews
Chapter Three: Lock, Stock, and Barrel
Chapter Four: The Biggest Little Invention
Chapter Five: Delicate Adjustments
Chapter Six: Mechanical Bent
Chapter Seven: Father of the Screw
GLOSSARY OF TOOLS
ACKNOWLEDGMENTS
ABOUT WITOLD RYBCZYNSKI
NOTES
TEXT ILLUSTRATION SOURCES
INDEX
to Shirley
Heurēka! [I've found it!]
—ARCHIMEDES
## CHAPTER ONE
## The Carpenter's Toolbox
THIS ALL STARTS with a telephone call from David Shipley, an editor at the New York Times. Would I write an article for a special millennium issue of the Sunday magazine? he asks. The end of the millennium is on many magazine editors' minds, and I have had a number of such requests. Shipley explains that the theme of the issue is The Best of the Millennium. That sounds interesting. "What do you want me to write about?" I ask.
"We're hoping that you can write a short essay about the best tool," he answers.
I am a bit let down. The best tool is hardly as weighty a subject as the best architect or the best city, topics I could really sink my teeth into. Still, I have been working on a long biography and would welcome a break. Writing about the best tool of the millennium might even be fun.
While David Shipley is speaking, I compose the essay in my head. There is so much to choose from: paper clips, fountain pens, eyeglasses. I have recently seen a portrait in the Pennsylvania Academy of the Fine Arts of Benjamin Franklin wearing round spectacles, a reminder that Franklin was the inventor of the bifocal. Yet eyeglasses are much older than the eighteenth century. The first reference to eyeglasses is in a sermon given by a Dominican friar in Florence in 1306. He mentions that eyeglasses were invented twenty years earlier, and that he has even spoken with the inventor, although he neglects to give his name. Medieval eyeglasses were only for farsighted people and were used for reading and writing. They were the first practical application of the new science of optics, paving the way for such far-reaching inventions as the telescope and the microscope. A key influence on literacy, astronomy, and biology, eyeglasses surely qualify as "the best tool of the millennium." This is going to be easy.
However, when I mention my idea to David, it becomes clear that he has something else in mind. He means tool in the literal sense—a handsaw or a hammer. So, not eyeglasses. He must hear the disappointment in my voice, and he points out that I once wrote a book about building my own house. That might make a good starting point, he suggests helpfully. All right, I say, I'll think about it.
—
In my case, "building my own house" meant actually building it. My wife and I, with the occasional help of friends, mixed concrete, sawed wood, plastered walls, and installed plumbing. We did everything ourselves except the electrical wiring. Ever since my boyhood experiences with recalcitrant train sets, I have been thwarted by electricity. Despite my father's patient explanations—he was an electrical engineer—and a college physics course, I never grasped the relationship between voltage, current, and resistance. Electricity, in fact, was a problem in our house-building project—there was none. We were building on a rural site about eight hundred feet from the road, and although we planned to bring in power, initially we could not afford the cost of a temporary line. Renting a gas-powered generator would be expensive, too—and noisy. I decided to build the framing and exterior of the house by hand. Once the basic structure was finished, which promised to take a year or two, we would bring in a line and hire a professional to install the electrical wiring.
Does one of my carpenter's tools qualify as the millennium's best? I discount power tools. I had used a portable circular saw, a drill, and a sander for finishing and cabinetwork, but these are chiefly laborsaving devices. Not that productivity isn't important. Ken Kern, the author of The Owner-Built Home, estimates that cutting all the two-by-fours for the frame of a small house would take seven full days using a handsaw, and only thirty minutes using a power saw. I appreciate the ease of cutting wood with power tools, but the result, while more quickly arrived at, is no different than if I use a handsaw. In any case, I enjoy working with my hands. One of the rewards of building something yourself—a house or a bookshelf—is the pleasure of using tools. Hand tools are true extensions of the human body, for they have evolved over centuries of trial and error. Power tools are more convenient, of course, but they lack precisely that sense of refinement. No doubt, if I spent my life hammering nails, I would feel differently about the virtues of a nail gun, say. Yet increasing the productivity of carpenters does not seem to me in the same category as the invention of entirely new devices such as eyeglasses.
That leaves my box of hand tools. The tools required for the construction of a small wood-frame house fall roughly into four categories: measurement, cutting and shaping, hammering, and drilling. My measuring tools include a try square, a bevel, a chalk line, a plumb bob, a spirit level, and a tape measure. A little reading informs me that almost all these tools predate our millennium; indeed, most predate the first millennium of the Christian age. A Roman builder, or mensor aedificorum, was familiar with the try square, the plumb line, and the chalk line—all tools that were developed by the ancient Egyptians. The level, or libella, also an Egyptian invention, consisted of a wood frame resembling the letter A, with a plumb bob suspended from the apex. To level, the string was lined up with a mark in the center of the crossbar. Not as compact as my spirit level, perhaps, but obviously just as serviceable since A-levels continued to be used until the mid-1800s. The spirit level, with its sealed tube containing an air bubble floating in alcohol, was invented in the mid-1600s. It was first exclusively a surveying instrument—it took another two hundred years to find its way into the carpenter's toolbox. For measuring length, the Roman mensor used a regula, or a wooden stick divided into feet, palms, twelfths or unciae (whence our inches), and digiti or finger widths. I have a yardstick, too, but most of my measuring is done with a retractable steel tape. That, at least, would impress my Roman counterpart, whose only compact measuring device was a one-foot bronze folding rule. Oak yardsticks were used in the Middle Ages, and folding rules, in ivory, brass, or boxwood, reappeared in the eighteenth century. I can't find the origins of the tape measure, but I would guess that it was developed sometime in the late 1800s. I would be lost without my twenty-five-foot retractable tape measure, but it does not seem to me to qualify as the best tool of the millennium.
I own several saws. The handsaw, too, is an ancient tool: archaeologists have found metal-toothed Egyptian saws dating back to 1500 B.C. They have broad blades, some as long as twenty inches, curved wooden handles, and irregular teeth. The blades are copper, a soft metal. To keep the blade from buckling, the Egyptian saw was pulled—not pushed. Pulling is less effective than pushing, since the carpenter cannot bear down on the cutting stroke, and sawing wood must have been a slow and laborious process.I The Romans made two important improvements. They used iron for the blades, which made them stiffer, and they set the teeth of the saw to project alternatively right and left, which had the effect of making the saw-cut—or kerf—slightly wider than the blade, allowing smooth movement.
The Romans also invented the stiffened backsaw, whose blade is reinforced at the top. This prevents straight-through cuts, but the tool is useful for cabinetwork, especially when used in combination with a miter box. The most ingenious Roman addition to cutting tools is the frame saw. A relatively inexpensive narrow blade is held in a wooden frame and is kept taut by tightening a cord. Wooden frame saws worked so well that they remained the most common type of saw well into the nineteenth century (the principle of the frame saw survives in the modern hacksaw). In the mid–seventeenth century, a new type of saw was introduced in Holland and England. It had a broad, unstayed blade and a wooden pistol-grip handle. The rigid blade, originally made by rolling steel strips, makes a more accurate cut than a frame saw, and there is no frame to interfere with deep cuts. This effective tool became the basic modern handsaw. My workhorse is a twenty-six-inch Disston crosscut handsaw, with a skew-back blade, first introduced in 1874 by Henry Disston, a Philadelphia saw-maker. The open handsaw is a definite contender for best tool, but while it is certainly an elegant solution to an old problem, I think that David expects something a little more momentous.
The chief shaping device of the carpenter is the plane. The box plane is nothing more than a holder for a chisel blade, but it marks an important moment in the evolution of hand tools. Unlike an adze or a chisel, which depend on the skill of the craftsman, the effectiveness of a plane is built-in; that is, the carpenter does not need to control the blade, he provides only the motive force. One historian has called the plane "the most important advance in the history of woodworking tools." That makes it sound like a worthy candidate for best tool of the millennium. Unfortunately, I find that the plane, too, is a Roman invention.
Chisels have more ancient origins. Bronze Age carpenters used chisels with both integral handles and socketed wooden handles in house and furniture construction. The first mallets, which resembled bowling pins, were pounded across the grain and had a short working life. Eventually, a handle was fitted to a separate head, whose harder end-grain made a more durable hammering surface. Heavy, long-handled mallets are called mauls. Eighteenth-century carpenters used a huge maul, known as the Commander, to drive together the joints of timber-framed houses and barns. The Commander has a head six inches in diameter and a foot long. I didn't have anything that big, but I did use a steel sledgehammer to coax stubborn joists and studs into place.
The most unusual hammer I own comes from a hardware market in Mexico City. Made in China, it is a "combination case opener," that is, a packing-crate opener. Like the specialized shingler's hammer, which combines a hammer and a hatchet, the case opener incorporates several tools: a hammer, a nail puller, a hatchet, and a crowbar. Mine must have been made in one of Mao's backyard furnaces, for shortly after I bought it, one of the metal claws broke off as I was pulling nails. Nevertheless, I still have it. While I am unsentimental about most possessions, I have never thrown away a tool.
I have always thought of combination tools as particularly modern gadgets—I am embarrassed to recollect that I once gave my father a screwdriver with a built-in flashlight as a Christmas present. In fact, the combination tool is ancient. The two oldest woodworking tools are the ax, for felling trees, and the adze—with its blade turned ninety degrees—used for dressing timber. A combination ax-adze was used by the Minoan civilization of Crete, which also invented the double-headed ax. The ax-adze was popular with Roman carpenters. The Romans, who invented forged iron nails, used another dual-purpose tool: the claw hammer. Pulling nails exerts heavy pressure on the handle, which risks being pulled out of its socket, or eye. Medieval English claw hammers sometimes had two metal straps that reinforced the connection to the handle. An American was responsible for the modern form of the claw hammer. In 1840, a Connecticut blacksmith, inspired by the adze, added a tapered neck that extended down the hammer handle, resulting in the so-called adze-eye hammer, which survives to this day.
Ancient Egyptian woodworkers used wooden pegs instead of nails. They made the holes with a bow drill. The bow drill, probably adapted from a fire-stick, has a cord wrapped around the drill and held taut by a bow. Holding the drill vertically, the carpenter moves the bow back and forth, like a cellist, pressing down on alternate turns. Because the carpenter exerts downward pressure with only one hand—and the cord can easily slip—the bow drill is ineffective for heavy drilling. (Bow drills continued to be used for delicate drilling until the nineteenth century.) Moreover, since each drilling stroke is followed by an idle return stroke, the bow drill wastes energy. Once again, it was the Romans who found a solution: the auger. The auger has a short wooden cross-handle, attached to a steel shaft whose tip is a spoon-shaped bit. The carpenter, holding the handle with both hands, can apply both great rotational force and heavy downward pressure. A particular variation of the auger, developed in the Middle Ages for drilling deep holes in ships' timbers, is called a breast auger. It is topped by a broad pad on which the carpenter rested the entire weight of his body.
A medieval workman with his tools, including a carpenter's brace. Detail from Bearing the Cross, part of an altarpiece painted by Meister Franke, 1424.
The auger is a great advance, but it has one drawback: the bit tends to freeze in the wood between turns. The great breakthrough in drilling tools occurred during the Middle Ages with the invention of the carpenter's brace. The brace holds the same spoon-shaped bits as an auger, but the handle is shaped in such a way that it is possible—for the first time in history—to drill holes with a continuous rotation. A rounded pad atop the brace enables the carpenter to push down on the bit as he turns with a smooth back-and-forth motion.
One of the earliest representations of a brace is contained in the right-hand panel of an altar triptych painted about 1425 by the Flemish artist Robert Campin and now hanging in the Metropolitan Museum of Art in New York. The subject is Saint Joseph in his workshop. Joseph is making mousetraps (this is an allegorical painting), and he is surrounded by tools—a hammer and nails, a chisel, pincers, a straight saw, and an auger. He is holding a carpenter's brace and is drilling a hole in a piece of wood that he awkwardly balances on the arm of his chair.
What is striking about the tool that Joseph is holding is that it is identical to the eighteenth-century wooden braces I have seen in collections of American tools, and basically not much different from the brace in my own toolbox (although mine is steel). Some tools, such as hammers and saws, evolve slowly over centuries; others, such as planes, seemingly spring to life fully formed. The brace seems to have been such a case—it bears no resemblance to the auger or the bow drill. The brace has no antecedents because it incorporates an entirely new scientific principle: the crank. The crank is a mechanical device with a unique characteristic: it changes reciprocal motion—the carpenter's arm, moving back and forth—into rotary motion—the turning bit. The historian Lynn White Jr. characterized the discovery of the crank as "second in importance only to the wheel itself." The crank made possible not only the carpenter's brace, but also hand-cranked mills and grinders, as well as a variety of water- and wind-driven machines such as stamping mills and pumps, and eventually steam engines.
There is no material or textual evidence that the crank existed in antiquity—as far as we know, it is a medieval European discovery. The oldest representation of a crank is in a fourteenth-century medieval treatise that shows a design for a boat with a manual crank drive that resembles the kind of recreational foot-driven paddle-boat that is a staple of summer-cottage lakes and city parks. A Bavarian book on military engineering published in 1405 includes a sketch of a milling machine turned by a hand crank. At about the same time, cranked lecterns (similar to modern dentists' adjustable tables) were used by scholars to swing books within convenient reading range. So, around 1400, cranks were in the air. Whether the carpenter's brace came first or was inspired by one of these other gadgets, there is no doubt that this simple tool was the first practical application of the crank on a broad scale. The origin of the name brace, incidentally, is obscure. The tool was first called a piercer, for it was used to drill starting holes that were then enlarged with an auger. One historian speculates that brace may refer to the metal braces that were sometimes added to reinforce the crank shape.
The carpenter's brace is a good tool and it definitely belongs to our millennium. But, as far as my essay is concerned, there is a problem: the brace is, well, boring. Despite the importance of the crank, the carpenter's brace itself never really developed further. The only nonwoodworking application occurred in the sixteenth century, when surgical braces, called trephines, were used to cut out a disk of bone from the skull. Otherwise, the brace seems to have had an uneventful history. It was merely a better way of drilling holes.
—
I have spent a week thinking and reading without making much progress. Since I am embarrassed to admit to David Shipley that I can't come up with a subject, it's beginning to look as if I will have to write about the unexciting carpenter's brace. This is not going to be an easy assignment; what had seemed like fun is turning into a chore. Dejected, I mention my predicament to my wife, Shirley. She thinks for a moment and answers, "There is one tool that I've always had at home. A screwdriver." I look at her skeptically. "Definitely, a screwdriver," she says. "Wherever I've lived, I've always had a screwdriver in the kitchen drawer. Preferably the kind that has several interchangeable heads, or whatever those end pieces are called." She adds conclusively, "You always need a screwdriver for something."
I had forgotten the screwdriver. I go back to my standard reference on hand tools, William Louis Goodman's History of Woodworking Tools, published in 1964. Goodman was a thirty-year veteran of teaching wood shop in an English boys' school. He was also a tool collector. I have the impression that he was someone who not only knew a lot about the origin of the Saxon adze, but could also give a handy personal demonstration of its proper use.
I look up screwdriver in Goodman's index—nothing. That's odd. Flipping through the book, I find an entire chapter on the carpenter's bench, a meditation on the origin of the glue pot, but nothing about screwdrivers. Then a chart catches my eye: "Woodworkers' Tool Kits at Various Periods." It lists the times when various carpentry tools were invented and confirms what I already know—most hand tools originated during the Roman period. The Middle Ages added the carpenter's brace; the Renaissance, some specialized planes. The next period, "1600 to 1800," saw the invention of the spokeshave, a sort of pulling knife used to make wheel spokes and chair spindles. Finally, in "1800 to 1962," I find the screwdriver. It is one of the last additions to the woodworker's toolbox.
Usually, my 1949 edition of the Encyclopaedia Britannica is informative, but the entry "Screwdriver" is a simple definition—no history. The "Tools" entry does not even mention screwdrivers. I check the on-line Britannica, which is more helpful: "The handled screwdriver is shown on the woodworker's bench after 1800 and appears in inventories of tool kits from that date." At least it isn't another Roman invention. I'm not convinced that the screwdriver is any more earthshaking than the carpenter's brace, and it is a laughably simple tool. Still, I am puzzled by its late appearance. It is definitely worth looking into.
* * *
I. Traditional Japanese saws likewise are pulled rather than pushed. With paper-thin blades, they are used chiefly for delicate cabinetwork.
## CHAPTER TWO
## Turnscrews
I START MY SEARCH for the origins of the screwdriver by consulting the Oxford English Dictionary. According to the citation, the first appearance in print of screwdriver was in 1812, in a book titled Mechanical Exercises. My university library has an original copy. It is a self-help manual for budding artisans written by a Glaswegian, Peter Nicholson. At the back of the book, in a list of definitions, I find the quote: "Screw Driver: a tool used to turn screws into their places." Simple enough. Unfortunately, the author does not include an illustration. Nor does he mention the screwdriver anywhere else in the book; he either thought that the tool was little used—or else he took it for granted.
In the introduction, Nicholson acknowledges his debt to Joseph Moxon, the author of the first systematic account in English of craftsmen's tools and methods, published more than a hundred years earlier. Moxon, a friend of the diarist Samuel Pepys, was a printer by trade. His London shop, "under the Sign of Atlas in Warwick Lane," sold not only books, but also maps, nautical charts, globes, and mathematical instruments. In 1678, to expand his business, Moxon began publishing how-to-do-it pamphlets for carpenters, bricklayers, and joiners. The booklets appeared monthly and sold for sixpence. In 1693, he compiled the series into a book. The 238 octavo pages, including eighteen copperplate engravings, was titled Mechanick Exercises.
The endearing subtitle of Moxon's book is "The Doctrine of Handy-Works." "I may safely tell you," the author advises in the preface, "that these are the Rules that every one that will endeavour to perform them must follow; and that by the true observing them, he may, according to his stock of Ingenuity and Diligence, sooner or later, inure his hand to the Cunning or Craft of working like a Handy-Craft." Moxon begins his book by discussing smithing, "which comprehends not only the Black-Smith's Trade, but takes in all the trades which use either Forge or File, from the Anchor-Smith, to the Watch-Maker; they all working by the same Rules, tho' not with equal exactness, and all using the same Tools." Moxon describes a screw-pin and a screw-plate, crude taps and dies used to make nuts and bolts to attach strap hinges to wooden doors. The bolts have square heads and are tightened with a wrench, which may be why Moxon does not mention a screwdriver, here or anywhere else in his book.
I keep looking. There are false leads. I come across a reference to an ancient Greek dedicatory epigram that describes the tools of a carpenter and includes not only a plane and a hammer but also "four screwdrivers." Since the author lived in the third century B.C., this would make the screwdriver ancient after all. I consult a classical scholar at the university. He points out that the Greek word translated as "screwdrivers" really means "tools to make holes," for the "dowels" that are mentioned in the same line. So, not screwdrivers—bow drills.
A comment in an addendum to a history of woodworking tools leads me to the entry on "Navigation" in the third edition of the Encyclopaedia Britannica. In an illustration of a sextant and its accessories—interchangeable lenses, a magnifying glass, a key for adjusting the central mirror—is a clearly labeled wood-handled screwdriver. The third edition was published in 1797, which is fifteen years earlier than Nicholson's Mechanical Exercises. I find an even older reference in the tenth edition of the Merriam-Webster's Collegiate Dictionary. The citation quotes a York County, Virginia, will: "1 doz. draw rings, screw driver, and gimlet." No illustration this time, but the date is April 28, 1779, thirty-three years before Nicholson. So, the OED is not infallible.
Raphael A. Salaman's Dictionary of Tools of 1975 is probably the most complete modern work of its kind. The British compilation includes several specialized screwdrivers: a slender electrician's screwdriver; a tiny jeweler's screwdriver; a stubby gunmaker's screwdriver; and a short, heavy undertaker's screwdriver for fastening coffin lids. Salaman dates the origin of the screwdriver slightly earlier than the Encyclopaedia Britannica: "Wood screws were not extensively used by carpenters until the mid eighteenth century, and consequently the Screwdriver does not appear to have been commonly employed until after that time." If screws were in use by 1750, I should be able to find a reference to screwdrivers earlier than 1779.
Something else catches my eye. Salaman writes that "although nowadays the generally accepted name is Screwdriver, it appears from the trade catalogues and other literature that, at least in the Midlands and the North of England, the usual name was Turnscrew." This is news to me. I can't find an entry for turnscrew in any of my dictionaries. Yet Salaman is unequivocal. This raises an interesting question. Turnscrew, if such a word really exists, would be a literal translation of tournevis, French for "screwdriver." Maybe the screwdriver was invented in France?
In an encyclopedia of arts and crafts published in Paris in 1772, I find an entry by A. J. Roubo, a master cabinetmaker, who describes in detail how screws—"sold ready made"—are countersunk in brass plates and moldings inlaid into furniture. "The head of the screw is turned by means of a screwdriver," he writes. The tournevis illustrated in an accompanying engraving is not the familiar hand tool but a flat-tipped bit for a carpenter's brace. The brace actually makes an excellent screwdriver, since the crank of the handle greatly increases the torque and the continuous turning motion prevents the screw from "freezing" in the wood. So, the first screwdriver may have been simply a modified drill bit. Maybe my essay should be about the brace and the screwdriver?
An obvious place to look for French technology is Diderot and d'Alembert's great Encyclopédie. My university's library again comes through with a complete set, all seventeen volumes, as well as eleven volumes of plates and the seven supplementary volumes. The librarian unlocks the glass case in the Rare Book Room and I heft the heavy folio over to a reading table. I open the old book carefully. The paper feels coarse. The authors of the Encyclopédie provide no fewer than three entries under Tourne-vis. First a general description, ending with the observation that "the screwdriver is a very useful tool." Then a brief mention of the arquebusier's screwdriver, used by soldiers to adjust matchlock guns. Last, a long paragraph on the cabinetmaker's screwdriver. The description of the latter is characteristically thorough: the steel of the blade must be tempered for strength; the tip is to be sharp so that it won't slip out of the slot in the head of the screw; a metal ferrule, or band, is required to reinforce the base of the wooden handle; and the handle itself must be slightly flattened so that it can be firmly held while screwing. The text closes with a reference to an illustration. Excitedly I find the correct volume and turn to a chapter devoted to tools used by cabinetmakers and workers in marquetry. There it is at the bottom of the page. An engraving of a short-bladed tool with a flat, oval wooden handle, just as described in the text. The folio was published in 1765, fourteen years before the Virginia will, which makes it the oldest evidence of a screwdriver that I have come across so far. I'm not sure what I expected, but I'm disappointed that the tool resembles an ordinary modern screwdriver. Can this really be the first screwdriver?
Tourne-vis from Diderot and d'Alembert's Encyclopédie, 1765.
Next to the engraving of the screwdriver in the Encyclopédie is an illustration of a curious tool that consists of a screw attached to a ring. It is identified as a tire-fond, which the authors explain is used by inlay workers and cabinetmakers to pull pieces of wood into place. On the same page is a description of a tire-bouchon (literally, cork-puller): "a kind of screw of iron or steel that is attached to a ring." For centuries, wine bottles were sealed with wooden bungs. In the mid-1600s, it was discovered that the elastic outer bark of the cork oak, which grows predominantly in Spain and Portugal, made a more effective stopper. However, the new, tight-fitting "corks" were difficult to draw. Someone—perhaps a thirsty cabinetmaker—found that the tire-fond made a convenient corkscrew. My old Dictionnaire Général de la Langue Française records the first use of tire-bouchon in 1718, two years before corkscrew appeared in English. For a moment I toy with nominating the corkscrew as the best tool of the millennium—certainly the most agreeable—but decide to continue my search.
My Dictionnaire states that the word tournevis was officially accepted by the Académie Française in 1740 and first appeared in print as early as 1723, which anticipates the first English-language reference by more than fifty years. That makes sense. I had read that Moxon copied many of his illustrations from earlier French publications. It is beginning to look as if the screwdriver might be a French invention.
The first screwdrivers were probably handmade by local blacksmiths. Yet, as the engraving in the Encyclopédie made clear, there was nothing primitive about these early tools. Not that the screwdriver is complicated—there are many traditional tools from which it could easily have been derived. For example, the Encyclopédie mentions that the tournevis was often confused with the tourne à gauche, a wood-handled steel spike that was used as a key to turn other tools. Awls, files, and chisels could also have provided models for the screwdriver. Or the earliest screwdriver may simply have been a modification of a broken or disused implement. The Colonial Williamsburg Foundation owns two such screwdrivers: one is made from the broken blade of a colchimarde, or small sword; the other is adapted from an old file, with a stubby wooden handle mounted transversely, like an auger handle. According to Henry C. Mercer, who in 1929 wrote the first history of American tools, auger-handled screwdrivers were commonly used in the eighteenth century to release the heavy iron screw bolts that connected rails to bedposts.
I often consult Mercer's Ancient Carpenters' Tools. Together with Goodman's History of Woodworking Tools it is one of the basic texts on the history of hand tools. Mercer's book includes several photographs of nineteenth-century screwdrivers from his extensive collection of early American tools and artifacts. Unfortunately, he has nothing new to say about the origin of the screwdriver. He has never heard of ancient Roman screwdrivers or seen medieval pictures of screwdrivers. He, too, writes that screwdrivers were not commonly used by carpenters before the nineteenth century. Nevertheless, Mercer conjectures that screwdrivers must have been used before 1700, and he speculates that Moxon may simply have overlooked the tool. I find myself agreeing with Mercer: if there were screws, there must have been screwdrivers.
Henry Chapman Mercer is an interesting figure. He was born in 1856 in Doylestown, the seat of Bucks County, Pennsylvania. He attended Harvard, where he studied art history under Charles Eliot Norton, then went on to law school. He was admitted to the bar, but thanks to a small inheritance, he was able to spend the next decade in leisurely European travel. His chief legacy of this idle period was an appreciation for the arts, an interest in antiquity, and a case of venereal disease that would prevent him from marrying. After his return to the United States, he worked as a curator of American archaeology at the University of Pennsylvania museum. At this time he appears to be an unremarkable type: the gentlemanly amateur. Photographs show a dapper young man with curly mustaches. "A good fellow: a member of the Rittenhouse Club: a collector and traveler: a man of means," is how one acquaintance described him. Then Mercer showed an independent streak. He developed an original theory of archaeology, reasoning that the past could best be understood not by examining prehistory but by working back from the present. He left the university, returned to Doylestown, and began collecting early American tools.
Mercer's interest in old crafts led him to traditional ceramics. He visited England and met a tile maker who had worked for William Morris, and on his return he established an art pottery that he called the Moravian Pottery and Tile Works. Mercer fell under the spell of the British Arts and Crafts movement. Many craft-based enterprises in furniture, metalwork, and weaving, as well as ceramics, were founded in America at this time, a reaction to the shoddy products of mass production and industrialization. Like Morris, whose handicraft business flourished, Mercer achieved not only artistic but also financial success. So-called Mercer tiles became famous and were used in prominent buildings throughout Philadelphia and the Northeast. Isabella Stewart Gardner's palatial Boston home, Fenway Court (now the Gardner Museum), owes much of its charm to a profusion of Mercer tiles.
In 1907, enriched by a second inheritance, Mercer built a home for himself. Fonthill was traditional in conception, but it was not built of traditional materials. Encouraged by his brother William, a sculptor who had been experimenting with cement, Mercer chose reinforced concrete as his primary building material. Frank Lloyd Wright would complete Unity Temple in Oak Park out of concrete the following year, but Mercer, who designed his house himself, used the new material differently—in a free-flowing and sculptural manner that recalls the Barcelona architect Antonio Gaudí. When Mercer, who personally oversaw the construction, completed his mansion—it took him four years—he followed it with a pottery works adjacent to the house, then turned his hand to building a museum to house his vast collection of tools and artifacts.
Doylestown is not far from where I live, and I decide to visit the Mercer Museum. It stands in the center of town. The building is a seven-story pile of gray concrete surmounted by clay-tiled towers, gables, and parapets. It resembles a baronial castle transplanted from the Transylvanian Alps. The unusual interior is dominated by a tall room rising to the roof and surrounded on all sides by stairs and galleries. This central space is crammed with an astonishing array of objects: high-back chairs suspended from the ceiling; rakes, hoes, and wagon wheels fixed to the walls; a wooden sleigh that floats through the air and almost crashes into a New Bedford whaleboat. The main floor contains carriages, wagons, and a cigar-store Indian standing next to a large apple press.
The guidebook informs me that there are fifty thousand objects in the museum. I had hoped to find a case with screwdrivers, but Mercer did not organize his collection according to simple categories. Instead, he created a series of small alcoves, each resembling a workshop dedicated to a different craft or occupation. I peer through the small-paned shop windows; the mullions, like everything else, are concrete. In the wheelwright's workshop I recognize a huge adze for routing axle holes; elsewhere, I glimpse a massive Commander maul. The watchmaker's shop contains several interesting miniature lathes powered by the same sort of bows that the Egyptians used to turn drills. In the woodworker's shop I see an assortment of wooden carpenter's braces as well as a giant five-foot-long plane for finishing floor planks. The room contains so many tools that the effect is dizzying—a vast nineteenth-century garage sale. Eventually, in the gunsmith's shop, I find a screwdriver. Like almost everything else, it is unlabeled.
It is December and I am the only visitor in the cavernous, cold building. Before leaving I drop into the museum library, run by the Bucks County Historical Society, to which Mercer presented the museum after its completion. Several people are working at long tables. It is the only part of the building that is heated; I will at least get warm and perhaps come across something useful. The card index has only two entries for screwdrivers, both books that I have read. There are several copies of Mercer's own book, as well as reprints of Moxon and other standard texts familiar to me.
A page from the tool catalog of William Marples & Sons, Sheffield, 1870.
Browsing through the stacks, I come across a book on nineteenth-century English tool manufacturers in Sheffield. This privately published book—its typed pages bound in a heavy leather cover—is relatively recent, but it is unlikely I would have found it elsewhere; it is one of only 750 copies printed. Inside are reproductions of pages from English tool manufacturers' catalogs. Sheffield, then the center of the British steel industry, produced probably the finest tools in the world. According to the author, Kenneth Roberts, the oldest surviving example of a Sheffield price list is dated 1828. There, among spokeshaves and squares, I find not one but a whole family of screwdrivers: three inches to fourteen inches long, in black or bright finishes, and in two patterns, Scotch (flat, tapered blades) and London (more elaborate, waisted blades). The prices vary from four shillings and sixpence to twenty-two shillings a dozen; evidently the list was for job-lot buyers. Later catalogs include illustrations of screwdrivers with flat, oval handles, just like the engraving in the Encyclopédie. What surprises me, however, is the terminology: sewing machine turnscrew, cabinet turnscrew, and a small pocket model, the Gent's Fancy Turnscrew. There is even a turnscrew bit, for driving screws with a carpenter's brace. There is no doubt about it. Salaman was right. Despite its absence from my dictionaries, turnscrew is a real word, perhaps an older word than screwdriver.
The Sheffield catalogs in Roberts's book demonstrate that by the early 1800s, the demand for screwdrivers was large enough to warrant factory production. The other evidence I had found suggests that the screwdriver appeared sometime in the previous century, perhaps in France. Turnscrew is a literal translation of the French word, and the paper trail runs out in 1723 with the tournevis entry in my Dictionnaire Général. I now have enough material to write a short essay for the New York Times, but I have scarcely solved the puzzle of the screwdriver.
## CHAPTER THREE
## Lock, Stock, and Barrel
THERE ARE TOOLS, such as the handsaw, that develop slowly and are refined over centuries. Others, such as the carpenter's brace, are adaptations of a new scientific principle. Then there are those inventions that appear seemingly out of the blue. The button, for example, a useful device that secures clothing against cold drafts, was unknown for most of mankind's history. The ancient Egyptians, Greeks, and Romans wore loose tunics, cloaks, and togas. Buttons were likewise absent in traditional dress throughout the Middle East, Africa, and South Asia. True, the climate in these places is mild, but northern dress was likewise buttonless. Eskimos and Vikings slipped their clothes over their heads and cinched them with belts and straps; Celts wrapped themselves in kilts; the Japanese used sashes to fasten their robes. The Romans did use buttons to ornament clothing, but the buttonhole eluded them. The ancient Chinese invented the toggle and loop, but never went on to the button and buttonhole, which are both simpler to make and more convenient to use. Then, suddenly, in the thirteenth century in northern Europe, the button appeared. Or, more precisely, the button and the buttonhole. The invention of this combination—so simple, yet so cunning—is a mystery. There was no scientific or technical breakthrough—buttons can easily be made from wood, horn, or bone; the buttonhole is merely a slit in the fabric. Yet the leap of imagination that this deceptively simple device required is impressive. Try to describe in words the odd flick-and-twist motion as you button and unbutton and you realize just how complicated it is. The other mystery of the button is the manner of its discovery. It is difficult to imagine the button evolving—it either exists or it doesn't. We don't know who invented the button and the buttonhole, but he—more likely she—was a genius.
Maybe the screwdriver, like the button, is a medieval invention. I examine a book of engravings and woodcuts by the sixteenth-century artist Albrecht Dürer. Dürer occasionally portrays tools. A woodcut of the Holy Family in Egypt has Joseph using an adze to hollow out a heavy plank. In a Crucifixion scene, a man turns a large auger to drill preparatory holes for the spikes while his mate wields a heavy hammer. The fullest depiction of tools is in the famous engraving Melancolia I. A winged female figure is surrounded by an assortment of woodworking tools: a pair of metal dividers, an open handsaw, iron pincers, a rule, a template, a claw hammer, and four wrought-iron nails. But no screwdriver. Melancolia I includes several magical and allegorical objects such as an alchemist's crucible, a millstone, and an hourglass, and art historians assume that the tools in Melancolia I were likewise chosen for their symbolic meanings. The hammer and four nails, for example, probably refer to the Crucifixion. Maybe the screwdriver simply lacked metaphorical weight.
The most famous technological treatise of the sixteenth century was Agostino Ramelli's Le diverse et artificiose machine (Various and ingenious machines), published in Paris in 1588. Ramelli was an Italian military engineer who had apprenticed with the Marquis of Marigano and moved to France to serve with the Catholic League in their war against the Huguenots. He had a colorful career. During the siege of La Rochelle he was wounded and captured, but escaped—or was exchanged—and a few months later successfully mined under a bastion and breached the fortification. His commander at La Rochelle was Henri d'Anjou, who became Henri III of France, and it was to the king that Ramelli dedicated his book. Capitano Ramelli, as he styled himself, was following in the footsteps of his celebrated countryman Leonardo da Vinci, and he was no less renowned; he is described by a French contemporary as "a true Daedalus as architect and the Archimedes of our age." The frontispiece of Ramelli's book shows a vigorous, bearded man holding a pair of dividers over a model of a fortification, his other, well-manicured hand resting on a steel cuirassier helmet. The author's portrait is flanked by allegorical figures symbolizing his two vocations: war and mathematics.
Ramelli's beautifully illustrated compilation of machines and technological devices was the most influential book of its kind. (Leonardo's notebooks, while celebrated today, were not published until several centuries after the author's death.) As is to be expected, the Capitano includes a number of siege engines, cunning pontoon bridges that unfold like accordions, scaling machines, and monstrous catapults. He also presents devices for clandestine break-ins: wrenches for tearing loose door bolts, giant clamps for forcing apart iron gratings and portcullises, and jacks for lifting doors off their hinges, "with great ease and little noise." The latter claim, at least, is doubtful, since there is no provision for keeping the massive door, once free of the hinge, from crashing to the ground.
The majority of the two hundred machines in his book are peaceful devices. Ramelli was fascinated by the problem of raising water and included a variety of waterwheels, pumps, and bucket conveyor belts. There are also domestic gadgets such as automatic fountains and hand-cranked machines for milling flour. The latter is important since it is the first known example of the use of rollers, rather than millstones. Ramelli's version of a revolving bookstand is particularly fascinating. Revolving bookstands were not unknown in Ramelli's day and were used by scholars consulting several heavy tomes in turn. While a conventional bookstand turned horizontally and held four books, Ramelli's six-foot-diameter bookwheel turned vertically, like a modern Ferris wheel, and could support no fewer than eight books. "This is a beautiful and ingenious machine, very useful and convenient for anyone who takes pleasure in study, especially those who are indisposed and tormented by gout," he points out with no false modesty. The bookwheel was a mechanical tour de force. To ensure that the open books remained at a constant angle while the wheel turned, he incorporated a complicated epicyclic gearing arrangement, a device that had previously been used only in astronomical clocks. Of course, gravity would have done the job equally well (as it does in a Ferris wheel), but the gearing system allowed Ramelli to demonstrate his considerable skill as a mathematician.
This splendid folly distracts me—I'm supposed to be looking for screwdrivers. As far as I can see, the heavy wooden bookwheel is held together with pegs. However, elsewhere in Ramelli's book, I do find screws. The iron legs of the hand-cranked flour mill are attached to a wooden base with slotted screws, one of which is shown partially unscrewed to reveal the threads. This is proof that screws—and presumably screwdrivers—were used more than a hundred years earlier than any of my previous sources had suggested.
Bookwheel, from Agostino Ramelli's Le diverse et artificiose machine, 1588.
Another celebrated medieval technical book is De Re Metallica. This treatise on mining and metallurgy was written by Georg Bauer, a Saxon scholar whose Latinized pen name was Georgius Agricola. Agricola, Germany's first mineralogist, laid the foundation for the systematic and scientific study of geology and mining. De Re Metallica, which appeared in 1556, shortly after his death, is heavily illustrated with woodcuts of mining and smelting machinery: pumps, mining hoists, and furnaces. Since many of the machines are made of wood, Agricola portrays a number of woodworking tools: axes and adzes for preparing heavy timber shoring; hammers and nails; mallets and chisels; and a long-handled auger for hollowing wooden logs into pipes.
He describes how to make the large bellows to be used for smelting iron. The woodcut illustrates the various components: the iron nozzle, the wooden boards, and the leather bellows. Ox hide is superior to horsehide, according to the author, who goes on to advise that "some people do not fix the hide to the bellows-boards and bows by iron nails, but by iron screws, screwed at the same time through strips laid over the hide." I read the passage twice. Yes, he definitely says iron screws, and there, nestled in the bottom left-hand corner of the engraving, is a neat drawing of a screw. The tapered, threaded body is topped by a flat, slotted head. Although the means of driving the screw are not shown, Agricola provides clear evidence of the use of screws as early as the middle of the sixteenth century.
Portable flour mill, from Agostino Ramelli's Le diverse et artificiose machine, 1588.
A technical work that predates both Agricola and Ramelli is the so-called Medieval Housebook. This handwritten manuscript, whose author and exact provenance are unknown, is thought to come from southern Germany. It has been described as a household manual for a knight's castle, a common genre at the time. In its present state the book consists of sixty-three parchment leaves, beautifully illustrated and covering a variety of subjects: jousting, hunting, warfare, courtship. Astrological horoscopes describe traits of people born under the sign of different planets: the regal Sun, amorous Venus, warlike Mars. Industrious Mercury is accompanied by a variety of craftsmen: an organ builder; a goldsmith, wearing eyeglasses and hammering out a beaker; and a clockmaker. I examine the drawing through a magnifying glass, helpfully provided by the Frick Collection in New York City, where a traveling exhibition of selected pages from the Housebook is on display. I'm hoping to find a screwdriver on the clockmaker's workbench, but no luck. The section on smelting includes a water-powered device for working bellows, but there is no indication that screws were used. Further on, several pages are devoted to the technology of war. I pore over each drawing in turn, under the watchful eye of an increasingly suspicious museum guard.
Among the intricate drawings of cannons, battle wagons, and scaling ladders, I find a collection of miscellaneous hardware: an auger, assorted manacles, and mysteriously shaped crowbars that the caption describes as tools for forcing apart iron gratings—ancestors of Ramelli's portcullis twisters. Although the Housebook drawing shows a wrench, there is no screwdriver. But there is something almost as good. Two of the devices—a leg iron and a pair of manacles—are fastened with slotted screws.
The exact date of the Housebook is unknown. Most scholars believe that it was written between 1475 and 1490, almost a century earlier than the books of Agricola and Ramelli, and more than three hundred years before the Encyclopédie. Since the author of the Housebook included a separate drawing of a screw, one might guess that screws were a novelty. Interestingly, the screws in the Housebook are used to join metal, not wood. Such screws must mate with threaded holes, so these fifteenth-century screws were made with a relatively high degree of precision.
I have not found a screwdriver, but I have found a very old screw. Surely slotted screws were used for something less specialized than attaching leg irons and manacles? I go back to Dürer. Although his religious and allegorical engravings rarely include mechanical devices, an exception is his last etching, made in 1518. The subject is a cannon. It is being towed through a pastoral countryside, the roofs of a peaceful village visible in the valley below. The contrast between the artillery piece and the idyllic landscape is dramatic. This is also a comment on the mechanization of war, for the scene includes a glum-looking group of Oriental warriors holding swords and pikes. Dürer renders the cannon, its wooden carriage, and the two-wheeled limber in great detail. However, the iron parts of the cannon, including a complicated elevating mechanism, are not attached to the wooden frame with screws but with heavy spikes.
Dürer's etching gives me an idea. Weapons have often been the source of technological invention. Radar and the jet engine, which both originated during the Second World War, are two modern examples. The most dramatic military innovation of the Renaissance was the gun. The first guns were bombards, short heavy mortars firing stone balls. Bombards were fixed to wooden platforms and were dragged from place to place only with great difficulty. Before the end of the fifteenth century, however, bell foundries cast bronze barrels, about eight feet long, that were light enough to be mounted on a wheeled carriage and were fully mobile. One of these innovative weapons is the subject of Dürer's etching.
Before casting full-size cannons, foundries experimented with small portable weapons. The oldest surviving example of such a "hand-cannon" is a one-foot-long bronze gun barrel, made in Sweden in the mid-1300s. The barrel is attached to a straight wooden stock that the gunner either pressed against his body with his elbow or rested atop his shoulder like a modern antitank gun. Italians called the new weapon arcobugio (literally, a hollow crossbow). The Spaniards, who were leaders in gun-making, called it arcabuz, whence the French and English arquebus.
Firing an arquebus was tricky. After loading the gun by the muzzle, the gunner had to balance the heavy weapon with one hand while holding a smoldering match to the touchhole or firing pan with the other. Even when a forked rest or tripod was used, it was difficult to aim properly. In addition, bringing one's hand close to the priming powder was dangerous since there was always the risk of a premature explosion. Groups of arquebusiers waving burning matches while pouring gunpowder on their priming pans were likely to cause as much damage to themselves as to the enemy.
A solution to the firing problem was developed in the early 1400s. A curved metal arm holding the match was attached to the stock. In the earliest versions, the gunner manually pivoted the arm, gradually moving the match to the touchhole. Eventually, the movement was accomplished by a spring-operated mechanism, the so-called matchlock. The arm holding the match was cocked back, and when a button was depressed, a spring brought it down to the pan. In a further refinement, pressure on a lever-shaped trigger—a feature adapted from the crossbow—slowly lowered the match into the pan. Now the gunner had both hands free to steady and aim the gun. The modern firearm had arrived—lock, stock, and barrel.
The arquebus quickly became popular. In 1471, the army of the duke of Burgundy counted 1,250 armored knights, 1,250 pikemen, 5,000 archers, and 1,250 arquebusiers. By 1527, in a French expeditionary force of eight hundred soldiers, more than half were arquebusiers. Gunners were common soldiers. Technological innovation often trickles down from the rich to the poor; firearms evolved in the opposite direction. The first arquebuses were disdained by the nobility as unwieldy, and too inaccurate for hunting. Only in the late 1500s did the gun become a gentleman's weapon.
I go to the arms and armor gallery of the Metropolitan Museum of Art in New York City to see these early firearms for myself. In a glass case I find a matchlock made in Italy in the 1570s. The gun is about three and a half feet long with an odd-shaped, curved wooden stock that resembles a field-hockey stick. This type of gun, known as a petronel, was developed by the French, who called it a poitrinal, since the stock was shaped to rest against the poitrine (chest). Petronels were short-lived—as a skeptical English soldier pointed out, "fewe or none could abide their recoyling"—and they were replaced by guns with so-called Spanish stocks, which rested against the shoulder.
A musketeer firing his matchlock, 1607.
The petronel in the Metropolitan is elaborately ornamented and was obviously intended for hunting. The steel barrel and lock are engraved, and the stock is inlaid with carved bone. As I look closely at the decorations, my eye is drawn to the lock. The slotted heads of two screws are plainly visible. The lock is screwed—definitely screwed—to the stock.
Screws were probably used instead of nails to ensure that the lock was not loosened by the vibration of successive detonations. This use must have happened early, certainly before the 1570s. Since there are no older matchlocks in the Metropolitan, I consult a well-known reference book, Pollard's History of Firearms. I find a detailed view of a matchlock in a drawing made in Nuremberg in 1505. The moving parts are fixed with rivets, but the mechanism itself is fastened to the stock with four screws, just like the petronel. In this exploded view the screws are shown in their entirety. They have round, slotted heads and threaded cores tapering to sharp points. The oldest depiction of a matchlock in Pollard's is from a fifteenth-century German manuscript. The stubby weapon resembles a modern sawn-off shotgun. The short barrel sits in a wooden stock whose slightly angled butt suggests that the principle of transforming some of the shock of recoil into vertical movement was beginning to be understood. The precise drawing shows the right side of the gun. The lock is similarly attached to the stock with two slotted screws. The manuscript is dated 1475, about the same period as the Medieval Housebook. Here, at last, is a widespread application of early screws.
View of matchlock, 1505.
During the 1500s, the matchlock was replaced by a new type of lock—the so-called wheel lock. The wheel, which was on a spring, was wound up, or "spanned." The key used to turn the wheel was called a spanner (which is what the English still call a wrench). When the trigger was pulled, the wheel turned rapidly against a piece of iron pyrites, producing a spark (the same principle as a modern cigarette lighter). The spark ignited the priming powder and the gun discharged. The piece of pyrites was held in a set of jaws that were tightened with small screws, and since it was necessary to regularly replace the worn pyrites, the gunner needed to have a screwdriver with him at all times. The solution was a combination tool: the end of the spanner handle was flattened to serve as a screwdriver. This must be the "arquebusier's screwdriver" mentioned in Diderot's Encyclopédie.
—
The matchlocks at the Metropolitan Museum are displayed in a small room that is part of a large area devoted to arms and armor. After examining the guns I decide to take a look at the armor. This is not research—I simply have fond boyhood memories of reading Ivanhoe and seeing the Knights of the Round Table at the movies. The centerpiece of the main gallery is a group of knights mounted on armored steeds. The armor, which was tinned to prevent rusting, is shiny. There are banners and colorful pennants, which give the display a jaunty, festive air; it is easy to forget that much of this is killing dress. The day I visit, the place is full of noisy, excited schoolchildren. I stop at a display case containing a utilitarian outfit, painted entirely black—not the Black Knight, just a cheap method of preventing rust. The beak-shaped helm has only a narrow slit for the eyes. "Neat!" the boy beside me exclaims to his companion. "It's just like Darth Vader."
The display is German armor from Dresden, dated between 1580 and 1590. This is slightly later than what is generally considered to have been the golden age of armor, which lasted from about 1450 to 1550. Contrary to the movies of my boyhood, King Arthur's knights, who lived in the sixth century, would have worn chain mail, not steel armor. Protective steel plates came into use only at the end of the thirteenth century. First the knees and shins were covered, then the arms, and by about 1400, the entire body was encased. The common method of connecting the steel plates was with iron, brass, or copper rivets. When a small amount of movement was required between two plates, the rivet was set in a slot instead of a hole. Removable pieces of armor were fastened with cotter pins, turning catches, and pivot hooks; major pieces, such as the breastplate and backplate, were buckled together with leather straps.
The Dresden suit is identified as jousting armor. Jousting, or tilting, originated in martial tournaments in which groups of mounted knights fought with lance, sword, and mace. By the sixteenth century, this rude free-for-all had evolved into a highly regulated sport. Two knights, each carrying a twelve-foot-long blunted wooden lance, rode at each other on either side of a low wooden barricade called the tilt. The aim was to unseat the opponent, have him shatter his lance, or score points by hitting different parts of the body. To protect the wearer, jousting armor was heavily reinforced and weighed more than a hundred pounds (field armor was lighter, weighing between forty and sixty pounds).
The black Dresden armor was for the scharfrennen, a particularly deadly German form of joust fought with sharpened lances and particularly popular with young men. Such combat required additional protection. The helm, called a rennhut, covered only the head and upper part of the face. The lower part of the face and the neck were protected by the renntartsche, a large molded plate that extended down to cover the left shoulder and was attached to the breastplate. A small shield, called a tilt targe, was fastened to the breastplate. Such "target" pieces were designed to fall off when struck; sometimes they were fitted with springs that caused them to fly dramatically into the air to the delight of the wildly applauding spectators.
Like most of the armor in the gallery, the steel plates of the Dresden suit are held together by rivets and buckled straps. Then I notice something: the renntartsche is screwed to the breastplate—the slotted heads, about half an inch in diameter, are plainly visible. Armorers, too, used screwdrivers! Since armor plate is relatively thin, these screws are probably mated with nuts, although I can't see them since they are hidden inside the suit. The Greenwich Armory outside London employed a dozen or more general armorers as well as a variety of specialists such as platers, millmen, helmsmiths, mail-makers, and locksmiths. It was probably the latter who fabricated the screws (medieval locks sometimes used threaded turning mechanisms).
We can be fairly sure how these screws and nuts were fabricated. In Mechanick Exercises, Moxon includes a section titled "The Making of Screws and Nuts," a process that could not have changed much since the Middle Ages. He describes how, after the head and shank are hammered out of a forged blank, the "screw-pin," that is, the thread, is cut with a die called a screw plate. The screw plate, made of tempered steel, has several threaded holes of different diameters. The blank is placed in a vise, and the screw plate is forced down hard and turned to cut the threads. (The corresponding nut is threaded with a tap, a tapered screw fitted with a handle.) "Screw the Nut in the Vise directly flat, that the hole may stand upright, and put the Screw-tap upright in the hole; then if your Screw-tap have a handle, turn it by the handle hard round in the Hole, so will the Screw-tap work it self into the Hole, and make Grooves in it to fit the Threds [sic] of the Screw-pin." Moxon's complicated instructions underline the combination of delicacy and brute strength that was needed to make a screw in this fashion.
Looking more closely at the Dresden armor, I see that the helm is attached to the backplate by large wing nuts. Since the highest points in a joust were accorded to a hit to the helm, special precautions had to be taken to protect the head. Field helms were close-fitting and worn over a coif of chain mail; the heavy jousting helm, on the other hand, did not touch the head. It was supported on the shoulders like a modern deep-sea diver's helmet and attached to the breastplate and backplate with leather straps to keep from getting knocked off. "In suits for the joust or tourney these adjustable fastenings could not always be depended upon," observes Charles Ffoulkes in a 1912 book on armor, "and the great helm . . . [was] often screwed on to the suit." Wing nuts, such as the ones on the Dresden armor, were a later refinement that allowed the exact angle of the helm to be closely adjusted. This was important. The so-called frog-mouth helm had a narrow, beaklike viewing slit, designed so that the knight could see out as he leaned forward in the saddle, riding toward his adversary. At the last minute, just before the moment of impact, he would straighten up and the lower part of the helm would protect his eyes from stray splinters. It required nerve: galloping down the list, aiming the heavy lance at one's opponent who was barely visible through the helm's shaking, narrow slot, then sudden darkness followed by the jarring crash of wood against steel.
Bracket for jousting helm and protective renntartsche, Dresden, sixteenth century.
Multipurpose armorer's tool, sixteenth century.
It is unclear exactly when screws were substituted for straps. Ffoulkes refers to a French military manual, written in 1446, that provides a detailed description of jousting armor. The text refers to most attachments as cloué (literally "nailed," as rivets were called arming nails), but in one place describes a piece as being rivez en dedens (fixed from the inside), which sounds like a screw and nut. I came across references to helms being screwed to breastplates as early as 1480. The oldest screw in the Metropolitan Museum is part of a steel breastplate that is identified as German or Austrian and dated 1480–90. If screws were used in the 1480s, that would make them the same age as the screws in the matchlock in Pollard's History of Firearms and the metal screws in the Housebook. Ffoulkes describes the heads of the screws as square or polygonal. However, all the screws I saw at the Metropolitan were slotted.
I look through Ffoulkes's chapter on "Tools, Appliances, Etc." According to the author, few armorer's tools have survived. He describes a display in the British Museum: "In the same case is a pair of armourer's pincers, which resemble the multum in parvo tools of today, for they include hammer, wire-cutter, nail-drawer, and turnscrew." He refers to a photograph. Excitedly, I turn to plate V. I had missed it earlier. Upon closer examination I can make out what looks like a pick at the end of one handle, and at the end of the other—a flat screwdriver blade. The caption beneath the photograph gives the date as the sixteenth century.
Another combination tool. I am disappointed that the oldest screwdriver resembles the kind of gimcrack household gadget that is sold by Hammacher Schlemmer. Although Ffoulkes calls this a turnscrew, like the screwdriver blade that was part of the arquebusier's spanner, it probably didn't have a special name. With so few screws, all that was needed was a part-time tool.
## CHAPTER FOUR
## The Biggest Little Invention
IN SEARCHING FOR the first screwdriver I have become interested in screws. When Agricola compared the screw to the nail as a way of constructing bellows, he observed that "there is no doubt that it the screw] surpasses it in excellence." In fact, the wrought-iron nail is a remarkable fastener. It bears little resemblance to the modern steel nail. The modern nail is round and pointed and forces itself between the wood fibers. Such nails are reasonably effective when driven into softwood (spruce, pine, fir), but will usually split hardwood (maple, birch, oak). Moreover, even in softwood the holding power of a round nail is weak, since it is kept in place only by the pressure of the fibers along two sides. The wrought-iron nail, on the other hand, is square or rectangular in cross-section with a hand-filed chisel point. The chisel point, driven across the grain, cuts through the wood fibers rather than forcing its way between them, just like a modern railroad spike. Such nails can be driven into the hardest wood without splitting it, and they are almost impossible to remove, as I discovered when I nailed a replica wrought-iron ship's nail into a board as an experiment.[I
Wrought-iron nails have limitations, however. If they are driven into a thin piece of wood, such as a door, their holding power is greatly reduced and their protruding ends must be clenched—bent over—to keep them fast. Wrought-iron nails are most effective—and easier to fabricate—when they are relatively large (at least an inch or two long). That is why the earliest screws replaced nails in small-scale applications such as fixing leather to a bellows board, or attaching a matchlock to a gunstock. Even a short screw has great holding power. Unlike a nail or a spike, a screw is not held by friction but by a mechanical bond: the interpenetration of the sharp spiral thread and the wood fibers. This bond is so strong that a well-set screw can be removed only by destroying the surrounding wood.
The problem with screws in the sixteenth century was that, compared to nails, they were expensive. A blacksmith could turn out nails relatively quickly. Taking a red-hot rod of forged iron, he squared, drew, and tapered the rod to a point, pushed the reheated nail through a heading tool, then with a heavy hammer formed the head. The whole procedure, which had been invented by the Romans and was still used in the 1800s (Thomas Jefferson's slaves produced nails this way at Monticello), took less than a minute, especially for an experienced "nailsmith." Making a screw was more complicated. A blank was forged, pointed, and headed, much like a nail, but round instead of square. Then a slot was cut into the head with a hacksaw. Finally, the thread was laboriously filed by hand.
Gunsmiths manufactured their own screws, just as armorers made their own bolts and wing nuts. What about clockmakers? Turret clocks appeared in Europe as early as the fourteenth century. The oldest clock of which we have detailed knowledge was built by an Italian, Giovanni De'Dondi. It is an astronomical clock of extraordinary complexity. The seven faces show the position of the ancient planets: the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn; in addition, one rotating dial indicates religious feast days, and another displays the number of daylight hours in the day. De'Dondi fashioned the bronze, brass, and copper parts by hand. It took him sixteen years to build the clock, which he finished in 1362. Although the original was destroyed by fire in the sixteenth century, the inventor left detailed instructions, and two working replicas were built in London in 1962. One of these now belongs to the National Museum of American History, and I catch up with it in Montreal, where it is part of a temporary exhibit. The exquisite seven-sided machine stands about four feet tall; the gearwheels are driven by suspended weights. I examine the mechanism. As far as I can see, all the connections are pegged mortises and tenons, a detail adapted from carpentry. The projecting tenons have slot-holes into which a wedge is driven. These wedges vary in size from tiny needlelike pins to one inch long. There must be several hundred such attachments, but I can't see a single screw.
According to Britten's Old Clocks and Watches and Their Makers, the standard work of horological history originally published in 1899, "screws were entirely unknown in clocks before 1550." Their introduction was a result of the demand for smaller and lighter domestic clocks, especially watches. According to Britten's, "Even the earliest watches generally possess at least one screw. These screws have dome-shaped heads and the slots are V-shaped. The thread is coarse and irregular."
—
By the mid-sixteenth century, applications for screws had grown to include miniature screws and bolts in watches, larger screws in guns, and heavy bolts in armor. Yet it was another two hundred years before demand grew enough that a screw industry developed. The Encyclopédie mentions that the region of Forez, near Lyon, specialized in screws, which were available in a variety of lengths—one-half inch to four or five inches. These screws were still so expensive that they were sold individually. According to the Encyclopédie, heads were either slotted or square.
In England, screw-making was concentrated in the Midlands. It was organized as a cottage industry. Forged-steel blanks with formed heads were made in large quantities by local blacksmiths and delivered to the so-called girder, who, with his family and an assistant or two, worked at home. The first step was to cut the slot, or "nick," into the head with a hacksaw. That was the easy part. Next the thread, or "worm," had to be filed by hand. Some girders used a spindle—a crude lathe—turning a crank with one hand and guiding a heavy cutter with the other, back and forth, back and forth. Whichever method was used, the work was slow and laborious, and since the worm was cut by eye, the result was a screw with imperfect, shallow threads. According to one contemporary observer, who had seen screw-girders at work, "The expensive and tedious character of these processes rendered it impossible for the screws to compete with nails, and consequently the sale was very small. The quality was also exceedingly bad, it being impossible to produce a well-cut thread by such means."
Both Moxon and the Encyclopédie mention that screws are used by locksmiths to fasten locks to doors. I also come across references to eighteenth-century carpenters using screws to attach hinges, particularly the novel garnet hinge. A garnet hinge resembles a |—, the vertical part being fastened to the doorjamb and the horizontal to the door. Garnet hinges, used with light cupboard doors and shutters, were screwed rather than nailed to the frame. Heavy doors, on the other hand, were hung on traditional strap hinges that extended the full width of the door and were nailed and clenched.
Strap and garnet hinges are still used today, but by far the most popular modern door hinge is the butt hinge, which is not mounted on the surface but mortised into the thick end—the butt—of the door. Butt hinges are aesthetically pleasing, being almost entirely hidden when the door is closed. They were used in France as early as the sixteenth century (butt hinges are illustrated by Ramelli), but were luxury objects, crafted by hand of brass or steel. In 1775, two Englishmen patented a design for mass-producing cast-iron butt hinges. Cast-iron butt hinges, cheaper than strap hinges, had one drawback: they could not be nailed. Nails worked themselves loose as the door was repeatedly opened and closed, and since the nails were in the butt of the door, they could not be clenched. Butt hinges had to be screwed.
By coincidence, at the very moment that butt hinges were being popularized, a technique for manufacturing good-quality, inexpensive screws was being perfected. Years earlier, Job and William Wyatt, two brothers from Staffordshire in the English Midlands, had set out to improve screw-making. In 1760, they patented a "method of cutting screws of iron commonly called wood-screws in a better manner than had been heretofore practiced." Their method involved three separate operations. First, while the forged blank of wrought iron was held in a rotating spindle, the countersunk head was shaped with a file. Next, with the spindle stopped, a revolving saw-blade cut a slot into the head. Finally, the blank was placed in a second spindle and the thread was cut. This was the most original part of the process. Instead of being guided by hand, the cutter was connected to a pin that tracked a lead screw. In other words, the operation was automatic. Now, instead of taking several minutes, a girder could turn out a screw—a much better screw—in six or seven seconds.
It took the Wyatt brothers sixteen years to raise the capital required to convert a disused water corn-mill north of Birmingham into the world's first screw factory. Then, for unexplained reasons, their enterprise failed. Maybe the brothers were poor businessmen, or maybe they were simply ahead of their time. A few years later, the factory's new owners, capitalizing on the new demand for screws created by the popularity of butt hinges, turned screw manufacturing into a phenomenal success. Their thirty employees produced sixteen thousand screws a day.
Machine-made screws were not simply produced more quickly, they were much better screws. Better and cheaper. In 1800, British screws cost less than tuppence a dozen. Eventually, steam power replaced waterpower in the screw factories, and a series of improvements further refined the manufacturing process. Over the next fifty years, the price dropped by almost half; in the following two decades, it dropped by half again. Inexpensive screws found a ready market. They proved useful not only for fastening butt hinges but for any application where pieces of thin wood needed to be firmly attached, which included boatbuilding, furniture-making, cabinetwork, and coachwork. Demand increased and production soared. British screw factories, which had annually produced less than one hundred thousand gross in 1800, sixty years later produced almost 7 million gross.
—
Take a close look at a modern screw. It is a remarkable little object. The thread begins at a gimlet point, sharp as a pin. This point gently tapers into the body of the screw, whose core is cylindrical. At the top, the core tapers into a smooth shank, the thread running out to nothing. The running-out is important since an abrupt termination of the thread would weaken the screw.
The first factory-made screws were not like this at all. For one thing, although handmade screws were pointed, manufactured screws had blunt ends and were not self-starting—it was always necessary first to drill a lead hole. The problem lay in the manufacturing process. Blunt screws could not simply be filed to a point—the thread itself had to come to a point, too. But lathes were incapable of cutting a tapering thread. Screw manufacturers tried angling the cutters, which produced screws that tapered along their entire length. Such screws had poor holding power, however, and carpenters refused to use them. What was needed was a machine that could cut a continuous thread in the body of the screw (a cylinder) and also in the gimlet point (a cone).
An inventive American mechanic found the solution. The first American screw factories had been established in Rhode Island in 1810, using adapted English machines. Providence became the center of the American screw industry, which by the mid-1830s was experiencing a boom in demand for its products. Beginning in 1837, a series of patents addressed the problem of manufacturing gimlet-pointed screws, but it took more than a decade of trial and error to get it right. In 1842, Cullen Whipple, a mechanic from Providence who worked for the New England Screw Company, invented a method of manufacturing screws on a machine that was entirely automatic. Seven years later he made a breakthrough and successfully patented a method of producing pointed screws. A slightly different technique was devised by Thomas J. Sloan, whose patent became the mainstay of the giant American Screw Company. Another New Englander, Charles D. Rogers, solved the problem of tapering the threaded core into the smooth shank. Such advances put American screw manufacturers firmly in the lead, and by the turn of the century, when the screw had achieved its final form, American methods of production dominated the globe.
—
Ever since the fifteenth century, screws had had either square or octagonal heads, or slots. The former were turned by a wrench, the latter by a screwdriver. There is no mystery as to the origin of the slot. A square head had to be accurate to fit the wrench; a slot was a shape that could be roughly filed or cut by hand. Screws with slotted heads could also be countersunk so they would not protrude beyond the surface—which was necessary to attach butt hinges. Once countersunk screws came into common use in the early 1800s, slotted heads—and flat-bladed screwdrivers—became standard. So, even as screws were entirely made by machine, the traditional slot remained. Yet slotted screws have several drawbacks. It is easy to "cam out," that is, to push the screwdriver out of the slot; the result is often damage to the material that is being fastened or injury to one's fingers—or both. The slot offers a tenuous purchase on the screw, and it is not uncommon to strip the slot when trying to tighten a new screw or loosen an old one. Finally, there are awkward situations—balancing on a stepladder, for example, or working in confined quarters—when one has to drive the screw with one hand. This is almost impossible to do with a slotted screw. The screw wobbles, the screwdriver slips, the screw falls to the ground and rolls away, the handyman curses—not for the first time—the inventor of this maddening device.
American screw manufacturers were well aware of these shortcomings. Between 1860 and 1890, there was a flurry of patents for magnetic screwdrivers, screw-holding gadgets, slots that did not extend across the face of the screw, double slots, and a variety of square, triangular, and hexagonal sockets or recesses. The latter held the most promise. Replacing the slot by a socket held the screwdriver snugly and prevented cam-out. The difficulty—once more—lay in manufacturing. Screw heads are formed by mechanically stamping a cold steel rod; punching a socket sufficiently deep to hold the screwdriver tended to either weaken the screw or deform the head.
The solution was discovered by a twenty-seven-year-old Canadian, Peter L. Robertson. Robertson was a so-called high-pitch man for a Philadelphia tool company, a traveling salesman who plied his wares on street corners and at country fairs in eastern Canada. He spent his spare time in his workshop, dabbling in mechanical inventions. He invented and promoted "Robertson's 20th Century Wrench-Brace," a combination tool that could be used as a brace, a monkey wrench, a screwdriver, a bench vise, and a rivet maker. He vainly patented an improved corkscrew, a new type of cuff links, even a better mousetrap. Then, in 1907, he received a patent for a socket-head screw.
Peter L. Robertson's 1907 patent for a socket-head screw.
Robertson later said that he got the idea for the socket head while demonstrating a spring-loaded screwdriver to a group of sidewalk gawkers in Montreal—the blade slipped out of the slot and injured his hand. The secret of his invention was the exact shape of the recess, which was square with chamfered edges, slightly tapering sides, and a pyramidal bottom. "It was early discovered that by the use of this form of punch, constructed with the exact angles indicated, cold metal would flow to the sides, and not be driven ahead of the tools, resulting beneficially in knitting the atoms into greater strength, and also assisting in the work of lateral extension, and without a waste or cutting away of any of the metal so treated, as is the case in the manufacture of the ordinary slotted head screw," he rather grandly explained.
An enthusiastic promoter, Robertson found financial backers, talked a small Ontario town, Milton, into giving him a tax-free loan and other concessions, and established his own screw factory. "The big fortunes are in the small inventions," he trumpeted to prospective investors. "This is considered by many as the biggest little invention of the 20th century so far." In truth, the square socket really was a big improvement. The special square-headed screwdriver fit snuggly—Robertson claimed an accuracy within one one-thousandth of an inch—and never cammed out. Craftsmen, especially furniture-makers and boatbuilders, appreciated the convenience of screws that were self-centering and could be driven with one hand. Industry liked socket-head screws, too, since they reduced product damage and speeded up production. The Fisher Body Company, which made wood bodies in Canada for Ford cars, became a large Robertson customer; so did the new Ford Model T plant in Windsor, Ontario, which soon accounted for a third of Robertson's output. Within five years of starting, Robertson built his own wire-drawing plant and powerhouse and employed seventy-five workers.
In 1913, Robertson decided to expand his business outside Canada. His father had been a Scottish immigrant, so Robertson set his sights on Britain. He established an independent English company to serve as a base for exporting to Germany and Russia. The venture was not a success. He was thwarted by a combination of undercapitalization, the First World War, the defeat of Germany, and the Russian Revolution. Moreover, it proved difficult to run businesses on two continents. After seven years, unhappy English shareholders replaced Robertson as managing director. The English company struggled along until it was liquidated in 1926. Meanwhile, Robertson turned to the United States. Negotiations with a large screw manufacturer in Buffalo broke down after it became clear that Robertson was unwilling to share control over production decisions. Henry Ford was interested, since his Canadian plants were reputedly saving as much as $2.60 per car using Robertson screws. However, Ford, too, wanted a measure of control that the stubborn Robertson was unwilling to grant. They met but no deal was struck. It was Robertson's last attempt to export his product. A lifelong bachelor, he spent the rest of his life in Milton, a big fish in a decidedly small pond.
Meanwhile, American automobile manufacturers followed Ford's lead and stuck to slotted screws. Yet the success of the new Robertson screw did not go unnoticed. In 1936 alone, there were more than twenty American patents for improved screws and screwdrivers. Several of these were granted to Henry F. Phillips, a forty-six-year-old businessman from Portland, Oregon. Like Robertson, Phillips had been a traveling salesman. He was also a promoter of new inventions, and acquired patents from a Portland inventor, John P. Thompson, for a socket screw. Thompson's socket was too deep to be practicable, but Phillips incorporated its distinctive shape—a cruciform—into an improved design of his own. Like Robertson, Phillips claimed that the socket was "particularly adapted for firm engagement with a correspondingly shaped driving tool or screwdriver, and in such a way that there will be no tendency of the driver to cam out of the recess." Unlike Robertson, however, Phillips did not start his own company but planned to license his patent to screw manufacturers.
All the major screw companies turned him down. "The manufacture and marketing of these articles do not promise sufficient commercial success" was a typical response. Phillips did not give up. Several years later a newly appointed president of the giant American Screw Company, which had prospered on the basis of Sloan's patent for manufacturing pointed screws, agreed to undertake the industrial development of the innovative socket screw. In his patents, Phillips emphasized that the screw was particularly suited to power-driven operations, which at the time chiefly meant automobile assembly lines. The American Screw Company convinced General Motors to test the new screw; it was used first in the 1936 Cadillac. The trial proved so effective that within two years all automobile companies save one had switched to socket screws, and by 1939 most screw manufacturers produced what were now called Phillips screws.
The Phillips screw has many of the same benefits as the Robertson screw (and the added advantage that it can be driven with a conventional screwdriver if necessary). "We estimate that our operators save between 30 and 60 percent of their time by using Phillips screws," wrote a satisfied builder of boats and gliders. "Our men claim they can accomplish at least 75 percent more work than with the old-fashioned type," maintained a manufacturer of garden furniture. Phillips screws—and the familiar cross-tipped screwdrivers—were now everywhere. The First World War had stymied Robertson; the Second World War ensured that the Phillips screw became an industry standard as it was widely adopted by wartime manufacturers. By the mid-1960s, when Phillips's patents expired, there were more than 160 domestic, and 80 foreign licensees.
The Phillips screw became the international socket screw; the Robertson screw is used only in Canada and by a select number of American woodworkers.II A few years ago, Consumer Reports tested Robertson and Phillips screwdrivers. "After driving hundreds of screws by hand and with a cordless drill fitted with a Robertson tip, we're convinced. Compared with slotted and Phillips-head screwdrivers, the Robertson worked faster, with less cam-out." The explanation is simple. Although Phillips designed his screw to have "firm engagement" with the screwdriver, in fact a cruciform recess is a less perfect fit than a square socket. Paradoxically, this very quality is what attracted automobile manufacturers to the Phillips screw. The point of an automated driver turning the screw with increasing force popped out of the recess when the screw was fully set, preventing overscrewing. Thus, a certain degree of cam-out was incorporated into the design from the beginning. However, what worked on the assembly line has bedeviled handymen ever since. Phillips screws are notorious for slippage, cam-out, and stripped sockets (especially if the screw or the screwdriver are improperly made). Here I must confess myself to be a confirmed Robertson user. The square-headed screwdriver sits snugly in the socket: you can shake a Robertson screwdriver, and the screw on the end will not fall off; drive a Robertson screw with a power drill, and the fully set screw simply stops the drill dead; no matter how old, rusty, or painted over, a Robertson screw can always be unscrewed. The "biggest little invention of the twentieth century"? Why not.
* * *
I. At the beginning of the nineteenth century, handmade nails were replaced by cut nails, stamped out of sheets of wrought iron (later steel), with a similar rectangular cross-section. Cut nails are sharpened by hand with a file.
II. Starting in the 1950s, Robertson screws began to be used by some American furniture manufacturers, by the mobile-home industry, and eventually by a growing number of craftsmen and hobbyists. The Robertson company itself was purchased by an American conglomerate in 1968.
## CHAPTER FIVE
## Delicate Adjustments
IN READING ABOUT the Wyatt brothers' factory in Staffordshire, I had been struck by the statement that their screw-making machines were operated by children. During the eighteenth century, children commonly worked in coal mines, workshops, and factories, but were usually given only menial tasks. Even a machine as simple as a screw girder's spindle required an experienced—not to say strong—operator. The Wyatt machines were obviously different. I had stumbled on a landmark of industrialization. At a remarkably early date—the industrial revolution would not get fully under way for another hundred years—the Wyatt brothers not only pioneered the use of multipurpose machines to achieve mass production, they were the first to put into place the guiding principle of industrialization. Their factory was the earliest example of an industrial process designed specifically to shift control over the quality of what was being produced from the skilled artisan to the machine itself.
The screw girder's spindle and the Wyatt brothers' screw-making machines are both examples of simple turning-lathes. In a lathe, the blank, or workpiece, is rotated around an axis, somewhat like a potter's wheel. However, while a potter creates a shape by building up clay, the turner removes material. As the workpiece turns, a sharp cutter is applied to the surface and, depending on the desired shape, removes inequalities until every part is equidistant from the axis. The lathe is an ancient tool that appears to have been invented in Europe, since the earliest surviving pieces of lathe work are an eighth-century B.C. Etruscan bowl, and a sixth-century B.C. bowl found in Upper Bavaria. Although these wooden objects were definitely turned, nothing is known of the lathes themselves. Turning technology eventually spread to the rest of the Mediterranean world, including Egypt, where the oldest depiction of a lathe, dating from the third century B.C., has been found in a bas-relief on a grave wall. The piece being turned, which appears to be a furniture leg, is held vertically. The turner's cutting tool resembles a chisel; his assistant rotates the piece by pulling a cord looped around the rotating axle, or mandrel. Since the workpiece rotates in alternate directions, the turner cuts only on every other turn.
The Egyptian bas-relief shows the turner and his assistant kneeling on the ground. It reminds me of my first visit to India, when I saw a carpenter at work squatting on the floor. Just as the world is divided into those who wrap and those who button up, or those who eat with their fingers and those who eat with utensils, it is divided into craftsmen who work kneeling, squatting, or sitting on the ground, and those who work erect—or sitting—at a bench. The ancient Egyptians belonged to the former category; the Romans, to the latter. Since the Romans invented the plane, they needed a flat surface to which the workpiece could be fastened, and the result was the first carpenter's bench.
Although Europeans in the Middle Ages often relaxed by sitting on cushions on the floor in the Oriental manner, they worked erect. This habit probably prompted the thirteenth-century European invention of the so-called pole lathe. The turner works standing up at a pole lathe. The workpiece rotates not vertically but horizontally. A cord is looped around the mandrel with one end attached to a hinged treadle, and the other fastened to a flexible pole, resembling a bowed fishing rod, that keeps the cord taut. The turner, alternately pressing and releasing the treadle with his foot, now has both hands free to guide the long-handled cutter, which he braces under his arm or over his shoulder for added stability. Like the Egyptian lathe, the pole lathe turns back and forth.
Screw-cutting lathe, from The Medieval Housebook of Wolfegg Castle, c. 1475–90.
The simple pole lathe was used by wood turners for a long time—working examples survived in England until the early 1900s. For turning metal, however, a more effective machine was required. Here the screw again plays a vital role, for the ancestor of the modern lathe is in fact a machine for cutting screws. It was invented almost three hundred years before the Wyatt brothers' screw-making lathes and appears in the Medieval Housebook, the fifteenth-century manuscript that I had consulted in the Frick Collection. The beautiful drawing is precise. The lathe, a radical departure from the pole lathe, consists of a heavy frame mounted on a solid workbench. The blank is held horizontally between two adjustable supports and rotated by turning a hand crank. One end of the blank is attached to a lead screw. As the blank turns, the lead screw advances through a threaded hole in one of the supports and pushes the blank through a box containing a sharp cutter that incises the thread. The operator has only to set up the blank in the jig, wedge the threaded support and the cutter-box in place, adjust the depth of the cutter, and turn the crank.
The Housebook lathe is made of wood, but it is a true machine tool; that is, it is a tool in which the machine—not the craftsman—controls the cutter. It anticipates many features of the modern bench lathe: the two supports (today called a headstock and a tailstock); the frame (ancestor of the modern slide-rest) that allows flexibility in the location of cutter-box and stocks; a continuous drive that can be connected by a belt drive to an external power source such as a waterwheel; a rotating lead screw that advances the blank by tiny increments; a design that integrates the lathe with the workbench; and heavy construction that assures rigidity and a relatively high degree of precision.
The drawing of the lathe appears on the same manuscript page of the Medieval Housebook as the manacles, wrenching tools, and slotted screws. The slotted screws, which are tapered, were obviously filed by hand; the lathe was used to turn the long wrought-iron screws that are part of the wrenching tools. It is several weeks since I visited the Frick, but I still have an illustrated catalog from the exhibition, and I examine the drawing of the lathe closely, trying to understand how it works. The pointed cutter, which must have been tempered steel, is threaded to enable the operator to adjust the depth of the cut. It would have taken many passes to cut a thread into a hard wrought-iron rod. After each pass, the workpiece was retracted, the cutter was tightened to cut a deeper groove, and the operation was repeated. A lengthy process, but one that probably produced a reasonably accurate screw.
The drawing of the lathe includes a short-handled tool lying on the workbench. At first I assume that it is some sort of chisel or gouge. But as the exact functioning of the lathe becomes clearer, it is obvious that a chisel plays no part in the process. The author of the Medieval Housebook is thorough and his drawings do not usually contain extraneous information. While beautifully composed, these are technical documents that carefully describe how the various machines work, and exactly what tools are needed to operate them. A view of a spinning wheel, for example, includes a couple of empty spools. So what is the function of the mysterious tool? One day, while I am puzzling over the drawing again, I realize that the blunt end is exactly the same size as the slot in the head of the cutter. Of course. It's not a chisel, it's used to adjust the cutter. It's a screwdriver.
Eureka! I've found it. The first screwdriver. No improvised gadget but a remarkably refined tool, complete with a pear-shaped wooden handle to give a good grip, and what appears to be a metal ferrule where the metal blade meets the handle. Since the Housebook was written during the last quarter of the fifteenth century, there is no doubt that a full-fledged screwdriver existed three hundred years before the tool portrayed in the Encyclopédie. This confirms what I had suspected: the screwdriver and the screw were invented at about the same time. My guess about fifteenth-century armorers and gunsmiths was not far off the mark either. The Housebook lathe is illustrated in a chapter devoted to the technology of war, so it is likely that screwdrivers appeared first in military workshops, though perhaps not in France, as I had assumed, but in Germany.I
—
For some reason, the potential of the Housebook lathe was not immediately recognized. Perhaps the unknown inventor did not publicize his lathe; as far as we know, the Housebook existed in only one copy, and medieval craftsmen were often possessive about their work. Yet it appears that Leonardo da Vinci, at least, was aware of the innovative lathe, for in the early 1500s he designed a number of screw-making machines, one of which bears a striking resemblance to the earlier machine. Characteristically, Leonardo made improvements. Instead of advancing the blank through the cutter, he made the cutter move along the rotating blank, as it does in a modern lathe. Further, by using different interchangeable gears (four are shown in his sketch), he could make the cutter advance at different rates. Since the blank was turning at a constant rate, if the cutter moved more slowly, the pitch (the distance between the threads) of the resulting screw was smaller; if the cutter moved more quickly, the pitch was larger. Thus, the same machine could make screws with four different pitches. As with so many of Leonardo's inventions, it is unclear if this remarkable machine was actually built.
Leonardo da Vinci's screw-cutting machine, c. 1500.
Jacques Besson's screw-cutting machine, 1579.
Although Agostino Ramelli worked in France, Leonardo's actual successor as engineer to the French court was Jacques Besson, who designed several screw-cutting lathes. Besson's lathes were extremely elaborate, turned not by means of a crank but by pulling on counterweighted cords. This produced the old-fashioned alternating rotation and also resulted in slippage and a loss of power. But efficiency was not uppermost in Besson's mind, for his machines were not intended for industrial workers but for hobbyists. Turning had become the gentleman's equivalent of needlepoint and remained in vogue as a pastime until the end of the eighteenth century. "It is an established fact that in present-day Europe this art is the most serious occupation of people of intelligence and merit," wrote Fr. Charles Plumier, who published the first treatise on the lathe, L'art de tourner, in 1701, "and, between amusements and reasonable pleasures, the one most highly regarded by those who seek in some honest exercise the means of avoiding those faults caused by excessive idleness." Hobbyists turned a variety of materials, not only wood, but also horn, copper, silver, and gold. Although the products of their labors were purely decorative, they took their machines seriously. The lathes could be simple bench models driven by a treadle, or complex machines with cams and other devices for making elaborate forms, including ornamental screws. The so-called guilloching lathe was capable of tracing complex intertwining curves onto flat disks such as watch cases and medallions. Louis XVI owned a guilloching lathe equipped with a mahogany bench, a gilded iron regulating device, and a tool-holding carriage of gilt bronze inset with the royal coat of arms.
Aristocrats used lathes to fill their idle hours, but for others, precision lathework was a purposeful occupation. In 1762, a London instrument-maker named Jesse Ramsden began a project that would revolutionize the lathe. Ramsden, born in Yorkshire in 1735, had originally been apprenticed to a cloth-maker. When he was twenty-three, he unexpectedly quit his trade and went to London to work for a maker of mathematical instruments. Four years later, he opened his own business. Now, to make his mark, he set out to solve a problem that plagued instrument-making: graduated scales. Linear scales, subdivided into standard measures, were a key ingredient of sextants, theodolites, and instruments used in astronomical observation. Graduated scales were traditionally made by hand and hence lacked accuracy. Ramsden designed a scale-dividing machine that was capable of engraving scales with great precision.
The machine incorporated long, fine-threaded regulating screws of microscopic accuracy. Regulating screws, the refined relatives of ordinary screws, convert rotation into minute horizontal movement by means of tracking pins or nuts. To ensure accuracy, the threads must meet stringent requirements: pitch must be constant; the cores must be exactly parallel and concentric; and the friction against the adjusting nut must be minimal but steady. In other words, these screws must be perfect.
Since screws of such precision were not available, Ramsden set out to make them himself. It was a daunting problem: how to produce a perfect screw using a lathe with an imperfect lead screw. Patiently, he produced a succession of screws of increasingly greater accuracy. In the process, he made important improvements to the lathe. At a time when most instrument-makers still used wooden pole lathes, he built bench lathes entirely of steel. He invented a triangular slide bar, which gave more accuracy, and he was also the first to use diamond-tipped cutting tools. He finally was able to produce a screw with a claimed accuracy of one four-thousandth of an inch. In all, it took him eleven years to build his dividing machine.
Jesse Ramsden's precision screw-cutting lathe, 1777.
Ramsden's achievement had enormous implications. Accurate regulating screws were incorporated into a variety of precision instruments and opened up new worlds to science as they facilitated the work not only of astronomers, but also of physicists, who depended on accurate regulating screws in microscopes. Other new worlds were opened up, too. The navigation instrument most affected by Ramsden's work was the sextant. A sextant incorporates an arclike graduated scale that spans sixty degrees (one-sixth of a circle, whence the Latin root sextus), a movable radial arm, and a fixed telescope. The navigator "shooting the sun" lines up the horizon in the telescope, then adjusts a mirror attached to the radial arm until he sees a reflection of the sun. The angle between the mirror and the telescope, which is read off the graduated scale, corresponds to the angle of the sun above the horizon. With this information, and the aid of published tables, the exact latitude can be computed. Thanks to Ramsden's scale-dividing machine, it was possible to know a ship's position to within ten seconds of latitude, or about a thousand feet. Such accuracy facilitated the feats of navigation and great voyages of discovery of explorers like Captain Cook.
—
Ramsden was working on his screw-cutting lathes in London at the same time the Wyatt brothers were organizing their screw factory in Staffordshire. The scientific instrument-maker's precision machines and the crude factory lathes both used regulating screws, but they existed in two different realms. These realms were soon to meet, thanks to the inventive genius of Henry Maudslay. Maudslay was born in 1771 in humble circumstances and was apprenticed to a blacksmith in the Royal Arsenal at Woolwich, near London. Unusually gifted as a metalworker, Maudslay came to the attention of Joseph Bramah, a prominent manufacturer and inventor. Bramah was looking for someone to make a prototype of his latest invention, an unpickable bank lock. The design, which incorporated numerous tumblers, was so complicated that it had confounded his own experienced craftsmen. Maudslay, then only eighteen, not only successfully built the prototype, but also designed and built the tools and machines needed to commercialize its production.II
The burly young blacksmith was a mechanical prodigy. Just as some people have a natural aptitude for chess or playing the violin, Maudslay could shape metal with a dexterity and precision that amazed his contemporaries. Moreover, he was able to intuit solutions to mechanical problems. For example, while he was building a lathe for Bramah, he invented the slide rest, a perfectly straight bar that supported a movable tool holder. Although a similar device had been suggested by Leonardo, it had never been implemented. The importance of the slide rest cannot be understated. In previous lathes, the turner guided the cutter by hand. The slide rest allowed the cutting tool to move smoothly and precisely along the length of the revolving workpiece. At first the device was greeted with scorn and nicknamed Maudslay's Go-Cart, but it proved so successful that it was soon widely copied (Maudslay rarely patented his inventions).
Henry Maudslay's first screw-cutting lathe, c. 1797.
After eight years with Bramah, and having risen to the position of foreman, Maudslay struck out on his own. While filling orders for customers—his first commission was a metal easel for an artist—he continued to tinker with precision lathes. His first breakthrough—in 1797—was a lathe for cutting long screws that incorporated a three-foot-long regulating screw. In a later version, following once more in Leonardo's footsteps, he added interchangeable gears to produce screws of different diameter and pitch.
Thanks to a precision regulating screw displayed in his shopwindow, Maudslay met an extraordinary Frenchman, Marc Isambard Brunel. A royalist, Brunel had fled the French Revolution for America, had worked in New York as an engineer and architect, and was now settled in London. A prolific inventor—and ex–naval officer—he had a plan for manufacturing wooden ships' pulleys for the British navy. He needed someone who could build the prototype machines to demonstrate the practicality of his process. With Maudslay's help, Brunel won the contract. The factory in Portsmouth, with forty-four of Maudslay's machines, took six years to build. It was the world's first example of a fully mechanized production line. Ten men produced 160,000 pulleys a year, the navy's entire annual requirement.
Maudslay and Brunel collaborated on another venture. In 1825, Brunel received the commission to build a twelve-hundred-foot tunnel under the Thames River. Previously, tunnels had been built with temporary timber shoring, but Brunel invented an extendable, waterproof, cast-iron shield that moved ahead of the construction as the excavation progressed. Maudslay built the device. His workshop also produced a variety of specialized machines—for printing, pressing, and minting. He invented a machine for punching holes in boilerplate (an operation previously carried out by hand) that greatly speeded up riveting. He was best known for his pioneering marine steam engines, with which he equipped at least forty vessels during his lifetime. When Isambard Kingdom Brunel, Marc's son and likewise an engineer, built the Great Western, the first steamship to cross the Atlantic, it was Maudslay's firm, now run by his son, that built the 750-horsepower engine, the largest in the world.
The key to the success of Maudslay's workshop was the precision lathe. The Housebook lathe had incorporated an early version of the slide rest; Leonardo had invented the moving cutter and interchangeable gears; and Plumier had described all-metal lathes. The eighteenth century saw many improvements to the lathe. In 1710, a Swede built a lathe for accurate cutting of iron screws; fifty years later, a Frenchman completed an industrial lathe with a traversing carriage; around 1796, a Rhode Island mechanic built an advanced lathe for screw-cutting; and, of course, Ramsden provided a striking example of precision screw-cutting. Yet it was Maudslay who synthesized all these features into a lathe capable of precision work on a large scale. In the process he produced the mother tool of the industrial age.
The heart of Maudslay's lathe was an extremely accurate regulating screw. He built a machine that cut threads of any pitch into soft metals such as tin and brass, then used these lead screws to make regulating screws of hard steel. "This beautiful and truly original contrivance became, in the hands of the inventor, the parent of a vast progeny of perfect screws, whose descendants, whether legitimate or not, are to be found in every workshop throughout the world, wherever first-class machinery is constructed," wrote a contemporary. It is important to understand the enormous impact of precision machine tools. It was not merely a question of supplanting manual labor. Steam engines, for example, simply could not have been built by hand—cylinders and piston rods required completely new standards of perfection. Precision opened the door to a mechanical world.
On Sundays, Maudslay would tour his quiet workshops to examine the work in progress. Chalk in hand, he would jot down his comments directly on his workmen's benches. He particularly singled out examples of mechanical accuracy—or its lack. The ideal of precision was perhaps Maudslay's greatest invention. He fabricated a regulating screw used in the manufacture of scientific instruments that was five feet long and two inches in diameter and was cut with fifty threads to the inch. He built himself a micrometer with a sixteen-inch-long screw that could measure to one ten-thousandth of an inch. It was used as the ultimate dimensioning standard by Maudslay's employees, who nicknamed it the Lord Chancellor. He provided each of his workers with a perfectly flat steel plate so that work in progress could periodically be placed on it to check if it was true. According to one of his assistants, these plates, which were filed and scraped by hand, were so smooth that "when placed over each other they would float upon the thin stratum of air between them until dislodged by time and pressure. When they adhered closely to each other, they could only be separated by sliding off each other."
Maudslay also championed uniformity in screws. Surprisingly, this was a radical idea. Previously, each nut and bolt had been fabricated as a unique matching pair. "Any intermixture that occurred between them led to endless trouble and expense, as well as inefficiency and confusion," observed one of his employees, "especially when parts of complex machines had to be taken to pieces for repairs." Maudslay adopted standardized taps and dies in his workshop, so that all nuts and bolts were made in a limited number of sizes. Now any nut would fit any bolt of the same size. This example inspired his pupil Joseph Whitworth, who in 1841 proposed a national standard for screw threads that eventually was adopted by all British manufacturers.iii
Whitworth was Maudslay's successor as Britain's great mechanical innovator. Unlike Maudslay, though, he was a manufacturer who also built specialized machine tools to order. The machine tools that came out of his factory were known throughout the world. They were versatile, dependable, and reasonably priced—and, incidentally, quite beautiful. It took a mechanical genius and a gifted craftsman like Maudslay to build the first precision lathe. Thanks to the machines that came from Whitworth's Manchester works, any well-equipped workshop could routinely achieve similar accuracy. The high standards that Maudslay had set for himself had become universal.
Maudslay died in 1831. He was interred in a cast-iron tomb of his own design. The inscription described him as "eminently distinguished as an engineer for mathematical accuracy and beauty of construction." True enough, but a more moving epitaph was provided by one of his old workmen: "It was a pleasure to see him handle a tool of any kind, but he was quite splendid with an eighteen-inch file."
* * *
I. The old German word for screwdriver is schraubendreher (screw-turner), which originally meant the craft of turning screws, but came to refer to the tool itself.
II. The prototype lock, prominently displayed in Bramah's shopwindow, remained unpicked for more than fifty years. It was finally cracked by an American locksmith—it took him sixteen days.
III. The Whitworth system was not international. When the United States developed its own, competing screw industry, it adopted a slightly different standard; continental Europe, which followed the metric system, likewise went its own way.
## CHAPTER SIX
## Mechanical Bent
MAUDSLAY HAD WHAT IS Often called a mechanical bent. So did the men he trained. Whitworth was the best known, but there were others: Joseph Clement, whom Charles Babbage commissioned to build his famous difference engine, the calculating machine that anticipated the modern computer; Richard Roberts, whose metal planing machine was capable of such precision that he used it to manufacture iron billiard tables; and Maudslay's personal assistant, James Nasmyth, who went on to invent the steam hammer and the pile driver. Like their master, these men generally came from modest backgrounds: Whitworth was the son of a schoolteacher; Clement's father was a weaver; Roberts's, a shoemaker. Moreover, they grew up not in cities but in small, remote villages or rural towns, without the least exposure to engineering. They often reached their calling by circuitous routes. Maudslay himself was apprenticed first to a carpenter; Roberts began as a quarryman; Clement was a slater's assistant. Despite such unpromising beginnings, all were attracted to the world of machines.
"My first essay at making a steam-engine was when I was fifteen," Nasmyth told his biographer. "I then made a real working steam-engine, 13/4 diameter cylinder, and 8-in. stroke, which not only could act, but really did some useful work; for I made it grind the oil colors which my father required for his painting." Nasmyth's background was different from that of his colleagues. He was born in a large city, Edinburgh, to a prosperous family—his father was the well-known Scottish landscape painter Alexander Nasmyth. James attended the High School in Edinburgh, the School of Art, and the university. In his spare time he continued his mechanical experiments with steam engines, casting parts in his bedroom, hanging around machine shops. The municipality was debating the wisdom of adopting steam-powered carriages for public roads, and Nasmyth achieved a small measure of local renown by building a vehicle that carried eight persons. This was more than two decades after George Stephenson built the first steam road-carriage, but it was still a remarkable accomplishment for a self-taught young man not yet twenty. Finally, having determined to pursue a career in mechanical engineering, Nasmyth decided to apprentice under the famed Henry Maudslay. He traveled to London and presented himself to the great man, bringing with him a working model of a steam engine. Maudslay, who no longer took pupils, spent twenty minutes examining the beautifully made engine, then took the young man on as his personal assistant.
Maudslay recognized in Nasmyth traits shared by all these men: an innate love of mechanics, a natural aptitude for working with metals, and above all a devotion to precision. Precision was an absolute standard. Maudslay, for example, produced regulating screws of much greater accuracy than was required by industry at the time; Whitworth built himself a micrometer that was accurate to one-millionth of an inch. These men are called engineers, but this designation is inadequate. For one thing, they were working in uncharted territory in which invention as well as technical proficiency was required. They were not simply designing replacements for traditional craft methods; they were inventing tools that were capable of previously unimagined accuracy. Moreover, these were also extremely skilled craftsmen. Indeed, the ability to actually make—with their own hands—what they conceived is an integral part of their achievements. "It is one thing to invent," observed Marc Brunel, "and another thing to make the invention work."
An affinity for steel and iron is a gift, like perfect pitch for a musician. These engineers had the artist's independence. Joseph Clement once received an order from America for a large regulating screw to be made "in the best possible manner." He produced an object of unparalleled accuracy and submitted a bill for several hundred pounds to his shocked client, who had expected to pay at most twenty (the case went before arbitrators and the American lost). In another case, Isambard Kingdom Brunel, who was in charge of building the Great Western Railway, commissioned Clement to design a piercing locomotive whistle. Delighted with the prototype, Brunel ordered a hundred. He, too, was shocked at the high price and declared that it was six times what he had previously been paying. "That may be," responded Clement, "but mine are more than six times better. You ordered a first-rate article, and you must be content to pay for it." He won that case, too.
Mechanical genius is less well understood and studied than artistic genius, yet it surely is analogous. "Is not invention the poetry of science?" asked E. M. Bataille, a French pioneer of the steam engine. "All great discoveries carry with them the indelible mark of poetic thought. It is necessary to be a poet to create." Nevertheless, while most of us would bridle at the suggestion that if Cézanne, say, had not lived, someone else would have created similar paintings, we readily accept the notion that the emergence of a new technology is inevitable or, at least, determined by necessity. My search for the best tool of the millennium suggests otherwise. Some tools were developed in direct response to a particularly vexing problem—this was the case with the Roman frame saw, or the socketed hammer. No doubt these devices would have appeared sooner or later. But the sudden and "mysterious" appearance of tools such as the carpenter's brace or the medieval bench lathe cannot be explained by necessity. Such tools are the result of leaps of an individual's creative imagination. They are the product of brilliant, inventive minds whose intuitive perceptions of complex mechanical relationships really are poetic.
—
The screwdriver is hardly poetic. The pragmatic way in which the arquebusier's spanner and the armorer's pincers were modified to include a screwdriver, or the casual combination of a screwdriver bit with a carpenter's brace, suggest expediency rather than invention. The screw itself, however, is a different matter. It is hard to imagine that even an inspired gunsmith or armorer—let alone a village blacksmith—simply happened on the screw by accident. To begin with, the thread of a screw describes a particularly complicated three-dimensional shape, often misnamed a spiral. In fact, a spiral is a curve that winds around a fixed point with a continuously increasing radius. A clock spring forms a spiral; so does a rope neatly coiled on the deck of a sailing ship. A helix, on the other hand, is a three-dimensional curve that twists around a cylinder at a constant inclined angle. So-called spiral staircases and spiral bindings are both examples of helixes. So, of course, are screws.
The helix occurs in nature in the form of the climbing vine and in some seashells.I But it requires a particular set of talents to invent a screw. First, it would take a skillful mathematician to describe the geometry of the helix. Then he—or someone else—would have to make the connection between theoretical mathematics and practical mechanics in order to imagine a use for such an unusual object. Finally, there would be the problem of how to actually fabricate a screw.
The builder of the Housebook lathe, whoever he was, resolved the problem of how to make a screw, but he did not invent the screw itself. An understanding of the principle of the screw predates the fifteenth century. The first documented use of the word screw, according to the Oxford English Dictionary, is in 1404. It occurs in a list of accounts: "Item 1 rabitstoke cum 2 scrwes" (the word was also spelled skrew, skrue, and scrue). A rabitstoke, I learn, is a plane for shaping complicated grooves, or rabbets; the two wooden screws, holding an adjustable fence, are part of the tool. Small wooden screws were also used to make bench vises and assorted clamps; large wooden screws adjusted the vertical and horizontal angle of cannons.
The most famous use of screws in the Middle Ages was in printing presses. Johannes Gutenberg played a pivotal role in the invention of movable type in the mid-1400s; unfortunately, there is no surviving description of his press. The earliest known representation of a printing press is about fifty years later. It consists of a heavy wood frame with a crosspiece through which a large screw is threaded. The screw is turned by means of a handspike, or lever, and pushes down a wooden board, which in turn presses the paper against the inked type.
The medieval printing press was likely an adaptation of a similar press that was used in paper-making. Stacks of damp sheets of paper, alternating with layers of felt cloth, were squeezed dry between two boards. But there might have been other models, for presses had many applications in medieval times. Linen presses, which were found in every large household, gave freshly woven cloth a smooth, lustrous finish. Olive and grape presses were used for pressing olive oil and wine. Apple presses made apple juice; seed presses squeezed oil from rapeseed and flax. All these presses used a vertical screw that was turned to exert downward pressure.
Printers at work, Frankfurt-am-Main, 1568.
The printing press and the paper press were medieval devices, but linen presses had been used since Roman times. A picture of a linen press with a heavy wooden frame and not one but two screws survives in a Pompeian fresco. The origins of olive and wine presses are ancient, too. The Roman architect Marcus Vitruvius Pollio, who lived sometime in the first century B.C., mentions olive-oil presses in The Ten Books on Architecture. In a passage on farmhouse planning, he describes a "pressing room" where olives are pressed to make oil. He writes that the room should be not less than forty feet long in order to accommodate the traditional beam press, but adds that such a large room is not required if the press is worked by "turning screws."
Medieval paper press.
Roman linen press, from a mural painting found in Pompeii.
The discovery of the screw press is described in detail in Historia Naturalis, published in A.D. 66 by Pliny the Elder. Pliny credits Hero of Alexandria with the invention. Hero was a mathematician (he discovered the formula for calculating the area of a triangle), but like most ancient mathematicians, he was also interested in mechanics. According to Pliny, Hero began his experiments with presses by improving the traditional beam press. The beam press consisted of a long wooden beam (prelum) whose end was inserted into a pocket in a wall. The beam was raised, and a bag of macerated olive pulp was placed beneath it, like a nut in a giant nutcracker. Then the beam was pulled down by means of a rope wound around a drum. To get rid of the clumsy rope, Hero inserted a large wooden screw, fixed to floor and ceiling, through the beam. As a nut was turned, the beam was forced down. This worked well, but Hero found that the nut had a tendency to jam. He then took a different tack. He attached a heavy stone to the bottom of the screw. Now, as the screw was turned, it lifted the stone, whose weight carried the beam down. "When you have hung up the stone and left it to itself," Hero wrote, "the prelum will do the pressing without your having to repeat the pressure several times."
Pliny describes the weighted beam press as "very much praised." Nevertheless, pulling up a weight in order to pull down a beam is hardly elegant, and Hero was not satisfied. Instead of pulling a weight up, he asked himself, what if the screw was used to push down? At the same time, what if the prelum was eliminated altogether? Thus Hero invented the direct-screw press, the ancestor of the printing press. In fact, his machine is virtually indistinguishable from later presses. "We fix on the table two uprights," Hero writes in his detailed and lucid description, "which carry the crosspiece. . . . The screw hole should be in the middle of the crosspiece. The screw is put through this hole and turned by means of handspikes till it reaches the lid which is laid on the galeagra the box containing the fruit] and presses it down and the juice flows."[II,
The screw-down press is a marvelous invention, not only because it is simple and compact but also because it is capable of enormous pressure. The downward force is a direct function of the ratio between the pitch of the screw and the circumference described by the handspike. For example, imagine a press of the type described by Hero whose large screw has a pitch of one inch, and which is turned by means of a handspike three feet long. If a man applies a force of, say, forty pounds to the handspike, the pressure exerted on the olive pulp will be more than nine thousand pounds. The ability of a single man—working without animals or waterpower—to exert this kind of force was unprecedented.
Hero's direct-screw press.
—
Hero invented a variety of machines in which he often incorporated a common mechanical device that the ancient Greeks called an "endless screw," today referred to as a worm gear. A worm gear is a combination of a long screw and a toothed wheel; each revolution of the screw advances the wheel a minute distance. The mechanical advantage depends on the pitch of the screw and the number of teeth in the wheel. Hero incorporated several endless screws into his hodometer, or "road measurer." The instrument was fixed atop a cart whose axle powered a train of endless screws that, at predetermined intervals, released a pebble into a box. The surveyor had only to count the pebbles to compute the distance traversed. Hero also invented the dioptra, an ancestor of the theodolite. The dioptra was mounted on a tripod or pedestal and the surveyor squinted down a sight whose horizontal and vertical orientation he adjusted by means of two worm gears, here used as regulating screws.
The screws used in ancient worm gears were usually bronze; screws for presses were wood. Both screws were made by tracing a helix onto a cylinder or rod and filing or carving the thread by hand. The template was a sheet of soft metal in the shape of a right-angled triangle. According to the ancient instructions, the triangle was wrapped around the rod so that one arm of the right angle was parallel to the axis, the hypotenuse automatically tracing a helix on the rod. I couldn't quite imagine this, so I thought I would try it using a piece of triangular paper and a broomstick handle. When I finished wrapping the triangular piece of paper around the stick, indeed, the edge of the paper made a neat helix. The problem was that there was no way to trace it without cutting through the paper. The instructions clearly stated that the template was reused. I realized that my mistake was to start wrapping with the vertical of the triangle against the broomstick. If I started with the point of the triangle, I could trace the hypotenuse, section by section, as I unwrapped the paper.
A worm gear.
Tracing a helix.
Another Greek device that used screws was called a "tortoise." The tortoise was a primitive nut made of a block of wood, drilled with a smooth hole, inside which was an iron or copper peg, called a tylos. The screw went into the hole, and as the tylos engaged the rotating thread, the tortoise "crawled" along the rotating screw. The tortoise is said to have originated in an apparatus for resetting fractured bones and is attributed to Andreas, a physician who lived in the third century B.C. This machine, which has an unfortunate resemblance to the torturer's rack, used the tortoise to gradually pull a harness that stretched the fractured limb. The tortoise was also used in adjustable obstetric instruments such as clamps and dilators.
Since the friction between the tylos and the threads of the screw was too great to allow much force to be exerted, the tortoise could be used only with relatively small devices. The massive screw of Hero's beam press would have jammed if he had used a tylos. He needed a different way of engaging the screw. Pondering the problem, Hero made another momentous discovery: the (male) screw has a (female) counterpart: the nut. We do not know exactly how he made this breakthrough. Perhaps he tried several pegs and intuited a continuous female thread. Maybe he thought it through mathematically. Or did it come in a flash of inspiration? Once he had the idea. It was a relatively simple matter to make a nut: using a Roman auger, he drilled a hole in a block of wood, split it in two, carved the female threads, and put the parts back together.
Hero's screw tap.
When it came to the screw-down press, however, Hero had to find a way to cut the threads inside a hole, leaving the heavy cross-beam intact. This was a challenge, but Hero was undeterred. He invented what is, as far as we know, the world's first screw tap. It was a box containing a wooden lead screw, guided by several tylos. The tip of the lead screw was fitted with an iron cutter. With the box firmly attached to the piece of wood in which a hole had been drilled, the lead screw was turned, and the cutter descended into the hole. "And we turn it till it comes into the plank, and we keep on turning it up and down, and we serve the wedge with blows again and again, until we have cut out the female screw with the furrow we wanted," instructed Hero. "And so we have made the female screw." In 1932, a Danish historian, Aage Gerhardt Drachmann, made a drawing of the screw tap from Hero's detailed written description. When a colleague challenged the practicability of the device and declared it "technically impossible," the intrepid Dane built a working model and successfully threaded a two-inch hole in a beechwood plank.
There is textual evidence that the Romans used screw taps with iron as well as wood. Josephus, a Jewish historian who lived in the first century A.D., writes of the temple in Jerusalem and describes eight-and-a-half-foot-long iron tie-rods that reinforced the supporting columns. "The head of each rod passed into the next by means of a cleverly made socket crafted in the form of a screw." Later, he elaborates: "They were held by these sockets, the male fitting into the female." These female sockets must have been threaded. Pappus of Alexandria, one of the last great Greek mathematicians, who lived in the fourth century, writes that "a screw is constructed having a helix with oblique threads in the drum made to fit in with another [emphasis added]," which might be a nut and bolt. Vitruvius is clearer. In describing a trispast, a crane that looked like a wooden A-frame, he writes that the two timbers "are fastened together at the upper end by a bolt." Oddly, archaeological evidence for nuts and bolts is extremely slim. Indeed, there is only one surviving Roman nut. Displayed in the Provincial Museum of Bonn, it is wrought iron, approximately one and a quarter inches square with a half-inch-diameter threaded hole. The nut was discovered in the 1890s among Roman relics dated A.D. 180–260 on the site of a fortified camp in Germany. No bolt was found.
If nuts and bolts were used only to assemble demountable structures such as the crane described by Vitruvius, that may explain why so few have been found. One thing is certain, the Romans—despite being skilled ironworkers and having invented nails—never made the connection between bolts and screws. Roman screws and screwdrivers are nowhere written about, and none have been discovered. "Necessity is the mother of invention" is an old Roman saying. Of course, the Romans had neither matchlocks nor butt hinges, so perhaps they felt no pressing need to develop a small, effective fastener. On the other hand, they did use bellows, and as Agricola pointed out, screws were superior to nails. Yet there is no such thing as a technological imperative. It would take another fourteen hundred years for the screw to appear. That is, it would take another fourteen hundred years for a mechanical poet to realize that the helix that could press olives, stretch broken limbs, and adjust surveying instruments could also serve as a kind of threaded nail.
* * *
I.The Latin for screw is cochlea, which is Greek for "snail" or "snail shell." The Latin for vine is vitis, which is the root of the French word vis, or screw, whence the English vise.
II.The technology of olive and grape presses was identical.
## CHAPTER SEVEN
## Father of the Screw
HERO OF ALEXANDRIA was a Greek. I had been taught that mechanical expertise was the preserve of the Romans, who invented the arch and the dome, never mind the auger and the plane. The Greeks were philosophers and artists. As a student, I had been to Greece, climbed the Acropolis, and visited museums. But, like many people, I had misinterpreted what I had seen. "So little has come down to us from the Greek Miracle, decisive for the birth of our sort of civilization, that we have become used to making a great deal of the things we have," wrote Derek J. de Solla Price, a Yale professor of the history of science. "Preservation has been highly selective so that we tend to see the Greeks in terms of only the more indestructible masses of building stone, statuary, and ceramic together with coins and a few grave goods that are the main holdings of our museums and archaeological sites." Indeed, the material evidence for Greek mechanical devices is so scant that, according to Price, it was long thought that the Greeks simply did not use complex machines, and that the surviving written descriptions of machines, by authors such as Hero, were merely speculative.
This belief was altered by a momentous discovery. Like many archaeological finds, it came about largely by chance. In 1900, two boats belonging to sponge fishermen were crossing the strait that lies between Crete and the Greek mainland and were swept off course by a squall. They sought shelter in the lee of an uninhabited islet called Antikythera. When the storm abated, the divers explored the unfamiliar waters, looking for sponges. Instead, at a depth of 140 feet, they discovered the remains of an ancient ship surrounded by scattered bronze and marble statues. They reported their find to the authorities, who organized an archaeological expedition.
The pottery dated the shipwreck between 80 and 50 B.C. The vessel appeared to have been a trader, sailing from somewhere in Asia Minor—perhaps the island of Rhodes—and bound for Rome. The salvaged material included many fragments encrusted with two thousand years of debris. The fragments were set aside while the archaeologists turned their attention to restoring the statues. Occasionally, the restorers went through the debris hoping to locate a missing piece of statue. Eight months into the work, during one such search, they made a startling discovery. One of the encrusted lumps had split apart, probably as the ancient wood inside shrank after being exposed to the atmosphere. The break revealed not a piece of statue but several corroded and crumbling bronze disks with inscriptions, as well as the traces of what appeared to be gearwheels. The mechanical device, whatever it was, had been contained in a wooden case about eight inches high, six inches wide, and four inches thick.
Preliminary cleaning revealed that the so-called Antikythera Mechanism was a machine of great complexity with many interlocking gearwheels. However, the heavy calcareous accretions on the fragile corroded fragments, many of which were fused together, made accurate reconstruction difficult. Some archaeologists believed that it was an ancient astrolabe; others argued that it was too complicated to be a navigation device and had to be some sort of clock. Since the oldest evidence of geared clockwork in Muslim and Chinese astronomical machines was no earlier than about A.D. 1000, to many scholars it seemed outlandish to suggest that the Greeks had this technology a thousand years earlier. Some argued that the mechanism was not ancient at all and had to be part of a later shipwreck on the same site. The last claim, at least, was laid to rest when it was ascertained that the disks were definitely bronze, a material used only in ancient times, more modern instruments being made of brass.
General plan of all gearing in the Antikythera Mechanism.
Decades later, as cleaning techniques improved, more of the inscriptions were deciphered and more of the mechanism was revealed. Yet the purpose of the machine remained a mystery. In 1959, Derek J. de Solla Price, who had been studying the Antikythera Mechanism, published a cover article in Scientific American titled "An Ancient Greek Computer." He speculated that the device was used to calculate the motion of stars and planets, which made it an ancient forebear of De'Dondi's planetary clock. Since the first known mechanical clock dated from the fourteenth century, this again brought counterclaims that such sophisticated technology could not have belonged to the ancient Greeks, and that the mechanism must be of later vintage. In 1971, Price and his Greek colleagues began to examine the fragments using the then new technology of gamma-radiographs and x-radiographs. They discerned layers of the mechanism previously hidden within the encrusted fragments. The final part of the puzzle fell into place when a missing crucial piece was found in the museum storeroom. It was now possible to reconstruct the machine.
According to Price, "The mechanism is like a great astronomical clock without an escapement, or like a modern analogue computer which uses mechanical parts to save tedious calculation." The front dial is inscribed with the signs of the zodiac, and a slip ring shows the months of the year; two back dials, one with three slip rings, one with four, indicate lunar and planetary phenomena. Inside, the movement consists of more than thirty interlocking toothed gearwheels assembled with pins and wedges—no screws. Most of these wheels are simple circular gears that transmit and modify rotary motion, the triangular teeth of one gear engaging the teeth of the other. Price also discovered a more complex set of gears that compound two different rates of revolution—the sidereal motions of the sun and the waxing and waning of the moon—to produce the cycles of the so-called synodic month. This is, in fact, the first known example of a differential gear. The differential in the axle of automobiles, which divides power between the driving wheels and allows the inside wheel to travel a shorter distance smoothly when the vehicle is turning a corner, was invented in 1827; the differential gear in the Antikythera Mechanism was made two thousand years ago. "It is a bit frightening to know that just before the fall of their great civilization the ancient Greeks had come so close to our age," writes Price, "not only in their thought, but also in their scientific technology."
The Antikythera Mechanism is the only complex mechanical instrument to survive from antiquity, yet we know that it was not unique. A similar device is described by Cicero, who witnessed a demonstration of a "celestial globe" in the first century B.C. "When Gallus set this globe in motion, it came about that the moon was as many revolutions behind the sun on the bronze instrument as in the heavens themselves, and therefore there was that same eclipse of the sun in that sphere, and the moon then met that point, which is the earth's shadow." Cicero was impressed. "I decided then that there was more genius in that Sicilian than human nature seems able to encompass." "That Sicilian" was Archimedes, the builder of the celestial globe, who had died about one hundred and fifty years earlier. Archimedes' globe was famous in the ancient world; it is also referred to by Plutarch and Ovid. Even eight hundred years after Archimedes' death, Claudianus wrote a poem in which Jupiter was mocked by the "skill of an old man of Syracuse [who] has copied the laws of the heavens, nature's reliability, and the ordinances of the gods." None of these authors provided technical details, however. We know from ancient references that Archimedes himself wrote a treatise titled On Sphere-Making, but it has long been lost. Price speculates that Archimedes probably used a complicated gear train of the type found in the Antikythera Mechanism, which appears to be a later copy of his celestial globe.
Archimedes was a citizen of Syracuse, a wealthy Greek city-state on the island of Sicily. He was born about 287 B.C., the son of an astronomer, and was sent as a young man to study mathematics in Alexandria with the successors of the great Euclid. On his return to Syracuse, he devoted himself to science. He became the foremost mathematician of the ancient world, devising a variety of proofs in both plane and solid geometry, including describing the geometry of the spiral. He wrote several treatises on the equilibrium of planes and established the mathematical foundation for the science of mechanics. In addition, he single-handedly invented the science of hydrostatics, the branch of physics that deals with fluids at rest and under pressure.
Archimedes left instructions that his sepulchral column include a depiction of his favorite proposition: the calculation of the exact ratio between a sphere and the cylinder that circumscribed it. He died at the age of seventy-five. One hundred and fifty years later, Cicero, while serving as Roman administrator of Sicily, sought out the tomb and, finding it neglected, had it restored. Cicero anticipated the interest in Archimedes by later historians such as Diodorus Siculus, Livy, and Plutarch. Of course, they wrote three hundred years (in the case of Plutarch, four hundred years) after the fact; by then, all that was left were stories. One of these concerned Archimedes' death. During the Second Punic War, Syracuse was attacked by the Roman army, and after a two-year siege the city fell. According to Plutarch, the victorious Roman general, Marcellus, who was an amateur mathematician, sent a soldier to fetch the renowned Archimedes. "As fate would have it, he [Archimedes] was intent on working out some problem with a diagram, and having fixed his mind and his eyes alike on his investigation, he never noticed the incursion of the Romans nor the capture of the city," Plutarch writes. "And when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him." The remorseful Marcellus is said to have personally erected the mathematician's tomb. He also appropriated two of Archimedes' celestial globes, one of which later came into the hands of the astronomer Gallus, who showed it to Cicero.
The most famous story told about Archimedes concerns his solution of the so-called wreath problem. Hieron, the king of Syracuse, commissioned a gold wreath as an offering to the gods. He provided gold to the jeweler, who in due time delivered the finished wreath. Hieron suspected that the gold had been diluted with silver, but could not prove it. The wreath was a consecrated object and could not be tampered with, so a chemical assay was out of the question. Since the goldsmith refused to confess, the king turned to Archimedes. The mathematician pondered the matter and devised a simple experiment. He weighed the wreath, and immersed similar weights of silver and gold in a vessel of water, measuring how much water each displaced. He discovered that silver displaces more water than gold (the specific gravity of silver is almost half that of gold). Since the immersed wreath caused more water to overflow than the equivalent weight of gold, he deduced the presence of silver and proved that the wreath was indeed impure. According to legend, the idea for the water experiment came to Archimedes as he was plunging himself into a tub in a public bath. Seeing the water overflowing triggered something in his mind. "Transported with joy, he jumped out of the tub and rushed home naked," writes Vitruvius, "crying out in a loud voice, 'Heure-ka! Heure-ka!' [I've found it! I've found it!]."
Plutarch wrote that Archimedes "regarded as sordid and ignoble the construction of instruments, and in general every art directed to use and profit, and he only strove after those things which, in their beauty and excellence, remain beyond all contact with the common needs of life." Yet there is no doubt that the mathematician had a mechanical bent, no less than Hero or Maudslay. Archimedes' reputation for cleverness and ingenuity was legendary. The Romans named a popular puzzle, which consisted of various-shaped pieces of ivory that had to be assembled into a square, Loculus Archimedius in his honor. His cleverness manifested itself in many practical inventions, all still in use: the compound pulley, whose several sheaves increased lifting power and allowed a single man to raise heavy weights; the windlass, a rope wound around a drum, which was used as a hoisting device aboard ships and in mines; and the ancestor of the balancing weighing scale, the steelyard. In addition to the celestial globe, he is said to have built a water clock and a hydraulic organ in which air was compressed by water.
Like Leonardo and Ramelli, Archimedes served as a military engineer. During the siege of Syracuse he was called on to build defensive weapons. He designed catapults that threw rocks weighing five hundred pounds, and complicated underwater obstacles that capsized ships. His most renowned weapon was a mirror that beamed the sun's rays and set the attackers' ships on fire. To prove the practicality of what had long been considered merely a colorful legend, in 1973, a Greek engineer named Ioannis Sakas built a working version of the ancient ray gun. He used seventy bronze-coated mirrors, which he aimed at a tarred plywood cutout of a ship. At a distance of 165 feet, approximating the "bow-shot" that is mentioned in the classical text, it took only a few minutes for fire to break out.
In 1981, the redoubtable Sakas tested another Archimedean invention, the architronito, or steam cannon. This device was credited to Archimedes by Leonardo da Vinci, whose sketchbooks show a gun barrel with a breech encased in a heated firebox. When water is released from a cistern into the white-hot barrel, the resulting steam creates sufficient pressure to eject the cannonball. Leonardo wrote that "this machine has driven a ball weighing one talent [about twenty pounds] six stadia [about three thousand feet]." Sakas's scale model successfully fired a cement-filled tennis ball a distance of two hundred feet.
According to Plutarch, after Archimedes had written a treatise titled "To Move a Given Weight by a Given Force," in which he claimed that any weight could be moved, Hieron challenged the mathematician to prove his assertion by moving a beached ship loaded with freight. Archimedes set up his apparatus, attached a line to the ship, "and then drew it along, smoothly and evenly as if it were floating in water, not with great labor, but sitting down at a distance." It was on that occasion that he made his famous claim: "Just give me somewhere to stand, and I shall move the earth." How could Archimedes move a vessel weighing as much as seventy-five tons? According to Plutarch, it was done with a compound pulley; a description by a Byzantine historian mentions a three-sheaved pulley; and Athenaeus, a Greek historian, writes that Archimedes used an endless screw. A. G. Drachmann suggested that it is not unreasonable to assume that the ship was moved by a combination of these machines. He calculated the mechanical advantage of a pulley of five sheaves, pulled by a windlass that was powered by a combination of endless screws, would be 1:125,000. That is, one pound of force on the rope would translate into a pulling force of more than sixty tons. Even assuming losses for friction, Drachmann argued, Archimedes alone—the different versions are in agreement on this point—could easily move the heavy ship a small distance.
Who was the inventor of the endless screw? Some modern historians credit Archytas of Terentum, a Pythagorean philosopher who lived at the time of Plato, around 400 B.C.; others point to Apollonius of Perge, a younger contemporary of Archimedes'. Drachmann champions Archimedes himself, citing not only Athenaeus' story of moving the ship, but also quoting Eustathius, a Greek scholar who wrote, "Screw is also the name of a sort of machine, which was first invented by Archimedes."
Drachmann's claim is all the more plausible since Archimedes' name is associated with another kind of screw, the water screw, a device for lifting water. The water screw consists of a giant screw about one foot in diameter and ten to fifteen feet long, encased in a watertight wooden tube. The tube, whose ends are left open, is installed at a low angle, with the lower end immersed in water. As the entire apparatus turns, powered by a person walking on cleats fastened to the exterior, the water entering the lower end is moved up by the helical partitions—or threads—of the screw and emerges at the top. The water screw turns slowly, but its capacity is substantial (the lower the angle, the greater the flow), and its mechanical efficiency has been estimated to be as much as 60 percent, which compares favorably with later water-lifting devices such as waterwheels and bucket conveyor belts.I,
The oldest references to the water screw, in the second century B.C., all credit Archimedes with the invention. According to Diodorus, Archimedes invented the water screw while he was a young man in Alexandria. That makes sense. The device is ideal for agricultural irrigation in Egypt: unlike large waterwheels, it can easily be moved from place to place; its lift is not great, but sufficient for the flat delta; and the simple design—there are no moving parts—resists clogging by the silted Nile River water.
Water screw technology spread from Egypt throughout the Mediterranean. Water screws were used for irrigation, but they had other applications. Archimedes is said to have used them to empty bilge water from one of Hieron's huge ships. Water screws were also used by the Romans to lift water in municipal water systems, and to pump out mines. Several well-preserved wooden water screws were discovered in the early 1900s in ancient Roman copper mines in Spain. The twelve-foot-long tubes, approximately one foot in diameter, are wrapped in pitched cloth and strengthened with rope; inside, the helical partitions are made of laminated strips of wood, glued and attached with copper nails. Four such water screws in series could lift water a vertical distance of about twenty feet. Diodorus describes how "with constant pumping by turns they throw up the water to the mouth of the pit and thus drain the mine; for this engine is so ingeniously contrived that a vast quantity of water is strangely and with little labor cast out."
Archimedean screw, from a later edition of Vitruvius, The Ten Books on Architecture, first century B.C.
Diodorus was impressed with the simplicity and effectiveness of the water screw since he compared it to other ancient water-lifting devices such as complicated bucket conveyor belts and waterwheels. A common type of waterwheel was the tympanum, a large hollow wheel, ten to fifteen feet in diameter, divided into eight pie-shaped compartments. As the wheel turned, water flowed into the lowest compartment when it was submerged, and out when the compartment reached the top position. It has been suggested that the tympanum may have inspired Archimedes. Indeed, if the tympanum shape is stretched and rotated along its central axis, it produces a cylindrical helix. This three-dimensional extrapolation, although hardly obvious, would not have been difficult for a skilled mathematician. The presumed authorship of Archimedes is supported by another curious fact. The only detailed description of a water screw in all Greek and Latin literature, which is by Vitruvius, specifies a water screw with eight helical partitions—the precise number that would be produced if the water screw were inspired by the tympanum. Vitruvius was presumably describing the original water screw; later Roman engineers, realizing that there is no mechanical advantage to eight partitions—and considerable added cost—reduced their number to two or three.
Whether or not Archimedes was inspired by the tympanum, the water screw is yet another example of a mechanical invention that owes its existence to human imagination rather than technological evolution. And imagination is fickle. The ancient Chinese, for example, did not know the water screw; indeed, they didn't know screws at all: the screw is the only major mechanical device that they did not independently invent. The Romans, on the other hand, knew about the screw when they invented the auger, yet they never realized that the same principle could solve a major drilling problem: the tendency of deep holes to become clogged with sawdust. Not until the early 1800s was the so-called spiral auger, whose helical shank cleared the sawdust as the bit turned, invented.
The water screw is not only a simple and ingenious machine, it is also, as far as we know, the first appearance in human history of the helix. The discovery of the screw represents a kind of miracle. Only a mathematical genius like Archimedes could have described the geometry of the helix in the first place, and only a mechanical genius like him could have conceived a practical application for this unusual shape. If he invented the water screw as a young man in Alexandria, and—as I like to think—later adapted the idea of the helix to the endless screw, then we must add a small but hardly trifling honor to his many distinguished achievements: Father of the Screw.
* * *
I. The Archimedean screw continues in use to this day. In modern screw-conveyors, the screw rotates inside the cylinder; in the ancient version, the entire cylinder rotated.
## GLOSSARY OF TOOLS
London pattern screwdriver
Scotch pattern screwdriver
Undertaker's screwdriver
Gent' fancy screwdriver
Carpenter's brace
Breast auger
Spiral bit auger
Cooper's adze
Wooden carpenter's brace with brass plates
Try square
Bevel
A-level
Spirit level
Maul
Combination case opener
Plane
Backsaw
Skew-back handsaw
Frame saw
## ACKNOWLEDGMENTS
Thanks, first, to David Shipley for asking the question. For help with the Greek quote, my appreciation to Prof. Ralph Rosen, chair of classical studies at the University of Pennsylvania. Robert A. Ruhloff was kind enough to send me information on wrought-iron spikes, including several interesting samples. Jamie Kendrick, Adam Barzilay, Maria Gonzalez, and Yi-Ting Liu provided capable research assistance. The Milton Historical Society supplied information on the redoubtable P. L. Robertson. The staff of the Fisher Fine Arts Library and the Van Pelt Library of the University of Pennsylvania were helpful, as always. I doubt that this small book would have seen the light of day without the encouragement of my editor, Nan Graham, and my agent, Carl Brandt, who both share my interest in tools and handy-work. Shirley Hallam, my wife, pointed me in the right direction, at the right time—as usual.
The Icehouse, Chestnut Hill, Pennsylvania
October 1999
WITOLD RYBCZYNSKI is the author of nine books, including Home, City Life, and the national bestseller A Clearing in the Distance, for which he won a Christopher Award and the J. Anthony Lukas Prize. He is a regular contributor to The Atlantic Monthly, The New Yorker, The New York Times Magazine, and The New York Review of Books. He teaches at the University of Pennsylvania.
ALSO BY WITOLD RYBCZYNSKI
Paper Heroes
Taming the Tiger
Home
The Most Beautiful House in the World
Waiting for the Weekend
Looking Around
A Place for Art
City Life
A Clearing in the Distance
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## NOTES
### Chapter One: The Carpenter's Toolbox
1. Edward Rosen, "The Invention of Eyeglasses: Part I," Journal of the History of Medicine (January 1956): 34–35.
2. Ken Kern, The Owner-Built Home (Oakhurst, Calif.: Owner-Builder Publications, 1972), 78.
3. W. L. Goodman, The History of Woodworking Tools (London: G. Bell & Sons, 1964), 199–201.
4. R. A. Salaman, Dictionary of Tools: used in the woodworking and allied trades, c. 1700–1970 (London: George Allen & Unwin Ltd., 1975), 299.
5. Lynn White Jr., "Technology and Invention in the Middle Ages," Speculum 15 (April 1940): 153.
6. For a dissenting view, see A. G. Drachmann, "The Crank in Graeco-Roman Antiquity," Changing Perspectives in the History of Science: Essays in Honour of Joseph Needham (London: Heinemann, 1973), 33–51.
7. Bertrand Gille, "Machines," in Charles Joseph Singer et al., eds., A History of Technology: Vol. II, The Mediterranean Civilizations and the Middle Ages c. 700 B.C. to c. A.D. 1500 (New York: Oxford University Press, 1957), 651.
8. Bertrand Gille, "The Fifteenth and Sixteenth Centuries in the Western World," in Maurice Dumas, ed., A History of Technology & Invention: Vol. II, The First Stages of Mechanization, trans. Eileen B. Hennessy (New York: Crown Publishers, 1969), 23.
9. Graham Hollister-Short, "Cranks and Scholars," History of Technology 17 (1995): 223–24.
10. Goodman, History of Woodworking Tools, 178.
11. Ibid., 9.
12. "Tools: Later development of hand tools: SCREW-BASED TOOLS: Screwdrivers and wrenches," Britannica Online, December 1998.
### Chapter Two: Turnscrews
1. Peter Nicholson, Mechanical Exercises: or, the elements and practice of Carpentry, Joinery, Bricklaying, Masonry, Slating, Plastering, Painting, Smithing, and Turning (London: J. Taylor, 1812), 353.
2. Joseph Moxon, Mechanick Exercises: or, the Doctrine of Handy-Works (London: J. Moxon, 1693), A5–6.
3. The Greek Anthology, trans. W. R. Patton (London: William Heinemann, 1916), 405.
4. "Navigation," Encyclopaedia Britannica, vol. 12 (Edinburgh: A. Bell and C. Macfarquhar, 1797), plate 343. The reference is pointed out by Joseph E. Sandford, "Carpenters' Tool Notes," in Henry C. Mercer, Ancient Carpenters' Tools: Together with Lumbermen's, Joiners' and Cabinet Makers' Tools in Use in the Eighteenth Century (Doylestown, Pa.: Bucks County Historical Society, 1975), 311.
5. A Dictionary of American English: on historical principles, vol. 4 (Chicago: University of Chicago Press, 1944), 2045.
6. R. A. Salaman, Dictionary of Tools: used in the woodworking and allied trades, c. 1700–1970 (London: George Allen & Unwin Ltd., 1975), 450.
7. Ibid., 449.
8. A. J. Roubo, "L'Art du Menuisier en Meubles," Description des Arts et Métiers, vol. 19 (Paris: Académie des Sciences, 1772), 944 (author's translation).
9. Encyclopédie: ou dictionnaire raisonné des sciences, des arts et des métiers, vol. 17 (Neuchastel: Samuel Faulche & Co., 1765), 484 (author's translation).
10. Adolphe Hatzfeld and Arsène Darmesteter, Dictionnaire Général de la Langue Française: du commencement du XVIIe siècle jusqu'à nos jours, vol. 2 (Paris: Librairie Delagrave, 1932), 2171.
11. James M. Gaynor and Nancy L. Hagedorn, Tools: Working Wood in Eighteenth-Century America (Williamsburg, Va.: Colonial Williamsburg Foundation), 11.
12. Linda F. Dyke, Henry Chapman Mercer: An Annotated Chronology (Doylestown, Pa.: Bucks County Historical Society, 1989), 11.
13. Kenneth D. Roberts, Some 19th Century English Woodworking Tools: Edge & Joiner Tools and Bit Braces (Fitzwilliam, N.H.: Ken Roberts Publishing Co., 1980).
14. See Witold Rybczynski, "One Good Turn," New York Times Magazine, April 18, 1999, 133.
### Chapter Three: Lock, Stock, and Barrel
1. Lynn White Jr., "The Act of Invention: Causes, Contexts, Continuities, and Consequences," Technology and Culture 3 (fall 1963): 486–500.
2. Martha Teach Gnudi, "Agostino Ramelli and Ambrose Bachot," Technology and Culture 15, no. 4 (October 1974): 619.
3. The Various and Ingenious Machines of Agostino Ramelli (1588), trans. Martha Teach Gnudi (Baltimore: Johns Hopkins University Press, 1976), 508.
4. Bert S. Hall, "A Revolving Bookcase by Agostino Ramelli," Technology and Culture 11, no. 4 (July 1970): 397.
5. Georgius Agricola, De Re Metallica, trans. H. C. Hoover and L. H. Hoover (New York: Dover Publications, 1950), 364.
6. Christoph Graf zu Waldburg Wolfegg, Venus and Mars: The World of the Medieval Housebook (Munich: Prestel-Verlag, 1998), 8.
7. Hugh B. C. Pollard, Pollard's History of Firearms, Claude Blair, ed. (New York: Macmillan, 1983), 29.
8. John Keegan, A History of Warfare (New York: Alfred A. Knopf, 1993), 329.
9. Fernand Braudel, The Structures of Everyday Life: Vol. I, The Limits of the Possible, trans. Siân Reynolds (New York: Harper & Row, 1981), 392.
10. Pollard, Pollard's History of Firearms, 55.
11. Ibid., 35.
12. Ibid., 18.
13. Joseph Moxon, Mechanick Exercises: or, the Doctrine of Handy-Works (London: J. Moxon, 1693), 33–34.
14. Charles John Ffoulkes, The Armourer and His Craft: From the XIth to the XVIth Century (New York: Benjamin Blom, 1967), 55.
15. Claude Blair, European Armour: circa 1066 to circa 1700 (London: B. T. Batsford Ltd., 1958), 162.
16. Ffoulkes, Armourer and His Craft, 24.
17. Ibid., plate V.
### Chapter Four: The Biggest Little Invention
1. Georgius Agricola, De Re Metallica, trans. H. C. Hoover and L. H. Hoover (New York: Dover Publications, 1950), 364.
2. G. H. Baillie, C. Clutton, and C. A. Ilbert, Britten's Old Clocks and Watches and Their Makers (New York: E. P. Dutton, 1956), 14.
3. Ibid., 64.
4. Joseph Chamberlain, "Manufacture of Iron Wood Screws," in British Association for the Advancement of Science, Committee on Local Industries, The Resources, Products, and Industrial History of Birmingham and the Midland Hardware District (London: R. Hardwicke, 1866), 605–6.
5. Henry C. Mercer, Ancient Carpenters' Tools: Together with Lumbermen's, Joiners' and Cabinet Makers' Tools in Use in the Eighteenth Century (Doylestown, Pa.: Bucks County Historical Society, 1975), 259.
6. Quoted by H. W. Dickinson, "Origin and Manufacture of Wood Screws," Transactions of the Newcomen Society 22 (1941–42): 80.
7. Ibid., 81.
8. Ibid., 89.
9. Ken Lamb, P.L.: Inventor of the Robertson Screw (Milton, Ont.: Milton Historical Society, 1998), 35.
10. Ibid., 16.
11. Henry F. Phillips and Thomas M. Fitzpatrick, "Screw," U.S. patent number 2,046,839, July 7, 1936.
12. American Screw Company to Henry F. Phillips, March 27, 1933.
13. Mead Gliders, Chicago, to American Screw Company, April 26, 1938.
14. Wentling Woodcrafters, Camden, N.J., to American Screw Company, June 15, 1938.
15. "The Phillips Screw Company" (unpublished paper, Phillips Screw Company, Wakefield, Mass.).
16. Consumer Reports 60, no. 11 (November 1995): 695.
### Chapter Five: Delicate Adjustments
1. L. T. C. Rolt, A Short History of Machine Tools (Cambridge, Mass.: MIT Press, 1965), 59.
2. Robert S. Woodbury, Studies in the History of Machine Tools (Cambridge, Mass.: MIT Press, 1972), 20–21.
3. Ibid., 49.
4. Christoph Graf zu Waldburg Wolfegg, Venus and Mars: The World of the Medieval Housebook (Munich: Prestel-Verlag, 1998), 88.
5. Jacques Besson, Theatrum Machinarum (Lyon: 1578), plate IX.
6. Charles Plumier, L'art de tourner (Lyon: 1701).
7. Maurice Daumas and André Garanger, "Industrial Mechanization," in A History of Technology & Invention, Maurice Daumas, ed., trans. Eileen B. Hennessy (New York: Crown Publishers, 1969), 271.
8. James Nasmyth, James Nasmyth, Engineer: An Autobiography (London: John Murray, 1885), 136.
9. Ibid., 144.
10. Ibid., 128.
11. L. T. C. Rolt, Great Engineers (London: G. Bell and Sons, 1962), 105.
12. Samuel Smiles, Industrial Biography: Iron-Workers and Tool-Makers (Boston: Ticknor & Fields, 1864), 282.
### Chapter Six: Mechanical Bent
1. Samuel Smiles, Industrial Biography: Iron-Workers and Tool-Makers (Boston: Ticknor & Fields, 1864), 337.
2. Ibid., 223.
3. Ibid., 312.
4. Ibid., 204 (author's translation).
5. W. L. Goodman, The History of Woodworking Tools (London: G. Bell & Sons, 1964), 105.
6. Vitruvius, The Ten Books on Architecture, trans. Morris Hicky Morgan (New York: Dover Publications, 1960), 184.
7. A. G. Drachmann, "Ancient Oil Mills and Presses," Kgl. Danske Videnskabernes Selskab, Archaeologisk-kunsthistoriske Meddelelser 1, no.1 (1932): 73.
8. Ibid., 76.
9. Bertrand Gille, "Machines," in A History of Technology, vol. 2, Charles Joseph Singer et al., eds. (New York: Oxford University Press, 1957), 631–32.
10. John James Hall, "The Evolution of the Screw: Its Theory and Practical Application," Horological Journal, July 1929, 269–70.
11. Quoted in John W. Humphrey et al., Greek and Roman Technology: A Sourcebook (London: Routledge, 1998), 56.
12. A. G. Drachmann, "Heron's Screwcutter," Journal of Hellenic Studies 56 (1936): 72–77.
13. Quoted in Humphrey et al., Greek and Roman Technology, 56.
14. Quoted in Hall, "Evolution of the Screw: Its Theory and Practical Application," Horological Journal, August 1929: 285.
15. Vitruvius, Ten Books, 285.
16. Henry C. Mercer, Ancient Carpenters' Tools: Together with Lumbermen's, Joiners' and Cabinet Makers' Tools in Use in the Eighteenth Century (Doylestown, Pa.: Bucks County Historical Society, 1975), 273.
### Chapter Seven: Father of the Screw
1. Derek J. de Solla Price, "Gears from the Greeks: The Antikythera Mechanism—a Calendar Computer from ca. 80 B.C.," Transactions of the American Philosophical Society 64, pt. 7 (November 1974): 51.
2. Derek J. de Solla Price, "Clockwork Before the Clock," Horological Journal (December 1955): 810–14.
3. Derek J. de Solla Price, "An Ancient Greek Computer," Scientific American, June 1959, 60–67.
4. Ibid., 66.
5. Ibid., 67.
6. Quoted in John W. Humphrey et al., Greek and Roman Technology: A Sourcebook (London: Routledge, 1998), 57–58.
7. Claudius Claudianus, Shorter Poems 51, in Humphrey et al., Greek and Roman Technology, 58.
8. Quoted in The Works of Archimedes, T. L. Heath, ed. (New York: Dover Publications, 1953), xviii.
9. Vitruvius, The Ten Books on Architecture, trans. Morris Hicky Morgan (New York: Dover Publications, 1960), 254.
10. Quoted in E. J. Dijksterhuis, Archimedes, trans. C. Dikshoorn (Copenhagen: Ejnar Munksgaard, 1956), 13.
11. New York Times, November 11, 1973.
12. Quoted in D. L. Simms, "Archimedes' Weapons of War and Leonardo," British Journal of the History of Science 21 (1988): 196.
13. The Times, May 15, 1981.
14. Quoted in A. G. Drachmann, "How Archimedes Expected to Move the Earth," Centaurus 5, no. 3–4 (1958): 278.
15. Quoted in Dijksterhuis, Archimedes, 15.
16. Drachmann, "How Archimedes Expected to Move the Earth," 280–81.
17. R. J. Forbes, "Hydraulic Engineering and Sanitation," A History of Technology, vol. 2, Charles Joseph Singer et al., eds. (New York: Oxford University Press, 1957), 677; A. G. Drachmann, The Mechanical Technology of Greek and Roman Antiquity (Copenhagen: Munksgaard, 1963), 204.
18. Quoted in A. G. Drachmann, "The Screw of Archimedes," Actes de VIIIe Congrès International d'Histoire des Sciences, Florence-Milan, 3–9 septembre 1956, vol. 3 (Florence: Vinci, 1958), 940–41.
19. John Peter Oleson, Greek and Roman Mechanical Water-Lifting Devices: The History of Technology (Toronto: University of Toronto, 1984), 297, 365.
20. Humphrey et al., Greek and Roman Technology, 317.
21. William Giles Nash, The Rio Tinto Mine: Its History and Romance (London: Simpkin Marshall Hamilton Kent & Co., 1904), 35.
22. Diodorus Siculus, The Historical Library of Diodorus the Sicilian; in Fifteen Books, trans. G. Booth (London: W. M'Dowall for J. Davis, 1814).
23. See Drachmann, Mechanical Technology, 154.
24. Vitruvius, Ten Books, 297.
25. Joseph Needham and Wang Ling, Science and Civilization in China, Introductory Orientations (New York: Cambridge University Press, 1954), 241.
## TEXT ILLUSTRATION SOURCES
Page : A History of Technology: Vol. II, The Mediterranean Civilizations and the Middle Ages c. 700 B.C. to c. A.D. 1500, eds. Charles Joseph Singer et al. (New York: Oxford University Press, 1956), 152; : Kenneth D. Roberts, Some 19th Century English Woodworking Tools: Edge & Joiner Tools and Bit Braces (Fitzwilliam, N.H.: Ken Roberts Publishing Co., 1980), 234; : The Various and Ingenious Machines of Agostino Ramelli (1588), trans. Martha Teach Gnudi (Baltimore: Johns Hopkins University Press, 1976), plate 188; : The Various and Ingenious Machines of Agostino Ramelli (1588), trans. Martha Teach Gnudi (Baltimore: Johns Hopkins University Press, 1976), plate 129; : Wapenhandelinghe van Roers, Musquetten end Speissen, Achtervolgende de Ordre van Syn Excellente Maurits, Prince van Orangie . . . Figuirlyck vutgebeelt door Jacob de Gheyn (Musket drill devised by Maurice of Orange) (The Hague, 1607); facsimile edition, New York: McGraw-Hill, 1971; : Hugh B. C. Pollard, Pollard's History of Firearms, ed. Claude Blair (New York: Macmillan, 1983), 35; : Charles John Ffoulkes, The Armourer and His Craft: From the XIth to the XVIth Century (New York: Benjamin Blom, 1967), 55. Redrawn by author; : Charles John Ffoulkes, The Armourer and His Craft: From the XIth to the XVIth Century (New York: Benjamin Blom, 1967), plate V. Redrawn by author; : Ken Lamb, P.L.: Inventor of the Robertson Screw (Milton, Ont.: Milton Historical Society, 1998), 152; : Christoph Graf zu Waldburg Wolfegg, Venus and Mars: The World of the Medieval Housebook (Munich: Prestel-Verlag, 1998), 88; : A History of Technology: Vol. II, The Mediterranean Civilizations and the Middle Ages c. 700 B.C. to c. A.D. 1500, eds. Charles Joseph Singer et al. (New York: Oxford University Press, 1956), 334; : Robert S. Woodbury, Studies in the History of Machine Tools (Cambridge, Mass.: MIT Press, 1972), Fig. 30; : T. K. Derry and Trevor I. Williams, A Short History of Technology (New York: Oxford University Press, 1960), 236; : A. G. Drachmann, Ancient Oil Mills and Presses (Copenhagen: Levin & Munksgaard, 1932), 159. Redrawn by author; : Derek J. de Solla Price, "Gears from the Greeks: The Antikythera Mechanism—a Calendar Computer from ca. 80 B.C.," Transactions of the American Philosophical Society 64, pt. 7 (November 1974): 37; : M. H. Morgan, Vitruvius, The Ten Books on Architecture (Cambridge, Mass.: Harvard University Press, 1946), 295. Pages 101, 115, 120, 122, 145–151 were drawn by the author.
## INDEX
Académie Française,
adzes, , , , , ,
cooper's,
Agricola, Georgius, –53, , ,
Alembert, Jean Le Rond d', –35
A-levels, –17,
American Screw Company, ,
Ancient Carpenters' Tools (Mercer), ,
Andreas,
Antikythera Mechanism, –32,
Apollonius of Perge,
archaeology, –18, , –28
Archimedes, , –48, –43
Archytas of Terentum,
armor, ,
chain mail, ,
Dresden, –65
golden age of,
helms of, , –66
jousting vs. field, ,
screws used in, –64, –67,
steel plates of, , , , –67
weight of,
armorer's pincers, ,
arquebus, –57
arquebusier's screwdriver, , –61
arquebusier's spanner, , , ,
art de tourner, L' (Plumier),
astrology,
astronomy, , , , , ,
Athenaeus, ,
augers, –23, , , ,
breast, –23,
long-handled,
spiral, ,
awls, –37
ax-adzes,
axes,
double-headed,
Babbage, Charles,
backsaws, ,
Bataille, E. M.,
Bearing the Cross (Franke),
bellows, , , , –25
bench vises,
Besson, Jacques, –96
bevels, ,
blacksmiths, , , –71, , ,
bolts, , , , ,
bombards,
bookstands, revolving, ,
bow drills, –22, , ,
braces,
cranks in, –25,
wooden, –24
see also carpenter's braces
Bramah, Joseph, –101
British Arts and Crafts movement,
Britten's Old Clocks and Watches and Their Makers,
Bronze Age, –20
Brunel, Isambard Kingdom, ,
Brunel, Marc Isambard, –3,
bucket conveyor belts, ,
buttons, –46,
cabinetmaker's screwdriver, –35
cabinet turnscrews,
cabinetwork, , –20, –35,
Campin, Robert,
cannons, ,
steam, –38
carpenter's benches, , ,
carpenter's braces, , –25, , , , , , , ,
catapults, ,
celestial globes, , , ,
ceramics, –39, –29
China, ancient, , , –43
chisels, –20, , –37, ,
Cicero, –33, ,
clamps, ,
Claudianus,
Clement, Joseph, –8, –10
clockmaking, , –72
clocks:
astronomical, , –72, –32
turret,
combination case openers, ,
Commander mauls, , –41
computers,
Consumer Reports,
Cook, James,
corks, –36
corkscrews, ,
cotter pins,
cranks, –25,
crowbars, ,
cuirassier helmet,
De'Dondi, Giovanni, –72,
De Re Metallica (Agricola), –53
Dictionary of Tools (Salaman), –32
Dictionnaire Général de la Langue Française, ,
Diderot, Denis, –35,
dies, , ,
Diodorus Siculus, , –41
dioptra,
Disston, Henry,
diverse et artificiose machine, Le (Ramelli), , –51,
dividers, ,
Drachmann, A. G., , –39
draw rings,
drills, , , ,
bow, –22, , ,
Dürer, Albrecht, –47, –56
Egypt, ancient, ,
tools of, –18, , , –89
electrician's screwdriver,
electricity,
Encyclopaedia Britannica, , ,
Encyclopédie (Diderot and d'Alembert), –35, , , , , , ,
engines,
difference,
jet,
steam, , , , , –9,
Euclid,
Eustathius,
eyeglasses, –14,
Ffoulkes, Charles, –67
files, –37
Fisher Body Company,
Ford, Henry, –83
foundries, –56
Franke, Meister,
Franklin, Benjamin,
Frick Collection, –54, ,
Gallus, –33,
Gardner, Isabella Stewart,
Gardner Museum,
gearwheels, –32
General Motors,
generators,
gent's fancy screwdriver,
Gent's Fancy Turnscrew,
gimlets,
girders, , ,
Goodman, William Louis, –27,
Great Western,
Greeks, ancient, , , –19, –22, –24, –40, –43
Greenwich Armory,
gunmaker's screwdriver,
guns, –61
antitank,
barrels of, , ,
loading and firing of, –57,
ray,
screws in, –61, ,
Gutenberg, Johannes,
hacksaws, , ,
hammers, , , , , , ,
adze-eye,
claw, ,
combination, ,
socketed,
steam,
see also mallets; mauls; sledgehammers
handsaws, –19,
ancient Egyptian, –18
open, ,
origins of, –19
skew-back, ,
see also backsaws; hacksaws; saws
hand tools, –27
artistic depictions and illustrations of, , –24, , , , –42, –47, , , , , , , –51
carpenter's, –27
combination, –21, , ,
cutting and shaping, , –20
drilling, , –25,
as extension of body,
four categories of,
hammering, , –21
measurement, –17
origins and histories of, –25, –27, –37
power, –16,
helixes, , –21, , ,
Hero of Alexandria, –19, –22, , ,
Hieron, King of Syracuse, , ,
hinges, –75, , ,
Historia Naturalis (Pliny the Elder), –17
History of Woodworking Tools (Goodman), –27,
hodometers,
houses, building of, –17, ,
instruments:
mathematical, , ,
medical, ,
navigational, , ,
surveying, , ,
iron, , , ,
screws, , –53,
smelting of,
jacks,
Japanese, n,
Jefferson, Thomas,
jeweler's screwdrivers,
Joseph, Saint, –24,
Josephus,
jousting, –63,
Kern, Ken,
Knights of the Round Table, ,
lathes, , , , –101
bench, ,
guilloching,
pole, –90
precision, –6
screw-cutting, –106
turning-, –97
Leonardo da Vinci, , , , , , , , –38
levels:
A-, –17,
spirit, , ,
locksmiths, ,
London pattern screwdriver, ,
Louis XVI, King of France,
machines, –51
calculating,
milling, , –51
mining and smelting, ,
scale-dividing, –99
scaling,
screw-making, –78, –106
specialized, –3,
water-driven, , , , , –38, –43
wind-driven, , , –39
wooden,
see also engines
mallets, ,
manacles, , ,
Marcellus, –35
matchlocks, –61, ,
mathematics, –35
Maudslay, Henry, –107, –9,
mauls, , –41,
Mechanical Exercises (Nicholson), ,
Mechanick Exercises (Moxon), , ,
Medieval Housebook, –54, , , –93, ,
Melancolia I (Dürer), –47
mensor aedificorum, ,
Mercer, Henry Chapman, –41
Mercer Museum, –41
Metropolitan Museum of Art, , –66,
micrometers,
microscopes, ,
Middle Ages, , ,
tools of, , , –23, –25, , –67,
mills, –51
flour,
hand-cranked, , –51
portable,
millstones, ,
mining, ,
Minoans,
mirrors,
miter boxes,
Moravian Pottery and Tile Works, –39
Morris, William, ,
mousetraps, ,
Moxon, Joseph, –30, , , , ,
multum in parvo tools,
nail gun,
nails, , ,
arming,
fabrication of, –71
iron, , , , –71
pulling of, , ,
screws vs., –70, , ,
Nasmyth, James, –9
New England Screw Company,
New York Times, ,
Nicholson, Peter, ,
nuts, , , , , ,
wing, , ,
wooden,
On Sphere-Making (Archimedes),
optics,
Ovid,
Owner-Built Home, The (Kern),
Oxford English Dictionary (OED), , ,
Pappus of Alexandria, –24
pegs, , , ,
petronels, –59
Phillips, Henry F., –86
physics, ,
pincers, ,
armorer's, ,
pipes, wooden,
pivot hooks,
planes, , , , , , ,
Plato,
Pliny the Elder, –17
plumb bobs, ,
Plumier, Charles, ,
Plutarch, , –35, ,
poitrinal,
Pollard's History of Firearms, –60,
portcullis twisters, ,
presses, –18
beam, –17, –22
linen, –14,
olive and fruit, , , ,
paper, ,
printing, ,
screw, –18,
Price, Derek J. de Solla, , –32
production lines,
pulleys:
compound, –37
ships', , –39
pumps, , ,
pyrites, iron,
radar,
Ramelli, Agostino, –51, , , , , ,
Ramsden, Jesse, –99,
regula,
Renaissance, ,
rennhut,
renntartsche, , –66
rivets, , ,
Roberts, Kenneth,
Roberts, Richard, –8
Robertson, Peter L., –83, , ,
Rogers, Charles D.,
rollers,
Roman Empire, , , , , , –35, , –41
tools of, , , , , , , , , , , , , –25,
Roubo, A. J., –33
rules,
folding,
Sakas, Ioannis, –38
Salaman, Raphael A., –32
sanders,
saws, –19, ,
circular,
frame, , , ,
origins of, –19
power,
revolving,
Roman frame,
see also handsaws
scharfrennen, –63
Scotch pattern screwdriver, ,
screw bolts,
screwdrivers, –27, , ,
as adaptations, ,
auger-handled,
automated, –86
blades of, , , , ,
definition of,
handles of, , , , ,
magnetic,
origins and history of, , –37, –44,
Phillips, ,
Robertson, , –86
specialized, –32,
spring-loaded,
see also specific screwdrivers
screw-pins,
screw-plates, ,
screws:
accuracy of, –98, –2, –6,
cam-out and slippage of, –79, ,
cost of, , , ,
countersunk,
cruciform vs. square socket, ,
endless, –19, –39,
fabrication of, –64, , , –86, –106,
gimlet-pointed, ,
handmade, ,
iron, , –53,
mechanical bonding of,
metal-joining, –55
modern, –86
origins and history of, , –55, –64, –67, –86, –43
ornamental,
Phillips, –86
regulating, –99, –2, –4, –10
removal of, ,
Robertson, –83
slotted-head, , , , –55, , , , , –79, ,
socket-head, –86
steel, ,
tapered, threaded bodies of, , , , , –77, –12
uniformity in,
water, –43
wooden, –13, –17, , –41
screw taps, , –24
sewing machine turnscrew,
sextants, , ,
shipbuilding, , , –82, ,
Shipley, David, –14, ,
sledgehammers,
slide rests, –101,
Sloan, Thomas J., ,
spanners, , ,
spindles, , , ,
spirals, –12
spirit levels, , ,
spokeshaves, ,
steam engines, , , , , –9,
steam hammers,
steel, , , , ,
steelyards,
Stephenson, George,
surveying instruments, , ,
tape measures, ,
taps, , ,
screw, , –24
telescopes, ,
Ten Books on Architecture, The (Vitruvius), ,
theodolite,
Thompson, John P.,
tilt targe,
tire-bouchon, ,
tire-fond, ,
toggle and loop,
"To Move a Given Weight by a Given Force" (Archimedes),
tortoise,
tournaments, martial, –63
tourne à gauche,
tournevis, –35, ,
trephines,
trispast,
try squares, ,
tunnels,
turners, –89, ,
turning catches,
turnscrews, , –44,
two-by-fours,
tylos, –22,
tympanum, –42
undertaker's screwdriver, ,
Vitruvius Pollio, Marcus, , , , ,
watchmaking, ,
waterwheels, , –42
wheel locks,
wheels,
book, ,
Ferris,
spinning,
Whipple, Cullen, –78
White, Lynn, Jr.,
Whitworth, Joseph, –6, –8,
William Marples & Sons,
windlasses, , –39
woodcutting, –16, , ,
worm gears, ,
wrenches, , , , ,
Wyatt, Job and William, , –88, ,
SCRIBNER
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Copyright © 2000 by Witold Rybczynski
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Set in Fournier
The Library of Congress has cataloged the Scribner edition as follows:
Rybczynski, Witold.
One good turn : a natural history of the screwdriver and the screw / Witold Rybczynski.
p. cm.
Includes bibliographical references.
1. Screwdrivers—History. 2. Screws—History. I. Title.
TJ1201.S34 R93 2000
621.9'72—dc21 00-036988
ISBN 0-684-86729-X
0-684-86730-3 (Pbk)
ISBN 978-0-743-21908-2 (eBook)
| {
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\section{Introduction}
\PARstart{W}{e} live in a digitized information world, but many
of the physical sources, such as sound and images, are by nature
analog. To transmit analog sources using digital communication
systems require the signals to first of all be A/D
(analog-to-digital) converted, which involves sampling and
quantization. Sampling, the operation that transforms a signal
from ``continuous in time'' to ``discrete in time'', is
reversible, i.e. the original continuous-time signal can be
reconstructed from the discrete samples loss-free, provided
the samples were taken at (or above) the Nyquist rate. However,
quantization, the operation that transforms a signal from
``continuous in amplitude'' to ``discrete in amplitude'', is
irreversible, i.e., the distortion caused by rounding off the
signal amplitude cannot be recovered after quantization.
To keep down the granularity error in general requires a large
number of quantization levels and/or high-dimension (vector) quantization. Since vector quantizers are very
challenging to design, and usually require the knowledge of
high-order source statistics (e.g. joint probability distribution of the $n$th order for $n$-dimension quantization)
which may not be easily available, real-world systems
tend to use simpler scalar quantizers with many
levels, at the cost of a large bandwidth expansion.
However, quantization error and bandwidth expansion are not all
the problems. Another practical issue concerns the labeling.
An $n$-level quantization scheme takes
$\lceil log_2 n\rceil$ bits to label each level. Regardless of
what labeling scheme is used (Gray, natural or
mixed label), different bits in the label will have different
levels of importance, but this natural hierarchy is not
reflected in the communication channel which treats all the bits
equal. For example, consider a 256-level monochrome image via natural labeling, where each pixel takes 8 bits to represent. An error in the least significant bit causes the pixel to be distorted by only one gray level, whereas an error in the most significant bit causes a drastic distortion of 128 levels!
To avoid (wasteful) over-protection of some bits and/or (disastrous) under-protection of others, one must employ unequal error protection (UEP),
but to provision the right protection to bits in accordance with their individual importance is nevertheless challenging.
Many design issues arise, such as how much more important and hence how much more protection one bit deserves in comparison to another,
what error correction codes (lengths, rates, error correction capabilities) are appropriate for each, and how to
balance the rates between quantization, error correction
and modulation. Most of these issues are difficult to quantify
or optimize.
While all of the above issues appear like a fact-of-life that one
has to accept, they actually need not be so. They stand in the way
only because we force analog signals into a {\it digitalized}
transmission paradigm. Consider an analog alternative that leaves
out quantization altogether and directly transmits discrete-time
continuous-valued analog signals (see Fig.
\ref{fig:analogDigital}), then all the quantization problems
would be gone \cite{bib:analogTransmit}. The only obstacle, however, is that efficient analog error correction codes (AECC) are hard to find.
One exciting result we wish to report here is that practical and
efficient AECC {\it can} be designed and analog transmission can
be made reliable {\it and} simple! A much under-studied topic
(especially compared to the prolific literature on digital error
correction codes),
the notion of {\it analog error correction codes}, or, {\it real
number codes}, actually dates back to the early 80's, when
Marshall and Wolf independently introduced the concept \cite{bib:Mars81}\cite{bib:Wolf83a}.
Early ideas of analog codes were a natural
outgrowth of digital codes, by extending conventional digital
codes from the finite field to the real-valued or complex-valued
field (i.e. symbols from a very large finite field can approximate
real values). This has resulted in, for example, discrete Fourier
transform (DFT) codes (a subclass of which become analog BCH codes
and analog Reed-Solomon codes) \cite{bib:Mars81}\cite{bib:Wolf83a}\cite{bib:Wolf83b},
discrete cosine transform (DCT) codes \cite{bib:DCT}, and graph-based analog codes \cite{bib:vardy}. Although
linear codes dominate the short literature of analog codes like
they do in digital codes, linear analog codes are not nearly as
powerful as linear digital codes\footnote{A
performance lower bound (in terms of mean square error) was
recently established for an arbitrary $(n,k)$ linear analog code, and
it is shown that a carefully-designed {\it nonlinear} analog code
can easily beat this bound
\cite{bib:linearAnalog}.}, and {\it nonlinear} analog codes are
true cause for excitement \cite{bib:linearAnalog}.
Nonlinear analog codes rely on nonlinear transforms to encode
analog data. Of particular interest is {\it chaotic analog codes}
(CAC), a special class of nonlinear AECC that make essential use
of chaotic systems to transform signals. Chaotic systems are
nonlinear dynamical systems with bounded state spaces exhibiting a
topological mixing feature \cite{bib:chaos}. They are widely existent in the
natural world and the engineering world (e.g. climate change,
mechanical vibration, acoustic signals and ecology systems are all chaotic systems), and many of them can be realized or emulated
using simple electric circuits (e.g. chua's circuit \cite{bib:chua}).
Despite the rich variety of formalities, chaotic systems share an important common
property of {\it high sensitivity to the initial state}. Popularly
dubbed the ``butterfly effect,'' this property states that a small
perturbation to the initial state of a chaotic system will cause a
huge difference later on. Although this butterfly effect is in
general viewed as a system penalty, it can actually be cleverly
exploited to satisfy the {\it distance expansion} property required by a
good error correction code. Specifically, if one treats the
initial state of a chaotic system as the source (to be encoded),
and treats some later states as the codeword (having been
encoded), then the chaotic system naturally enacts an error
correction encoder that successfully magnifies the small
differences among the source sequences (i.e. distance expansion).
This elegant feature was first exploited by Chen and Wornell in
the late nineties, when they proposed the first-ever chaotic
analog code, the {\it tent map code} \cite{bib:tentmapcode}.
Using a single {\it tent map} (a simple one-dimension
chaotic function) as the encoder, they demonstrated the
feasibility of constructing error correction codes using chaotic systems. Sadly,
however, their code did not perform nearly as well as
digital codes, and hence the wonderful idea exposed therein slept
for a decade before it was recently picked up by Xie, Tan, Ng
and Li \cite{bib:cat}. Leveraging the successful experience from
digital turbo codes, i.e. building long, powerful codes by
concatenating shorter, weaker codes, \cite{bib:cat} succeeded in
constructing {\it chaotic analog turbo} (CAT) codes by parallelly
concatenating two tent maps. Just like turbo codes significantly outperform convolutional codes, CAT codes significantly outperform tent map codes.
The work of \cite{bib:mirror} further extends the parallel concatenation idea to 2-dimension chaotic maps and proposed {\it mirrored baker's map codes}.
In this paper we present a further generalization of the
idea of constructing a long, powerful system using a set of shorter, weaker
components. The key is to carefully leverage the strength of one another to cover up their individual weaknesses. Specifically, we
develop a class of {\it tail-biting analog codes} (TAC) based on
2-dimensional chaotic maps. Previous work has considered a two-branch mirrored construction \cite{bib:mirror}. Here we present a constructive example that
engages three branches of {\it baker's maps} in a looped
tail-biting manner, and discuss its maximum-likelihood (ML) decoding
algorithm.
To support our proposal of analog image transmission, we apply our analog
codes in image transmission, and compare it with
the state-of-the-art digital systems (i.e. turbo codes). Simulations reveal
a surprisingly good performance achieved by our analog codes, which, for practical purposes, is considerably better than digital turbo codes!
The result is particularly exciting, considering that 1) the proposed analog coding system incurs considerably less complexity, memory and delay than the turbo coding system, and 2) turbo codes, the well-known class of capacity-approaching codes, represent the culmination of 70 years of mature digital coding research, whereas the research of analog codes is still at a very early stage.
\begin{figure}[htb]
\vspace{-0.2in}
\centerline{
\includegraphics[width=2.8in]{fig/fig_system.jpg}
\vspace{-0.2in}
}
\caption{A single analog error correction code in lieu of the combination of quantization, digital error correction code and digital modulation.}
\label{fig:analogDigital}
\end{figure}
\section{Principles and System Model}
\label{sec:system}
Error correction is based on a simple but profound idea of {\it
distance expansion}. Through a linear or nonlinear map (i.e.
encoding), the {\it source space} in which elements have
relatively small separations and are therefore easily distorted,
is transformed to a {\it code space} of a larger
dimension, where elements have (much)
larger separation and can therefore tolerate (much) larger
perturbation.
Distance expansion is generally achieved by adding redundancy and hence incurs
bandwidth expansion.
An $(n,k)$ code that encodes a length-$k$ source sequence to a
length-$n$ codeword has increased the bandwidth consumption from
$k$ units to $n$ units. The code rate, defined as
$r\!=\!k/n\!<\!1$, provides a measure of the amount of bandwidth
expansion.
Chaotic systems are described by nonlinear chaotic functions whose
Lyapunov exponents $>1$. A discrete-time chaotic function
describes the time evolution of the state vector $\mathbf{z}$,
\begin{align}
\mathbf{z}[i] =F(\mathbf{z}[i-1]),
\end{align}
where $\mathbf{z}[0]$ denotes the initial state (seed). A rate $1/n$
code can be realized by feeding source symbols to the chaotic
function as the seed $\mathbf{z}[0]$, and collecting $(n\!-\!1)$
consecutive states.
The proposed tail-biting analog codes are based on 2-dimension chaotic maps, whose state vector $\mathbf{z}$ has a dimension of 2. Previous chaotic analog codes, such as the tent map code proposed in \cite{bib:tentmapcode} and the chaotic analog turbo code proposed \cite{bib:cat}, are based on 1-dimension chaotic maps (e.g. the tent map), and hence have an source block size of only 1, that is, the resultant code is always an $(n,1)$ code which encodes one source symbol to a codeword of $n$ encoded symbols. From the information theory, a larger block size in general provides a richer context, a better ``diversity'' and hence a better performance. However, high-order-dimension discrete-time chaotic functions, those that have relatively simple structures and hence allow for practical detection with manageable complexity, are very difficult to find. For this reason, we resort to 2-dimension chaotic maps, and employ a looped tail-biting structure to connect them. That is, we can take $k$ branches of 2-dimension chaotic map, take a block of $k$ source symbols, $u_1, u_2, ..., u_k$, and feed the source symbols as initial states to the $k$ branches: \vspace{-0.6cm}
\begin{align}
\{u_1,u_2\} & \mbox{\ for Branch 1},\nonumber \\
\{u_2,u_3\} & \mbox{\ for Branch 2},\nonumber \\
\cdots & \cdots \nonumber \\
\{u_{k-1},u_k\} & \mbox{\ for Branch\ }k\!-\!1,\nonumber \\
\{u_{k},u_1\} & \mbox{\ for Branch\ }k.\nonumber
\end{align}
\vspace{-0.2cm}
Below we detail an
example of triple branches.
\section{Triple-Branch Tail-Biting Baker's Map Codes}
\label{sec:decoding}
In what follows,
we will use regular fonts to denote scalars (e.g. $x_1$), and bold
fonts to denote vectors and matrices (e.g. $\mathbf{x_1}$).
$\mathbf{x_1}{}_m^l$ denotes the vector $(x_1[m], x_1[m\!+\!1],
\cdots, x_1[l])$, and $\mathbf{x_1}$ is short for
$\mathbf{x_1}{}_0^{n-1}$.
\subsection{Encoder}
We consider using folded baker's map, a simple, 2-dimension
chaotic map from a unit square to itself, as the base function.
The baker's map is named after a kneading operation that
bakers apply to dough: the dough is cut in half, and one half is
folded over and stacked on the other, and compressed. It is
nonlinear but piece-wise linear, and presents a 2-dimension
analogy of the tent map:
\begin{align}
&\ \{ x[i],y[i]\} \nonumber\\
=& F(\,\{x[i\!-\!1],y[i\!-\!1]\}\,) \nonumber \\
=&\left\{\!\!\!\!
\begin{array}{ll}
\{ 2x[i\!-\!1]\!+\!1, \, (y[i\!-\!1]\!-\!1)/2\}, &\mbox{if\ } -\!1\!\le\! x[i\!-\!1]\!<\!0 \\
\{1\!-\!2x[i\!-\!1], \, (1\!-\!y[i\!-\!1])/2\}, &\mbox{if\ } 0\!\le \! x[i\!-\!1]\!\le \!1
\end{array}
\right.\label{eqn:baker_func0}
\end{align}
Now consider building analog codes by engaging three baker's maps in a
looped tail-biting manner, as shown in Fig.
\ref{fig:3branchBaker}. A block of three real-valued symbols (e.g.
pixels in an image), $\{u_{1},u_{2},u_{3}\}$, is paired and fed into the three
branches as the initial states: $\{ u_{1},u_{2}\}$, $\{
u_{2},u_{3}\}$ and $\{ u_{3},u_{1}\}$. Each branch encoder
recursively performs the baker's map $F(\{x,y\})$ in
(\ref{eqn:baker_func0}), to generate additional $(n\!-\!1)$ pairs
of states (in addition to the initial states):
\begin{align}
\{x_{j}[i],y_{j}[i]\}
=&F(\{x_{j}[i\!-\!1],y_{j}[i\!-\!1]\})\nonumber \\
=&F^{i}(\{x_{j}[0],y_{j}[0]\}),
\label{eqn:baker_recursive}
\end{align}
where the subscript $j=1,2,3$ denotes the $j$th branch, and
$i=1,2,\cdots, n\!-\!1$ denotes the time index of the states.
\begin{figure}[htb]
\centerline{
\includegraphics[width=2.4in]{fig/fig_encoder.jpg}
}
\caption{The proposed triple-branch tail-biting analog codes.}
\label{fig:3branchBaker}
\end{figure}
The collection of all the states from $0$ to $(n\!-\!1)$,
$\{\mathbf{x_{1}}{}^{n-1}_{0},
\mathbf{y_{1}}{}^{n-1}_{0}\}$, $\{\mathbf{x_{2}}{}^{n-1}_{0},
\mathbf{y_{2}}{}^{n-1}_{0}\}$ and $\{\mathbf{x_{3}}{}^{n-1}_{0},
\mathbf{y_{3}}{}^{n-1}_{0}\}$, forms the sub-codeword for the first, second, and third branch, respectively. Altogether $6n$ symbols are generated\footnote{The
$6n$ symbols include two copies of the source symbols $u_1,u_2,u_3$. It
is also possible to transmit the systematic symbols only once,
which will lead to a codeword of $(6n\!-\!3)$ symbols and code rate
$1/(2n-1)$.}, corresponding to the three source symbols $\{
u_1,u_2, u_3\}$. Hence, the code is a $(6n, 3)$ systematic code
with a rate of $r=1/(2n)$.
\subsection{Transmission of Analog Symbols through CASK and CQAM}
The codewords are transmitted through the noisy channel. Each
analog symbol takes value from -1 to 1, and is
modulated as variations in the amplitude of a carrier wave, in a
way similar to digital amplitude shift keying (ASK).
The only difference is that an $m$-ary ASK only allows
a fixed set of $m$ discrete amplitude values to be valid (e.g.
$-\Delta$, $-\Delta/3$, $\Delta/3$ and $\Delta$ for 4-ary ASK),
whereas in this {\it continuous ASK} (CASK), amplitudes may be any
real value between $-\Delta$ and $\Delta$. Hence,
our CASK may be regarded as an $\infty$-ary ASK.
From the communication theory, two $m$-array ASK modulations can
be packed to form a $m^2$-ary quadrature amplitude modulation
(QAM). When the two carrier waves use sinusoids that are out of
phase with each other by $90^o$ (termed $I$-channel and
$Q$-channel respectively), then the $m^2$-ary QAM achieves twice
the data rate ($2 log_2 m$ bits/symbol) on the same bandwidth as a
single $m$-ary ASK. Likewise, we can also pack two CASK to form a
{\it continuous QAM} (CQAM) to double our data rate. From the
perspective of signal space, this is to take two of our
real-valued symbols to form a complex-valued symbol, and projected it
onto an $\infty$-ary QAM.
Mathematically, the noisy reception at the decoder is:
\begin{equation}\label{eq:noisy cw}
\left\{\!\!\!
\begin{array}{ll}
I\mbox{-channel:} & \mathbf{Rx_{j}}{}^{n-1}_{0} = \mathbf{x_{j}}{}^{n-1}_{0} + \mathbf{n_{xj}}{}^{n-1}_{0} \\
Q\mbox{-channel:} & \mathbf{Ry_{j}}{}^{n-1}_{0} = \mathbf{y_{j}}{}^{n-1}_{0} + \mathbf{n_{yj}}{}^{n-1}_{0} \\
\end{array}
\right. \ j\!=\!1,2,3,
\end{equation}
where $\mathbf{n_{xj}}{}_{0}^{n-1}$ and $\mathbf{n_{yj}}{}_{0}^{n-1}$ are
noise sequences. If we adopt the additive white Gaussian noise
(AWGN) channel model, then these noise samples follow independent
and identically distributed (i.i.d.) Gaussian distribution $\sim
{\cal N}(0, N_0/2)$.
\subsection{Decoder}
When two copies of the systematic symbols,
$x_1[0]\!=\!y_3[0]\!=\!u_1$, $x_2[0]\!=\!y_1[0]\!=\!u_2$,
$x_3[0]\!=\!y_2[0]\!=\!u_3$, are transmitted, it is advisable to first perform
maximum ratio combining (MRC) before proceeding to the actual decoding.
On a homogeneous channel such as AWGN channel, MRC is equivalent
to equal gain combining (EGC):
\begin{align}
Rx'_{1}[0]= Ry'_{3}[0]= \frac{Rx_{1}[0]+Ry_{3}[0]}{2},\\
Rx'_{2}[0]=Ry'_1[0]= \frac{ Rx_{2}[0]+Ry_{1}[0]}{2}, \\
Rx'_{3}[0]=Ry'_2[0]= \frac{ Rx_{3}[0]+Ry_{2}[0]}{2},
\end{align}
where the apostrophe $'$ denotes the symbols after MRC. For
convenience, we abuse the notation, and omit the apostrophe in
$Rx_j[0]$ and $Ry_j[0]$ in the following discussion.
The maximum-likelihood decoder tries to make the best decision of
the initial states, $u_1, u_2, u_0$, based on the noisy
observation of a sequence of states. From the definition of
baker's map in (\ref{eqn:baker_func0}), a later $x$-state $x_j[i]$
can be deduced unequivocally from a previous one $x_j[i-1]$, but
not the other way around; and the same holds for the $y$-states.
The ambiguity in the backward derivation is caused by the unknown
sign of the previous $x$-state $x_i[i-1]$. Hence, to facilitate
decoding, we introduce a sign sequence $\mathbf{s_j}{}_0^{n-1}$ for
$\mathbf{x_j}{}_0^{n-1}$ (the signs of $\mathbf{y_j}{}_0^{n-1}$ are
irrelevant):
\begin{align}
s_j[i]=sign(x_j[i]), \ \ i\!=\!0,1,...,n\!-\!1,\ \ j\!=\!1,2,3.
\end{align}
With the sign sequence established, we can
establish a one-to-one mapping between $x_j[i]$ and $x_j[i\!-\!1]$ and between $y_j[i]$ and $y_j[i\!-\!1]$
\begin{align}
\hspace{1cm}\left\{
\begin{array}{l}
x_j[i\!-\!1]= -\frac{1}{2}s_j[i\!-\!1](x_j[i]-1), \\
y_j[i\!-\!1]=-2s_j[i\!-\!1]y_j[i]+1.
\end{array}
\right.
\end{align}
Recall that the baker's map is a piece-wise linear function. With
each time evolution, the number of segments doubles, but linearity
preserves within each (new) segment.
Hence, one can rewrite the encoding function in
(\ref{eqn:baker_func0}) by directly establishing a linear relation
between the $i$th state $\{x_{j}[i],y_{j}[i]\}$ with the initial
state $\{x_{j}[0],y_{j}[0]\}$ in each segment:
\begin{align}
\left\{
\begin{array}{l}
x_{j}[i] = a_{j}[i]x_{j}[0]+b_{j}[i],\\
y_{j}[i] = c_{j}[i]y_{j}[0]+d_{j}[i],
\end{array}
\right. j\!=\!1,2,3.
\label{eqn:linear}
\end{align}
In general, the values of the parameters
$a_{j}[i],b_j[i],c_j[i],d_j[i]$ not only depend on the time index
$i$ but also on which segment $x_j[i]$ and $y_j[i]$ fall in.
Observe that the sign sequence $\mathbf{s_j}{}_0^{n-1}$ actually
serves as the natural label for all the segments, namely, $s_j[0]\in\{-1,+1\}$ specifies the two segments at
time index $i=1$, $(s_j[0]s_j[1])\in\{-1-1,-1+1,+1-1,+1+1\}$ specifies
the four segments at time index $i=2$, and so on. Thus, carefully
arranging the sign sequence, we can derive the parameters
$a_{j}[i],b_j[i],c_j[i],d_j[i]$ in a unified recursive form across all the segments:
\begin{align}
\left\{
\begin{array}{l}
a_{j}[i] = -2s_{j}[i\!-\!1]a_{j}[i\!-\!1],\\
b_{j}[i] = 1-2s_{j}[i\!-\!1]b_{j}[i\!-\!1],\\
c_{j}[i] = - 0.5\, s_{j}[i\!-\!1]c_{j}[i\!-\!1],\\
d_{j}[i] = 0.5\, ( s_{j}[i\!-\!1]-s_{j}d_{j}[i\!-\!1]),\\
\end{array}
\right. j\!=\!1,2,3.
\label{eqn:abcd}
\end{align}
Given the linear relation in (\ref{eqn:linear}) and (\ref{eqn:abcd}), an efficient ML decoding can be derived to obtain an estimation of the information bits $\{ \tilde{u}_{1}, \tilde{u}_{2},
\tilde{u}_{3}\}$.
\begin{align}
& \{\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}\}\nonumber \\
=& \underset{-1\leq u_{1},u_{2},u_{3}\le 1}{\arg\max}\!\!
\Pr\big(\mathbf{Rx_{1}}{}^{n-1}_{0},\mathbf{Ry_{1}}{}^{n-1}_{0},\mathbf{Rx_{2}}{}^{n-1}_{0}, \mathbf{Ry_{2}}{}^{n-1}_{0},\nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{Rx_{3}}{}^{n-1}_{0}, \mathbf{Ry_{3}}{}^{n-1}_{0} \mid u_{1},u_{2},u_{3}\big) \nonumber \\
= & \!\underset{-1\leq u_{1},u_{2},u_{3}\leq
1}{\arg\max} \prod_{i=0}^{n-1}\! \Big\{\!\!
\Pr\big(Rx_{1}[i]\mid u_1) \Pr\big(Ry_{1}[i] \mid u_2\big)\nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \cdot
\Pr\big(Rx_{2}[i]\mid u_2) \Pr\big(Ry_{2}[i] \mid u_3\big) \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \Pr\big(Rx_{3}[i]\mid u_3\big) \Pr\big(Ry_{3}[i] \mid u_1 \big)\Big\} \label{eqn:ML_1} \\
= & \underset{-1\leq u_{1},u_{2},u_{3}\leq
1}{\arg\min} \sum_{i=0}^{n-1}\Big\{
\big(Rx_{1}[i]\!-\!x_{1}[i]\big)^2 \!+\!
(Ry_{1}[i]\!-\!y_{1}[i]\big)^2 \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ + \big( Rx_{2}[i]\!-\!x_{2}[i]\big)^2 \!+\!
(Ry_{2}[i]\!-\!y_{2}[i]\big)^2
\nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ + \big( Rx_{3}[i]\!-\!x_{3}[i]\big)^2 \!+\!
\big(Ry_{3}[i]\!-\!y_{3}[i]\big)^2 \Big\},
\label{eqn:ML}
\end{align}
where the equality in (\ref{eqn:ML_1}) is due to the independence
of the noise (i.e. memoryless channel), and the equality in
(\ref{eqn:ML}) is due to the Gaussianity of the noise. Using the
segmented linear function expressions in (\ref{eqn:linear}) and
noting that $x_1[0]=y_3[0]=u_1$, $x_2[0]=y_1[0]=u_2$, and
$x_3[0]=y_2[0]=u_3$, we can further simplify (\ref{eqn:ML}) to:
\begin{align}
&\{\tilde{u}_{1},\tilde{u}_{2},\tilde{u}_{3}\}
=\underset{-1\leq u_{1},u_{2},u_{3}\leq
1,\ \mathbf{s_{1}}{}_{0}^{n-2},\mathbf{s_{2}}{}_{0}^{n-2},\mathbf{s_{3}}{}_{0}^{n-2}}{\arg\min}
\sum_{i=0}^{n-1}\nonumber \\
& \Big\{
\big(Rx_{1}[i]\!-\!a_{1}[i]u_{1}\!-\!b_{1}[i]\big)^{2} +
\big(Ry_{1}[i]\!-\!c_{1}[i]u_{2}\!-\!d_{1}[i]\big)^{2} \nonumber \\
& \big(Rx_{2}[i]\!-\!a_{2}[i]u_{2}\!-\!b_{2}[i]\big)^{2} +
\big(Ry_{1}[i]\!-\!c_{2}[i]u_{3}\!-\!d_{2}[i]\big)^{2} \nonumber \\
& \big(Rx_{3}[i]\!-\!a_{3}[i]u_{3}\!-\!b_{3}[i]\big)^{2} +
\big(Ry_{3}[i]\!-\!c_{3}[i]u_{1}\!-\!d_{3}[i]\big)^{2} \Big\}
\label{eqn:ML_2}
\end{align}
where the parameters $a_{j}[i],b_{j}[i], c_{j}[i],d_{j}[i]$ are
given in (\ref{eqn:abcd}).
The quadratic minimization problem in (\ref{eqn:ML_2}) can be solved
by taking the derivatives with respect to $u_{1}$, $u_{2}$ and
$u_{3}$, respectively. The ``global'' optimal solutions
$u_{1}^{\ast}$, $u_{2}^{\ast}$ and $u_{3}^{\ast}$ that minimize
(\ref{eqn:ML_2}) are:
\begin{align}
u_{1}^{\ast}\! =\! \frac{\sum_{i=0}^{n-1} (Rx_{1}[i]a_{1}[i]\!+\!Ry_{3}[i]c_{3}[i]\!-\!a_{1}[i]b_{1}[i]\!-\!c_{3}[i]d_{3}[i])}{\sum_{i=0}^{n-1}(a_{1}^{2}[i]+c_{3}^{2}[i])}\nonumber\\
u_{2}^{\ast} \! = \! \frac{\sum_{i=0}^{n-1}(Rx_{2}[i]a_{2}[i]\!+\!Ry_{1}[i]c_{1}[i]\!-\!a_{2}[i]b_{2}[i]\!-\!c_{1}[i]d_{1}[i])}{\sum_{i=0}^{n-1}(a_{2}^{2}[i]+c_{1}^{2}[i])}\nonumber\\
u_{3}^{\ast} \!=\!
\frac{\sum_{i=0}^{n-1}(Rx_{3}[i]a_{3}[i]\!+\!Ry_{2}[i]c_{2}[i]\!-\!a_{3}[i]b_{3}[i]\!-\!c_{2}[i]d_{2}[i])}{\sum_{i=0}^{n-1}(a_{3}^{2}[i]+c_{2}^{2}[i])}
\label{eqn:MLresult1}
\end{align}
It should be noted that the ``global'' optimal decisions
$u^{\ast}_1, u^{\ast}_2, u^{\ast}_3$ in (\ref{eqn:MLresult1}) are
not always the feasible solution, since they may fall outside the
support range, i.e. the respective linear segment of length
$\frac{1}{2^{n-1}}$.
To account for the boundary conditions, note that, for $j=1,2,3$,
\begin{equation}
\label{eqn:fn}
-1\le x_{j}[n-1]=a_{j}[n-1]u_{j}\!+\!b_{j}[n-1]\le 1,
\end{equation}
which leads to
\begin{align}
& \min\left( \frac{-\!b_{j}[n\!-\!1]\!+\!1}{a_{j}[n\!-\!1]}, \ \frac{-\!b_{j}[n\!-\!1]\!-\!1}{a_{j}[n\!-\!1]}\right)
\le u_j \nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \max\left(\frac{-\!b_{j}[n\!-\!1]\!+\!1}{a_{j}[n\!-\!1]} , \frac{-b_{j}[n\!-\!1]\!-\!1}{a_{j}[n\!-\!1]}\right).
\label{eqn:boundary}
\end{align}
Combining the quadratic minimal solution in (\ref{eqn:MLresult1}) and
the boundary solution in (\ref{eqn:boundary}), the ML decoder will produce the following
decision:
\begin{align}
\tilde{u}_{j}\!= &
\left\{\!
\begin{array}{l}
\min\Big(\frac{-b_{j}[n-1]+1}{a_{j}[n-1]}, \frac{-b_{j}[n-1]-1}{a_{j}[n-1]}\Big), \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\mbox{if\ } u_{j}^{\ast}\!\!<\!\!\min\Big(\frac{-b_{j}[n-1]+1}{a_{j}[n-1]},\frac{-b_{j}[n-1]-1}{a_{j}[n-1]}\Big), \\
\max\Big(\frac{-b_{j}[n-1]+1}{a_{j}[n-1]}, \frac{-b_{j}[n-1]-1}{a_{j}[n-1]}\Big),\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if\ }
u_{j}^{\ast}\!\!>\!\!\max\Big(\frac{-b_{j}[n-1]+1}{a_{j}[n-1]},\frac{-b_{j}[n-1]-1}{a_{j}[n-1]}\Big), \\
u_{j}^{\ast}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{otherwise},
\end{array}
\right.\nonumber\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{for\ }j=1,2,3.
\label{eqn:MLresult2}
\end{align}
\section{Image Transmission via Analog Codes}
\label{sec:simulation}
Analog codes are most useful for transmitting analog sources as shown in
Fig. \ref{fig:analogDigital}, but
they can also be used to transmit digital data, and
especially digitized images.
To illustrate, consider a monochrome image, the $256\!\times\!256$ (pixel) Lena, where each pixel is represented by a byte valued between 0 and
255. In the conventional digital transmission paradigm, all the
bits are assembled into a bit-stream ($65536\!\times\! 8$ bits altogether), and digitally coded and
modulated. In the proposed analog transmission paradigm, the
pixels are viewed as stream of real-valued analog symbols ($65536$ symbols in the range of $[-1,1]$, i.e. each pixel $0\le x\le 255$ is linearly scaled to $[-1,1]$ via $x\to \frac{x\!-\!128}{128}$), and then coded
by an analog code. We simulate and compare the analog system with
some of the best-known digital systems:
1). The analog system consists of the proposed $(12,3)$ 3-branch
tail-biting baker's map code with code rate 1/4. The encoded
symbols are transmitted via the continuous ASK modulation ($65536\!\times\! 4$ total symbols).
2). The digital system consists of a digital turbo code, one of
the best error correction codes known to date, and the ASK modulation. We
consider $(4096,2048)$ 16-state turbo codes with constituent
convolutional codes $(1,\frac{1+D+D^2+D^3}{1+D+D^3})$ and code
rate 1/2, and 16-ary ASK. Thus, with the default 8-bit quantization per pixel, the overall bandwidth expansion is $\frac{1}{8}\frac{1}{2}\log_216\!=\!1/4$, which is the same as the analog system (i.e. $65536\!\times\! 8\!\times\! 2/4=65536\!\times\!4$ symbols). At the decoder, the 16-ary ASK is softly demodulated, and the resultant log-likelihood ration (LLR) is passed to the soft iterative turbo decoder which uses the BCJR algorithm as the sub-decoders. Six iterations are performed before the decoder outputs hard decisions.
The transmission quality is evaluated using the mean square
error (MSE) between the original $m\times l$ monochrome image $I$
and the reconstructed image $K$,
\begin{align}
\mbox{MSE}\!=\frac{1}{ml} \sum_{i=0}^{m-1}\sum_{j=0}^{l-1} (I_{i,j} - K_{i,j})^2,
\end{align}
as well as the peak signal-to-noise ratio (PSNR),
\begin{align}
\mbox{PSNR}=20\log_{10} (I_{max}/\sqrt{\mbox{MSE}})\ \ \ (dB),
\end{align}
where $I_{max}$ is the maximum possible pixel value of the image
(e.g. 255 for 8-bit quantized monochrome images).
Fig. \ref{fig:psnr} plots the PSNR performance (dB) of the digital system and the analog system on AWGN channels with signal-to-noise ratio (SNR) measured in terms of $E_p/N_0$ (dB), where $E_p$ is the average energy per pixel in the original image.
Because of the bandwidth expansion in digital system (1 pixel becomes 8 bits), the digital turbo code cannot get to its waterfall region until after a rather high $E_p/N_0$ of $22$ dB. As a consequence,
the proposed analog code noticeably outperforms the digital turbo code for a wide range of channel conditions. In general, a PSNR of 30 dB or more is reckoned as good quality image. This is achieved by the analog system at $E_p/N_0=14$ dB, but is not achieved by the digital system until $E_p/N_0>22$ dB!
\begin{figure}[htb]
\vspace{-0.2cm}
\centerline{
\includegraphics[width=3in,height=2in]{fig/PSNR_performance.jpg}
\vspace{-0.2cm}
}
\caption{PSNR (dB) of the Lena Picture transmitted by the analog system and the digital system (digital turbo codes)}
\label{fig:psnr}
\vspace{-0.2cm}
\end{figure}
To provide a visual feel of the transmission quality, Fig. \ref{fig:lena} further demonstrates the reconstructed
images for the two systems at $E_p/N_0$ of 10, 14, 18, 22 and 24 dB, respectively. We see that the digital system (right column) still experiences annoying pepper-and-slat errors at a high $E_p/N_0$ of 22 dB, whereas the analog system (left column) can deliver quality image for as low as 14 dB.
The advantages of the analog system are rather obvious, including the capability of delivering good quality on poor channel conditions, graceful performance degradation, and considerably lower complexity. The last is particularly noteworthy. In the digital system, soft-demodulation and soft turbo decoding are both very complex and time-consuming, and involve nonlinear operations (e.g. the $max^*$ operation in the BCJR decoding). In comparison, the proposed analog code entails only a few simple linear operations. Further, the digital turbo code has a considerably longer block length (2048 bits or 256 pixels) than the analog system (3 pixels), and hence requires considerably longer memory consumption and delay.
Our simulations are currently run over raw or TIFF (tagged image file format) images, a popular lossless format especially for high color-depth images. Although many consumer electronics use lossy JPEG format, raw images which allow editing and re-saving without losing image quality play an important role in image archive and especially in bio-medical applications. Further, the proposed analog codes can also be extended to compressed images, as there is no fundamental conflict between analog error correction coding and compression. For example, in a JPEG image, the DCT (discrete cosine transform) coefficients (which are by nature real-valued) are each represented as a binary bit stream in the digital coding systems; but they can be directly taken in as analog symbols in the analog coding systems. Analog coding can also provide data accuracy as required; for example, if an analog code operating on sources bounded between $[-1,1]$ can provide an MSE $\Delta(<1)$, then it can on average guarantee the accuracy of $|\frac{1}{2}\log_2\Delta|$ binary bits after the decimal. Hence, with careful arrangement, analog coding schemes can also find good use in transmitting compressed images (including the control data and the efficients).
\section{Conclusion}
\label{sec:conclusion}
We have developed an efficient triple-branch tail-biting baker's map analog code. Using a simple linear-operation-based maximum likelihood decoding scheme, we apply the code to image transmission, and show that, despite its considerably lower complexity, memory consumption and delay, the analog code actually significantly outperforms turbo-code-based digital systems! We conclude by promoting analog coding as a new and potentially very rewarding technology for transmitting images as well as other analog sources.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,704 |
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